Field Observation and Theoretical Study on an Existing...
Transcript of Field Observation and Theoretical Study on an Existing...
Research ArticleField Observation and Theoretical Study on an ExistingTunnel Underpassed by New Twin Tunnels
Qiongfang Zhang 12
1MOE Key Laboratory of Soft Soils and Geoenvironmental Engineering Zhejiang University Hangzhou 310058 China2Research Center of Coastal and Urban Geotechnical Engineering Zhejiang University Hangzhou 310058 China
Correspondence should be addressed to Qiongfang Zhang yangziduozisinacom
Received 10 August 2017 Accepted 16 November 2017 Published 20 March 2018
Academic Editor Andrea Benedetto
Copyright copy 2018 Qiongfang Zhang)is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
)e methodology of the existing displacement control is illustrated taking the shield of twin tunnels of Line 4 underpassing theupline tunnel of existing metro Line 1 for example Vertical horizontal and convergence displacement of the existing tunnel ismonitored and analyzed in detail in this paper Shield parameters are predefined and adjusted based on the feedback of thedisplacement of Line 1 Short-term displacement of the existing tunnel is greatly influenced by the relative distance between theshield face and the existing tunnel and shield parameters )e shapes of horizontal and convergence displacement curves aresimilar Line 1 is reinforced and a new analysis method is firstly proposed for the design of reinforcement of the existing tunnelwhich is verified by the analytical methods derived from prior studies )e results show that the change of reinforcement stiffnesshas a greater effect on the normalized bending moment and the normalized shear force of the existing tunnel and reinforcementof 25 rings on either side of the intersection point is the best choice in this case )e proposed model can be widely applicable forreinforcement design and safety check of the existing tunnel
1 Introduction
)e interaction between new shield construction and existingtunnel has become a common and important issue with therapid development of underground traffic system which hasbeen studied in the past using a variety of approaches fieldobservations model tests analytical methods and finite el-ement modeling Kim and Liu et al [1 2] presented a goodsummary comparison of the studies and only studies thatillustrates the shield underpassing or parallel underpassingthe existing tunnel Yamaguchi et al [3] presented succes-sively the numerical model and then analyzed three config-urations of the twin tunnels in Japan aligned horizontallyvertically and inclined )e construction of the upper tunnelat first leads to both higher settlement and bending moment)emaximum soil settlement was obtained for vertical-alignedtunnels while horizontal-aligned tunnels caused the lowestsettlement Addenbrooke and Potts [4] analyzed the influenceof tunnel position tunnel spacing rest period and sequenceof excavation on the interaction between the two tunnels
It concentrated on the shape of the settlement profile and thevolume loss induced by the two tunnels )e shield under-passing the existing tunnel is highly site specific )e soilcondition the tunnel buried depth and the relative position ofthe new tunnel and existing tunnel all affect the response of theexisting tunnel Field observations remain the commonlyrecognized approach for understanding the interaction be-havior between the new tunnel and existing tunnel Usually theunderpassing of a shield with a loss of soil often causes set-tlement of the existing tunnel at last [5ndash7] Chehade andShahrour [8] used the finite element method to investigate theinfluence of the relative position of tunnels and the con-struction procedure on the soil settlement )e results showedthat the settlements of the existing tunnel for the verticalparallel tunnels were larger than those for the horizontalparallel tunnels Li and Yuan [9] studied the twin tunnelspassing under a double-decked tunnel at an angle of 55 inweathered granite gneiss Only settlements were found and thehorizontal displacement was smaller than the vertical dis-placement in the existing tunnel Despite a number of studies
HindawiAdvances in Civil EngineeringVolume 2018 Article ID 1598672 16 pageshttpsdoiorg10115520181598672
having been carried out current understanding of the in-teraction between the two tunnels is still limited
In order to ensure the stability of the existing tunnellocal thickening is needed at the sides of the existing concretelining )ere are a plenty of research about the conventionalreinforcement methods such as inner steel plate re-inforcement or new reinforcement approaches such as FRCor composite concrete but they all focus on the performanceof reinforcement on the single ring )e effect of re-inforcement on the longitudinal behavior of the tunnel is notyet clear in current analysis )e analytical method oflongitudinal displacement of the tunnel due to adjacentexcavation andmultiple tunnelling is researched a lot (1) theelastic continuum models developed by Vorster et al [10](2) theWinkler model [11] and (3) the two-parameter elasticmodels When variable stiffness of the existing tunnel needsto be considered the methods are no longer applicableSelvadurai [12] divided the beam on foundation into Nelements and used the method of initial parameters toanalyze the beamrsquos displacement In the paper we assumethe existing tunnel as a beam resting on a two-parameterfoundation combining the initial parameter method and thetransfer matrix method to analyze the reinforcement effecton the longitudinal behavior of the existing tunnel
)e major objectives of this paper are (1) to investigatethe influence of the tunnel driving parameters and therelative distance between the shield and the existing tunnelon the existing tunnel based on the interpretation of the fieldmeasured data and (2) to study the effects of differentstiffness and ranges of reinforcement on the existing tunnelusing analytical methods
2 Project Overview
)e location of the Line 4 tunnels and the upline of Line 1 areshown in Figure 1 )e upline of Line 1 is the existing tunnelbuilt in 2012 Figure 2 shows a plan view of tunnel alignmentand arrangement of the monitoring points in the existingtunnel )e shield of the northbound tunnel of Line 4 startsfrom Guanhe Station passes under the existing tunnel witha small angle of 23deg and reaches East railway station at last)en the shield machine is reassembled with tunnels ina reversed direction )e northbound tunnel with a length of329675m consists of 275 rings numbered starting with zerofrom Guanhe Station to East railway station while thesouthbound tunnel with a length of 323834m consists of 270rings numbered starting with zero fromEast railway station toGuanhe Station )e lateral distance between the northboundtunnel and the southbound tunnel is 94ndash150m
Figure 3 shows the longitudinal profile of soil andtunnels )e vertical distance between the existing tunneland Line 4 is 212m)e existing tunnel and new tunnels areall built using an articulated shield tunnelling machine withan outer diameter of 64m and a length of 85m )e spokesplus panel-type cutter head are used and the aperture ratioof cutter head is 40 Each ring of new tunnels consists of sixprecast concrete segments )e outer diameter the innerdiameter and the thickness of the segment are 62m 55mand 035m respectively
21 Soil Condition )e engineering properties of the rockand the soils in this site are very complicated Figure 4 showsgeotechnical parameters of soil in this site)e soil layers fromtop to bottom are (1) fill (3) silty clay (4) clay (6) the muddysilty clay and (8) soft silty clay and gravel layer )e existingtunnel and new tunnels are located in the muddy silty claylayer at a depth of 20m and 25m respectively Along themetro line an extensive geotechnical investigation is carriedout )e standard penetration test (SPT) vane shear test(VST) piezocone test (CPTU) flat plate dilatometer test(DMT) and water pumping test (WPT) are conducted alongLine 1 )e maximum water content of clay and muddy siltyclay is about 40 which is close to the liquid limit value )eaverage undrained shear strength of undisturbed clay andremolded clay which is obtained by VST is 705 kPa and109 kPa respectively )e coefficient of the at-rest earthpressure (K0) of soft clay is obtained by DMT
22 Hydrological Geology )e shallow ground water is porephreatic water mainly found in layers from (1) to (8) )eelevation of the water surface is 2232m )e laboratorypenetration test and field steady flow test can be seen inTable 1 It can be seen that the permeability coefficient ofthe clay layer is very small and the permeability coefficient ofthe gravel layer is 1ndash2md )e seepage induced from waterhead difference might influence the stability of tunnels
3 Shield Driving Parameters
Figure 5 presents advance-time curve of the shield Beforeunderpassing the existing tunnel the shield advances with a veryslow rate )e tunnel advancing rate is 7-8 ringsday and thepenetration rate of the shield ranges from 20 to 25mmmiddotminminus1
Figure 6 shows the applied tunnel face pressure Inpractice the tunnel face pressure should be adjusted with thetheoretical value as well as the feedback displacement of theexisting tunnel )e existing tunnel redistributes the soilstress and the applied tunnel face pressure should be ad-justed by (1) as follows
p2 K0cprimeh + pw + Q1 minusQ2 plusmn 20 kPa (1)
where K0 is the coefficient of earth pressure at rest cprime is theeffective gravity of soil h is the thickness of overburdendepth (m) pw is the water pressure Q1 is the weight of theexisting tunnel (kN) Q2 is the weight of the soil with thesame internal volume of the existing tunnel (kN) and 20 kPais the pressure fluctuation
)e applied tunnel face pressure of the southboundtunnel cannot be calculated by (1) which is due to the re-distribution of the earth pressure after the construction ofthe northbound tunnel nearby and should be adjustedaccording to the displacement of the existing tunnel
Figure 7 shows the tail void grouting volume of the shieldof the northbound and southbound tunnels of Line 4 Shieldparameter adjustments play an important role in the dis-placement controls reducing the tail void grouting volumeand applied tunnel face pressure when Line 1 heaves Op-positemeasures are takenwhen there is a subsidence in Line 1
2 Advances in Civil Engineering
4 Observation Results and Discussions
41 Monitoring Arrangement of Line 1 Arrangement ofthe monitoring rings of the existing tunnel can be seenin Figure 2 e total station Topcon MS05AX xed onthe tunnel sidewall arranged from rings 404 to 559 inthe existing tunnel is an automatic real-time measuringsystem which is used to monitor vertical and horizontal
displacement One monitor section is set every two rings inthe most aected zone (from rings 447 to 499 in the existingtunnel) and one monitor section is set every ve rings inthe rest part An LECAI D5 hand-held distance nder isused to monitor the converge displacement every 5 ringsfrom 404 to 679 in the existing tunnel e arrangement ofmonitoring points at the cross section of the existing tunnelis shown in Figure 8
Figure 1 Location of the new tunnels and the existing tunnel in Hangzhou A Guanhe Station B East railway station
409
404 414 419 424 429 434 439
444 447449451453455457459461463465467469471473475477479481483485487489491493495497499 509
514 519
524
529
534
539
544
549
554
559
564
569
574
579
584
589
594
599
604
609
614
619
624
629
634
639
644
649
679
15101520253035404550556065707580859095100
105
110
115
120
125
130
135
140
145
150155
165170175180
18519019520021021522022523023540
45
5055
60
65
70
75
80
85
90
95
100
105
110
115
120
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135
140
145
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155
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175
180
185
190
195
200
205
210
215
220
225
230
235
240
245
250
255
260
265
270
275
East railway station
Guanhe Station
786 m
966 m835 m
35302520
674
669
664
659
654205
160
504
240245250255
e northbound tunnel of Line 4
e southbound tunnel of Line 4
Reinforcement of the existing tunnel
Zha-nong-kou Station
e upline tunnel of existing Line 1
Da-ju-yuan Station
A B
C
e auto monitoring point of vertical and horizontal displacement
e manual monitoring point of horizontal displacement
e manual monitoring point of vertical displacment
e manual monitoring point of convergence displacement
Figure 2 Plan view of tunnel alignment and location of the monitored rings
Line 4
65
Elev
atio
n (m
)
1-1 sandy silt
2-1 silt
3-1mucky silty
clay
3-2 clay
4-1 sandy clay
5-1 clay
GuanheStation
East railwaystation
5-2mucky silty
clay
Line 1
141210 8 6 4 2 0minus2minus4minus6minus8
minus10minus12minus14minus16minus18minus20minus22minus24minus26minus28minus30
minus15077 minus14954
208 212
K20+
898
754
K21+
053
342
K21+
094
361
K21+
608
099
232West square (B period)
West square (D period)Bottom of the basement floor 2100
Communication pipeline
K21+
015
389
Water pipeline
K21+
144
111
Groundwater table
Chainage (m)
Bottom of the basement floor 2900
K20+
900
K21+
000
K21+
100
K21+
200
K21+
300
K21+
400
K21+
500
K21+
600
K20+
971
955
K21+
301
630
Initi
al p
oint
Term
inal
poi
nt
Figure 3 Longitudinal prole of soils and tunnels
Advances in Civil Engineering 3
42 Displacement of the Existing Tunnel e location ofpoint A as shown in Figure 2 is corresponding to the in-tersection point of a plan view of the existing tunnel andnorthbound tunnel of Line 4 e location of point B iscorresponding to the intersection point of the plan view ofthe existing tunnel and southbound tunnel of Line 4 emiddle point of points A and B is point C
421 Vertical Displacement of the Existing Tunnel Figure 9shows time-varying vertical displacement in the moni-toring rings of the existing tunnel e x-axis is the ringnumber of the existing tunnel A positive value of theordinate denotes heave while a negative value denotessettlement of the existing tunnel e selected monitoringrings start to heave when the shield face is 0ndash10m awayfrom the selected monitoring rings which is mainly due tolarge applied tunnel face pressure bulk addictive thrust(Figure 6) and the friction force between the shield shelland soil mass When the shield tail is far beyond the se-lected monitoring rings a reduction of heave in the selectedmonitoring points is observed which is mainly due to theclosure of the tail void Only small settlements and heavesare measured after the construction of the northboundtunnel During the 3-month shutdown of the shield ad-ditional settlement ranges from 2mm to 3mm can beobserved in rings from 430 to 520 Additional settlements
ranges from 2 to 4mm were measured in rings from 460 to540 during the southbound tunnelrsquos construction elong-term additional settlements monitored up to 140 days(from 201461 to 20141019) range from 2 to 4mm inrings from 460 to 510 e settlements of rings from 430 to520 range from 2mm to 12mm and the settlement curve ofthe upline of Line 1 is ldquoUrdquo shaped after the long-termmonitoring e settlement curve is approximately sym-metric about the dashed line C after the long-term mon-itoringemaximum settlement is 12mmwhich is locatedin ring 487
0 4 8 12 03 04 05 06 0716 18 20 22
50
45
40
35
30
25
20
15
10
5
020 30 40 50 0 10 20 300 10 20 30 40 025 030 035 040
Es (MPa)w () vk0
1 Fill
3 silt
4 clay
6 muckysilty clay
8 so siltyclay
Depth of metro 1 tunnel axis
r unit weightw water contentwp plastic limit
wL liquid limitv void ratioEs elasticity modulusCU consolidation undrainage conditionCU effective value in consolidation undrainage conditionUU unconsolidation undrainage condition
gravel
r (kNm)
Dep
th (m
)
wpwL
w
Soil layer
uucucu
Depth of metro 4 tunnel axis
uucucu
c (kPa) φordm
Figure 4 Soil prole and geotechnical parameters
Table 1 Laboratory penetration test and eld steady centow test
Soil layerLaboratory
penetration test (cms) Field steady centow test (cms)
Kv Kh K
3ndash2 479times10minus5 781times10minus5 421times10minus3
3ndash3 255times10minus4 230times10minus4 157times10minus3
3ndash5 963times10minus5 174times10minus4 mdash3ndash6 329times10minus4 301times10minus4 mdash4ndash3 239times10minus7 784times10minus7 6ndash1 206times10minus7 303times10minus7 mdash6ndash2 262times10minus7 128times10minus7 mdash8ndash1 188times10minus7 511times10minus7 mdash
4 Advances in Civil Engineering
422 Horizontal Displacement of the Existing Tunnel ehorizontal displacement of the existing tunnel with respectto the location of the shield is illustrated in Figure 10 Apositive horizontal displacement denotes northward trans-verse tunnel movement away from the original tunnelcenterline while a negative horizontal displacement denotessouthward transverse tunnel movement away from theoriginal tunnel centerline
ere are northward displacement on the left side ofpoint C and southward displacement on the right side ofpoint Ce horizontal displacement curve is approximatelysymmetric about point C after the long-term monitoringe maximum northward displacement is 10mm in the ringnear the intersection point A and the maximum southwarddisplacement is minus105mm in the ring near the intersectionpoint B after the completion of the two tunnelsrsquo construc-tion During the construction of the northbound tunnelrings from 453 to 471 move southward which is likely due tothe additional bulkhead additive thrust and the squeezingforce provided by the shield shell When the shield tail leavesring 459 rings from 439 to 459 move southward slowlyOnly 2 to 5mm additional southward displacement ismeasured in rings from 487 to 560 and nearly no dis-placement is observed on the left side of point B during theconstruction of the southbound tunnel which is mainly dueto the northbound tunnelrsquos barrier eect No change ofhorizontal displacement is observed in the long-term con-ditions (from 2014528 to 20141019)
423 Convergence Displacement of the ExistingTunnel Figure 11 shows the convergence displacement ofthe existing tunnel with respect to the locations of the shield
A negative value indicates the reduction horizontal diameterof the existing tunnel while a positive value indicates theaddition horizontal diameter of the existing tunnel It can beobserved that the convergence displacement is not obviouswhen the shield reaches ring 48 (22 rings away from theintersection point A) During the process of the shielddriving from ring 64 to ring 87 a signicantly additionalincrease in the negative convergence displacement in ringsfrom 440 to 490 is observed and themaximum displacementoccurs in the intersection point A e reason might be thatthe shield face squeezes one side of the existing tunnel
20950
21000
21050
21100
21150
21200
21250
21300268
270
e r
ing
num
ber o
f the
no
rthb
ound
tunn
el
70
Guanhe Station
paused for 106 days
56 days
e tunnel face of the northbound tunnele tunnel face of the southbound tunnel
Tunn
el ch
aina
ge (m
)
Date
46 days
East railway station
170
250
200
150
100
50
0
-50
0
50
100
150
200
250
e r
ing
num
ber o
f the
sout
hbou
nd tu
nnel
2013
12
3
2013
12
13
2013
12
23
2014
12
2014
11
2
2014
12
2
2014
13
020
144
10
2014
41
4
2014
42
0
2014
43
0
2014
51
0
2014
52
0
2014
53
0
Figure 5 Advance-time curve of the shield
430 440 450 460 470 480 490 500 510 515
020
022
024
026
028
030
032
034
036
038
040
Northbound tunnelSouthbound tunnel
225 220 210 200 190 180 170 160 150 143
The ring number of the northbound tunnel of Line 4
The corresponding ring number of the upline tunnel of Line 1
The ring number of the southbound tunnel of Line 4
50 60 70 80 90 100 110 120 130
The a
pplie
d tu
nnel
face
pre
ssur
e (M
Pa)
Figure 6 e applied tunnel face pressure
Advances in Civil Engineering 5
leading to the reduction of the horizontal diameter of theexisting tunnel With the tunnelling of the southboundtunnel there are a reduction negative convergence dis-placement on the left side of point B and an addition positiveconvergence displacement on the right side of point B Ascan be seen from Figures 10 and 11 the shapes of theconvergence displacement curve and the horizontal dis-placement curve are similar because the causes of conver-gence displacement and horizontal displacement aretheoretically the same
5 Reinforcement Schemeof theExistingTunnel
Figure 12 shows the reinforcement in the existing tunnelRadial reinforcement by an arc-shaped supporting steelplate connected to the tunnel segment and longitudinalreinforcement by channel section steel to provide lon-gitudinal tensile stress are conducted in the existingtunnel Radial steels and longitudinal steels are connected
by welding and so the reinforcement becomes a wholeone 25 rings are reinforced at rst on either side of theintersection point A before the underpassing of thenorthbound tunnel and the whole reinforcement is
Northbound tunnelSouthbound tunnel
430 440 450 460 470 480 490 500 510 518
35
40
45
50
55
60
65
70
75
80
e corresponding ring number of the upline tunnel of Line 1
45 50 60 70 80 90 100 110 120 130 135e ring number of the northbound tunnel of Line 4
220 210 200 190 180 170 160 150 140e ring number of the southbound tunnel of Line 4
e g
rout
ing
volu
me (
m3 )
Figure 7 e grouting volume
e manual monitoring points ofconvergence displacement
Track bed
Take one point as the automatic monitoringpoint of the vertical and horizontal displacement
Centerline
e manual monitoring pointsof horizontal displacement
Figure 8 Arrangement of the monitoring points at the crosssection of the existing tunnel
400 420 440 460 480 500 520 540 560 580 600 620 640 660 670minus12
minus10
minus8
minus6
minus4
minus2
0
2
4
6 C
e construction ofnorthbound tunnel
e construction ofsouthbound tunnel
201453 Ring 135 2014515 Ring 226
Long-term monitoring201465
20146152014714
20141019
B
2013129 Ring 4820131227 Ring 72 2014112 Ring 1762014128 Ring 275
e shield is shut down2014227
Ver
tical
disp
lace
men
t (m
m)
e ring number of upline tunnel of Line 1
A
Figure 9 Time-varying vertical displacement of the existingtunnel
400 420 440 460 480 500 520 540 560 580 600 620 640 660 670
minus14minus12minus10minus8minus6minus4minus2
02468
10121416
C
e construction of thesouthbound tunnel
2014420 Ring 362014502 Ring 1242014528 Ring 270
Long-term monitoring20141019
e construction of thenorthbound tunnel
2013129 Ring 4820131227 Ring 72201415 Ring 1252014128 Rring 275
e ring number of upline tunnel of Line 1
Hor
izon
tal d
ispla
cem
ent (
mm
)
A B
Figure 10 Time-varying horizontal displacement of the existingtunnel
6 Advances in Civil Engineering
completed after the construction of the northboundtunnel e reinforcement range is from ring 424 to ring510 (ie 25 rings on either sides of the intersection pointsA and B)
6 Theoretical Analysis of the ReinforcementDesign
e deformed tunnel lining reinforced by inner bondingsteel plates has been partially or entirely used in manybuilt tunnels or currently under construction To better
understand the benets coming from the reinforcementmethod the behavior of the tunnel is investigated
Figure 13 shows a schematic view of the existingtunnel-soil tunnelling interaction e analysis methoddemonstrated in this paper can be divided into two stepsrstly estimating the greeneld displacement induced bythe tunnelling Secondly calculating the responses of theexisting tunnel subjected to the soil displacement emethod of analysis is based on three assumptions (1) theabove existing tunnel does not aect the displacement ofsoil due to tunnelling (2) the soil foundation is assumedas the Winkler or the Pasternak foundation and (3) thesoil displacement is calculated by superimposing theindependent settlement predicted for each individualtunnelling
In this paper the tunnels are considered as an innitebeam on the Winkler foundation an innite beam on thePasternak foundation and a nite beam on the Pasternakfoundation (the new method) Comparing the decentectionrotation angle normalized bendingmoment and shear forceof the existing tunnel with constant stiness based on thethree models the new method is veried en dierentstiness of the reinforcement is taken into considerationand the optimal reinforcement range of the existing tunnel isdiscussed
61 e Subsurface Soil Displacement due to Tunnelling Forthe theoretical analysis the alignment of the new tunnels andthe existing tunnel is assumed to be straight Figure 14 showsa schematic diagram of the new tunnels and the existingtunnel in this case e greeneld settlement s(x) due to thetunnelling can be replaced by the equivalent distributed loadq(x) acting on the beam on the third assumption as follows
q(x) ks(x) (2)
In this study subsurface settlement s(x) at the depth of zinduced by tunnelling is calculated based on closed-formanalytical solutions presented by Loganathan and Poulos[13] as follows
s(x) ε0R2 zminusHx2 +(zminusH)2
+(3minus 4v)z +H
x2 +(z +H)2minus2z x2 minus(z +H)2( )
x2 +(z +H)2( )2
middot e minus138x2(H+R)2minus069z2H2( ) (3)
Conv
erge
nce d
ispla
cem
ent (
mm
)
400 420 440 460 480 500 520 540 560 580 600 620 640 660minus7minus6minus5minus4minus3minus2minus1
012345
e construction ofsouthbound tunnel
201453 Ring 135201459 Ring 1912014528 Ring 270
e shield is shut down201461
e construction ofnorthbound tunnel
2013129 Ring 4820131222 Ring 6420131230 Ring 87
e shield is shut down201424
e ring number of the upline of Line 1
A B
Figure 11 Convergence displacement of the existing tunnel
Figure 12 Reinforcement in the existing tunnel
xk1
k2
e existing tunnel
Sz(x)
Figure 13 A schematic view of the existing tunnel-soil tunnellinginteraction
Advances in Civil Engineering 7
where s(x) subsurface settlement of the soils due to tun-nelling R tunnel radius z depth below ground surfaceH depth of the axis of the new tunnel vPoissonrsquos ratiox lateral distance from tunnel centerline and ε0 equivalentground loss ratio which is dened as
ε0 4Rg + g2
4R2 (4)
g Gp + ulowast3D + w asymp Gp (5)
where g is the gap parameter (Lee et al [14]) ulowast3D is thethree-dimensional elastoplastic deformation at the tunnelface w is the workmanship factor and Gp represents thephysical gap between the outer skin of the shield and thelining which is given as
Gp 2Δ + δ (6)
where Δ is the thickness of the tail piece and δ is the clearancerequired for erection of the tail piece e parameter w can beneglected due to considerable experience with the equipmentand good tunnelling technique of construction workers basedon the study of Rowe and Lee [15]
62 Analytical Methods Derived from Prior Studies Attewellet al [16] proposed a tunnel-soil interaction model andsolved the analytical solution of the longitudinal displace-ment of an innite tunnel
d4w(x)dx4
+ 4λ4p(x) 4λ4q(x) (7)
where q(x) is the concentrated load acting on the originpoint of the beam EI is the centexural stiness of a tunnel
which is recommended by Ye et al [17] and k is the subgrademodulus that represents the pipe-soil interaction
If the soil foundation is assumed as the Winklerfoundation the foundation counterforce p(x) can beexpressed as
p(x) kw(x) (8)
e two-parameter model of foundation can capture theshear resistance of soil Based on the assumption of theplane strain condition the displacement of ground in they-direction is neglected If the soil foundation is assumed asthe Pasternak foundation the foundation counterforce p(x)can be expressed as
p(x) minusGnabla2w(x) + kw(x) (9)
where G is the coelaquocient of the shear element in thePasternakrsquos model with the dimension of force per unitlength
Attewell et al [16] presented a solution for the con-centrated load q on an innite beam resting on the Winklerfoundation and computed the decentection as
w(x) 1
8EIλ3qeminusλx( cos λx + sin λx) (10)
It can be concluded that each of the concentrated loadks(τ) at point τ contributes the following amount to thedecentection of the existing tunnel on the Winkler foundation
w(x) 1
8EIλ3int+infin
minusinfin2Rks tminus L1( )sin α( ) eminusλ|xminusτ|
(cos λ|xminus τ| + sin λ|xminus τ|)dτ(11)
where L1 is the distance between the coordinate origin of theexisting tunnel and the intersection point between the newtunnel and the existing tunnel and α is the intersection anglebetween the new tunnel and existing tunnel e decentectionrotational angle bending moment and shear force of theexisting tunnel which is assumed as an innite beam on theWinkler foundation and the Pasternak foundation are il-lustrated in Appendix A
63 A Brief Description of the NewMethod In this study wewant to gure out the eects of the reinforcement on theexisting tunnel the stiness varies in this case and theexisting tunnel assumed as an innite beam is not suitableAssuming the existing tunnel with nite length and in-troducing (7) to (9) the basic dierential equation gov-erning the centexure of the beam resting on the Pasternakfoundation can be written in the form with an assumptionof Gλ2klt 1
EiIid4wi(x)dx4minusGibi
d2wi(x)dx4
+ kibiwi(x) biqi(x) (12)
where b1i bi(1 +(Gikiradic
bi)) subscript i represents the ithelement of the beam bi is the width of the beam and b1i isthe modied width of the beam Assuming a solution of (12)in the form of w(x) eλx the general solution of verticaldisplacement of the beam is as follows
y
e southbound tunnel
e northbound tunnel
e e
xistin
g tun
nel
x
L 1L 2
0x 1
x 2x 3
x ix i+
1
x nminus2
x nminus1
x n
O2
O1
xx
Figure 14 A schematic diagram of tunnels for the analytical methods
8 Advances in Civil Engineering
w(x) sum4
i1CiFi(x) (13)
where Ci (i 1 2 3 4) are undetermined parameters whichare dependent on the boundary conditions and Fi (x)(i 1 2 3 4) are the basic functions of the general solutionwhich are given as follows
F1i(x) cos a1icix( ) sinh a2icix( ) (14)
F2i(x) cos a1icix( ) cosh a2icix( ) (15)
F3i(x) sin a1icix( ) cosh a2icix( ) (16)
F4i(x) sin a1icix( ) sinh a2icix( ) (17)
where ci kib14EiIi4radic
a1i 1minusGic2i kiradic
a2i1 + Gic2i kiradic
e relationship of the decentection rotational angle
bending moment and shear force of the beam proposed byLancaster and Mitchell [18] is as follows
θi(x) wprimei(x)Mi(x) minusEiIiwPrime(x)
Qi(x) minusEiIiw(x) + Gpib1iwprimei(x)
(18)
where θi(x) Mi(x) and Qi(x) are the amplitudes of ro-tational angle bending moment and shear force of thesection area respectively
Figure 15 shows that dierent stiness is replaced byequivalent stepped stiness According to dierent stinessand distributed load the local coordinate system is establishedas illustrated in Figure 16 Selvadurai [12] suggests an initialparameter method to solve the parameters of the beam efour parameters at the origin of the coordinates arewi(0i) θi(0i)Mi(0i) and Qi(0i) and four parameters at theend of the element are wi(xi) θi(xi) Mi(xi) and Qi(xi)C1 C2 C3 and C4 can be expressed by wi(0i) θi(0i)Mi(0i) and Qi(0i) by introducing xi 0 to (13) and (18)
wi(xi) θi(xi)Mi(xi) and Qi(xi) can be rewritten in thematrix-array form as
wi xi( )θi xi( )Mi xi( )Vi xi( )1
k11 k12 k13 k14 k15
k21 k22 k23 k24 k25
k31 k32 k33 k34 k35
k41 k42 k43 k44 k45
0 0 0 0 1
middot
wi 0i( )θi 0i( )Mi 0i( )Vi 0i( )1
(19)
where [kij] is a stiness submatrix with 5 times 5 order ij 1 2 5M and V are respectively the bending momentand the generalized shear force Formula (19) can be sim-plied as
Ni xi( ) KijNi 0i( ) (20)
where Ni(xi) [Wi(xi) θi(xi)Mi(xi) Qi(xi) 1]T andNi(0i) [Wi(0i) θi(0i)Mi(0i) Qi(0i) 1]T
Figure 17 shows that curved distributed load is replacedby equivalent triangular and square distributed load whichcan be expressed as
qiminus1 +qi qiminus1 + ξqi minus qiminus1Li
( ) (21)
where qiminus1 and qi are respectively the load of the startingpoint and the ending point of the ith element and ξ is theposition of arbitrary load From (19) we can obtain
k15 minusbintx0 qiminus1 +qi( ) aminus12i F1i[xminus ξ]minus aminus11i F3i[xminus ξ]( )dξ
4c3i EiIi
(22)In order to avoid numerical errors in the calculation
process the integration of ki5 is shown in Appendix B erest of kij can be referred in the work by Selvadurai [12]
Figure 18 shows the tunnel matrix transfer diagramAssuming that the tunnel is composed of N elements thex-coordinate of beginning points of the ith element isxi(i 0 1 2 n) and the length of each element is Liwhere Li xi minus ximinus1
E0I0
0 x1 x2 x3 ximinus1 xi xnminus1 xn x
E1I1E2I2
Eiminus1Iiminus1Enminus1Inminus1
Figure 15 Sketch of equivalent stepped stiness
qi(x)
pi
Qi Qi+1Mi Mi+1
Oimi
xi
Figure 16 Forces and local coordinate system
q0q1 q2
qiminus1 qnminus1
x1 x2 x3 ximinus1 xi xnminus1 xn x
q (x)
0
Figure 17 Replacement of variable load with trapezoid load
Advances in Civil Engineering 9
For the rst element
N1(x) K1(x) middotN1 01( ) x isin 0 L1( ) (23)
For the end point of the rst element
N1 x1( ) K1 x1 minus 0( ) middotN1 01( ) (24)
For the end point of the ith element
Ni xi( ) Ki xi( ) middotKiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( ) middotN1 01( )(25)
For x at the ith element section where x is a globalcoordinate
Ni(x)Ki xminus ximinus1( ) middotKiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( ) middotN1 01( )(26)
Suppose
Ai(x) Ki xminusximinus1( ) middot Kiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( )(27)
en (20) can be expressed as
Ni(x) Ai(x) middotN1 01( ) (28)
us the relation between the vectorNn(xn) at the pointxn and the vector N1(01) at the start point x0 is
Nn xn( ) An xn( ) middotN1 01( ) (29)
where
An xn( ) prod1
inKi Li( ) (30)
e boundary condition of the existing tunnel assumedas an innite long tunnel considering the shear stress of soilcan be written as
M(0) 0
V(0) 2
k
G
radic
middot G middot R middot w(0)
V(L) 0
V(L) minus2
k
G
radic
middot G middot R middot w(L)
(31)
By applying the boundary conditions to (19) the de-centection rotational angle bending moment and shear forceof the overall beam can be obtained
64 Case Study
641 Parameter Selection To enable a direct compari-son assumed parameters of the soil and existing tunnelare selected in this study Poissonrsquos ratio of the soil is0388 e grouting process is regarded as the inverseprocess of soil loss due to tunnelling and g can beexpressed as g (2Δ + δ)(1minus η) where η is the injectionratio
For the Pasternak foundation as the thickness of theelastic layer is assumed to be 25 B the subgrade elasticmodulus k and shear stiness G are suggested by Xu [19] asfollows
k 5E0β
16 1minus v05pt2( )times 122
G 13E0B
2
32 1 + v0( )βtimes 085
(32)
where β is a modied parameter (β 129) and E0 and v0 arerespectively the elastic modulus and Poissonrsquos ratio forelastic foundation which are given as follows
E0 Es
1minus v2s
v0 vs
1minus vs
(33)
where the empirical relation between the Youngrsquos modulusEs and the compression modulus Es01minus02 of soft soils is builtup by Yang and Zhao [20]
E0 (25 sim 35)Es01minus02 (34)
As illustrated in Figure 14 points O1 and O2 are re-spectively the intersection points A and B in practicalcoordinate transformation expressions of the axis xprime and xPrimeare respectively
xprime xminusL1( )sin αxPrime xminusL2( )sin α
(35)
where L1 and L2 are the distance from the beginningpoint of the existing tunnel to points O1 and O2 re-spectively e length of the existing tunnel is assumedas 250 m and the ring spacing is 12 m So the tunnel isdivided into 209 sections and each section is 12 m inlength
N1(01) Ni(xi)N1(x1) = N2(02) N2(x2)L1
L1
E1I1
G1 k1 G2 k2 G3 k3 Gi ki Gn knGnminus1 knminus1
E2I2 E3I3 EiIi EnInEnminus1Inminus1
L2 L3 Li LnLnminus1
L2 Li
x x x
xx1 x2 x3 xi-1 xnminus2 xnminus1 xnxi0
Ni(0i)
Figure 18 e matrix transfer diagram
10 Advances in Civil Engineering
7 Discussions
e normalized bending moment and shear force are de-ned as
Mi MiL0EiIi
Qi QiL
20
EiIi
(36)
where Mi and Qi are respectively the normalized bendingmoment and shear force Mi and Qi are respectively thecalculated bending moment and shear force and L0 is thelength of the existing tunnel
Figure 19 shows the comparison of the decentection ro-tational angle bending moment and shear force of theexisting tunnel with a constant stiness which is based onthree kinds of models an innite tunnel on the Winklerfoundation an innite tunnel on the Pasternak foundationand a nite tunnel on the Winkler foundation Note that the
368 380 400 420 440 460 480 500 520 540 560 580 600 612
minus0007
minus0006
minus0005
minus0004
minus0003
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)
minus0002
minus0001
0000
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundatione observed data on 201465
e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
minus000015
minus000010
minus000005
000000
000005
000010
000015
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
(b)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
e b
endi
ng m
omen
t of t
he ex
istin
g tu
nnel
(kN
middotm)
387 400 420 440 460 480 500 520 540 560 580 595
0
minus6000
minus5000
minus4000
minus3000
minus2000
minus1000
1000
2000
3000
4000
e ring number of the existing tunnel
(c)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
387 400 420 440 460 480 500 520 540 560 580 595
minus600
minus400
minus200
0
200
400
600
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(kN
)
(d)
Figure 19 (a) Comparison of the decentection of the existing tunnel using dierent models (b) Comparison of the rotational angle ofthe existing tunnel using dierent models (c) Comparison of the normalized bending moment of the existing tunnel using dierentmodels (d) Comparison of the normalized shear force of the existing tunnel using dierent models
Advances in Civil Engineering 11
existing tunnel is only subject to soil deformation force inthis case e displacement and the decentection of beam areparticular similar concepts owing to the fact that the existingtunnel is securely attached to the foundation generallywithout rigid body motion Fairly good agreement can beobserved between the innite tunnel on the Pasternakfoundation and the nite tunnel on the Pasternak founda-tion which shows that the method proposed in this paper iscorrect and can be used to analyze similar problems Becauseof considering soil shear stress vertical stress and de-formation are aected and weakened the displacement of
the existing tunnel on the Pasternak foundation is smallerthan that on the Winkler foundation without consideringthe shear stress of soil Using the Winkler foundation modelis in good agreement with the observed data It can be seenfrom (3) that the displacement of the existing tunnel isproportional to the grouting elaquociency (ie η) Due to theuncertainty of η it cannot be decided which one is the bestanalytical method It is clear that the maximum normalizedbending moment occurs at the points O1 and O2 which arethe nearest points between the existing tunnel and the newtunnels
minus0003
minus0002
minus0001
0000387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
minus000010
minus000005
000000
000005
000010
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e ring number of the existing tunnel
(b)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10
minus0003
minus0002
minus0001
0000
0001
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
xL
(c)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10xL
minus008
minus006
minus004
minus002
000
002
004
006
008
The n
orm
alise
d sh
ear f
orce
of t
he ex
istin
g tu
nnel
(d)
Figure 20 (a) e eects of dierent reinforcement stiness on the decentection of the existing tunnel (b) e eects of dierent re-inforcement stiness on the slope of the existing tunnel (c)e eects of dierent reinforcement stiness on the normalized moment of theexisting tunnel (d) e eects of dierent reinforcement stiness on the normalized shear force of the existing tunnel
12 Advances in Civil Engineering
e reinforce range of the existing tunnel is assumed as25 rings on either side of the intersection point A and thevarious stiness is taken into consideration using themethod proposed in this paper Figure 20 shows the in-centuence of dierent stiness of the reinforcement on theexisting tunnel (ie 4 EI EI and 14 EI) It is evident thatthe reinforcement can reduce the displacement rotationalangle normalized bending moment and shear force of theexisting tunnel while tunnel damage (reduce the stinessof reinforcement) has the opposite eect It is evident thatthe change of reinforcement stiness has greater eect onthe normalized bending moment and the normalized shearforce of the existing tunnel particularly at the incentection
points of the normalized bending moment and normalizedshear force curve
e reinforce stiness of the reinforcement range isassumed as 4 EI and the various reinforcement ranges istaken into consideration using the method proposed in thispaper Figure 21 shows the incentuence of dierent re-inforcement ranges on the existing tunnel when thenorthbound tunnel of Line 4 underpasses e normalizedbending moment on either side of the intersection point istoo large when the reinforcement range is 69ndash892m andthe normalized shear force on either side of the intersectionpoint is relatively too large when the reinforcement range isless than 592ndash992m So it is necessary to reinforce larger
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00035
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
The d
eflec
tions
of t
he ex
istin
g tu
nnel
(m)
xL
(a)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus000010
minus000005
000000
000005
000010
The s
lope
of t
he ex
istin
g tu
nnel
(rad
)
xL
(b)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
00005
00010
00015
xL
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
(c)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus010
minus008
minus006
minus004
minus002
000
002
004
006
008
010
xLTh
e nor
mal
ised
shea
r for
ce o
f the
exist
ing
tunn
el
(d)
Figure 21 (a)e eects of dierent reinforcement ranges on the decentection of the existing tunnel (b)e eects of dierent reinforcementranges on the slope of the existing tunnel (c)e eects of dierent reinforcement ranges on the normalized moment of the existing tunnel(d) e eects of dierent reinforcement ranges on the normalized shear force of the existing tunnel
Advances in Civil Engineering 13
than the range of 492ndash1092m in the existing tunnel in thiscase that is larger than 30m (25 rings) on either side of theintersection point )ere is no big difference in the bendingmoment and shear force of the existing tunnel when thereinforcement ranges are 492ndash1092m 392ndash1192m and292ndash1292m In order to reduce the costs of reinforcementreinforcement in the 30m range (ie 25 rings) on eitherside of the intersection point is the best choice whichverifies the actual design
8 Conclusions
Based on the abovementioned statements the response of theexisting tunnel is analyzed and analytical methods are proposedfor verification and the reinforcement design Main conclusionsderived from the analysis are as follows
(1) )e displacement of the existing tunnel changes withthe relative position of the shield and the existingtunnel Shield parameters have a major impact on theexisting tunnel )e heaves of the existing tunnelmight be caused by the large applied face pressure andbulkhead addictive thrust When the shield tail isdriving beyond the selected monitoring rings theselected monitoring rings settle rapidly due to theclosure of the shield tail void
(2) )e southbound tunnel of Line 4 has less effect onthe left side of the intersection point A on theexisting tunnel because of the northbound tunnelrsquosbarrier effect A variety of measures of displacementprevention of Line 1 such as the control of shield
parameters reinforcement of the existing tunnel canensure the normal operation of metro Line 1
(3) )e analytical method proposed in the paper is ver-ified by the methods derived from prior studies It ispossible to take the different stiffness and curved loadsinto account by applying the stepped stiffness andtrapezoidal load while using a local coordinate systemin the derivation of the new method )e change ofreinforcement stiffness has greater effect on thenormalized bending moment and the normalizedshear force of the existing tunnel particularly at theinflection points of the normalized bending momentand normalized shear force curve )e analyticalmethod can be applied for the reinforcement designand safety check of the built tunnel
Appendices
A1 An Infinite Beam on the WinklerFoundation
θ(x) 1113946+infin
minusinfin
minusλ2s((tminus L)sin α)eminusλ(xminust) sin(λ(xminus t))
b
λ2s((tminus L)sin α)eλ(xminust)sin(λ(tminus x))
b
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎭
dt
(A1)
M(x) 14λ
1113946+infin
minusinfinks((tminusL)sin α)e
minusλ|xminust|( cos(λ|xminus t|)minus sin(λ|xminus t|))dt (A2)
Q(x) 1113946+infin
minusinfin
minus12
ks((tminusL)sin α)eminusλ(xminust)cos(λ(xminus t))dt xminus tgt 0
12
ks((tminusL)sin α)eminusλ(tminusx)cos(λ(tminusx))dt xminus tle 0
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
⎫⎪⎪⎪⎬
⎪⎪⎪⎭
dt (A3)
A2 An Infinite Beam on the PasternakFoundation
w(x) λ2bk
1113946+infin
minusinfinks((tminus L)sin α)De
minusa1λ|xminusτ| 1a1
cos a2λ|xminus τ|( 1113857 +1a2
sin a2λ|xminus τ|( 11138571113888 1113889dτ
θ(x) 1113946+infin
minusinfin
minusλ2s((tminusL))eminusa1λ(xminust)sin a2(xminus t)( 1113857
a1a2b
λ2s((tminus L))ea1λ(xminust)sin a2(tminus x)( 1113857
a1a2b
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A4)
14 Advances in Civil Engineering
M(x) 14λ
1113946+infin
minusinfinks((tminus L)sin α)e
minusa1λ|xminust| 1a1
cos a2λ|xminus t|( 1113857minus1a2
sin a2λ|xminus t|( 11138571113888 1113889dt
Q(x) 1113946+infin
minusinfin
minus12
ks((tminus L)sin α)eminusa1λ(xminust) cos a2λ(xminus t)( 1113857minus
a21 minus a 2
22a1a2
1113888 1113889sin a2λ(xminus t)( 11138571113890 1113891dt xminus tgt 0
12
ks((tminusL)sin α)eminusa1λ(tminusx) cos a2λ(tminusx)( 1113857minus
a21 minus a2
2
2a1a21113888 1113889sin a2λ(tminusx)( 11138571113890 1113891dt xminus tle 0
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A5)
B1 Coefficients of the Matrix ki5
k15 minusbiqiminus1
4a1ia2ici4EiIi a1i
2 + a2i2( 1113857
minus2a1ia2i + 2a1ia2iF2i
+ a1i2 minus a2i
2( 1113857F4i
⎛⎜⎝ ⎞⎟⎠
+bi qi minus qiminus1( 1113857
4a1ia2ici5EiIiL3 a1i
2 + a2i2( 1113857
2
a2i minus3a21i + a2
2i( 1113857F3i
+a1i 2a2i a21i + a2
2i( 1113857cix + a1i2 minus 3a2i
2( 1113857F1i( 1113857
⎛⎜⎝ ⎞⎟⎠
(B1)
k25 biqiminus1 a2iF3i minus a1iF1i( 1113857
2a1a2 a21 + a 2
2( 1113857c3EiIi
+bi qi minus qiminus1( 1113857 2a1ia2i minus 2a1ia2iF2i minus a1i
2 minus a2i2( 1113857F4i( 1113857( 1113857
2a1ia2i a1i2 + a2i
2( 11138572ci
4EiIix1113872 1113873 (B2)
k35 minusbiqiminus1F4i
2a1ia2ici2 minus
bi qi minus qiminus1( 1113857 a2iF3i minus a1iF1i( 1113857
2a1ia2ici3 a1i
2 + a2i2( 1113857x
(B3)
k45 minusbiqiminus1
a1i2 + a2i
2( 1113857ci
1113888a1i 1 + 2a1i2a22i minus 2a2i
41113872 1113873F3i + a2i 1 + 2a1i
2a22i minus 2a1i
41113872 1113873F1i1113889
minusbi qi minus qiminus1( 1113857
a21i + a2i
2( 11138572c 2
i x
4a1i2a2i
2 minus 1( 1113857 a1i2 minus a2i
2( 1113857 + a2i2 minus 4a1i
4a2i2 + a1i
2 minus1 + 4a2i4( 1113857( 1113857F2i
minus2a1ia2i a 21i minus a 2
2i( 11138572 minus 11113872 1113873F4i
⎛⎜⎝ ⎞⎟⎠
(B4)
Conflicts of Interest
)e author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
)e author appreciates the help from the staff of Hangzhouand Hong Kong Tunnel Company Ltd during in-strumentation setup and data acquisition )is study issupported by the National Natural Science Foundation ofChina (no 41702313)
References
[1] S H Kim Model testing and analysis of interactions betweentunnels in clay [PhD 3esis] Department of EngineeringScience University of Oxford Oxford UK 1996
[2] H Y Liu J C Small J P Carter and D J Williams ldquoEffectsof tunnelling on existing support systems of perpendicularlypassing tunnelsrdquo Computers and Geotechnics vol 36 no 5pp 880ndash894 2009
[3] I Yamaguchi I Yamazaki and Y Kiritani ldquoStudy ofgroundndashtunnel interactions of four shield tunnels driven inclose proximity in relation to design and construction ofparallel shield tunnelsrdquo Tunnelling and Underground SpaceTechnology vol 13 no 3 pp 289ndash304 1998
[4] T I Addenbrooke and D M Potts ldquoTwin tunnel interactionsurface and subsurface effectsrdquo International Journal ofGeomechanics vol 1 no 2 pp 249ndash271 2001
[5] J P Kimmance S Lawrence O Hassan and N J PurchaseldquoObservations of deformations created in existing tunnels byadjacent and cross cutting excavationsrdquo in Proceedings ofGeotechnical Aspects of Underground Construction in SoftGround pp 707ndash712 London UK July 1996
[6] E Leca and B New ldquoSettlements induced by tunnelling in softgroundrdquo Tunnelling and Underground Space Technologyvol 22 no 2 pp 119ndash149 2007
Advances in Civil Engineering 15
[7] S M Liao J H Liu R L Wang and Z M Li ldquoShieldtunnelling and environment protection in Shanghai softgroundrdquo Tunnelling and Underground Space Technologyvol 24 no 4 pp 454ndash465 2009
[8] F H Chehade and I Shahrour ldquoNumerical analysis of theinteraction between twin-tunnels influence of the relativeposition and construction procedurerdquo Tunnelling and Un-derground Space Technology vol 23 no 2 pp 210ndash214 2008
[9] X G Li and D J Yuan ldquoResponse of a double-decked metrotunnel to shield driving of twin closely underpasses tunnelsrdquoTunnelling and Underground Space Technology vol 28 no 1pp 18ndash30 2012
[10] T E Vorster A Klar K Soga and R J Mair ldquoEstimating theeffects of tunnelling on existing pipelinesrdquo Journal of Geo-technical and Geoenvironmental Engineering vol 131 no 11pp 1399ndash1410 2005
[11] Z G Zhang M S Huang and W D Wang ldquoEvaluation ofdeformation response for adjacent tunnels due to soilunloading in excavation engineeringrdquo Tunnelling and Un-derground Space Technology vol 38 no 3 pp 244ndash253 2013
[12] A P S Selvadurai Elastic Analysis of Soil-Foundation InteractionElsevier Scientific Pub Co Amsterdam Netherlands 1979
[13] N Loganathan and H G Poulos ldquoAnalytical prediction fortunnelling-induced ground movements in claysrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124no 9 pp 846ndash856 1998
[14] K M Lee R K Rowe and K Y Lo ldquoSubsidence owing totunnelling I Estimating the gap parameterrdquo CanadianGeotechnical Journal vol 29 no 6 pp 929ndash940 1992
[15] R K Rowe and KM Lee ldquoSubsidence owing to tunnelling IIEvaluation of a prediction techniquerdquo Canadian GeotechnicalJournal vol 29 no 6 pp 941ndash954 1992
[16] P B Attewell J Yeates and A R Selby Soil MovementsInduced by Tunnelling and their Effects on Pipelines andStructures Methuen Inc New York NY USA 1986
[17] F Ye C He H H Zhu et al ldquoAnalysis of longitudinalequivalent stiffness of shield tunnel considering lateral per-formancerdquo Chinese Journal of Geotechnical Engineeringvol 33 no 12 pp 1870ndash1876 2011 in Chinese
[18] P R Lancaster and D Mitchell ldquoProblems in Bending of Barsand Plates [M]rdquo Advanced Solid Mechanics Macmillan UK1980
[19] L Xu ldquoStudy on vertical settlement of shield tunnel in soft soilrdquoPhD dissertation Tongji University Shanghai China 2005
[20] M Yang and X Zhao ldquoAn approach for a single pile in layeredsoilrdquo Tongji University Natural Science vol 20 no 4pp 421ndash428 1992 in Chinese
16 Advances in Civil Engineering
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Control Scienceand Engineering
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International Journal of
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Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
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Submit your manuscripts atwwwhindawicom
having been carried out current understanding of the in-teraction between the two tunnels is still limited
In order to ensure the stability of the existing tunnellocal thickening is needed at the sides of the existing concretelining )ere are a plenty of research about the conventionalreinforcement methods such as inner steel plate re-inforcement or new reinforcement approaches such as FRCor composite concrete but they all focus on the performanceof reinforcement on the single ring )e effect of re-inforcement on the longitudinal behavior of the tunnel is notyet clear in current analysis )e analytical method oflongitudinal displacement of the tunnel due to adjacentexcavation andmultiple tunnelling is researched a lot (1) theelastic continuum models developed by Vorster et al [10](2) theWinkler model [11] and (3) the two-parameter elasticmodels When variable stiffness of the existing tunnel needsto be considered the methods are no longer applicableSelvadurai [12] divided the beam on foundation into Nelements and used the method of initial parameters toanalyze the beamrsquos displacement In the paper we assumethe existing tunnel as a beam resting on a two-parameterfoundation combining the initial parameter method and thetransfer matrix method to analyze the reinforcement effecton the longitudinal behavior of the existing tunnel
)e major objectives of this paper are (1) to investigatethe influence of the tunnel driving parameters and therelative distance between the shield and the existing tunnelon the existing tunnel based on the interpretation of the fieldmeasured data and (2) to study the effects of differentstiffness and ranges of reinforcement on the existing tunnelusing analytical methods
2 Project Overview
)e location of the Line 4 tunnels and the upline of Line 1 areshown in Figure 1 )e upline of Line 1 is the existing tunnelbuilt in 2012 Figure 2 shows a plan view of tunnel alignmentand arrangement of the monitoring points in the existingtunnel )e shield of the northbound tunnel of Line 4 startsfrom Guanhe Station passes under the existing tunnel witha small angle of 23deg and reaches East railway station at last)en the shield machine is reassembled with tunnels ina reversed direction )e northbound tunnel with a length of329675m consists of 275 rings numbered starting with zerofrom Guanhe Station to East railway station while thesouthbound tunnel with a length of 323834m consists of 270rings numbered starting with zero fromEast railway station toGuanhe Station )e lateral distance between the northboundtunnel and the southbound tunnel is 94ndash150m
Figure 3 shows the longitudinal profile of soil andtunnels )e vertical distance between the existing tunneland Line 4 is 212m)e existing tunnel and new tunnels areall built using an articulated shield tunnelling machine withan outer diameter of 64m and a length of 85m )e spokesplus panel-type cutter head are used and the aperture ratioof cutter head is 40 Each ring of new tunnels consists of sixprecast concrete segments )e outer diameter the innerdiameter and the thickness of the segment are 62m 55mand 035m respectively
21 Soil Condition )e engineering properties of the rockand the soils in this site are very complicated Figure 4 showsgeotechnical parameters of soil in this site)e soil layers fromtop to bottom are (1) fill (3) silty clay (4) clay (6) the muddysilty clay and (8) soft silty clay and gravel layer )e existingtunnel and new tunnels are located in the muddy silty claylayer at a depth of 20m and 25m respectively Along themetro line an extensive geotechnical investigation is carriedout )e standard penetration test (SPT) vane shear test(VST) piezocone test (CPTU) flat plate dilatometer test(DMT) and water pumping test (WPT) are conducted alongLine 1 )e maximum water content of clay and muddy siltyclay is about 40 which is close to the liquid limit value )eaverage undrained shear strength of undisturbed clay andremolded clay which is obtained by VST is 705 kPa and109 kPa respectively )e coefficient of the at-rest earthpressure (K0) of soft clay is obtained by DMT
22 Hydrological Geology )e shallow ground water is porephreatic water mainly found in layers from (1) to (8) )eelevation of the water surface is 2232m )e laboratorypenetration test and field steady flow test can be seen inTable 1 It can be seen that the permeability coefficient ofthe clay layer is very small and the permeability coefficient ofthe gravel layer is 1ndash2md )e seepage induced from waterhead difference might influence the stability of tunnels
3 Shield Driving Parameters
Figure 5 presents advance-time curve of the shield Beforeunderpassing the existing tunnel the shield advances with a veryslow rate )e tunnel advancing rate is 7-8 ringsday and thepenetration rate of the shield ranges from 20 to 25mmmiddotminminus1
Figure 6 shows the applied tunnel face pressure Inpractice the tunnel face pressure should be adjusted with thetheoretical value as well as the feedback displacement of theexisting tunnel )e existing tunnel redistributes the soilstress and the applied tunnel face pressure should be ad-justed by (1) as follows
p2 K0cprimeh + pw + Q1 minusQ2 plusmn 20 kPa (1)
where K0 is the coefficient of earth pressure at rest cprime is theeffective gravity of soil h is the thickness of overburdendepth (m) pw is the water pressure Q1 is the weight of theexisting tunnel (kN) Q2 is the weight of the soil with thesame internal volume of the existing tunnel (kN) and 20 kPais the pressure fluctuation
)e applied tunnel face pressure of the southboundtunnel cannot be calculated by (1) which is due to the re-distribution of the earth pressure after the construction ofthe northbound tunnel nearby and should be adjustedaccording to the displacement of the existing tunnel
Figure 7 shows the tail void grouting volume of the shieldof the northbound and southbound tunnels of Line 4 Shieldparameter adjustments play an important role in the dis-placement controls reducing the tail void grouting volumeand applied tunnel face pressure when Line 1 heaves Op-positemeasures are takenwhen there is a subsidence in Line 1
2 Advances in Civil Engineering
4 Observation Results and Discussions
41 Monitoring Arrangement of Line 1 Arrangement ofthe monitoring rings of the existing tunnel can be seenin Figure 2 e total station Topcon MS05AX xed onthe tunnel sidewall arranged from rings 404 to 559 inthe existing tunnel is an automatic real-time measuringsystem which is used to monitor vertical and horizontal
displacement One monitor section is set every two rings inthe most aected zone (from rings 447 to 499 in the existingtunnel) and one monitor section is set every ve rings inthe rest part An LECAI D5 hand-held distance nder isused to monitor the converge displacement every 5 ringsfrom 404 to 679 in the existing tunnel e arrangement ofmonitoring points at the cross section of the existing tunnelis shown in Figure 8
Figure 1 Location of the new tunnels and the existing tunnel in Hangzhou A Guanhe Station B East railway station
409
404 414 419 424 429 434 439
444 447449451453455457459461463465467469471473475477479481483485487489491493495497499 509
514 519
524
529
534
539
544
549
554
559
564
569
574
579
584
589
594
599
604
609
614
619
624
629
634
639
644
649
679
15101520253035404550556065707580859095100
105
110
115
120
125
130
135
140
145
150155
165170175180
18519019520021021522022523023540
45
5055
60
65
70
75
80
85
90
95
100
105
110
115
120
125
130
135
140
145
150
155
160
165
170
175
180
185
190
195
200
205
210
215
220
225
230
235
240
245
250
255
260
265
270
275
East railway station
Guanhe Station
786 m
966 m835 m
35302520
674
669
664
659
654205
160
504
240245250255
e northbound tunnel of Line 4
e southbound tunnel of Line 4
Reinforcement of the existing tunnel
Zha-nong-kou Station
e upline tunnel of existing Line 1
Da-ju-yuan Station
A B
C
e auto monitoring point of vertical and horizontal displacement
e manual monitoring point of horizontal displacement
e manual monitoring point of vertical displacment
e manual monitoring point of convergence displacement
Figure 2 Plan view of tunnel alignment and location of the monitored rings
Line 4
65
Elev
atio
n (m
)
1-1 sandy silt
2-1 silt
3-1mucky silty
clay
3-2 clay
4-1 sandy clay
5-1 clay
GuanheStation
East railwaystation
5-2mucky silty
clay
Line 1
141210 8 6 4 2 0minus2minus4minus6minus8
minus10minus12minus14minus16minus18minus20minus22minus24minus26minus28minus30
minus15077 minus14954
208 212
K20+
898
754
K21+
053
342
K21+
094
361
K21+
608
099
232West square (B period)
West square (D period)Bottom of the basement floor 2100
Communication pipeline
K21+
015
389
Water pipeline
K21+
144
111
Groundwater table
Chainage (m)
Bottom of the basement floor 2900
K20+
900
K21+
000
K21+
100
K21+
200
K21+
300
K21+
400
K21+
500
K21+
600
K20+
971
955
K21+
301
630
Initi
al p
oint
Term
inal
poi
nt
Figure 3 Longitudinal prole of soils and tunnels
Advances in Civil Engineering 3
42 Displacement of the Existing Tunnel e location ofpoint A as shown in Figure 2 is corresponding to the in-tersection point of a plan view of the existing tunnel andnorthbound tunnel of Line 4 e location of point B iscorresponding to the intersection point of the plan view ofthe existing tunnel and southbound tunnel of Line 4 emiddle point of points A and B is point C
421 Vertical Displacement of the Existing Tunnel Figure 9shows time-varying vertical displacement in the moni-toring rings of the existing tunnel e x-axis is the ringnumber of the existing tunnel A positive value of theordinate denotes heave while a negative value denotessettlement of the existing tunnel e selected monitoringrings start to heave when the shield face is 0ndash10m awayfrom the selected monitoring rings which is mainly due tolarge applied tunnel face pressure bulk addictive thrust(Figure 6) and the friction force between the shield shelland soil mass When the shield tail is far beyond the se-lected monitoring rings a reduction of heave in the selectedmonitoring points is observed which is mainly due to theclosure of the tail void Only small settlements and heavesare measured after the construction of the northboundtunnel During the 3-month shutdown of the shield ad-ditional settlement ranges from 2mm to 3mm can beobserved in rings from 430 to 520 Additional settlements
ranges from 2 to 4mm were measured in rings from 460 to540 during the southbound tunnelrsquos construction elong-term additional settlements monitored up to 140 days(from 201461 to 20141019) range from 2 to 4mm inrings from 460 to 510 e settlements of rings from 430 to520 range from 2mm to 12mm and the settlement curve ofthe upline of Line 1 is ldquoUrdquo shaped after the long-termmonitoring e settlement curve is approximately sym-metric about the dashed line C after the long-term mon-itoringemaximum settlement is 12mmwhich is locatedin ring 487
0 4 8 12 03 04 05 06 0716 18 20 22
50
45
40
35
30
25
20
15
10
5
020 30 40 50 0 10 20 300 10 20 30 40 025 030 035 040
Es (MPa)w () vk0
1 Fill
3 silt
4 clay
6 muckysilty clay
8 so siltyclay
Depth of metro 1 tunnel axis
r unit weightw water contentwp plastic limit
wL liquid limitv void ratioEs elasticity modulusCU consolidation undrainage conditionCU effective value in consolidation undrainage conditionUU unconsolidation undrainage condition
gravel
r (kNm)
Dep
th (m
)
wpwL
w
Soil layer
uucucu
Depth of metro 4 tunnel axis
uucucu
c (kPa) φordm
Figure 4 Soil prole and geotechnical parameters
Table 1 Laboratory penetration test and eld steady centow test
Soil layerLaboratory
penetration test (cms) Field steady centow test (cms)
Kv Kh K
3ndash2 479times10minus5 781times10minus5 421times10minus3
3ndash3 255times10minus4 230times10minus4 157times10minus3
3ndash5 963times10minus5 174times10minus4 mdash3ndash6 329times10minus4 301times10minus4 mdash4ndash3 239times10minus7 784times10minus7 6ndash1 206times10minus7 303times10minus7 mdash6ndash2 262times10minus7 128times10minus7 mdash8ndash1 188times10minus7 511times10minus7 mdash
4 Advances in Civil Engineering
422 Horizontal Displacement of the Existing Tunnel ehorizontal displacement of the existing tunnel with respectto the location of the shield is illustrated in Figure 10 Apositive horizontal displacement denotes northward trans-verse tunnel movement away from the original tunnelcenterline while a negative horizontal displacement denotessouthward transverse tunnel movement away from theoriginal tunnel centerline
ere are northward displacement on the left side ofpoint C and southward displacement on the right side ofpoint Ce horizontal displacement curve is approximatelysymmetric about point C after the long-term monitoringe maximum northward displacement is 10mm in the ringnear the intersection point A and the maximum southwarddisplacement is minus105mm in the ring near the intersectionpoint B after the completion of the two tunnelsrsquo construc-tion During the construction of the northbound tunnelrings from 453 to 471 move southward which is likely due tothe additional bulkhead additive thrust and the squeezingforce provided by the shield shell When the shield tail leavesring 459 rings from 439 to 459 move southward slowlyOnly 2 to 5mm additional southward displacement ismeasured in rings from 487 to 560 and nearly no dis-placement is observed on the left side of point B during theconstruction of the southbound tunnel which is mainly dueto the northbound tunnelrsquos barrier eect No change ofhorizontal displacement is observed in the long-term con-ditions (from 2014528 to 20141019)
423 Convergence Displacement of the ExistingTunnel Figure 11 shows the convergence displacement ofthe existing tunnel with respect to the locations of the shield
A negative value indicates the reduction horizontal diameterof the existing tunnel while a positive value indicates theaddition horizontal diameter of the existing tunnel It can beobserved that the convergence displacement is not obviouswhen the shield reaches ring 48 (22 rings away from theintersection point A) During the process of the shielddriving from ring 64 to ring 87 a signicantly additionalincrease in the negative convergence displacement in ringsfrom 440 to 490 is observed and themaximum displacementoccurs in the intersection point A e reason might be thatthe shield face squeezes one side of the existing tunnel
20950
21000
21050
21100
21150
21200
21250
21300268
270
e r
ing
num
ber o
f the
no
rthb
ound
tunn
el
70
Guanhe Station
paused for 106 days
56 days
e tunnel face of the northbound tunnele tunnel face of the southbound tunnel
Tunn
el ch
aina
ge (m
)
Date
46 days
East railway station
170
250
200
150
100
50
0
-50
0
50
100
150
200
250
e r
ing
num
ber o
f the
sout
hbou
nd tu
nnel
2013
12
3
2013
12
13
2013
12
23
2014
12
2014
11
2
2014
12
2
2014
13
020
144
10
2014
41
4
2014
42
0
2014
43
0
2014
51
0
2014
52
0
2014
53
0
Figure 5 Advance-time curve of the shield
430 440 450 460 470 480 490 500 510 515
020
022
024
026
028
030
032
034
036
038
040
Northbound tunnelSouthbound tunnel
225 220 210 200 190 180 170 160 150 143
The ring number of the northbound tunnel of Line 4
The corresponding ring number of the upline tunnel of Line 1
The ring number of the southbound tunnel of Line 4
50 60 70 80 90 100 110 120 130
The a
pplie
d tu
nnel
face
pre
ssur
e (M
Pa)
Figure 6 e applied tunnel face pressure
Advances in Civil Engineering 5
leading to the reduction of the horizontal diameter of theexisting tunnel With the tunnelling of the southboundtunnel there are a reduction negative convergence dis-placement on the left side of point B and an addition positiveconvergence displacement on the right side of point B Ascan be seen from Figures 10 and 11 the shapes of theconvergence displacement curve and the horizontal dis-placement curve are similar because the causes of conver-gence displacement and horizontal displacement aretheoretically the same
5 Reinforcement Schemeof theExistingTunnel
Figure 12 shows the reinforcement in the existing tunnelRadial reinforcement by an arc-shaped supporting steelplate connected to the tunnel segment and longitudinalreinforcement by channel section steel to provide lon-gitudinal tensile stress are conducted in the existingtunnel Radial steels and longitudinal steels are connected
by welding and so the reinforcement becomes a wholeone 25 rings are reinforced at rst on either side of theintersection point A before the underpassing of thenorthbound tunnel and the whole reinforcement is
Northbound tunnelSouthbound tunnel
430 440 450 460 470 480 490 500 510 518
35
40
45
50
55
60
65
70
75
80
e corresponding ring number of the upline tunnel of Line 1
45 50 60 70 80 90 100 110 120 130 135e ring number of the northbound tunnel of Line 4
220 210 200 190 180 170 160 150 140e ring number of the southbound tunnel of Line 4
e g
rout
ing
volu
me (
m3 )
Figure 7 e grouting volume
e manual monitoring points ofconvergence displacement
Track bed
Take one point as the automatic monitoringpoint of the vertical and horizontal displacement
Centerline
e manual monitoring pointsof horizontal displacement
Figure 8 Arrangement of the monitoring points at the crosssection of the existing tunnel
400 420 440 460 480 500 520 540 560 580 600 620 640 660 670minus12
minus10
minus8
minus6
minus4
minus2
0
2
4
6 C
e construction ofnorthbound tunnel
e construction ofsouthbound tunnel
201453 Ring 135 2014515 Ring 226
Long-term monitoring201465
20146152014714
20141019
B
2013129 Ring 4820131227 Ring 72 2014112 Ring 1762014128 Ring 275
e shield is shut down2014227
Ver
tical
disp
lace
men
t (m
m)
e ring number of upline tunnel of Line 1
A
Figure 9 Time-varying vertical displacement of the existingtunnel
400 420 440 460 480 500 520 540 560 580 600 620 640 660 670
minus14minus12minus10minus8minus6minus4minus2
02468
10121416
C
e construction of thesouthbound tunnel
2014420 Ring 362014502 Ring 1242014528 Ring 270
Long-term monitoring20141019
e construction of thenorthbound tunnel
2013129 Ring 4820131227 Ring 72201415 Ring 1252014128 Rring 275
e ring number of upline tunnel of Line 1
Hor
izon
tal d
ispla
cem
ent (
mm
)
A B
Figure 10 Time-varying horizontal displacement of the existingtunnel
6 Advances in Civil Engineering
completed after the construction of the northboundtunnel e reinforcement range is from ring 424 to ring510 (ie 25 rings on either sides of the intersection pointsA and B)
6 Theoretical Analysis of the ReinforcementDesign
e deformed tunnel lining reinforced by inner bondingsteel plates has been partially or entirely used in manybuilt tunnels or currently under construction To better
understand the benets coming from the reinforcementmethod the behavior of the tunnel is investigated
Figure 13 shows a schematic view of the existingtunnel-soil tunnelling interaction e analysis methoddemonstrated in this paper can be divided into two stepsrstly estimating the greeneld displacement induced bythe tunnelling Secondly calculating the responses of theexisting tunnel subjected to the soil displacement emethod of analysis is based on three assumptions (1) theabove existing tunnel does not aect the displacement ofsoil due to tunnelling (2) the soil foundation is assumedas the Winkler or the Pasternak foundation and (3) thesoil displacement is calculated by superimposing theindependent settlement predicted for each individualtunnelling
In this paper the tunnels are considered as an innitebeam on the Winkler foundation an innite beam on thePasternak foundation and a nite beam on the Pasternakfoundation (the new method) Comparing the decentectionrotation angle normalized bendingmoment and shear forceof the existing tunnel with constant stiness based on thethree models the new method is veried en dierentstiness of the reinforcement is taken into considerationand the optimal reinforcement range of the existing tunnel isdiscussed
61 e Subsurface Soil Displacement due to Tunnelling Forthe theoretical analysis the alignment of the new tunnels andthe existing tunnel is assumed to be straight Figure 14 showsa schematic diagram of the new tunnels and the existingtunnel in this case e greeneld settlement s(x) due to thetunnelling can be replaced by the equivalent distributed loadq(x) acting on the beam on the third assumption as follows
q(x) ks(x) (2)
In this study subsurface settlement s(x) at the depth of zinduced by tunnelling is calculated based on closed-formanalytical solutions presented by Loganathan and Poulos[13] as follows
s(x) ε0R2 zminusHx2 +(zminusH)2
+(3minus 4v)z +H
x2 +(z +H)2minus2z x2 minus(z +H)2( )
x2 +(z +H)2( )2
middot e minus138x2(H+R)2minus069z2H2( ) (3)
Conv
erge
nce d
ispla
cem
ent (
mm
)
400 420 440 460 480 500 520 540 560 580 600 620 640 660minus7minus6minus5minus4minus3minus2minus1
012345
e construction ofsouthbound tunnel
201453 Ring 135201459 Ring 1912014528 Ring 270
e shield is shut down201461
e construction ofnorthbound tunnel
2013129 Ring 4820131222 Ring 6420131230 Ring 87
e shield is shut down201424
e ring number of the upline of Line 1
A B
Figure 11 Convergence displacement of the existing tunnel
Figure 12 Reinforcement in the existing tunnel
xk1
k2
e existing tunnel
Sz(x)
Figure 13 A schematic view of the existing tunnel-soil tunnellinginteraction
Advances in Civil Engineering 7
where s(x) subsurface settlement of the soils due to tun-nelling R tunnel radius z depth below ground surfaceH depth of the axis of the new tunnel vPoissonrsquos ratiox lateral distance from tunnel centerline and ε0 equivalentground loss ratio which is dened as
ε0 4Rg + g2
4R2 (4)
g Gp + ulowast3D + w asymp Gp (5)
where g is the gap parameter (Lee et al [14]) ulowast3D is thethree-dimensional elastoplastic deformation at the tunnelface w is the workmanship factor and Gp represents thephysical gap between the outer skin of the shield and thelining which is given as
Gp 2Δ + δ (6)
where Δ is the thickness of the tail piece and δ is the clearancerequired for erection of the tail piece e parameter w can beneglected due to considerable experience with the equipmentand good tunnelling technique of construction workers basedon the study of Rowe and Lee [15]
62 Analytical Methods Derived from Prior Studies Attewellet al [16] proposed a tunnel-soil interaction model andsolved the analytical solution of the longitudinal displace-ment of an innite tunnel
d4w(x)dx4
+ 4λ4p(x) 4λ4q(x) (7)
where q(x) is the concentrated load acting on the originpoint of the beam EI is the centexural stiness of a tunnel
which is recommended by Ye et al [17] and k is the subgrademodulus that represents the pipe-soil interaction
If the soil foundation is assumed as the Winklerfoundation the foundation counterforce p(x) can beexpressed as
p(x) kw(x) (8)
e two-parameter model of foundation can capture theshear resistance of soil Based on the assumption of theplane strain condition the displacement of ground in they-direction is neglected If the soil foundation is assumed asthe Pasternak foundation the foundation counterforce p(x)can be expressed as
p(x) minusGnabla2w(x) + kw(x) (9)
where G is the coelaquocient of the shear element in thePasternakrsquos model with the dimension of force per unitlength
Attewell et al [16] presented a solution for the con-centrated load q on an innite beam resting on the Winklerfoundation and computed the decentection as
w(x) 1
8EIλ3qeminusλx( cos λx + sin λx) (10)
It can be concluded that each of the concentrated loadks(τ) at point τ contributes the following amount to thedecentection of the existing tunnel on the Winkler foundation
w(x) 1
8EIλ3int+infin
minusinfin2Rks tminus L1( )sin α( ) eminusλ|xminusτ|
(cos λ|xminus τ| + sin λ|xminus τ|)dτ(11)
where L1 is the distance between the coordinate origin of theexisting tunnel and the intersection point between the newtunnel and the existing tunnel and α is the intersection anglebetween the new tunnel and existing tunnel e decentectionrotational angle bending moment and shear force of theexisting tunnel which is assumed as an innite beam on theWinkler foundation and the Pasternak foundation are il-lustrated in Appendix A
63 A Brief Description of the NewMethod In this study wewant to gure out the eects of the reinforcement on theexisting tunnel the stiness varies in this case and theexisting tunnel assumed as an innite beam is not suitableAssuming the existing tunnel with nite length and in-troducing (7) to (9) the basic dierential equation gov-erning the centexure of the beam resting on the Pasternakfoundation can be written in the form with an assumptionof Gλ2klt 1
EiIid4wi(x)dx4minusGibi
d2wi(x)dx4
+ kibiwi(x) biqi(x) (12)
where b1i bi(1 +(Gikiradic
bi)) subscript i represents the ithelement of the beam bi is the width of the beam and b1i isthe modied width of the beam Assuming a solution of (12)in the form of w(x) eλx the general solution of verticaldisplacement of the beam is as follows
y
e southbound tunnel
e northbound tunnel
e e
xistin
g tun
nel
x
L 1L 2
0x 1
x 2x 3
x ix i+
1
x nminus2
x nminus1
x n
O2
O1
xx
Figure 14 A schematic diagram of tunnels for the analytical methods
8 Advances in Civil Engineering
w(x) sum4
i1CiFi(x) (13)
where Ci (i 1 2 3 4) are undetermined parameters whichare dependent on the boundary conditions and Fi (x)(i 1 2 3 4) are the basic functions of the general solutionwhich are given as follows
F1i(x) cos a1icix( ) sinh a2icix( ) (14)
F2i(x) cos a1icix( ) cosh a2icix( ) (15)
F3i(x) sin a1icix( ) cosh a2icix( ) (16)
F4i(x) sin a1icix( ) sinh a2icix( ) (17)
where ci kib14EiIi4radic
a1i 1minusGic2i kiradic
a2i1 + Gic2i kiradic
e relationship of the decentection rotational angle
bending moment and shear force of the beam proposed byLancaster and Mitchell [18] is as follows
θi(x) wprimei(x)Mi(x) minusEiIiwPrime(x)
Qi(x) minusEiIiw(x) + Gpib1iwprimei(x)
(18)
where θi(x) Mi(x) and Qi(x) are the amplitudes of ro-tational angle bending moment and shear force of thesection area respectively
Figure 15 shows that dierent stiness is replaced byequivalent stepped stiness According to dierent stinessand distributed load the local coordinate system is establishedas illustrated in Figure 16 Selvadurai [12] suggests an initialparameter method to solve the parameters of the beam efour parameters at the origin of the coordinates arewi(0i) θi(0i)Mi(0i) and Qi(0i) and four parameters at theend of the element are wi(xi) θi(xi) Mi(xi) and Qi(xi)C1 C2 C3 and C4 can be expressed by wi(0i) θi(0i)Mi(0i) and Qi(0i) by introducing xi 0 to (13) and (18)
wi(xi) θi(xi)Mi(xi) and Qi(xi) can be rewritten in thematrix-array form as
wi xi( )θi xi( )Mi xi( )Vi xi( )1
k11 k12 k13 k14 k15
k21 k22 k23 k24 k25
k31 k32 k33 k34 k35
k41 k42 k43 k44 k45
0 0 0 0 1
middot
wi 0i( )θi 0i( )Mi 0i( )Vi 0i( )1
(19)
where [kij] is a stiness submatrix with 5 times 5 order ij 1 2 5M and V are respectively the bending momentand the generalized shear force Formula (19) can be sim-plied as
Ni xi( ) KijNi 0i( ) (20)
where Ni(xi) [Wi(xi) θi(xi)Mi(xi) Qi(xi) 1]T andNi(0i) [Wi(0i) θi(0i)Mi(0i) Qi(0i) 1]T
Figure 17 shows that curved distributed load is replacedby equivalent triangular and square distributed load whichcan be expressed as
qiminus1 +qi qiminus1 + ξqi minus qiminus1Li
( ) (21)
where qiminus1 and qi are respectively the load of the startingpoint and the ending point of the ith element and ξ is theposition of arbitrary load From (19) we can obtain
k15 minusbintx0 qiminus1 +qi( ) aminus12i F1i[xminus ξ]minus aminus11i F3i[xminus ξ]( )dξ
4c3i EiIi
(22)In order to avoid numerical errors in the calculation
process the integration of ki5 is shown in Appendix B erest of kij can be referred in the work by Selvadurai [12]
Figure 18 shows the tunnel matrix transfer diagramAssuming that the tunnel is composed of N elements thex-coordinate of beginning points of the ith element isxi(i 0 1 2 n) and the length of each element is Liwhere Li xi minus ximinus1
E0I0
0 x1 x2 x3 ximinus1 xi xnminus1 xn x
E1I1E2I2
Eiminus1Iiminus1Enminus1Inminus1
Figure 15 Sketch of equivalent stepped stiness
qi(x)
pi
Qi Qi+1Mi Mi+1
Oimi
xi
Figure 16 Forces and local coordinate system
q0q1 q2
qiminus1 qnminus1
x1 x2 x3 ximinus1 xi xnminus1 xn x
q (x)
0
Figure 17 Replacement of variable load with trapezoid load
Advances in Civil Engineering 9
For the rst element
N1(x) K1(x) middotN1 01( ) x isin 0 L1( ) (23)
For the end point of the rst element
N1 x1( ) K1 x1 minus 0( ) middotN1 01( ) (24)
For the end point of the ith element
Ni xi( ) Ki xi( ) middotKiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( ) middotN1 01( )(25)
For x at the ith element section where x is a globalcoordinate
Ni(x)Ki xminus ximinus1( ) middotKiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( ) middotN1 01( )(26)
Suppose
Ai(x) Ki xminusximinus1( ) middot Kiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( )(27)
en (20) can be expressed as
Ni(x) Ai(x) middotN1 01( ) (28)
us the relation between the vectorNn(xn) at the pointxn and the vector N1(01) at the start point x0 is
Nn xn( ) An xn( ) middotN1 01( ) (29)
where
An xn( ) prod1
inKi Li( ) (30)
e boundary condition of the existing tunnel assumedas an innite long tunnel considering the shear stress of soilcan be written as
M(0) 0
V(0) 2
k
G
radic
middot G middot R middot w(0)
V(L) 0
V(L) minus2
k
G
radic
middot G middot R middot w(L)
(31)
By applying the boundary conditions to (19) the de-centection rotational angle bending moment and shear forceof the overall beam can be obtained
64 Case Study
641 Parameter Selection To enable a direct compari-son assumed parameters of the soil and existing tunnelare selected in this study Poissonrsquos ratio of the soil is0388 e grouting process is regarded as the inverseprocess of soil loss due to tunnelling and g can beexpressed as g (2Δ + δ)(1minus η) where η is the injectionratio
For the Pasternak foundation as the thickness of theelastic layer is assumed to be 25 B the subgrade elasticmodulus k and shear stiness G are suggested by Xu [19] asfollows
k 5E0β
16 1minus v05pt2( )times 122
G 13E0B
2
32 1 + v0( )βtimes 085
(32)
where β is a modied parameter (β 129) and E0 and v0 arerespectively the elastic modulus and Poissonrsquos ratio forelastic foundation which are given as follows
E0 Es
1minus v2s
v0 vs
1minus vs
(33)
where the empirical relation between the Youngrsquos modulusEs and the compression modulus Es01minus02 of soft soils is builtup by Yang and Zhao [20]
E0 (25 sim 35)Es01minus02 (34)
As illustrated in Figure 14 points O1 and O2 are re-spectively the intersection points A and B in practicalcoordinate transformation expressions of the axis xprime and xPrimeare respectively
xprime xminusL1( )sin αxPrime xminusL2( )sin α
(35)
where L1 and L2 are the distance from the beginningpoint of the existing tunnel to points O1 and O2 re-spectively e length of the existing tunnel is assumedas 250 m and the ring spacing is 12 m So the tunnel isdivided into 209 sections and each section is 12 m inlength
N1(01) Ni(xi)N1(x1) = N2(02) N2(x2)L1
L1
E1I1
G1 k1 G2 k2 G3 k3 Gi ki Gn knGnminus1 knminus1
E2I2 E3I3 EiIi EnInEnminus1Inminus1
L2 L3 Li LnLnminus1
L2 Li
x x x
xx1 x2 x3 xi-1 xnminus2 xnminus1 xnxi0
Ni(0i)
Figure 18 e matrix transfer diagram
10 Advances in Civil Engineering
7 Discussions
e normalized bending moment and shear force are de-ned as
Mi MiL0EiIi
Qi QiL
20
EiIi
(36)
where Mi and Qi are respectively the normalized bendingmoment and shear force Mi and Qi are respectively thecalculated bending moment and shear force and L0 is thelength of the existing tunnel
Figure 19 shows the comparison of the decentection ro-tational angle bending moment and shear force of theexisting tunnel with a constant stiness which is based onthree kinds of models an innite tunnel on the Winklerfoundation an innite tunnel on the Pasternak foundationand a nite tunnel on the Winkler foundation Note that the
368 380 400 420 440 460 480 500 520 540 560 580 600 612
minus0007
minus0006
minus0005
minus0004
minus0003
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)
minus0002
minus0001
0000
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundatione observed data on 201465
e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
minus000015
minus000010
minus000005
000000
000005
000010
000015
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
(b)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
e b
endi
ng m
omen
t of t
he ex
istin
g tu
nnel
(kN
middotm)
387 400 420 440 460 480 500 520 540 560 580 595
0
minus6000
minus5000
minus4000
minus3000
minus2000
minus1000
1000
2000
3000
4000
e ring number of the existing tunnel
(c)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
387 400 420 440 460 480 500 520 540 560 580 595
minus600
minus400
minus200
0
200
400
600
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(kN
)
(d)
Figure 19 (a) Comparison of the decentection of the existing tunnel using dierent models (b) Comparison of the rotational angle ofthe existing tunnel using dierent models (c) Comparison of the normalized bending moment of the existing tunnel using dierentmodels (d) Comparison of the normalized shear force of the existing tunnel using dierent models
Advances in Civil Engineering 11
existing tunnel is only subject to soil deformation force inthis case e displacement and the decentection of beam areparticular similar concepts owing to the fact that the existingtunnel is securely attached to the foundation generallywithout rigid body motion Fairly good agreement can beobserved between the innite tunnel on the Pasternakfoundation and the nite tunnel on the Pasternak founda-tion which shows that the method proposed in this paper iscorrect and can be used to analyze similar problems Becauseof considering soil shear stress vertical stress and de-formation are aected and weakened the displacement of
the existing tunnel on the Pasternak foundation is smallerthan that on the Winkler foundation without consideringthe shear stress of soil Using the Winkler foundation modelis in good agreement with the observed data It can be seenfrom (3) that the displacement of the existing tunnel isproportional to the grouting elaquociency (ie η) Due to theuncertainty of η it cannot be decided which one is the bestanalytical method It is clear that the maximum normalizedbending moment occurs at the points O1 and O2 which arethe nearest points between the existing tunnel and the newtunnels
minus0003
minus0002
minus0001
0000387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
minus000010
minus000005
000000
000005
000010
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e ring number of the existing tunnel
(b)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10
minus0003
minus0002
minus0001
0000
0001
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
xL
(c)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10xL
minus008
minus006
minus004
minus002
000
002
004
006
008
The n
orm
alise
d sh
ear f
orce
of t
he ex
istin
g tu
nnel
(d)
Figure 20 (a) e eects of dierent reinforcement stiness on the decentection of the existing tunnel (b) e eects of dierent re-inforcement stiness on the slope of the existing tunnel (c)e eects of dierent reinforcement stiness on the normalized moment of theexisting tunnel (d) e eects of dierent reinforcement stiness on the normalized shear force of the existing tunnel
12 Advances in Civil Engineering
e reinforce range of the existing tunnel is assumed as25 rings on either side of the intersection point A and thevarious stiness is taken into consideration using themethod proposed in this paper Figure 20 shows the in-centuence of dierent stiness of the reinforcement on theexisting tunnel (ie 4 EI EI and 14 EI) It is evident thatthe reinforcement can reduce the displacement rotationalangle normalized bending moment and shear force of theexisting tunnel while tunnel damage (reduce the stinessof reinforcement) has the opposite eect It is evident thatthe change of reinforcement stiness has greater eect onthe normalized bending moment and the normalized shearforce of the existing tunnel particularly at the incentection
points of the normalized bending moment and normalizedshear force curve
e reinforce stiness of the reinforcement range isassumed as 4 EI and the various reinforcement ranges istaken into consideration using the method proposed in thispaper Figure 21 shows the incentuence of dierent re-inforcement ranges on the existing tunnel when thenorthbound tunnel of Line 4 underpasses e normalizedbending moment on either side of the intersection point istoo large when the reinforcement range is 69ndash892m andthe normalized shear force on either side of the intersectionpoint is relatively too large when the reinforcement range isless than 592ndash992m So it is necessary to reinforce larger
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00035
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
The d
eflec
tions
of t
he ex
istin
g tu
nnel
(m)
xL
(a)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus000010
minus000005
000000
000005
000010
The s
lope
of t
he ex
istin
g tu
nnel
(rad
)
xL
(b)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
00005
00010
00015
xL
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
(c)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus010
minus008
minus006
minus004
minus002
000
002
004
006
008
010
xLTh
e nor
mal
ised
shea
r for
ce o
f the
exist
ing
tunn
el
(d)
Figure 21 (a)e eects of dierent reinforcement ranges on the decentection of the existing tunnel (b)e eects of dierent reinforcementranges on the slope of the existing tunnel (c)e eects of dierent reinforcement ranges on the normalized moment of the existing tunnel(d) e eects of dierent reinforcement ranges on the normalized shear force of the existing tunnel
Advances in Civil Engineering 13
than the range of 492ndash1092m in the existing tunnel in thiscase that is larger than 30m (25 rings) on either side of theintersection point )ere is no big difference in the bendingmoment and shear force of the existing tunnel when thereinforcement ranges are 492ndash1092m 392ndash1192m and292ndash1292m In order to reduce the costs of reinforcementreinforcement in the 30m range (ie 25 rings) on eitherside of the intersection point is the best choice whichverifies the actual design
8 Conclusions
Based on the abovementioned statements the response of theexisting tunnel is analyzed and analytical methods are proposedfor verification and the reinforcement design Main conclusionsderived from the analysis are as follows
(1) )e displacement of the existing tunnel changes withthe relative position of the shield and the existingtunnel Shield parameters have a major impact on theexisting tunnel )e heaves of the existing tunnelmight be caused by the large applied face pressure andbulkhead addictive thrust When the shield tail isdriving beyond the selected monitoring rings theselected monitoring rings settle rapidly due to theclosure of the shield tail void
(2) )e southbound tunnel of Line 4 has less effect onthe left side of the intersection point A on theexisting tunnel because of the northbound tunnelrsquosbarrier effect A variety of measures of displacementprevention of Line 1 such as the control of shield
parameters reinforcement of the existing tunnel canensure the normal operation of metro Line 1
(3) )e analytical method proposed in the paper is ver-ified by the methods derived from prior studies It ispossible to take the different stiffness and curved loadsinto account by applying the stepped stiffness andtrapezoidal load while using a local coordinate systemin the derivation of the new method )e change ofreinforcement stiffness has greater effect on thenormalized bending moment and the normalizedshear force of the existing tunnel particularly at theinflection points of the normalized bending momentand normalized shear force curve )e analyticalmethod can be applied for the reinforcement designand safety check of the built tunnel
Appendices
A1 An Infinite Beam on the WinklerFoundation
θ(x) 1113946+infin
minusinfin
minusλ2s((tminus L)sin α)eminusλ(xminust) sin(λ(xminus t))
b
λ2s((tminus L)sin α)eλ(xminust)sin(λ(tminus x))
b
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎭
dt
(A1)
M(x) 14λ
1113946+infin
minusinfinks((tminusL)sin α)e
minusλ|xminust|( cos(λ|xminus t|)minus sin(λ|xminus t|))dt (A2)
Q(x) 1113946+infin
minusinfin
minus12
ks((tminusL)sin α)eminusλ(xminust)cos(λ(xminus t))dt xminus tgt 0
12
ks((tminusL)sin α)eminusλ(tminusx)cos(λ(tminusx))dt xminus tle 0
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
⎫⎪⎪⎪⎬
⎪⎪⎪⎭
dt (A3)
A2 An Infinite Beam on the PasternakFoundation
w(x) λ2bk
1113946+infin
minusinfinks((tminus L)sin α)De
minusa1λ|xminusτ| 1a1
cos a2λ|xminus τ|( 1113857 +1a2
sin a2λ|xminus τ|( 11138571113888 1113889dτ
θ(x) 1113946+infin
minusinfin
minusλ2s((tminusL))eminusa1λ(xminust)sin a2(xminus t)( 1113857
a1a2b
λ2s((tminus L))ea1λ(xminust)sin a2(tminus x)( 1113857
a1a2b
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A4)
14 Advances in Civil Engineering
M(x) 14λ
1113946+infin
minusinfinks((tminus L)sin α)e
minusa1λ|xminust| 1a1
cos a2λ|xminus t|( 1113857minus1a2
sin a2λ|xminus t|( 11138571113888 1113889dt
Q(x) 1113946+infin
minusinfin
minus12
ks((tminus L)sin α)eminusa1λ(xminust) cos a2λ(xminus t)( 1113857minus
a21 minus a 2
22a1a2
1113888 1113889sin a2λ(xminus t)( 11138571113890 1113891dt xminus tgt 0
12
ks((tminusL)sin α)eminusa1λ(tminusx) cos a2λ(tminusx)( 1113857minus
a21 minus a2
2
2a1a21113888 1113889sin a2λ(tminusx)( 11138571113890 1113891dt xminus tle 0
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A5)
B1 Coefficients of the Matrix ki5
k15 minusbiqiminus1
4a1ia2ici4EiIi a1i
2 + a2i2( 1113857
minus2a1ia2i + 2a1ia2iF2i
+ a1i2 minus a2i
2( 1113857F4i
⎛⎜⎝ ⎞⎟⎠
+bi qi minus qiminus1( 1113857
4a1ia2ici5EiIiL3 a1i
2 + a2i2( 1113857
2
a2i minus3a21i + a2
2i( 1113857F3i
+a1i 2a2i a21i + a2
2i( 1113857cix + a1i2 minus 3a2i
2( 1113857F1i( 1113857
⎛⎜⎝ ⎞⎟⎠
(B1)
k25 biqiminus1 a2iF3i minus a1iF1i( 1113857
2a1a2 a21 + a 2
2( 1113857c3EiIi
+bi qi minus qiminus1( 1113857 2a1ia2i minus 2a1ia2iF2i minus a1i
2 minus a2i2( 1113857F4i( 1113857( 1113857
2a1ia2i a1i2 + a2i
2( 11138572ci
4EiIix1113872 1113873 (B2)
k35 minusbiqiminus1F4i
2a1ia2ici2 minus
bi qi minus qiminus1( 1113857 a2iF3i minus a1iF1i( 1113857
2a1ia2ici3 a1i
2 + a2i2( 1113857x
(B3)
k45 minusbiqiminus1
a1i2 + a2i
2( 1113857ci
1113888a1i 1 + 2a1i2a22i minus 2a2i
41113872 1113873F3i + a2i 1 + 2a1i
2a22i minus 2a1i
41113872 1113873F1i1113889
minusbi qi minus qiminus1( 1113857
a21i + a2i
2( 11138572c 2
i x
4a1i2a2i
2 minus 1( 1113857 a1i2 minus a2i
2( 1113857 + a2i2 minus 4a1i
4a2i2 + a1i
2 minus1 + 4a2i4( 1113857( 1113857F2i
minus2a1ia2i a 21i minus a 2
2i( 11138572 minus 11113872 1113873F4i
⎛⎜⎝ ⎞⎟⎠
(B4)
Conflicts of Interest
)e author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
)e author appreciates the help from the staff of Hangzhouand Hong Kong Tunnel Company Ltd during in-strumentation setup and data acquisition )is study issupported by the National Natural Science Foundation ofChina (no 41702313)
References
[1] S H Kim Model testing and analysis of interactions betweentunnels in clay [PhD 3esis] Department of EngineeringScience University of Oxford Oxford UK 1996
[2] H Y Liu J C Small J P Carter and D J Williams ldquoEffectsof tunnelling on existing support systems of perpendicularlypassing tunnelsrdquo Computers and Geotechnics vol 36 no 5pp 880ndash894 2009
[3] I Yamaguchi I Yamazaki and Y Kiritani ldquoStudy ofgroundndashtunnel interactions of four shield tunnels driven inclose proximity in relation to design and construction ofparallel shield tunnelsrdquo Tunnelling and Underground SpaceTechnology vol 13 no 3 pp 289ndash304 1998
[4] T I Addenbrooke and D M Potts ldquoTwin tunnel interactionsurface and subsurface effectsrdquo International Journal ofGeomechanics vol 1 no 2 pp 249ndash271 2001
[5] J P Kimmance S Lawrence O Hassan and N J PurchaseldquoObservations of deformations created in existing tunnels byadjacent and cross cutting excavationsrdquo in Proceedings ofGeotechnical Aspects of Underground Construction in SoftGround pp 707ndash712 London UK July 1996
[6] E Leca and B New ldquoSettlements induced by tunnelling in softgroundrdquo Tunnelling and Underground Space Technologyvol 22 no 2 pp 119ndash149 2007
Advances in Civil Engineering 15
[7] S M Liao J H Liu R L Wang and Z M Li ldquoShieldtunnelling and environment protection in Shanghai softgroundrdquo Tunnelling and Underground Space Technologyvol 24 no 4 pp 454ndash465 2009
[8] F H Chehade and I Shahrour ldquoNumerical analysis of theinteraction between twin-tunnels influence of the relativeposition and construction procedurerdquo Tunnelling and Un-derground Space Technology vol 23 no 2 pp 210ndash214 2008
[9] X G Li and D J Yuan ldquoResponse of a double-decked metrotunnel to shield driving of twin closely underpasses tunnelsrdquoTunnelling and Underground Space Technology vol 28 no 1pp 18ndash30 2012
[10] T E Vorster A Klar K Soga and R J Mair ldquoEstimating theeffects of tunnelling on existing pipelinesrdquo Journal of Geo-technical and Geoenvironmental Engineering vol 131 no 11pp 1399ndash1410 2005
[11] Z G Zhang M S Huang and W D Wang ldquoEvaluation ofdeformation response for adjacent tunnels due to soilunloading in excavation engineeringrdquo Tunnelling and Un-derground Space Technology vol 38 no 3 pp 244ndash253 2013
[12] A P S Selvadurai Elastic Analysis of Soil-Foundation InteractionElsevier Scientific Pub Co Amsterdam Netherlands 1979
[13] N Loganathan and H G Poulos ldquoAnalytical prediction fortunnelling-induced ground movements in claysrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124no 9 pp 846ndash856 1998
[14] K M Lee R K Rowe and K Y Lo ldquoSubsidence owing totunnelling I Estimating the gap parameterrdquo CanadianGeotechnical Journal vol 29 no 6 pp 929ndash940 1992
[15] R K Rowe and KM Lee ldquoSubsidence owing to tunnelling IIEvaluation of a prediction techniquerdquo Canadian GeotechnicalJournal vol 29 no 6 pp 941ndash954 1992
[16] P B Attewell J Yeates and A R Selby Soil MovementsInduced by Tunnelling and their Effects on Pipelines andStructures Methuen Inc New York NY USA 1986
[17] F Ye C He H H Zhu et al ldquoAnalysis of longitudinalequivalent stiffness of shield tunnel considering lateral per-formancerdquo Chinese Journal of Geotechnical Engineeringvol 33 no 12 pp 1870ndash1876 2011 in Chinese
[18] P R Lancaster and D Mitchell ldquoProblems in Bending of Barsand Plates [M]rdquo Advanced Solid Mechanics Macmillan UK1980
[19] L Xu ldquoStudy on vertical settlement of shield tunnel in soft soilrdquoPhD dissertation Tongji University Shanghai China 2005
[20] M Yang and X Zhao ldquoAn approach for a single pile in layeredsoilrdquo Tongji University Natural Science vol 20 no 4pp 421ndash428 1992 in Chinese
16 Advances in Civil Engineering
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4 Observation Results and Discussions
41 Monitoring Arrangement of Line 1 Arrangement ofthe monitoring rings of the existing tunnel can be seenin Figure 2 e total station Topcon MS05AX xed onthe tunnel sidewall arranged from rings 404 to 559 inthe existing tunnel is an automatic real-time measuringsystem which is used to monitor vertical and horizontal
displacement One monitor section is set every two rings inthe most aected zone (from rings 447 to 499 in the existingtunnel) and one monitor section is set every ve rings inthe rest part An LECAI D5 hand-held distance nder isused to monitor the converge displacement every 5 ringsfrom 404 to 679 in the existing tunnel e arrangement ofmonitoring points at the cross section of the existing tunnelis shown in Figure 8
Figure 1 Location of the new tunnels and the existing tunnel in Hangzhou A Guanhe Station B East railway station
409
404 414 419 424 429 434 439
444 447449451453455457459461463465467469471473475477479481483485487489491493495497499 509
514 519
524
529
534
539
544
549
554
559
564
569
574
579
584
589
594
599
604
609
614
619
624
629
634
639
644
649
679
15101520253035404550556065707580859095100
105
110
115
120
125
130
135
140
145
150155
165170175180
18519019520021021522022523023540
45
5055
60
65
70
75
80
85
90
95
100
105
110
115
120
125
130
135
140
145
150
155
160
165
170
175
180
185
190
195
200
205
210
215
220
225
230
235
240
245
250
255
260
265
270
275
East railway station
Guanhe Station
786 m
966 m835 m
35302520
674
669
664
659
654205
160
504
240245250255
e northbound tunnel of Line 4
e southbound tunnel of Line 4
Reinforcement of the existing tunnel
Zha-nong-kou Station
e upline tunnel of existing Line 1
Da-ju-yuan Station
A B
C
e auto monitoring point of vertical and horizontal displacement
e manual monitoring point of horizontal displacement
e manual monitoring point of vertical displacment
e manual monitoring point of convergence displacement
Figure 2 Plan view of tunnel alignment and location of the monitored rings
Line 4
65
Elev
atio
n (m
)
1-1 sandy silt
2-1 silt
3-1mucky silty
clay
3-2 clay
4-1 sandy clay
5-1 clay
GuanheStation
East railwaystation
5-2mucky silty
clay
Line 1
141210 8 6 4 2 0minus2minus4minus6minus8
minus10minus12minus14minus16minus18minus20minus22minus24minus26minus28minus30
minus15077 minus14954
208 212
K20+
898
754
K21+
053
342
K21+
094
361
K21+
608
099
232West square (B period)
West square (D period)Bottom of the basement floor 2100
Communication pipeline
K21+
015
389
Water pipeline
K21+
144
111
Groundwater table
Chainage (m)
Bottom of the basement floor 2900
K20+
900
K21+
000
K21+
100
K21+
200
K21+
300
K21+
400
K21+
500
K21+
600
K20+
971
955
K21+
301
630
Initi
al p
oint
Term
inal
poi
nt
Figure 3 Longitudinal prole of soils and tunnels
Advances in Civil Engineering 3
42 Displacement of the Existing Tunnel e location ofpoint A as shown in Figure 2 is corresponding to the in-tersection point of a plan view of the existing tunnel andnorthbound tunnel of Line 4 e location of point B iscorresponding to the intersection point of the plan view ofthe existing tunnel and southbound tunnel of Line 4 emiddle point of points A and B is point C
421 Vertical Displacement of the Existing Tunnel Figure 9shows time-varying vertical displacement in the moni-toring rings of the existing tunnel e x-axis is the ringnumber of the existing tunnel A positive value of theordinate denotes heave while a negative value denotessettlement of the existing tunnel e selected monitoringrings start to heave when the shield face is 0ndash10m awayfrom the selected monitoring rings which is mainly due tolarge applied tunnel face pressure bulk addictive thrust(Figure 6) and the friction force between the shield shelland soil mass When the shield tail is far beyond the se-lected monitoring rings a reduction of heave in the selectedmonitoring points is observed which is mainly due to theclosure of the tail void Only small settlements and heavesare measured after the construction of the northboundtunnel During the 3-month shutdown of the shield ad-ditional settlement ranges from 2mm to 3mm can beobserved in rings from 430 to 520 Additional settlements
ranges from 2 to 4mm were measured in rings from 460 to540 during the southbound tunnelrsquos construction elong-term additional settlements monitored up to 140 days(from 201461 to 20141019) range from 2 to 4mm inrings from 460 to 510 e settlements of rings from 430 to520 range from 2mm to 12mm and the settlement curve ofthe upline of Line 1 is ldquoUrdquo shaped after the long-termmonitoring e settlement curve is approximately sym-metric about the dashed line C after the long-term mon-itoringemaximum settlement is 12mmwhich is locatedin ring 487
0 4 8 12 03 04 05 06 0716 18 20 22
50
45
40
35
30
25
20
15
10
5
020 30 40 50 0 10 20 300 10 20 30 40 025 030 035 040
Es (MPa)w () vk0
1 Fill
3 silt
4 clay
6 muckysilty clay
8 so siltyclay
Depth of metro 1 tunnel axis
r unit weightw water contentwp plastic limit
wL liquid limitv void ratioEs elasticity modulusCU consolidation undrainage conditionCU effective value in consolidation undrainage conditionUU unconsolidation undrainage condition
gravel
r (kNm)
Dep
th (m
)
wpwL
w
Soil layer
uucucu
Depth of metro 4 tunnel axis
uucucu
c (kPa) φordm
Figure 4 Soil prole and geotechnical parameters
Table 1 Laboratory penetration test and eld steady centow test
Soil layerLaboratory
penetration test (cms) Field steady centow test (cms)
Kv Kh K
3ndash2 479times10minus5 781times10minus5 421times10minus3
3ndash3 255times10minus4 230times10minus4 157times10minus3
3ndash5 963times10minus5 174times10minus4 mdash3ndash6 329times10minus4 301times10minus4 mdash4ndash3 239times10minus7 784times10minus7 6ndash1 206times10minus7 303times10minus7 mdash6ndash2 262times10minus7 128times10minus7 mdash8ndash1 188times10minus7 511times10minus7 mdash
4 Advances in Civil Engineering
422 Horizontal Displacement of the Existing Tunnel ehorizontal displacement of the existing tunnel with respectto the location of the shield is illustrated in Figure 10 Apositive horizontal displacement denotes northward trans-verse tunnel movement away from the original tunnelcenterline while a negative horizontal displacement denotessouthward transverse tunnel movement away from theoriginal tunnel centerline
ere are northward displacement on the left side ofpoint C and southward displacement on the right side ofpoint Ce horizontal displacement curve is approximatelysymmetric about point C after the long-term monitoringe maximum northward displacement is 10mm in the ringnear the intersection point A and the maximum southwarddisplacement is minus105mm in the ring near the intersectionpoint B after the completion of the two tunnelsrsquo construc-tion During the construction of the northbound tunnelrings from 453 to 471 move southward which is likely due tothe additional bulkhead additive thrust and the squeezingforce provided by the shield shell When the shield tail leavesring 459 rings from 439 to 459 move southward slowlyOnly 2 to 5mm additional southward displacement ismeasured in rings from 487 to 560 and nearly no dis-placement is observed on the left side of point B during theconstruction of the southbound tunnel which is mainly dueto the northbound tunnelrsquos barrier eect No change ofhorizontal displacement is observed in the long-term con-ditions (from 2014528 to 20141019)
423 Convergence Displacement of the ExistingTunnel Figure 11 shows the convergence displacement ofthe existing tunnel with respect to the locations of the shield
A negative value indicates the reduction horizontal diameterof the existing tunnel while a positive value indicates theaddition horizontal diameter of the existing tunnel It can beobserved that the convergence displacement is not obviouswhen the shield reaches ring 48 (22 rings away from theintersection point A) During the process of the shielddriving from ring 64 to ring 87 a signicantly additionalincrease in the negative convergence displacement in ringsfrom 440 to 490 is observed and themaximum displacementoccurs in the intersection point A e reason might be thatthe shield face squeezes one side of the existing tunnel
20950
21000
21050
21100
21150
21200
21250
21300268
270
e r
ing
num
ber o
f the
no
rthb
ound
tunn
el
70
Guanhe Station
paused for 106 days
56 days
e tunnel face of the northbound tunnele tunnel face of the southbound tunnel
Tunn
el ch
aina
ge (m
)
Date
46 days
East railway station
170
250
200
150
100
50
0
-50
0
50
100
150
200
250
e r
ing
num
ber o
f the
sout
hbou
nd tu
nnel
2013
12
3
2013
12
13
2013
12
23
2014
12
2014
11
2
2014
12
2
2014
13
020
144
10
2014
41
4
2014
42
0
2014
43
0
2014
51
0
2014
52
0
2014
53
0
Figure 5 Advance-time curve of the shield
430 440 450 460 470 480 490 500 510 515
020
022
024
026
028
030
032
034
036
038
040
Northbound tunnelSouthbound tunnel
225 220 210 200 190 180 170 160 150 143
The ring number of the northbound tunnel of Line 4
The corresponding ring number of the upline tunnel of Line 1
The ring number of the southbound tunnel of Line 4
50 60 70 80 90 100 110 120 130
The a
pplie
d tu
nnel
face
pre
ssur
e (M
Pa)
Figure 6 e applied tunnel face pressure
Advances in Civil Engineering 5
leading to the reduction of the horizontal diameter of theexisting tunnel With the tunnelling of the southboundtunnel there are a reduction negative convergence dis-placement on the left side of point B and an addition positiveconvergence displacement on the right side of point B Ascan be seen from Figures 10 and 11 the shapes of theconvergence displacement curve and the horizontal dis-placement curve are similar because the causes of conver-gence displacement and horizontal displacement aretheoretically the same
5 Reinforcement Schemeof theExistingTunnel
Figure 12 shows the reinforcement in the existing tunnelRadial reinforcement by an arc-shaped supporting steelplate connected to the tunnel segment and longitudinalreinforcement by channel section steel to provide lon-gitudinal tensile stress are conducted in the existingtunnel Radial steels and longitudinal steels are connected
by welding and so the reinforcement becomes a wholeone 25 rings are reinforced at rst on either side of theintersection point A before the underpassing of thenorthbound tunnel and the whole reinforcement is
Northbound tunnelSouthbound tunnel
430 440 450 460 470 480 490 500 510 518
35
40
45
50
55
60
65
70
75
80
e corresponding ring number of the upline tunnel of Line 1
45 50 60 70 80 90 100 110 120 130 135e ring number of the northbound tunnel of Line 4
220 210 200 190 180 170 160 150 140e ring number of the southbound tunnel of Line 4
e g
rout
ing
volu
me (
m3 )
Figure 7 e grouting volume
e manual monitoring points ofconvergence displacement
Track bed
Take one point as the automatic monitoringpoint of the vertical and horizontal displacement
Centerline
e manual monitoring pointsof horizontal displacement
Figure 8 Arrangement of the monitoring points at the crosssection of the existing tunnel
400 420 440 460 480 500 520 540 560 580 600 620 640 660 670minus12
minus10
minus8
minus6
minus4
minus2
0
2
4
6 C
e construction ofnorthbound tunnel
e construction ofsouthbound tunnel
201453 Ring 135 2014515 Ring 226
Long-term monitoring201465
20146152014714
20141019
B
2013129 Ring 4820131227 Ring 72 2014112 Ring 1762014128 Ring 275
e shield is shut down2014227
Ver
tical
disp
lace
men
t (m
m)
e ring number of upline tunnel of Line 1
A
Figure 9 Time-varying vertical displacement of the existingtunnel
400 420 440 460 480 500 520 540 560 580 600 620 640 660 670
minus14minus12minus10minus8minus6minus4minus2
02468
10121416
C
e construction of thesouthbound tunnel
2014420 Ring 362014502 Ring 1242014528 Ring 270
Long-term monitoring20141019
e construction of thenorthbound tunnel
2013129 Ring 4820131227 Ring 72201415 Ring 1252014128 Rring 275
e ring number of upline tunnel of Line 1
Hor
izon
tal d
ispla
cem
ent (
mm
)
A B
Figure 10 Time-varying horizontal displacement of the existingtunnel
6 Advances in Civil Engineering
completed after the construction of the northboundtunnel e reinforcement range is from ring 424 to ring510 (ie 25 rings on either sides of the intersection pointsA and B)
6 Theoretical Analysis of the ReinforcementDesign
e deformed tunnel lining reinforced by inner bondingsteel plates has been partially or entirely used in manybuilt tunnels or currently under construction To better
understand the benets coming from the reinforcementmethod the behavior of the tunnel is investigated
Figure 13 shows a schematic view of the existingtunnel-soil tunnelling interaction e analysis methoddemonstrated in this paper can be divided into two stepsrstly estimating the greeneld displacement induced bythe tunnelling Secondly calculating the responses of theexisting tunnel subjected to the soil displacement emethod of analysis is based on three assumptions (1) theabove existing tunnel does not aect the displacement ofsoil due to tunnelling (2) the soil foundation is assumedas the Winkler or the Pasternak foundation and (3) thesoil displacement is calculated by superimposing theindependent settlement predicted for each individualtunnelling
In this paper the tunnels are considered as an innitebeam on the Winkler foundation an innite beam on thePasternak foundation and a nite beam on the Pasternakfoundation (the new method) Comparing the decentectionrotation angle normalized bendingmoment and shear forceof the existing tunnel with constant stiness based on thethree models the new method is veried en dierentstiness of the reinforcement is taken into considerationand the optimal reinforcement range of the existing tunnel isdiscussed
61 e Subsurface Soil Displacement due to Tunnelling Forthe theoretical analysis the alignment of the new tunnels andthe existing tunnel is assumed to be straight Figure 14 showsa schematic diagram of the new tunnels and the existingtunnel in this case e greeneld settlement s(x) due to thetunnelling can be replaced by the equivalent distributed loadq(x) acting on the beam on the third assumption as follows
q(x) ks(x) (2)
In this study subsurface settlement s(x) at the depth of zinduced by tunnelling is calculated based on closed-formanalytical solutions presented by Loganathan and Poulos[13] as follows
s(x) ε0R2 zminusHx2 +(zminusH)2
+(3minus 4v)z +H
x2 +(z +H)2minus2z x2 minus(z +H)2( )
x2 +(z +H)2( )2
middot e minus138x2(H+R)2minus069z2H2( ) (3)
Conv
erge
nce d
ispla
cem
ent (
mm
)
400 420 440 460 480 500 520 540 560 580 600 620 640 660minus7minus6minus5minus4minus3minus2minus1
012345
e construction ofsouthbound tunnel
201453 Ring 135201459 Ring 1912014528 Ring 270
e shield is shut down201461
e construction ofnorthbound tunnel
2013129 Ring 4820131222 Ring 6420131230 Ring 87
e shield is shut down201424
e ring number of the upline of Line 1
A B
Figure 11 Convergence displacement of the existing tunnel
Figure 12 Reinforcement in the existing tunnel
xk1
k2
e existing tunnel
Sz(x)
Figure 13 A schematic view of the existing tunnel-soil tunnellinginteraction
Advances in Civil Engineering 7
where s(x) subsurface settlement of the soils due to tun-nelling R tunnel radius z depth below ground surfaceH depth of the axis of the new tunnel vPoissonrsquos ratiox lateral distance from tunnel centerline and ε0 equivalentground loss ratio which is dened as
ε0 4Rg + g2
4R2 (4)
g Gp + ulowast3D + w asymp Gp (5)
where g is the gap parameter (Lee et al [14]) ulowast3D is thethree-dimensional elastoplastic deformation at the tunnelface w is the workmanship factor and Gp represents thephysical gap between the outer skin of the shield and thelining which is given as
Gp 2Δ + δ (6)
where Δ is the thickness of the tail piece and δ is the clearancerequired for erection of the tail piece e parameter w can beneglected due to considerable experience with the equipmentand good tunnelling technique of construction workers basedon the study of Rowe and Lee [15]
62 Analytical Methods Derived from Prior Studies Attewellet al [16] proposed a tunnel-soil interaction model andsolved the analytical solution of the longitudinal displace-ment of an innite tunnel
d4w(x)dx4
+ 4λ4p(x) 4λ4q(x) (7)
where q(x) is the concentrated load acting on the originpoint of the beam EI is the centexural stiness of a tunnel
which is recommended by Ye et al [17] and k is the subgrademodulus that represents the pipe-soil interaction
If the soil foundation is assumed as the Winklerfoundation the foundation counterforce p(x) can beexpressed as
p(x) kw(x) (8)
e two-parameter model of foundation can capture theshear resistance of soil Based on the assumption of theplane strain condition the displacement of ground in they-direction is neglected If the soil foundation is assumed asthe Pasternak foundation the foundation counterforce p(x)can be expressed as
p(x) minusGnabla2w(x) + kw(x) (9)
where G is the coelaquocient of the shear element in thePasternakrsquos model with the dimension of force per unitlength
Attewell et al [16] presented a solution for the con-centrated load q on an innite beam resting on the Winklerfoundation and computed the decentection as
w(x) 1
8EIλ3qeminusλx( cos λx + sin λx) (10)
It can be concluded that each of the concentrated loadks(τ) at point τ contributes the following amount to thedecentection of the existing tunnel on the Winkler foundation
w(x) 1
8EIλ3int+infin
minusinfin2Rks tminus L1( )sin α( ) eminusλ|xminusτ|
(cos λ|xminus τ| + sin λ|xminus τ|)dτ(11)
where L1 is the distance between the coordinate origin of theexisting tunnel and the intersection point between the newtunnel and the existing tunnel and α is the intersection anglebetween the new tunnel and existing tunnel e decentectionrotational angle bending moment and shear force of theexisting tunnel which is assumed as an innite beam on theWinkler foundation and the Pasternak foundation are il-lustrated in Appendix A
63 A Brief Description of the NewMethod In this study wewant to gure out the eects of the reinforcement on theexisting tunnel the stiness varies in this case and theexisting tunnel assumed as an innite beam is not suitableAssuming the existing tunnel with nite length and in-troducing (7) to (9) the basic dierential equation gov-erning the centexure of the beam resting on the Pasternakfoundation can be written in the form with an assumptionof Gλ2klt 1
EiIid4wi(x)dx4minusGibi
d2wi(x)dx4
+ kibiwi(x) biqi(x) (12)
where b1i bi(1 +(Gikiradic
bi)) subscript i represents the ithelement of the beam bi is the width of the beam and b1i isthe modied width of the beam Assuming a solution of (12)in the form of w(x) eλx the general solution of verticaldisplacement of the beam is as follows
y
e southbound tunnel
e northbound tunnel
e e
xistin
g tun
nel
x
L 1L 2
0x 1
x 2x 3
x ix i+
1
x nminus2
x nminus1
x n
O2
O1
xx
Figure 14 A schematic diagram of tunnels for the analytical methods
8 Advances in Civil Engineering
w(x) sum4
i1CiFi(x) (13)
where Ci (i 1 2 3 4) are undetermined parameters whichare dependent on the boundary conditions and Fi (x)(i 1 2 3 4) are the basic functions of the general solutionwhich are given as follows
F1i(x) cos a1icix( ) sinh a2icix( ) (14)
F2i(x) cos a1icix( ) cosh a2icix( ) (15)
F3i(x) sin a1icix( ) cosh a2icix( ) (16)
F4i(x) sin a1icix( ) sinh a2icix( ) (17)
where ci kib14EiIi4radic
a1i 1minusGic2i kiradic
a2i1 + Gic2i kiradic
e relationship of the decentection rotational angle
bending moment and shear force of the beam proposed byLancaster and Mitchell [18] is as follows
θi(x) wprimei(x)Mi(x) minusEiIiwPrime(x)
Qi(x) minusEiIiw(x) + Gpib1iwprimei(x)
(18)
where θi(x) Mi(x) and Qi(x) are the amplitudes of ro-tational angle bending moment and shear force of thesection area respectively
Figure 15 shows that dierent stiness is replaced byequivalent stepped stiness According to dierent stinessand distributed load the local coordinate system is establishedas illustrated in Figure 16 Selvadurai [12] suggests an initialparameter method to solve the parameters of the beam efour parameters at the origin of the coordinates arewi(0i) θi(0i)Mi(0i) and Qi(0i) and four parameters at theend of the element are wi(xi) θi(xi) Mi(xi) and Qi(xi)C1 C2 C3 and C4 can be expressed by wi(0i) θi(0i)Mi(0i) and Qi(0i) by introducing xi 0 to (13) and (18)
wi(xi) θi(xi)Mi(xi) and Qi(xi) can be rewritten in thematrix-array form as
wi xi( )θi xi( )Mi xi( )Vi xi( )1
k11 k12 k13 k14 k15
k21 k22 k23 k24 k25
k31 k32 k33 k34 k35
k41 k42 k43 k44 k45
0 0 0 0 1
middot
wi 0i( )θi 0i( )Mi 0i( )Vi 0i( )1
(19)
where [kij] is a stiness submatrix with 5 times 5 order ij 1 2 5M and V are respectively the bending momentand the generalized shear force Formula (19) can be sim-plied as
Ni xi( ) KijNi 0i( ) (20)
where Ni(xi) [Wi(xi) θi(xi)Mi(xi) Qi(xi) 1]T andNi(0i) [Wi(0i) θi(0i)Mi(0i) Qi(0i) 1]T
Figure 17 shows that curved distributed load is replacedby equivalent triangular and square distributed load whichcan be expressed as
qiminus1 +qi qiminus1 + ξqi minus qiminus1Li
( ) (21)
where qiminus1 and qi are respectively the load of the startingpoint and the ending point of the ith element and ξ is theposition of arbitrary load From (19) we can obtain
k15 minusbintx0 qiminus1 +qi( ) aminus12i F1i[xminus ξ]minus aminus11i F3i[xminus ξ]( )dξ
4c3i EiIi
(22)In order to avoid numerical errors in the calculation
process the integration of ki5 is shown in Appendix B erest of kij can be referred in the work by Selvadurai [12]
Figure 18 shows the tunnel matrix transfer diagramAssuming that the tunnel is composed of N elements thex-coordinate of beginning points of the ith element isxi(i 0 1 2 n) and the length of each element is Liwhere Li xi minus ximinus1
E0I0
0 x1 x2 x3 ximinus1 xi xnminus1 xn x
E1I1E2I2
Eiminus1Iiminus1Enminus1Inminus1
Figure 15 Sketch of equivalent stepped stiness
qi(x)
pi
Qi Qi+1Mi Mi+1
Oimi
xi
Figure 16 Forces and local coordinate system
q0q1 q2
qiminus1 qnminus1
x1 x2 x3 ximinus1 xi xnminus1 xn x
q (x)
0
Figure 17 Replacement of variable load with trapezoid load
Advances in Civil Engineering 9
For the rst element
N1(x) K1(x) middotN1 01( ) x isin 0 L1( ) (23)
For the end point of the rst element
N1 x1( ) K1 x1 minus 0( ) middotN1 01( ) (24)
For the end point of the ith element
Ni xi( ) Ki xi( ) middotKiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( ) middotN1 01( )(25)
For x at the ith element section where x is a globalcoordinate
Ni(x)Ki xminus ximinus1( ) middotKiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( ) middotN1 01( )(26)
Suppose
Ai(x) Ki xminusximinus1( ) middot Kiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( )(27)
en (20) can be expressed as
Ni(x) Ai(x) middotN1 01( ) (28)
us the relation between the vectorNn(xn) at the pointxn and the vector N1(01) at the start point x0 is
Nn xn( ) An xn( ) middotN1 01( ) (29)
where
An xn( ) prod1
inKi Li( ) (30)
e boundary condition of the existing tunnel assumedas an innite long tunnel considering the shear stress of soilcan be written as
M(0) 0
V(0) 2
k
G
radic
middot G middot R middot w(0)
V(L) 0
V(L) minus2
k
G
radic
middot G middot R middot w(L)
(31)
By applying the boundary conditions to (19) the de-centection rotational angle bending moment and shear forceof the overall beam can be obtained
64 Case Study
641 Parameter Selection To enable a direct compari-son assumed parameters of the soil and existing tunnelare selected in this study Poissonrsquos ratio of the soil is0388 e grouting process is regarded as the inverseprocess of soil loss due to tunnelling and g can beexpressed as g (2Δ + δ)(1minus η) where η is the injectionratio
For the Pasternak foundation as the thickness of theelastic layer is assumed to be 25 B the subgrade elasticmodulus k and shear stiness G are suggested by Xu [19] asfollows
k 5E0β
16 1minus v05pt2( )times 122
G 13E0B
2
32 1 + v0( )βtimes 085
(32)
where β is a modied parameter (β 129) and E0 and v0 arerespectively the elastic modulus and Poissonrsquos ratio forelastic foundation which are given as follows
E0 Es
1minus v2s
v0 vs
1minus vs
(33)
where the empirical relation between the Youngrsquos modulusEs and the compression modulus Es01minus02 of soft soils is builtup by Yang and Zhao [20]
E0 (25 sim 35)Es01minus02 (34)
As illustrated in Figure 14 points O1 and O2 are re-spectively the intersection points A and B in practicalcoordinate transformation expressions of the axis xprime and xPrimeare respectively
xprime xminusL1( )sin αxPrime xminusL2( )sin α
(35)
where L1 and L2 are the distance from the beginningpoint of the existing tunnel to points O1 and O2 re-spectively e length of the existing tunnel is assumedas 250 m and the ring spacing is 12 m So the tunnel isdivided into 209 sections and each section is 12 m inlength
N1(01) Ni(xi)N1(x1) = N2(02) N2(x2)L1
L1
E1I1
G1 k1 G2 k2 G3 k3 Gi ki Gn knGnminus1 knminus1
E2I2 E3I3 EiIi EnInEnminus1Inminus1
L2 L3 Li LnLnminus1
L2 Li
x x x
xx1 x2 x3 xi-1 xnminus2 xnminus1 xnxi0
Ni(0i)
Figure 18 e matrix transfer diagram
10 Advances in Civil Engineering
7 Discussions
e normalized bending moment and shear force are de-ned as
Mi MiL0EiIi
Qi QiL
20
EiIi
(36)
where Mi and Qi are respectively the normalized bendingmoment and shear force Mi and Qi are respectively thecalculated bending moment and shear force and L0 is thelength of the existing tunnel
Figure 19 shows the comparison of the decentection ro-tational angle bending moment and shear force of theexisting tunnel with a constant stiness which is based onthree kinds of models an innite tunnel on the Winklerfoundation an innite tunnel on the Pasternak foundationand a nite tunnel on the Winkler foundation Note that the
368 380 400 420 440 460 480 500 520 540 560 580 600 612
minus0007
minus0006
minus0005
minus0004
minus0003
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)
minus0002
minus0001
0000
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundatione observed data on 201465
e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
minus000015
minus000010
minus000005
000000
000005
000010
000015
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
(b)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
e b
endi
ng m
omen
t of t
he ex
istin
g tu
nnel
(kN
middotm)
387 400 420 440 460 480 500 520 540 560 580 595
0
minus6000
minus5000
minus4000
minus3000
minus2000
minus1000
1000
2000
3000
4000
e ring number of the existing tunnel
(c)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
387 400 420 440 460 480 500 520 540 560 580 595
minus600
minus400
minus200
0
200
400
600
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(kN
)
(d)
Figure 19 (a) Comparison of the decentection of the existing tunnel using dierent models (b) Comparison of the rotational angle ofthe existing tunnel using dierent models (c) Comparison of the normalized bending moment of the existing tunnel using dierentmodels (d) Comparison of the normalized shear force of the existing tunnel using dierent models
Advances in Civil Engineering 11
existing tunnel is only subject to soil deformation force inthis case e displacement and the decentection of beam areparticular similar concepts owing to the fact that the existingtunnel is securely attached to the foundation generallywithout rigid body motion Fairly good agreement can beobserved between the innite tunnel on the Pasternakfoundation and the nite tunnel on the Pasternak founda-tion which shows that the method proposed in this paper iscorrect and can be used to analyze similar problems Becauseof considering soil shear stress vertical stress and de-formation are aected and weakened the displacement of
the existing tunnel on the Pasternak foundation is smallerthan that on the Winkler foundation without consideringthe shear stress of soil Using the Winkler foundation modelis in good agreement with the observed data It can be seenfrom (3) that the displacement of the existing tunnel isproportional to the grouting elaquociency (ie η) Due to theuncertainty of η it cannot be decided which one is the bestanalytical method It is clear that the maximum normalizedbending moment occurs at the points O1 and O2 which arethe nearest points between the existing tunnel and the newtunnels
minus0003
minus0002
minus0001
0000387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
minus000010
minus000005
000000
000005
000010
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e ring number of the existing tunnel
(b)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10
minus0003
minus0002
minus0001
0000
0001
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
xL
(c)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10xL
minus008
minus006
minus004
minus002
000
002
004
006
008
The n
orm
alise
d sh
ear f
orce
of t
he ex
istin
g tu
nnel
(d)
Figure 20 (a) e eects of dierent reinforcement stiness on the decentection of the existing tunnel (b) e eects of dierent re-inforcement stiness on the slope of the existing tunnel (c)e eects of dierent reinforcement stiness on the normalized moment of theexisting tunnel (d) e eects of dierent reinforcement stiness on the normalized shear force of the existing tunnel
12 Advances in Civil Engineering
e reinforce range of the existing tunnel is assumed as25 rings on either side of the intersection point A and thevarious stiness is taken into consideration using themethod proposed in this paper Figure 20 shows the in-centuence of dierent stiness of the reinforcement on theexisting tunnel (ie 4 EI EI and 14 EI) It is evident thatthe reinforcement can reduce the displacement rotationalangle normalized bending moment and shear force of theexisting tunnel while tunnel damage (reduce the stinessof reinforcement) has the opposite eect It is evident thatthe change of reinforcement stiness has greater eect onthe normalized bending moment and the normalized shearforce of the existing tunnel particularly at the incentection
points of the normalized bending moment and normalizedshear force curve
e reinforce stiness of the reinforcement range isassumed as 4 EI and the various reinforcement ranges istaken into consideration using the method proposed in thispaper Figure 21 shows the incentuence of dierent re-inforcement ranges on the existing tunnel when thenorthbound tunnel of Line 4 underpasses e normalizedbending moment on either side of the intersection point istoo large when the reinforcement range is 69ndash892m andthe normalized shear force on either side of the intersectionpoint is relatively too large when the reinforcement range isless than 592ndash992m So it is necessary to reinforce larger
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00035
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
The d
eflec
tions
of t
he ex
istin
g tu
nnel
(m)
xL
(a)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus000010
minus000005
000000
000005
000010
The s
lope
of t
he ex
istin
g tu
nnel
(rad
)
xL
(b)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
00005
00010
00015
xL
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
(c)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus010
minus008
minus006
minus004
minus002
000
002
004
006
008
010
xLTh
e nor
mal
ised
shea
r for
ce o
f the
exist
ing
tunn
el
(d)
Figure 21 (a)e eects of dierent reinforcement ranges on the decentection of the existing tunnel (b)e eects of dierent reinforcementranges on the slope of the existing tunnel (c)e eects of dierent reinforcement ranges on the normalized moment of the existing tunnel(d) e eects of dierent reinforcement ranges on the normalized shear force of the existing tunnel
Advances in Civil Engineering 13
than the range of 492ndash1092m in the existing tunnel in thiscase that is larger than 30m (25 rings) on either side of theintersection point )ere is no big difference in the bendingmoment and shear force of the existing tunnel when thereinforcement ranges are 492ndash1092m 392ndash1192m and292ndash1292m In order to reduce the costs of reinforcementreinforcement in the 30m range (ie 25 rings) on eitherside of the intersection point is the best choice whichverifies the actual design
8 Conclusions
Based on the abovementioned statements the response of theexisting tunnel is analyzed and analytical methods are proposedfor verification and the reinforcement design Main conclusionsderived from the analysis are as follows
(1) )e displacement of the existing tunnel changes withthe relative position of the shield and the existingtunnel Shield parameters have a major impact on theexisting tunnel )e heaves of the existing tunnelmight be caused by the large applied face pressure andbulkhead addictive thrust When the shield tail isdriving beyond the selected monitoring rings theselected monitoring rings settle rapidly due to theclosure of the shield tail void
(2) )e southbound tunnel of Line 4 has less effect onthe left side of the intersection point A on theexisting tunnel because of the northbound tunnelrsquosbarrier effect A variety of measures of displacementprevention of Line 1 such as the control of shield
parameters reinforcement of the existing tunnel canensure the normal operation of metro Line 1
(3) )e analytical method proposed in the paper is ver-ified by the methods derived from prior studies It ispossible to take the different stiffness and curved loadsinto account by applying the stepped stiffness andtrapezoidal load while using a local coordinate systemin the derivation of the new method )e change ofreinforcement stiffness has greater effect on thenormalized bending moment and the normalizedshear force of the existing tunnel particularly at theinflection points of the normalized bending momentand normalized shear force curve )e analyticalmethod can be applied for the reinforcement designand safety check of the built tunnel
Appendices
A1 An Infinite Beam on the WinklerFoundation
θ(x) 1113946+infin
minusinfin
minusλ2s((tminus L)sin α)eminusλ(xminust) sin(λ(xminus t))
b
λ2s((tminus L)sin α)eλ(xminust)sin(λ(tminus x))
b
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎭
dt
(A1)
M(x) 14λ
1113946+infin
minusinfinks((tminusL)sin α)e
minusλ|xminust|( cos(λ|xminus t|)minus sin(λ|xminus t|))dt (A2)
Q(x) 1113946+infin
minusinfin
minus12
ks((tminusL)sin α)eminusλ(xminust)cos(λ(xminus t))dt xminus tgt 0
12
ks((tminusL)sin α)eminusλ(tminusx)cos(λ(tminusx))dt xminus tle 0
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
⎫⎪⎪⎪⎬
⎪⎪⎪⎭
dt (A3)
A2 An Infinite Beam on the PasternakFoundation
w(x) λ2bk
1113946+infin
minusinfinks((tminus L)sin α)De
minusa1λ|xminusτ| 1a1
cos a2λ|xminus τ|( 1113857 +1a2
sin a2λ|xminus τ|( 11138571113888 1113889dτ
θ(x) 1113946+infin
minusinfin
minusλ2s((tminusL))eminusa1λ(xminust)sin a2(xminus t)( 1113857
a1a2b
λ2s((tminus L))ea1λ(xminust)sin a2(tminus x)( 1113857
a1a2b
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A4)
14 Advances in Civil Engineering
M(x) 14λ
1113946+infin
minusinfinks((tminus L)sin α)e
minusa1λ|xminust| 1a1
cos a2λ|xminus t|( 1113857minus1a2
sin a2λ|xminus t|( 11138571113888 1113889dt
Q(x) 1113946+infin
minusinfin
minus12
ks((tminus L)sin α)eminusa1λ(xminust) cos a2λ(xminus t)( 1113857minus
a21 minus a 2
22a1a2
1113888 1113889sin a2λ(xminus t)( 11138571113890 1113891dt xminus tgt 0
12
ks((tminusL)sin α)eminusa1λ(tminusx) cos a2λ(tminusx)( 1113857minus
a21 minus a2
2
2a1a21113888 1113889sin a2λ(tminusx)( 11138571113890 1113891dt xminus tle 0
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A5)
B1 Coefficients of the Matrix ki5
k15 minusbiqiminus1
4a1ia2ici4EiIi a1i
2 + a2i2( 1113857
minus2a1ia2i + 2a1ia2iF2i
+ a1i2 minus a2i
2( 1113857F4i
⎛⎜⎝ ⎞⎟⎠
+bi qi minus qiminus1( 1113857
4a1ia2ici5EiIiL3 a1i
2 + a2i2( 1113857
2
a2i minus3a21i + a2
2i( 1113857F3i
+a1i 2a2i a21i + a2
2i( 1113857cix + a1i2 minus 3a2i
2( 1113857F1i( 1113857
⎛⎜⎝ ⎞⎟⎠
(B1)
k25 biqiminus1 a2iF3i minus a1iF1i( 1113857
2a1a2 a21 + a 2
2( 1113857c3EiIi
+bi qi minus qiminus1( 1113857 2a1ia2i minus 2a1ia2iF2i minus a1i
2 minus a2i2( 1113857F4i( 1113857( 1113857
2a1ia2i a1i2 + a2i
2( 11138572ci
4EiIix1113872 1113873 (B2)
k35 minusbiqiminus1F4i
2a1ia2ici2 minus
bi qi minus qiminus1( 1113857 a2iF3i minus a1iF1i( 1113857
2a1ia2ici3 a1i
2 + a2i2( 1113857x
(B3)
k45 minusbiqiminus1
a1i2 + a2i
2( 1113857ci
1113888a1i 1 + 2a1i2a22i minus 2a2i
41113872 1113873F3i + a2i 1 + 2a1i
2a22i minus 2a1i
41113872 1113873F1i1113889
minusbi qi minus qiminus1( 1113857
a21i + a2i
2( 11138572c 2
i x
4a1i2a2i
2 minus 1( 1113857 a1i2 minus a2i
2( 1113857 + a2i2 minus 4a1i
4a2i2 + a1i
2 minus1 + 4a2i4( 1113857( 1113857F2i
minus2a1ia2i a 21i minus a 2
2i( 11138572 minus 11113872 1113873F4i
⎛⎜⎝ ⎞⎟⎠
(B4)
Conflicts of Interest
)e author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
)e author appreciates the help from the staff of Hangzhouand Hong Kong Tunnel Company Ltd during in-strumentation setup and data acquisition )is study issupported by the National Natural Science Foundation ofChina (no 41702313)
References
[1] S H Kim Model testing and analysis of interactions betweentunnels in clay [PhD 3esis] Department of EngineeringScience University of Oxford Oxford UK 1996
[2] H Y Liu J C Small J P Carter and D J Williams ldquoEffectsof tunnelling on existing support systems of perpendicularlypassing tunnelsrdquo Computers and Geotechnics vol 36 no 5pp 880ndash894 2009
[3] I Yamaguchi I Yamazaki and Y Kiritani ldquoStudy ofgroundndashtunnel interactions of four shield tunnels driven inclose proximity in relation to design and construction ofparallel shield tunnelsrdquo Tunnelling and Underground SpaceTechnology vol 13 no 3 pp 289ndash304 1998
[4] T I Addenbrooke and D M Potts ldquoTwin tunnel interactionsurface and subsurface effectsrdquo International Journal ofGeomechanics vol 1 no 2 pp 249ndash271 2001
[5] J P Kimmance S Lawrence O Hassan and N J PurchaseldquoObservations of deformations created in existing tunnels byadjacent and cross cutting excavationsrdquo in Proceedings ofGeotechnical Aspects of Underground Construction in SoftGround pp 707ndash712 London UK July 1996
[6] E Leca and B New ldquoSettlements induced by tunnelling in softgroundrdquo Tunnelling and Underground Space Technologyvol 22 no 2 pp 119ndash149 2007
Advances in Civil Engineering 15
[7] S M Liao J H Liu R L Wang and Z M Li ldquoShieldtunnelling and environment protection in Shanghai softgroundrdquo Tunnelling and Underground Space Technologyvol 24 no 4 pp 454ndash465 2009
[8] F H Chehade and I Shahrour ldquoNumerical analysis of theinteraction between twin-tunnels influence of the relativeposition and construction procedurerdquo Tunnelling and Un-derground Space Technology vol 23 no 2 pp 210ndash214 2008
[9] X G Li and D J Yuan ldquoResponse of a double-decked metrotunnel to shield driving of twin closely underpasses tunnelsrdquoTunnelling and Underground Space Technology vol 28 no 1pp 18ndash30 2012
[10] T E Vorster A Klar K Soga and R J Mair ldquoEstimating theeffects of tunnelling on existing pipelinesrdquo Journal of Geo-technical and Geoenvironmental Engineering vol 131 no 11pp 1399ndash1410 2005
[11] Z G Zhang M S Huang and W D Wang ldquoEvaluation ofdeformation response for adjacent tunnels due to soilunloading in excavation engineeringrdquo Tunnelling and Un-derground Space Technology vol 38 no 3 pp 244ndash253 2013
[12] A P S Selvadurai Elastic Analysis of Soil-Foundation InteractionElsevier Scientific Pub Co Amsterdam Netherlands 1979
[13] N Loganathan and H G Poulos ldquoAnalytical prediction fortunnelling-induced ground movements in claysrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124no 9 pp 846ndash856 1998
[14] K M Lee R K Rowe and K Y Lo ldquoSubsidence owing totunnelling I Estimating the gap parameterrdquo CanadianGeotechnical Journal vol 29 no 6 pp 929ndash940 1992
[15] R K Rowe and KM Lee ldquoSubsidence owing to tunnelling IIEvaluation of a prediction techniquerdquo Canadian GeotechnicalJournal vol 29 no 6 pp 941ndash954 1992
[16] P B Attewell J Yeates and A R Selby Soil MovementsInduced by Tunnelling and their Effects on Pipelines andStructures Methuen Inc New York NY USA 1986
[17] F Ye C He H H Zhu et al ldquoAnalysis of longitudinalequivalent stiffness of shield tunnel considering lateral per-formancerdquo Chinese Journal of Geotechnical Engineeringvol 33 no 12 pp 1870ndash1876 2011 in Chinese
[18] P R Lancaster and D Mitchell ldquoProblems in Bending of Barsand Plates [M]rdquo Advanced Solid Mechanics Macmillan UK1980
[19] L Xu ldquoStudy on vertical settlement of shield tunnel in soft soilrdquoPhD dissertation Tongji University Shanghai China 2005
[20] M Yang and X Zhao ldquoAn approach for a single pile in layeredsoilrdquo Tongji University Natural Science vol 20 no 4pp 421ndash428 1992 in Chinese
16 Advances in Civil Engineering
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42 Displacement of the Existing Tunnel e location ofpoint A as shown in Figure 2 is corresponding to the in-tersection point of a plan view of the existing tunnel andnorthbound tunnel of Line 4 e location of point B iscorresponding to the intersection point of the plan view ofthe existing tunnel and southbound tunnel of Line 4 emiddle point of points A and B is point C
421 Vertical Displacement of the Existing Tunnel Figure 9shows time-varying vertical displacement in the moni-toring rings of the existing tunnel e x-axis is the ringnumber of the existing tunnel A positive value of theordinate denotes heave while a negative value denotessettlement of the existing tunnel e selected monitoringrings start to heave when the shield face is 0ndash10m awayfrom the selected monitoring rings which is mainly due tolarge applied tunnel face pressure bulk addictive thrust(Figure 6) and the friction force between the shield shelland soil mass When the shield tail is far beyond the se-lected monitoring rings a reduction of heave in the selectedmonitoring points is observed which is mainly due to theclosure of the tail void Only small settlements and heavesare measured after the construction of the northboundtunnel During the 3-month shutdown of the shield ad-ditional settlement ranges from 2mm to 3mm can beobserved in rings from 430 to 520 Additional settlements
ranges from 2 to 4mm were measured in rings from 460 to540 during the southbound tunnelrsquos construction elong-term additional settlements monitored up to 140 days(from 201461 to 20141019) range from 2 to 4mm inrings from 460 to 510 e settlements of rings from 430 to520 range from 2mm to 12mm and the settlement curve ofthe upline of Line 1 is ldquoUrdquo shaped after the long-termmonitoring e settlement curve is approximately sym-metric about the dashed line C after the long-term mon-itoringemaximum settlement is 12mmwhich is locatedin ring 487
0 4 8 12 03 04 05 06 0716 18 20 22
50
45
40
35
30
25
20
15
10
5
020 30 40 50 0 10 20 300 10 20 30 40 025 030 035 040
Es (MPa)w () vk0
1 Fill
3 silt
4 clay
6 muckysilty clay
8 so siltyclay
Depth of metro 1 tunnel axis
r unit weightw water contentwp plastic limit
wL liquid limitv void ratioEs elasticity modulusCU consolidation undrainage conditionCU effective value in consolidation undrainage conditionUU unconsolidation undrainage condition
gravel
r (kNm)
Dep
th (m
)
wpwL
w
Soil layer
uucucu
Depth of metro 4 tunnel axis
uucucu
c (kPa) φordm
Figure 4 Soil prole and geotechnical parameters
Table 1 Laboratory penetration test and eld steady centow test
Soil layerLaboratory
penetration test (cms) Field steady centow test (cms)
Kv Kh K
3ndash2 479times10minus5 781times10minus5 421times10minus3
3ndash3 255times10minus4 230times10minus4 157times10minus3
3ndash5 963times10minus5 174times10minus4 mdash3ndash6 329times10minus4 301times10minus4 mdash4ndash3 239times10minus7 784times10minus7 6ndash1 206times10minus7 303times10minus7 mdash6ndash2 262times10minus7 128times10minus7 mdash8ndash1 188times10minus7 511times10minus7 mdash
4 Advances in Civil Engineering
422 Horizontal Displacement of the Existing Tunnel ehorizontal displacement of the existing tunnel with respectto the location of the shield is illustrated in Figure 10 Apositive horizontal displacement denotes northward trans-verse tunnel movement away from the original tunnelcenterline while a negative horizontal displacement denotessouthward transverse tunnel movement away from theoriginal tunnel centerline
ere are northward displacement on the left side ofpoint C and southward displacement on the right side ofpoint Ce horizontal displacement curve is approximatelysymmetric about point C after the long-term monitoringe maximum northward displacement is 10mm in the ringnear the intersection point A and the maximum southwarddisplacement is minus105mm in the ring near the intersectionpoint B after the completion of the two tunnelsrsquo construc-tion During the construction of the northbound tunnelrings from 453 to 471 move southward which is likely due tothe additional bulkhead additive thrust and the squeezingforce provided by the shield shell When the shield tail leavesring 459 rings from 439 to 459 move southward slowlyOnly 2 to 5mm additional southward displacement ismeasured in rings from 487 to 560 and nearly no dis-placement is observed on the left side of point B during theconstruction of the southbound tunnel which is mainly dueto the northbound tunnelrsquos barrier eect No change ofhorizontal displacement is observed in the long-term con-ditions (from 2014528 to 20141019)
423 Convergence Displacement of the ExistingTunnel Figure 11 shows the convergence displacement ofthe existing tunnel with respect to the locations of the shield
A negative value indicates the reduction horizontal diameterof the existing tunnel while a positive value indicates theaddition horizontal diameter of the existing tunnel It can beobserved that the convergence displacement is not obviouswhen the shield reaches ring 48 (22 rings away from theintersection point A) During the process of the shielddriving from ring 64 to ring 87 a signicantly additionalincrease in the negative convergence displacement in ringsfrom 440 to 490 is observed and themaximum displacementoccurs in the intersection point A e reason might be thatthe shield face squeezes one side of the existing tunnel
20950
21000
21050
21100
21150
21200
21250
21300268
270
e r
ing
num
ber o
f the
no
rthb
ound
tunn
el
70
Guanhe Station
paused for 106 days
56 days
e tunnel face of the northbound tunnele tunnel face of the southbound tunnel
Tunn
el ch
aina
ge (m
)
Date
46 days
East railway station
170
250
200
150
100
50
0
-50
0
50
100
150
200
250
e r
ing
num
ber o
f the
sout
hbou
nd tu
nnel
2013
12
3
2013
12
13
2013
12
23
2014
12
2014
11
2
2014
12
2
2014
13
020
144
10
2014
41
4
2014
42
0
2014
43
0
2014
51
0
2014
52
0
2014
53
0
Figure 5 Advance-time curve of the shield
430 440 450 460 470 480 490 500 510 515
020
022
024
026
028
030
032
034
036
038
040
Northbound tunnelSouthbound tunnel
225 220 210 200 190 180 170 160 150 143
The ring number of the northbound tunnel of Line 4
The corresponding ring number of the upline tunnel of Line 1
The ring number of the southbound tunnel of Line 4
50 60 70 80 90 100 110 120 130
The a
pplie
d tu
nnel
face
pre
ssur
e (M
Pa)
Figure 6 e applied tunnel face pressure
Advances in Civil Engineering 5
leading to the reduction of the horizontal diameter of theexisting tunnel With the tunnelling of the southboundtunnel there are a reduction negative convergence dis-placement on the left side of point B and an addition positiveconvergence displacement on the right side of point B Ascan be seen from Figures 10 and 11 the shapes of theconvergence displacement curve and the horizontal dis-placement curve are similar because the causes of conver-gence displacement and horizontal displacement aretheoretically the same
5 Reinforcement Schemeof theExistingTunnel
Figure 12 shows the reinforcement in the existing tunnelRadial reinforcement by an arc-shaped supporting steelplate connected to the tunnel segment and longitudinalreinforcement by channel section steel to provide lon-gitudinal tensile stress are conducted in the existingtunnel Radial steels and longitudinal steels are connected
by welding and so the reinforcement becomes a wholeone 25 rings are reinforced at rst on either side of theintersection point A before the underpassing of thenorthbound tunnel and the whole reinforcement is
Northbound tunnelSouthbound tunnel
430 440 450 460 470 480 490 500 510 518
35
40
45
50
55
60
65
70
75
80
e corresponding ring number of the upline tunnel of Line 1
45 50 60 70 80 90 100 110 120 130 135e ring number of the northbound tunnel of Line 4
220 210 200 190 180 170 160 150 140e ring number of the southbound tunnel of Line 4
e g
rout
ing
volu
me (
m3 )
Figure 7 e grouting volume
e manual monitoring points ofconvergence displacement
Track bed
Take one point as the automatic monitoringpoint of the vertical and horizontal displacement
Centerline
e manual monitoring pointsof horizontal displacement
Figure 8 Arrangement of the monitoring points at the crosssection of the existing tunnel
400 420 440 460 480 500 520 540 560 580 600 620 640 660 670minus12
minus10
minus8
minus6
minus4
minus2
0
2
4
6 C
e construction ofnorthbound tunnel
e construction ofsouthbound tunnel
201453 Ring 135 2014515 Ring 226
Long-term monitoring201465
20146152014714
20141019
B
2013129 Ring 4820131227 Ring 72 2014112 Ring 1762014128 Ring 275
e shield is shut down2014227
Ver
tical
disp
lace
men
t (m
m)
e ring number of upline tunnel of Line 1
A
Figure 9 Time-varying vertical displacement of the existingtunnel
400 420 440 460 480 500 520 540 560 580 600 620 640 660 670
minus14minus12minus10minus8minus6minus4minus2
02468
10121416
C
e construction of thesouthbound tunnel
2014420 Ring 362014502 Ring 1242014528 Ring 270
Long-term monitoring20141019
e construction of thenorthbound tunnel
2013129 Ring 4820131227 Ring 72201415 Ring 1252014128 Rring 275
e ring number of upline tunnel of Line 1
Hor
izon
tal d
ispla
cem
ent (
mm
)
A B
Figure 10 Time-varying horizontal displacement of the existingtunnel
6 Advances in Civil Engineering
completed after the construction of the northboundtunnel e reinforcement range is from ring 424 to ring510 (ie 25 rings on either sides of the intersection pointsA and B)
6 Theoretical Analysis of the ReinforcementDesign
e deformed tunnel lining reinforced by inner bondingsteel plates has been partially or entirely used in manybuilt tunnels or currently under construction To better
understand the benets coming from the reinforcementmethod the behavior of the tunnel is investigated
Figure 13 shows a schematic view of the existingtunnel-soil tunnelling interaction e analysis methoddemonstrated in this paper can be divided into two stepsrstly estimating the greeneld displacement induced bythe tunnelling Secondly calculating the responses of theexisting tunnel subjected to the soil displacement emethod of analysis is based on three assumptions (1) theabove existing tunnel does not aect the displacement ofsoil due to tunnelling (2) the soil foundation is assumedas the Winkler or the Pasternak foundation and (3) thesoil displacement is calculated by superimposing theindependent settlement predicted for each individualtunnelling
In this paper the tunnels are considered as an innitebeam on the Winkler foundation an innite beam on thePasternak foundation and a nite beam on the Pasternakfoundation (the new method) Comparing the decentectionrotation angle normalized bendingmoment and shear forceof the existing tunnel with constant stiness based on thethree models the new method is veried en dierentstiness of the reinforcement is taken into considerationand the optimal reinforcement range of the existing tunnel isdiscussed
61 e Subsurface Soil Displacement due to Tunnelling Forthe theoretical analysis the alignment of the new tunnels andthe existing tunnel is assumed to be straight Figure 14 showsa schematic diagram of the new tunnels and the existingtunnel in this case e greeneld settlement s(x) due to thetunnelling can be replaced by the equivalent distributed loadq(x) acting on the beam on the third assumption as follows
q(x) ks(x) (2)
In this study subsurface settlement s(x) at the depth of zinduced by tunnelling is calculated based on closed-formanalytical solutions presented by Loganathan and Poulos[13] as follows
s(x) ε0R2 zminusHx2 +(zminusH)2
+(3minus 4v)z +H
x2 +(z +H)2minus2z x2 minus(z +H)2( )
x2 +(z +H)2( )2
middot e minus138x2(H+R)2minus069z2H2( ) (3)
Conv
erge
nce d
ispla
cem
ent (
mm
)
400 420 440 460 480 500 520 540 560 580 600 620 640 660minus7minus6minus5minus4minus3minus2minus1
012345
e construction ofsouthbound tunnel
201453 Ring 135201459 Ring 1912014528 Ring 270
e shield is shut down201461
e construction ofnorthbound tunnel
2013129 Ring 4820131222 Ring 6420131230 Ring 87
e shield is shut down201424
e ring number of the upline of Line 1
A B
Figure 11 Convergence displacement of the existing tunnel
Figure 12 Reinforcement in the existing tunnel
xk1
k2
e existing tunnel
Sz(x)
Figure 13 A schematic view of the existing tunnel-soil tunnellinginteraction
Advances in Civil Engineering 7
where s(x) subsurface settlement of the soils due to tun-nelling R tunnel radius z depth below ground surfaceH depth of the axis of the new tunnel vPoissonrsquos ratiox lateral distance from tunnel centerline and ε0 equivalentground loss ratio which is dened as
ε0 4Rg + g2
4R2 (4)
g Gp + ulowast3D + w asymp Gp (5)
where g is the gap parameter (Lee et al [14]) ulowast3D is thethree-dimensional elastoplastic deformation at the tunnelface w is the workmanship factor and Gp represents thephysical gap between the outer skin of the shield and thelining which is given as
Gp 2Δ + δ (6)
where Δ is the thickness of the tail piece and δ is the clearancerequired for erection of the tail piece e parameter w can beneglected due to considerable experience with the equipmentand good tunnelling technique of construction workers basedon the study of Rowe and Lee [15]
62 Analytical Methods Derived from Prior Studies Attewellet al [16] proposed a tunnel-soil interaction model andsolved the analytical solution of the longitudinal displace-ment of an innite tunnel
d4w(x)dx4
+ 4λ4p(x) 4λ4q(x) (7)
where q(x) is the concentrated load acting on the originpoint of the beam EI is the centexural stiness of a tunnel
which is recommended by Ye et al [17] and k is the subgrademodulus that represents the pipe-soil interaction
If the soil foundation is assumed as the Winklerfoundation the foundation counterforce p(x) can beexpressed as
p(x) kw(x) (8)
e two-parameter model of foundation can capture theshear resistance of soil Based on the assumption of theplane strain condition the displacement of ground in they-direction is neglected If the soil foundation is assumed asthe Pasternak foundation the foundation counterforce p(x)can be expressed as
p(x) minusGnabla2w(x) + kw(x) (9)
where G is the coelaquocient of the shear element in thePasternakrsquos model with the dimension of force per unitlength
Attewell et al [16] presented a solution for the con-centrated load q on an innite beam resting on the Winklerfoundation and computed the decentection as
w(x) 1
8EIλ3qeminusλx( cos λx + sin λx) (10)
It can be concluded that each of the concentrated loadks(τ) at point τ contributes the following amount to thedecentection of the existing tunnel on the Winkler foundation
w(x) 1
8EIλ3int+infin
minusinfin2Rks tminus L1( )sin α( ) eminusλ|xminusτ|
(cos λ|xminus τ| + sin λ|xminus τ|)dτ(11)
where L1 is the distance between the coordinate origin of theexisting tunnel and the intersection point between the newtunnel and the existing tunnel and α is the intersection anglebetween the new tunnel and existing tunnel e decentectionrotational angle bending moment and shear force of theexisting tunnel which is assumed as an innite beam on theWinkler foundation and the Pasternak foundation are il-lustrated in Appendix A
63 A Brief Description of the NewMethod In this study wewant to gure out the eects of the reinforcement on theexisting tunnel the stiness varies in this case and theexisting tunnel assumed as an innite beam is not suitableAssuming the existing tunnel with nite length and in-troducing (7) to (9) the basic dierential equation gov-erning the centexure of the beam resting on the Pasternakfoundation can be written in the form with an assumptionof Gλ2klt 1
EiIid4wi(x)dx4minusGibi
d2wi(x)dx4
+ kibiwi(x) biqi(x) (12)
where b1i bi(1 +(Gikiradic
bi)) subscript i represents the ithelement of the beam bi is the width of the beam and b1i isthe modied width of the beam Assuming a solution of (12)in the form of w(x) eλx the general solution of verticaldisplacement of the beam is as follows
y
e southbound tunnel
e northbound tunnel
e e
xistin
g tun
nel
x
L 1L 2
0x 1
x 2x 3
x ix i+
1
x nminus2
x nminus1
x n
O2
O1
xx
Figure 14 A schematic diagram of tunnels for the analytical methods
8 Advances in Civil Engineering
w(x) sum4
i1CiFi(x) (13)
where Ci (i 1 2 3 4) are undetermined parameters whichare dependent on the boundary conditions and Fi (x)(i 1 2 3 4) are the basic functions of the general solutionwhich are given as follows
F1i(x) cos a1icix( ) sinh a2icix( ) (14)
F2i(x) cos a1icix( ) cosh a2icix( ) (15)
F3i(x) sin a1icix( ) cosh a2icix( ) (16)
F4i(x) sin a1icix( ) sinh a2icix( ) (17)
where ci kib14EiIi4radic
a1i 1minusGic2i kiradic
a2i1 + Gic2i kiradic
e relationship of the decentection rotational angle
bending moment and shear force of the beam proposed byLancaster and Mitchell [18] is as follows
θi(x) wprimei(x)Mi(x) minusEiIiwPrime(x)
Qi(x) minusEiIiw(x) + Gpib1iwprimei(x)
(18)
where θi(x) Mi(x) and Qi(x) are the amplitudes of ro-tational angle bending moment and shear force of thesection area respectively
Figure 15 shows that dierent stiness is replaced byequivalent stepped stiness According to dierent stinessand distributed load the local coordinate system is establishedas illustrated in Figure 16 Selvadurai [12] suggests an initialparameter method to solve the parameters of the beam efour parameters at the origin of the coordinates arewi(0i) θi(0i)Mi(0i) and Qi(0i) and four parameters at theend of the element are wi(xi) θi(xi) Mi(xi) and Qi(xi)C1 C2 C3 and C4 can be expressed by wi(0i) θi(0i)Mi(0i) and Qi(0i) by introducing xi 0 to (13) and (18)
wi(xi) θi(xi)Mi(xi) and Qi(xi) can be rewritten in thematrix-array form as
wi xi( )θi xi( )Mi xi( )Vi xi( )1
k11 k12 k13 k14 k15
k21 k22 k23 k24 k25
k31 k32 k33 k34 k35
k41 k42 k43 k44 k45
0 0 0 0 1
middot
wi 0i( )θi 0i( )Mi 0i( )Vi 0i( )1
(19)
where [kij] is a stiness submatrix with 5 times 5 order ij 1 2 5M and V are respectively the bending momentand the generalized shear force Formula (19) can be sim-plied as
Ni xi( ) KijNi 0i( ) (20)
where Ni(xi) [Wi(xi) θi(xi)Mi(xi) Qi(xi) 1]T andNi(0i) [Wi(0i) θi(0i)Mi(0i) Qi(0i) 1]T
Figure 17 shows that curved distributed load is replacedby equivalent triangular and square distributed load whichcan be expressed as
qiminus1 +qi qiminus1 + ξqi minus qiminus1Li
( ) (21)
where qiminus1 and qi are respectively the load of the startingpoint and the ending point of the ith element and ξ is theposition of arbitrary load From (19) we can obtain
k15 minusbintx0 qiminus1 +qi( ) aminus12i F1i[xminus ξ]minus aminus11i F3i[xminus ξ]( )dξ
4c3i EiIi
(22)In order to avoid numerical errors in the calculation
process the integration of ki5 is shown in Appendix B erest of kij can be referred in the work by Selvadurai [12]
Figure 18 shows the tunnel matrix transfer diagramAssuming that the tunnel is composed of N elements thex-coordinate of beginning points of the ith element isxi(i 0 1 2 n) and the length of each element is Liwhere Li xi minus ximinus1
E0I0
0 x1 x2 x3 ximinus1 xi xnminus1 xn x
E1I1E2I2
Eiminus1Iiminus1Enminus1Inminus1
Figure 15 Sketch of equivalent stepped stiness
qi(x)
pi
Qi Qi+1Mi Mi+1
Oimi
xi
Figure 16 Forces and local coordinate system
q0q1 q2
qiminus1 qnminus1
x1 x2 x3 ximinus1 xi xnminus1 xn x
q (x)
0
Figure 17 Replacement of variable load with trapezoid load
Advances in Civil Engineering 9
For the rst element
N1(x) K1(x) middotN1 01( ) x isin 0 L1( ) (23)
For the end point of the rst element
N1 x1( ) K1 x1 minus 0( ) middotN1 01( ) (24)
For the end point of the ith element
Ni xi( ) Ki xi( ) middotKiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( ) middotN1 01( )(25)
For x at the ith element section where x is a globalcoordinate
Ni(x)Ki xminus ximinus1( ) middotKiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( ) middotN1 01( )(26)
Suppose
Ai(x) Ki xminusximinus1( ) middot Kiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( )(27)
en (20) can be expressed as
Ni(x) Ai(x) middotN1 01( ) (28)
us the relation between the vectorNn(xn) at the pointxn and the vector N1(01) at the start point x0 is
Nn xn( ) An xn( ) middotN1 01( ) (29)
where
An xn( ) prod1
inKi Li( ) (30)
e boundary condition of the existing tunnel assumedas an innite long tunnel considering the shear stress of soilcan be written as
M(0) 0
V(0) 2
k
G
radic
middot G middot R middot w(0)
V(L) 0
V(L) minus2
k
G
radic
middot G middot R middot w(L)
(31)
By applying the boundary conditions to (19) the de-centection rotational angle bending moment and shear forceof the overall beam can be obtained
64 Case Study
641 Parameter Selection To enable a direct compari-son assumed parameters of the soil and existing tunnelare selected in this study Poissonrsquos ratio of the soil is0388 e grouting process is regarded as the inverseprocess of soil loss due to tunnelling and g can beexpressed as g (2Δ + δ)(1minus η) where η is the injectionratio
For the Pasternak foundation as the thickness of theelastic layer is assumed to be 25 B the subgrade elasticmodulus k and shear stiness G are suggested by Xu [19] asfollows
k 5E0β
16 1minus v05pt2( )times 122
G 13E0B
2
32 1 + v0( )βtimes 085
(32)
where β is a modied parameter (β 129) and E0 and v0 arerespectively the elastic modulus and Poissonrsquos ratio forelastic foundation which are given as follows
E0 Es
1minus v2s
v0 vs
1minus vs
(33)
where the empirical relation between the Youngrsquos modulusEs and the compression modulus Es01minus02 of soft soils is builtup by Yang and Zhao [20]
E0 (25 sim 35)Es01minus02 (34)
As illustrated in Figure 14 points O1 and O2 are re-spectively the intersection points A and B in practicalcoordinate transformation expressions of the axis xprime and xPrimeare respectively
xprime xminusL1( )sin αxPrime xminusL2( )sin α
(35)
where L1 and L2 are the distance from the beginningpoint of the existing tunnel to points O1 and O2 re-spectively e length of the existing tunnel is assumedas 250 m and the ring spacing is 12 m So the tunnel isdivided into 209 sections and each section is 12 m inlength
N1(01) Ni(xi)N1(x1) = N2(02) N2(x2)L1
L1
E1I1
G1 k1 G2 k2 G3 k3 Gi ki Gn knGnminus1 knminus1
E2I2 E3I3 EiIi EnInEnminus1Inminus1
L2 L3 Li LnLnminus1
L2 Li
x x x
xx1 x2 x3 xi-1 xnminus2 xnminus1 xnxi0
Ni(0i)
Figure 18 e matrix transfer diagram
10 Advances in Civil Engineering
7 Discussions
e normalized bending moment and shear force are de-ned as
Mi MiL0EiIi
Qi QiL
20
EiIi
(36)
where Mi and Qi are respectively the normalized bendingmoment and shear force Mi and Qi are respectively thecalculated bending moment and shear force and L0 is thelength of the existing tunnel
Figure 19 shows the comparison of the decentection ro-tational angle bending moment and shear force of theexisting tunnel with a constant stiness which is based onthree kinds of models an innite tunnel on the Winklerfoundation an innite tunnel on the Pasternak foundationand a nite tunnel on the Winkler foundation Note that the
368 380 400 420 440 460 480 500 520 540 560 580 600 612
minus0007
minus0006
minus0005
minus0004
minus0003
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)
minus0002
minus0001
0000
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundatione observed data on 201465
e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
minus000015
minus000010
minus000005
000000
000005
000010
000015
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
(b)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
e b
endi
ng m
omen
t of t
he ex
istin
g tu
nnel
(kN
middotm)
387 400 420 440 460 480 500 520 540 560 580 595
0
minus6000
minus5000
minus4000
minus3000
minus2000
minus1000
1000
2000
3000
4000
e ring number of the existing tunnel
(c)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
387 400 420 440 460 480 500 520 540 560 580 595
minus600
minus400
minus200
0
200
400
600
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(kN
)
(d)
Figure 19 (a) Comparison of the decentection of the existing tunnel using dierent models (b) Comparison of the rotational angle ofthe existing tunnel using dierent models (c) Comparison of the normalized bending moment of the existing tunnel using dierentmodels (d) Comparison of the normalized shear force of the existing tunnel using dierent models
Advances in Civil Engineering 11
existing tunnel is only subject to soil deformation force inthis case e displacement and the decentection of beam areparticular similar concepts owing to the fact that the existingtunnel is securely attached to the foundation generallywithout rigid body motion Fairly good agreement can beobserved between the innite tunnel on the Pasternakfoundation and the nite tunnel on the Pasternak founda-tion which shows that the method proposed in this paper iscorrect and can be used to analyze similar problems Becauseof considering soil shear stress vertical stress and de-formation are aected and weakened the displacement of
the existing tunnel on the Pasternak foundation is smallerthan that on the Winkler foundation without consideringthe shear stress of soil Using the Winkler foundation modelis in good agreement with the observed data It can be seenfrom (3) that the displacement of the existing tunnel isproportional to the grouting elaquociency (ie η) Due to theuncertainty of η it cannot be decided which one is the bestanalytical method It is clear that the maximum normalizedbending moment occurs at the points O1 and O2 which arethe nearest points between the existing tunnel and the newtunnels
minus0003
minus0002
minus0001
0000387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
minus000010
minus000005
000000
000005
000010
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e ring number of the existing tunnel
(b)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10
minus0003
minus0002
minus0001
0000
0001
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
xL
(c)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10xL
minus008
minus006
minus004
minus002
000
002
004
006
008
The n
orm
alise
d sh
ear f
orce
of t
he ex
istin
g tu
nnel
(d)
Figure 20 (a) e eects of dierent reinforcement stiness on the decentection of the existing tunnel (b) e eects of dierent re-inforcement stiness on the slope of the existing tunnel (c)e eects of dierent reinforcement stiness on the normalized moment of theexisting tunnel (d) e eects of dierent reinforcement stiness on the normalized shear force of the existing tunnel
12 Advances in Civil Engineering
e reinforce range of the existing tunnel is assumed as25 rings on either side of the intersection point A and thevarious stiness is taken into consideration using themethod proposed in this paper Figure 20 shows the in-centuence of dierent stiness of the reinforcement on theexisting tunnel (ie 4 EI EI and 14 EI) It is evident thatthe reinforcement can reduce the displacement rotationalangle normalized bending moment and shear force of theexisting tunnel while tunnel damage (reduce the stinessof reinforcement) has the opposite eect It is evident thatthe change of reinforcement stiness has greater eect onthe normalized bending moment and the normalized shearforce of the existing tunnel particularly at the incentection
points of the normalized bending moment and normalizedshear force curve
e reinforce stiness of the reinforcement range isassumed as 4 EI and the various reinforcement ranges istaken into consideration using the method proposed in thispaper Figure 21 shows the incentuence of dierent re-inforcement ranges on the existing tunnel when thenorthbound tunnel of Line 4 underpasses e normalizedbending moment on either side of the intersection point istoo large when the reinforcement range is 69ndash892m andthe normalized shear force on either side of the intersectionpoint is relatively too large when the reinforcement range isless than 592ndash992m So it is necessary to reinforce larger
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00035
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
The d
eflec
tions
of t
he ex
istin
g tu
nnel
(m)
xL
(a)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus000010
minus000005
000000
000005
000010
The s
lope
of t
he ex
istin
g tu
nnel
(rad
)
xL
(b)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
00005
00010
00015
xL
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
(c)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus010
minus008
minus006
minus004
minus002
000
002
004
006
008
010
xLTh
e nor
mal
ised
shea
r for
ce o
f the
exist
ing
tunn
el
(d)
Figure 21 (a)e eects of dierent reinforcement ranges on the decentection of the existing tunnel (b)e eects of dierent reinforcementranges on the slope of the existing tunnel (c)e eects of dierent reinforcement ranges on the normalized moment of the existing tunnel(d) e eects of dierent reinforcement ranges on the normalized shear force of the existing tunnel
Advances in Civil Engineering 13
than the range of 492ndash1092m in the existing tunnel in thiscase that is larger than 30m (25 rings) on either side of theintersection point )ere is no big difference in the bendingmoment and shear force of the existing tunnel when thereinforcement ranges are 492ndash1092m 392ndash1192m and292ndash1292m In order to reduce the costs of reinforcementreinforcement in the 30m range (ie 25 rings) on eitherside of the intersection point is the best choice whichverifies the actual design
8 Conclusions
Based on the abovementioned statements the response of theexisting tunnel is analyzed and analytical methods are proposedfor verification and the reinforcement design Main conclusionsderived from the analysis are as follows
(1) )e displacement of the existing tunnel changes withthe relative position of the shield and the existingtunnel Shield parameters have a major impact on theexisting tunnel )e heaves of the existing tunnelmight be caused by the large applied face pressure andbulkhead addictive thrust When the shield tail isdriving beyond the selected monitoring rings theselected monitoring rings settle rapidly due to theclosure of the shield tail void
(2) )e southbound tunnel of Line 4 has less effect onthe left side of the intersection point A on theexisting tunnel because of the northbound tunnelrsquosbarrier effect A variety of measures of displacementprevention of Line 1 such as the control of shield
parameters reinforcement of the existing tunnel canensure the normal operation of metro Line 1
(3) )e analytical method proposed in the paper is ver-ified by the methods derived from prior studies It ispossible to take the different stiffness and curved loadsinto account by applying the stepped stiffness andtrapezoidal load while using a local coordinate systemin the derivation of the new method )e change ofreinforcement stiffness has greater effect on thenormalized bending moment and the normalizedshear force of the existing tunnel particularly at theinflection points of the normalized bending momentand normalized shear force curve )e analyticalmethod can be applied for the reinforcement designand safety check of the built tunnel
Appendices
A1 An Infinite Beam on the WinklerFoundation
θ(x) 1113946+infin
minusinfin
minusλ2s((tminus L)sin α)eminusλ(xminust) sin(λ(xminus t))
b
λ2s((tminus L)sin α)eλ(xminust)sin(λ(tminus x))
b
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎭
dt
(A1)
M(x) 14λ
1113946+infin
minusinfinks((tminusL)sin α)e
minusλ|xminust|( cos(λ|xminus t|)minus sin(λ|xminus t|))dt (A2)
Q(x) 1113946+infin
minusinfin
minus12
ks((tminusL)sin α)eminusλ(xminust)cos(λ(xminus t))dt xminus tgt 0
12
ks((tminusL)sin α)eminusλ(tminusx)cos(λ(tminusx))dt xminus tle 0
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
⎫⎪⎪⎪⎬
⎪⎪⎪⎭
dt (A3)
A2 An Infinite Beam on the PasternakFoundation
w(x) λ2bk
1113946+infin
minusinfinks((tminus L)sin α)De
minusa1λ|xminusτ| 1a1
cos a2λ|xminus τ|( 1113857 +1a2
sin a2λ|xminus τ|( 11138571113888 1113889dτ
θ(x) 1113946+infin
minusinfin
minusλ2s((tminusL))eminusa1λ(xminust)sin a2(xminus t)( 1113857
a1a2b
λ2s((tminus L))ea1λ(xminust)sin a2(tminus x)( 1113857
a1a2b
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A4)
14 Advances in Civil Engineering
M(x) 14λ
1113946+infin
minusinfinks((tminus L)sin α)e
minusa1λ|xminust| 1a1
cos a2λ|xminus t|( 1113857minus1a2
sin a2λ|xminus t|( 11138571113888 1113889dt
Q(x) 1113946+infin
minusinfin
minus12
ks((tminus L)sin α)eminusa1λ(xminust) cos a2λ(xminus t)( 1113857minus
a21 minus a 2
22a1a2
1113888 1113889sin a2λ(xminus t)( 11138571113890 1113891dt xminus tgt 0
12
ks((tminusL)sin α)eminusa1λ(tminusx) cos a2λ(tminusx)( 1113857minus
a21 minus a2
2
2a1a21113888 1113889sin a2λ(tminusx)( 11138571113890 1113891dt xminus tle 0
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A5)
B1 Coefficients of the Matrix ki5
k15 minusbiqiminus1
4a1ia2ici4EiIi a1i
2 + a2i2( 1113857
minus2a1ia2i + 2a1ia2iF2i
+ a1i2 minus a2i
2( 1113857F4i
⎛⎜⎝ ⎞⎟⎠
+bi qi minus qiminus1( 1113857
4a1ia2ici5EiIiL3 a1i
2 + a2i2( 1113857
2
a2i minus3a21i + a2
2i( 1113857F3i
+a1i 2a2i a21i + a2
2i( 1113857cix + a1i2 minus 3a2i
2( 1113857F1i( 1113857
⎛⎜⎝ ⎞⎟⎠
(B1)
k25 biqiminus1 a2iF3i minus a1iF1i( 1113857
2a1a2 a21 + a 2
2( 1113857c3EiIi
+bi qi minus qiminus1( 1113857 2a1ia2i minus 2a1ia2iF2i minus a1i
2 minus a2i2( 1113857F4i( 1113857( 1113857
2a1ia2i a1i2 + a2i
2( 11138572ci
4EiIix1113872 1113873 (B2)
k35 minusbiqiminus1F4i
2a1ia2ici2 minus
bi qi minus qiminus1( 1113857 a2iF3i minus a1iF1i( 1113857
2a1ia2ici3 a1i
2 + a2i2( 1113857x
(B3)
k45 minusbiqiminus1
a1i2 + a2i
2( 1113857ci
1113888a1i 1 + 2a1i2a22i minus 2a2i
41113872 1113873F3i + a2i 1 + 2a1i
2a22i minus 2a1i
41113872 1113873F1i1113889
minusbi qi minus qiminus1( 1113857
a21i + a2i
2( 11138572c 2
i x
4a1i2a2i
2 minus 1( 1113857 a1i2 minus a2i
2( 1113857 + a2i2 minus 4a1i
4a2i2 + a1i
2 minus1 + 4a2i4( 1113857( 1113857F2i
minus2a1ia2i a 21i minus a 2
2i( 11138572 minus 11113872 1113873F4i
⎛⎜⎝ ⎞⎟⎠
(B4)
Conflicts of Interest
)e author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
)e author appreciates the help from the staff of Hangzhouand Hong Kong Tunnel Company Ltd during in-strumentation setup and data acquisition )is study issupported by the National Natural Science Foundation ofChina (no 41702313)
References
[1] S H Kim Model testing and analysis of interactions betweentunnels in clay [PhD 3esis] Department of EngineeringScience University of Oxford Oxford UK 1996
[2] H Y Liu J C Small J P Carter and D J Williams ldquoEffectsof tunnelling on existing support systems of perpendicularlypassing tunnelsrdquo Computers and Geotechnics vol 36 no 5pp 880ndash894 2009
[3] I Yamaguchi I Yamazaki and Y Kiritani ldquoStudy ofgroundndashtunnel interactions of four shield tunnels driven inclose proximity in relation to design and construction ofparallel shield tunnelsrdquo Tunnelling and Underground SpaceTechnology vol 13 no 3 pp 289ndash304 1998
[4] T I Addenbrooke and D M Potts ldquoTwin tunnel interactionsurface and subsurface effectsrdquo International Journal ofGeomechanics vol 1 no 2 pp 249ndash271 2001
[5] J P Kimmance S Lawrence O Hassan and N J PurchaseldquoObservations of deformations created in existing tunnels byadjacent and cross cutting excavationsrdquo in Proceedings ofGeotechnical Aspects of Underground Construction in SoftGround pp 707ndash712 London UK July 1996
[6] E Leca and B New ldquoSettlements induced by tunnelling in softgroundrdquo Tunnelling and Underground Space Technologyvol 22 no 2 pp 119ndash149 2007
Advances in Civil Engineering 15
[7] S M Liao J H Liu R L Wang and Z M Li ldquoShieldtunnelling and environment protection in Shanghai softgroundrdquo Tunnelling and Underground Space Technologyvol 24 no 4 pp 454ndash465 2009
[8] F H Chehade and I Shahrour ldquoNumerical analysis of theinteraction between twin-tunnels influence of the relativeposition and construction procedurerdquo Tunnelling and Un-derground Space Technology vol 23 no 2 pp 210ndash214 2008
[9] X G Li and D J Yuan ldquoResponse of a double-decked metrotunnel to shield driving of twin closely underpasses tunnelsrdquoTunnelling and Underground Space Technology vol 28 no 1pp 18ndash30 2012
[10] T E Vorster A Klar K Soga and R J Mair ldquoEstimating theeffects of tunnelling on existing pipelinesrdquo Journal of Geo-technical and Geoenvironmental Engineering vol 131 no 11pp 1399ndash1410 2005
[11] Z G Zhang M S Huang and W D Wang ldquoEvaluation ofdeformation response for adjacent tunnels due to soilunloading in excavation engineeringrdquo Tunnelling and Un-derground Space Technology vol 38 no 3 pp 244ndash253 2013
[12] A P S Selvadurai Elastic Analysis of Soil-Foundation InteractionElsevier Scientific Pub Co Amsterdam Netherlands 1979
[13] N Loganathan and H G Poulos ldquoAnalytical prediction fortunnelling-induced ground movements in claysrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124no 9 pp 846ndash856 1998
[14] K M Lee R K Rowe and K Y Lo ldquoSubsidence owing totunnelling I Estimating the gap parameterrdquo CanadianGeotechnical Journal vol 29 no 6 pp 929ndash940 1992
[15] R K Rowe and KM Lee ldquoSubsidence owing to tunnelling IIEvaluation of a prediction techniquerdquo Canadian GeotechnicalJournal vol 29 no 6 pp 941ndash954 1992
[16] P B Attewell J Yeates and A R Selby Soil MovementsInduced by Tunnelling and their Effects on Pipelines andStructures Methuen Inc New York NY USA 1986
[17] F Ye C He H H Zhu et al ldquoAnalysis of longitudinalequivalent stiffness of shield tunnel considering lateral per-formancerdquo Chinese Journal of Geotechnical Engineeringvol 33 no 12 pp 1870ndash1876 2011 in Chinese
[18] P R Lancaster and D Mitchell ldquoProblems in Bending of Barsand Plates [M]rdquo Advanced Solid Mechanics Macmillan UK1980
[19] L Xu ldquoStudy on vertical settlement of shield tunnel in soft soilrdquoPhD dissertation Tongji University Shanghai China 2005
[20] M Yang and X Zhao ldquoAn approach for a single pile in layeredsoilrdquo Tongji University Natural Science vol 20 no 4pp 421ndash428 1992 in Chinese
16 Advances in Civil Engineering
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422 Horizontal Displacement of the Existing Tunnel ehorizontal displacement of the existing tunnel with respectto the location of the shield is illustrated in Figure 10 Apositive horizontal displacement denotes northward trans-verse tunnel movement away from the original tunnelcenterline while a negative horizontal displacement denotessouthward transverse tunnel movement away from theoriginal tunnel centerline
ere are northward displacement on the left side ofpoint C and southward displacement on the right side ofpoint Ce horizontal displacement curve is approximatelysymmetric about point C after the long-term monitoringe maximum northward displacement is 10mm in the ringnear the intersection point A and the maximum southwarddisplacement is minus105mm in the ring near the intersectionpoint B after the completion of the two tunnelsrsquo construc-tion During the construction of the northbound tunnelrings from 453 to 471 move southward which is likely due tothe additional bulkhead additive thrust and the squeezingforce provided by the shield shell When the shield tail leavesring 459 rings from 439 to 459 move southward slowlyOnly 2 to 5mm additional southward displacement ismeasured in rings from 487 to 560 and nearly no dis-placement is observed on the left side of point B during theconstruction of the southbound tunnel which is mainly dueto the northbound tunnelrsquos barrier eect No change ofhorizontal displacement is observed in the long-term con-ditions (from 2014528 to 20141019)
423 Convergence Displacement of the ExistingTunnel Figure 11 shows the convergence displacement ofthe existing tunnel with respect to the locations of the shield
A negative value indicates the reduction horizontal diameterof the existing tunnel while a positive value indicates theaddition horizontal diameter of the existing tunnel It can beobserved that the convergence displacement is not obviouswhen the shield reaches ring 48 (22 rings away from theintersection point A) During the process of the shielddriving from ring 64 to ring 87 a signicantly additionalincrease in the negative convergence displacement in ringsfrom 440 to 490 is observed and themaximum displacementoccurs in the intersection point A e reason might be thatthe shield face squeezes one side of the existing tunnel
20950
21000
21050
21100
21150
21200
21250
21300268
270
e r
ing
num
ber o
f the
no
rthb
ound
tunn
el
70
Guanhe Station
paused for 106 days
56 days
e tunnel face of the northbound tunnele tunnel face of the southbound tunnel
Tunn
el ch
aina
ge (m
)
Date
46 days
East railway station
170
250
200
150
100
50
0
-50
0
50
100
150
200
250
e r
ing
num
ber o
f the
sout
hbou
nd tu
nnel
2013
12
3
2013
12
13
2013
12
23
2014
12
2014
11
2
2014
12
2
2014
13
020
144
10
2014
41
4
2014
42
0
2014
43
0
2014
51
0
2014
52
0
2014
53
0
Figure 5 Advance-time curve of the shield
430 440 450 460 470 480 490 500 510 515
020
022
024
026
028
030
032
034
036
038
040
Northbound tunnelSouthbound tunnel
225 220 210 200 190 180 170 160 150 143
The ring number of the northbound tunnel of Line 4
The corresponding ring number of the upline tunnel of Line 1
The ring number of the southbound tunnel of Line 4
50 60 70 80 90 100 110 120 130
The a
pplie
d tu
nnel
face
pre
ssur
e (M
Pa)
Figure 6 e applied tunnel face pressure
Advances in Civil Engineering 5
leading to the reduction of the horizontal diameter of theexisting tunnel With the tunnelling of the southboundtunnel there are a reduction negative convergence dis-placement on the left side of point B and an addition positiveconvergence displacement on the right side of point B Ascan be seen from Figures 10 and 11 the shapes of theconvergence displacement curve and the horizontal dis-placement curve are similar because the causes of conver-gence displacement and horizontal displacement aretheoretically the same
5 Reinforcement Schemeof theExistingTunnel
Figure 12 shows the reinforcement in the existing tunnelRadial reinforcement by an arc-shaped supporting steelplate connected to the tunnel segment and longitudinalreinforcement by channel section steel to provide lon-gitudinal tensile stress are conducted in the existingtunnel Radial steels and longitudinal steels are connected
by welding and so the reinforcement becomes a wholeone 25 rings are reinforced at rst on either side of theintersection point A before the underpassing of thenorthbound tunnel and the whole reinforcement is
Northbound tunnelSouthbound tunnel
430 440 450 460 470 480 490 500 510 518
35
40
45
50
55
60
65
70
75
80
e corresponding ring number of the upline tunnel of Line 1
45 50 60 70 80 90 100 110 120 130 135e ring number of the northbound tunnel of Line 4
220 210 200 190 180 170 160 150 140e ring number of the southbound tunnel of Line 4
e g
rout
ing
volu
me (
m3 )
Figure 7 e grouting volume
e manual monitoring points ofconvergence displacement
Track bed
Take one point as the automatic monitoringpoint of the vertical and horizontal displacement
Centerline
e manual monitoring pointsof horizontal displacement
Figure 8 Arrangement of the monitoring points at the crosssection of the existing tunnel
400 420 440 460 480 500 520 540 560 580 600 620 640 660 670minus12
minus10
minus8
minus6
minus4
minus2
0
2
4
6 C
e construction ofnorthbound tunnel
e construction ofsouthbound tunnel
201453 Ring 135 2014515 Ring 226
Long-term monitoring201465
20146152014714
20141019
B
2013129 Ring 4820131227 Ring 72 2014112 Ring 1762014128 Ring 275
e shield is shut down2014227
Ver
tical
disp
lace
men
t (m
m)
e ring number of upline tunnel of Line 1
A
Figure 9 Time-varying vertical displacement of the existingtunnel
400 420 440 460 480 500 520 540 560 580 600 620 640 660 670
minus14minus12minus10minus8minus6minus4minus2
02468
10121416
C
e construction of thesouthbound tunnel
2014420 Ring 362014502 Ring 1242014528 Ring 270
Long-term monitoring20141019
e construction of thenorthbound tunnel
2013129 Ring 4820131227 Ring 72201415 Ring 1252014128 Rring 275
e ring number of upline tunnel of Line 1
Hor
izon
tal d
ispla
cem
ent (
mm
)
A B
Figure 10 Time-varying horizontal displacement of the existingtunnel
6 Advances in Civil Engineering
completed after the construction of the northboundtunnel e reinforcement range is from ring 424 to ring510 (ie 25 rings on either sides of the intersection pointsA and B)
6 Theoretical Analysis of the ReinforcementDesign
e deformed tunnel lining reinforced by inner bondingsteel plates has been partially or entirely used in manybuilt tunnels or currently under construction To better
understand the benets coming from the reinforcementmethod the behavior of the tunnel is investigated
Figure 13 shows a schematic view of the existingtunnel-soil tunnelling interaction e analysis methoddemonstrated in this paper can be divided into two stepsrstly estimating the greeneld displacement induced bythe tunnelling Secondly calculating the responses of theexisting tunnel subjected to the soil displacement emethod of analysis is based on three assumptions (1) theabove existing tunnel does not aect the displacement ofsoil due to tunnelling (2) the soil foundation is assumedas the Winkler or the Pasternak foundation and (3) thesoil displacement is calculated by superimposing theindependent settlement predicted for each individualtunnelling
In this paper the tunnels are considered as an innitebeam on the Winkler foundation an innite beam on thePasternak foundation and a nite beam on the Pasternakfoundation (the new method) Comparing the decentectionrotation angle normalized bendingmoment and shear forceof the existing tunnel with constant stiness based on thethree models the new method is veried en dierentstiness of the reinforcement is taken into considerationand the optimal reinforcement range of the existing tunnel isdiscussed
61 e Subsurface Soil Displacement due to Tunnelling Forthe theoretical analysis the alignment of the new tunnels andthe existing tunnel is assumed to be straight Figure 14 showsa schematic diagram of the new tunnels and the existingtunnel in this case e greeneld settlement s(x) due to thetunnelling can be replaced by the equivalent distributed loadq(x) acting on the beam on the third assumption as follows
q(x) ks(x) (2)
In this study subsurface settlement s(x) at the depth of zinduced by tunnelling is calculated based on closed-formanalytical solutions presented by Loganathan and Poulos[13] as follows
s(x) ε0R2 zminusHx2 +(zminusH)2
+(3minus 4v)z +H
x2 +(z +H)2minus2z x2 minus(z +H)2( )
x2 +(z +H)2( )2
middot e minus138x2(H+R)2minus069z2H2( ) (3)
Conv
erge
nce d
ispla
cem
ent (
mm
)
400 420 440 460 480 500 520 540 560 580 600 620 640 660minus7minus6minus5minus4minus3minus2minus1
012345
e construction ofsouthbound tunnel
201453 Ring 135201459 Ring 1912014528 Ring 270
e shield is shut down201461
e construction ofnorthbound tunnel
2013129 Ring 4820131222 Ring 6420131230 Ring 87
e shield is shut down201424
e ring number of the upline of Line 1
A B
Figure 11 Convergence displacement of the existing tunnel
Figure 12 Reinforcement in the existing tunnel
xk1
k2
e existing tunnel
Sz(x)
Figure 13 A schematic view of the existing tunnel-soil tunnellinginteraction
Advances in Civil Engineering 7
where s(x) subsurface settlement of the soils due to tun-nelling R tunnel radius z depth below ground surfaceH depth of the axis of the new tunnel vPoissonrsquos ratiox lateral distance from tunnel centerline and ε0 equivalentground loss ratio which is dened as
ε0 4Rg + g2
4R2 (4)
g Gp + ulowast3D + w asymp Gp (5)
where g is the gap parameter (Lee et al [14]) ulowast3D is thethree-dimensional elastoplastic deformation at the tunnelface w is the workmanship factor and Gp represents thephysical gap between the outer skin of the shield and thelining which is given as
Gp 2Δ + δ (6)
where Δ is the thickness of the tail piece and δ is the clearancerequired for erection of the tail piece e parameter w can beneglected due to considerable experience with the equipmentand good tunnelling technique of construction workers basedon the study of Rowe and Lee [15]
62 Analytical Methods Derived from Prior Studies Attewellet al [16] proposed a tunnel-soil interaction model andsolved the analytical solution of the longitudinal displace-ment of an innite tunnel
d4w(x)dx4
+ 4λ4p(x) 4λ4q(x) (7)
where q(x) is the concentrated load acting on the originpoint of the beam EI is the centexural stiness of a tunnel
which is recommended by Ye et al [17] and k is the subgrademodulus that represents the pipe-soil interaction
If the soil foundation is assumed as the Winklerfoundation the foundation counterforce p(x) can beexpressed as
p(x) kw(x) (8)
e two-parameter model of foundation can capture theshear resistance of soil Based on the assumption of theplane strain condition the displacement of ground in they-direction is neglected If the soil foundation is assumed asthe Pasternak foundation the foundation counterforce p(x)can be expressed as
p(x) minusGnabla2w(x) + kw(x) (9)
where G is the coelaquocient of the shear element in thePasternakrsquos model with the dimension of force per unitlength
Attewell et al [16] presented a solution for the con-centrated load q on an innite beam resting on the Winklerfoundation and computed the decentection as
w(x) 1
8EIλ3qeminusλx( cos λx + sin λx) (10)
It can be concluded that each of the concentrated loadks(τ) at point τ contributes the following amount to thedecentection of the existing tunnel on the Winkler foundation
w(x) 1
8EIλ3int+infin
minusinfin2Rks tminus L1( )sin α( ) eminusλ|xminusτ|
(cos λ|xminus τ| + sin λ|xminus τ|)dτ(11)
where L1 is the distance between the coordinate origin of theexisting tunnel and the intersection point between the newtunnel and the existing tunnel and α is the intersection anglebetween the new tunnel and existing tunnel e decentectionrotational angle bending moment and shear force of theexisting tunnel which is assumed as an innite beam on theWinkler foundation and the Pasternak foundation are il-lustrated in Appendix A
63 A Brief Description of the NewMethod In this study wewant to gure out the eects of the reinforcement on theexisting tunnel the stiness varies in this case and theexisting tunnel assumed as an innite beam is not suitableAssuming the existing tunnel with nite length and in-troducing (7) to (9) the basic dierential equation gov-erning the centexure of the beam resting on the Pasternakfoundation can be written in the form with an assumptionof Gλ2klt 1
EiIid4wi(x)dx4minusGibi
d2wi(x)dx4
+ kibiwi(x) biqi(x) (12)
where b1i bi(1 +(Gikiradic
bi)) subscript i represents the ithelement of the beam bi is the width of the beam and b1i isthe modied width of the beam Assuming a solution of (12)in the form of w(x) eλx the general solution of verticaldisplacement of the beam is as follows
y
e southbound tunnel
e northbound tunnel
e e
xistin
g tun
nel
x
L 1L 2
0x 1
x 2x 3
x ix i+
1
x nminus2
x nminus1
x n
O2
O1
xx
Figure 14 A schematic diagram of tunnels for the analytical methods
8 Advances in Civil Engineering
w(x) sum4
i1CiFi(x) (13)
where Ci (i 1 2 3 4) are undetermined parameters whichare dependent on the boundary conditions and Fi (x)(i 1 2 3 4) are the basic functions of the general solutionwhich are given as follows
F1i(x) cos a1icix( ) sinh a2icix( ) (14)
F2i(x) cos a1icix( ) cosh a2icix( ) (15)
F3i(x) sin a1icix( ) cosh a2icix( ) (16)
F4i(x) sin a1icix( ) sinh a2icix( ) (17)
where ci kib14EiIi4radic
a1i 1minusGic2i kiradic
a2i1 + Gic2i kiradic
e relationship of the decentection rotational angle
bending moment and shear force of the beam proposed byLancaster and Mitchell [18] is as follows
θi(x) wprimei(x)Mi(x) minusEiIiwPrime(x)
Qi(x) minusEiIiw(x) + Gpib1iwprimei(x)
(18)
where θi(x) Mi(x) and Qi(x) are the amplitudes of ro-tational angle bending moment and shear force of thesection area respectively
Figure 15 shows that dierent stiness is replaced byequivalent stepped stiness According to dierent stinessand distributed load the local coordinate system is establishedas illustrated in Figure 16 Selvadurai [12] suggests an initialparameter method to solve the parameters of the beam efour parameters at the origin of the coordinates arewi(0i) θi(0i)Mi(0i) and Qi(0i) and four parameters at theend of the element are wi(xi) θi(xi) Mi(xi) and Qi(xi)C1 C2 C3 and C4 can be expressed by wi(0i) θi(0i)Mi(0i) and Qi(0i) by introducing xi 0 to (13) and (18)
wi(xi) θi(xi)Mi(xi) and Qi(xi) can be rewritten in thematrix-array form as
wi xi( )θi xi( )Mi xi( )Vi xi( )1
k11 k12 k13 k14 k15
k21 k22 k23 k24 k25
k31 k32 k33 k34 k35
k41 k42 k43 k44 k45
0 0 0 0 1
middot
wi 0i( )θi 0i( )Mi 0i( )Vi 0i( )1
(19)
where [kij] is a stiness submatrix with 5 times 5 order ij 1 2 5M and V are respectively the bending momentand the generalized shear force Formula (19) can be sim-plied as
Ni xi( ) KijNi 0i( ) (20)
where Ni(xi) [Wi(xi) θi(xi)Mi(xi) Qi(xi) 1]T andNi(0i) [Wi(0i) θi(0i)Mi(0i) Qi(0i) 1]T
Figure 17 shows that curved distributed load is replacedby equivalent triangular and square distributed load whichcan be expressed as
qiminus1 +qi qiminus1 + ξqi minus qiminus1Li
( ) (21)
where qiminus1 and qi are respectively the load of the startingpoint and the ending point of the ith element and ξ is theposition of arbitrary load From (19) we can obtain
k15 minusbintx0 qiminus1 +qi( ) aminus12i F1i[xminus ξ]minus aminus11i F3i[xminus ξ]( )dξ
4c3i EiIi
(22)In order to avoid numerical errors in the calculation
process the integration of ki5 is shown in Appendix B erest of kij can be referred in the work by Selvadurai [12]
Figure 18 shows the tunnel matrix transfer diagramAssuming that the tunnel is composed of N elements thex-coordinate of beginning points of the ith element isxi(i 0 1 2 n) and the length of each element is Liwhere Li xi minus ximinus1
E0I0
0 x1 x2 x3 ximinus1 xi xnminus1 xn x
E1I1E2I2
Eiminus1Iiminus1Enminus1Inminus1
Figure 15 Sketch of equivalent stepped stiness
qi(x)
pi
Qi Qi+1Mi Mi+1
Oimi
xi
Figure 16 Forces and local coordinate system
q0q1 q2
qiminus1 qnminus1
x1 x2 x3 ximinus1 xi xnminus1 xn x
q (x)
0
Figure 17 Replacement of variable load with trapezoid load
Advances in Civil Engineering 9
For the rst element
N1(x) K1(x) middotN1 01( ) x isin 0 L1( ) (23)
For the end point of the rst element
N1 x1( ) K1 x1 minus 0( ) middotN1 01( ) (24)
For the end point of the ith element
Ni xi( ) Ki xi( ) middotKiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( ) middotN1 01( )(25)
For x at the ith element section where x is a globalcoordinate
Ni(x)Ki xminus ximinus1( ) middotKiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( ) middotN1 01( )(26)
Suppose
Ai(x) Ki xminusximinus1( ) middot Kiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( )(27)
en (20) can be expressed as
Ni(x) Ai(x) middotN1 01( ) (28)
us the relation between the vectorNn(xn) at the pointxn and the vector N1(01) at the start point x0 is
Nn xn( ) An xn( ) middotN1 01( ) (29)
where
An xn( ) prod1
inKi Li( ) (30)
e boundary condition of the existing tunnel assumedas an innite long tunnel considering the shear stress of soilcan be written as
M(0) 0
V(0) 2
k
G
radic
middot G middot R middot w(0)
V(L) 0
V(L) minus2
k
G
radic
middot G middot R middot w(L)
(31)
By applying the boundary conditions to (19) the de-centection rotational angle bending moment and shear forceof the overall beam can be obtained
64 Case Study
641 Parameter Selection To enable a direct compari-son assumed parameters of the soil and existing tunnelare selected in this study Poissonrsquos ratio of the soil is0388 e grouting process is regarded as the inverseprocess of soil loss due to tunnelling and g can beexpressed as g (2Δ + δ)(1minus η) where η is the injectionratio
For the Pasternak foundation as the thickness of theelastic layer is assumed to be 25 B the subgrade elasticmodulus k and shear stiness G are suggested by Xu [19] asfollows
k 5E0β
16 1minus v05pt2( )times 122
G 13E0B
2
32 1 + v0( )βtimes 085
(32)
where β is a modied parameter (β 129) and E0 and v0 arerespectively the elastic modulus and Poissonrsquos ratio forelastic foundation which are given as follows
E0 Es
1minus v2s
v0 vs
1minus vs
(33)
where the empirical relation between the Youngrsquos modulusEs and the compression modulus Es01minus02 of soft soils is builtup by Yang and Zhao [20]
E0 (25 sim 35)Es01minus02 (34)
As illustrated in Figure 14 points O1 and O2 are re-spectively the intersection points A and B in practicalcoordinate transformation expressions of the axis xprime and xPrimeare respectively
xprime xminusL1( )sin αxPrime xminusL2( )sin α
(35)
where L1 and L2 are the distance from the beginningpoint of the existing tunnel to points O1 and O2 re-spectively e length of the existing tunnel is assumedas 250 m and the ring spacing is 12 m So the tunnel isdivided into 209 sections and each section is 12 m inlength
N1(01) Ni(xi)N1(x1) = N2(02) N2(x2)L1
L1
E1I1
G1 k1 G2 k2 G3 k3 Gi ki Gn knGnminus1 knminus1
E2I2 E3I3 EiIi EnInEnminus1Inminus1
L2 L3 Li LnLnminus1
L2 Li
x x x
xx1 x2 x3 xi-1 xnminus2 xnminus1 xnxi0
Ni(0i)
Figure 18 e matrix transfer diagram
10 Advances in Civil Engineering
7 Discussions
e normalized bending moment and shear force are de-ned as
Mi MiL0EiIi
Qi QiL
20
EiIi
(36)
where Mi and Qi are respectively the normalized bendingmoment and shear force Mi and Qi are respectively thecalculated bending moment and shear force and L0 is thelength of the existing tunnel
Figure 19 shows the comparison of the decentection ro-tational angle bending moment and shear force of theexisting tunnel with a constant stiness which is based onthree kinds of models an innite tunnel on the Winklerfoundation an innite tunnel on the Pasternak foundationand a nite tunnel on the Winkler foundation Note that the
368 380 400 420 440 460 480 500 520 540 560 580 600 612
minus0007
minus0006
minus0005
minus0004
minus0003
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)
minus0002
minus0001
0000
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundatione observed data on 201465
e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
minus000015
minus000010
minus000005
000000
000005
000010
000015
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
(b)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
e b
endi
ng m
omen
t of t
he ex
istin
g tu
nnel
(kN
middotm)
387 400 420 440 460 480 500 520 540 560 580 595
0
minus6000
minus5000
minus4000
minus3000
minus2000
minus1000
1000
2000
3000
4000
e ring number of the existing tunnel
(c)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
387 400 420 440 460 480 500 520 540 560 580 595
minus600
minus400
minus200
0
200
400
600
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(kN
)
(d)
Figure 19 (a) Comparison of the decentection of the existing tunnel using dierent models (b) Comparison of the rotational angle ofthe existing tunnel using dierent models (c) Comparison of the normalized bending moment of the existing tunnel using dierentmodels (d) Comparison of the normalized shear force of the existing tunnel using dierent models
Advances in Civil Engineering 11
existing tunnel is only subject to soil deformation force inthis case e displacement and the decentection of beam areparticular similar concepts owing to the fact that the existingtunnel is securely attached to the foundation generallywithout rigid body motion Fairly good agreement can beobserved between the innite tunnel on the Pasternakfoundation and the nite tunnel on the Pasternak founda-tion which shows that the method proposed in this paper iscorrect and can be used to analyze similar problems Becauseof considering soil shear stress vertical stress and de-formation are aected and weakened the displacement of
the existing tunnel on the Pasternak foundation is smallerthan that on the Winkler foundation without consideringthe shear stress of soil Using the Winkler foundation modelis in good agreement with the observed data It can be seenfrom (3) that the displacement of the existing tunnel isproportional to the grouting elaquociency (ie η) Due to theuncertainty of η it cannot be decided which one is the bestanalytical method It is clear that the maximum normalizedbending moment occurs at the points O1 and O2 which arethe nearest points between the existing tunnel and the newtunnels
minus0003
minus0002
minus0001
0000387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
minus000010
minus000005
000000
000005
000010
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e ring number of the existing tunnel
(b)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10
minus0003
minus0002
minus0001
0000
0001
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
xL
(c)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10xL
minus008
minus006
minus004
minus002
000
002
004
006
008
The n
orm
alise
d sh
ear f
orce
of t
he ex
istin
g tu
nnel
(d)
Figure 20 (a) e eects of dierent reinforcement stiness on the decentection of the existing tunnel (b) e eects of dierent re-inforcement stiness on the slope of the existing tunnel (c)e eects of dierent reinforcement stiness on the normalized moment of theexisting tunnel (d) e eects of dierent reinforcement stiness on the normalized shear force of the existing tunnel
12 Advances in Civil Engineering
e reinforce range of the existing tunnel is assumed as25 rings on either side of the intersection point A and thevarious stiness is taken into consideration using themethod proposed in this paper Figure 20 shows the in-centuence of dierent stiness of the reinforcement on theexisting tunnel (ie 4 EI EI and 14 EI) It is evident thatthe reinforcement can reduce the displacement rotationalangle normalized bending moment and shear force of theexisting tunnel while tunnel damage (reduce the stinessof reinforcement) has the opposite eect It is evident thatthe change of reinforcement stiness has greater eect onthe normalized bending moment and the normalized shearforce of the existing tunnel particularly at the incentection
points of the normalized bending moment and normalizedshear force curve
e reinforce stiness of the reinforcement range isassumed as 4 EI and the various reinforcement ranges istaken into consideration using the method proposed in thispaper Figure 21 shows the incentuence of dierent re-inforcement ranges on the existing tunnel when thenorthbound tunnel of Line 4 underpasses e normalizedbending moment on either side of the intersection point istoo large when the reinforcement range is 69ndash892m andthe normalized shear force on either side of the intersectionpoint is relatively too large when the reinforcement range isless than 592ndash992m So it is necessary to reinforce larger
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00035
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
The d
eflec
tions
of t
he ex
istin
g tu
nnel
(m)
xL
(a)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus000010
minus000005
000000
000005
000010
The s
lope
of t
he ex
istin
g tu
nnel
(rad
)
xL
(b)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
00005
00010
00015
xL
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
(c)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus010
minus008
minus006
minus004
minus002
000
002
004
006
008
010
xLTh
e nor
mal
ised
shea
r for
ce o
f the
exist
ing
tunn
el
(d)
Figure 21 (a)e eects of dierent reinforcement ranges on the decentection of the existing tunnel (b)e eects of dierent reinforcementranges on the slope of the existing tunnel (c)e eects of dierent reinforcement ranges on the normalized moment of the existing tunnel(d) e eects of dierent reinforcement ranges on the normalized shear force of the existing tunnel
Advances in Civil Engineering 13
than the range of 492ndash1092m in the existing tunnel in thiscase that is larger than 30m (25 rings) on either side of theintersection point )ere is no big difference in the bendingmoment and shear force of the existing tunnel when thereinforcement ranges are 492ndash1092m 392ndash1192m and292ndash1292m In order to reduce the costs of reinforcementreinforcement in the 30m range (ie 25 rings) on eitherside of the intersection point is the best choice whichverifies the actual design
8 Conclusions
Based on the abovementioned statements the response of theexisting tunnel is analyzed and analytical methods are proposedfor verification and the reinforcement design Main conclusionsderived from the analysis are as follows
(1) )e displacement of the existing tunnel changes withthe relative position of the shield and the existingtunnel Shield parameters have a major impact on theexisting tunnel )e heaves of the existing tunnelmight be caused by the large applied face pressure andbulkhead addictive thrust When the shield tail isdriving beyond the selected monitoring rings theselected monitoring rings settle rapidly due to theclosure of the shield tail void
(2) )e southbound tunnel of Line 4 has less effect onthe left side of the intersection point A on theexisting tunnel because of the northbound tunnelrsquosbarrier effect A variety of measures of displacementprevention of Line 1 such as the control of shield
parameters reinforcement of the existing tunnel canensure the normal operation of metro Line 1
(3) )e analytical method proposed in the paper is ver-ified by the methods derived from prior studies It ispossible to take the different stiffness and curved loadsinto account by applying the stepped stiffness andtrapezoidal load while using a local coordinate systemin the derivation of the new method )e change ofreinforcement stiffness has greater effect on thenormalized bending moment and the normalizedshear force of the existing tunnel particularly at theinflection points of the normalized bending momentand normalized shear force curve )e analyticalmethod can be applied for the reinforcement designand safety check of the built tunnel
Appendices
A1 An Infinite Beam on the WinklerFoundation
θ(x) 1113946+infin
minusinfin
minusλ2s((tminus L)sin α)eminusλ(xminust) sin(λ(xminus t))
b
λ2s((tminus L)sin α)eλ(xminust)sin(λ(tminus x))
b
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎭
dt
(A1)
M(x) 14λ
1113946+infin
minusinfinks((tminusL)sin α)e
minusλ|xminust|( cos(λ|xminus t|)minus sin(λ|xminus t|))dt (A2)
Q(x) 1113946+infin
minusinfin
minus12
ks((tminusL)sin α)eminusλ(xminust)cos(λ(xminus t))dt xminus tgt 0
12
ks((tminusL)sin α)eminusλ(tminusx)cos(λ(tminusx))dt xminus tle 0
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
⎫⎪⎪⎪⎬
⎪⎪⎪⎭
dt (A3)
A2 An Infinite Beam on the PasternakFoundation
w(x) λ2bk
1113946+infin
minusinfinks((tminus L)sin α)De
minusa1λ|xminusτ| 1a1
cos a2λ|xminus τ|( 1113857 +1a2
sin a2λ|xminus τ|( 11138571113888 1113889dτ
θ(x) 1113946+infin
minusinfin
minusλ2s((tminusL))eminusa1λ(xminust)sin a2(xminus t)( 1113857
a1a2b
λ2s((tminus L))ea1λ(xminust)sin a2(tminus x)( 1113857
a1a2b
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A4)
14 Advances in Civil Engineering
M(x) 14λ
1113946+infin
minusinfinks((tminus L)sin α)e
minusa1λ|xminust| 1a1
cos a2λ|xminus t|( 1113857minus1a2
sin a2λ|xminus t|( 11138571113888 1113889dt
Q(x) 1113946+infin
minusinfin
minus12
ks((tminus L)sin α)eminusa1λ(xminust) cos a2λ(xminus t)( 1113857minus
a21 minus a 2
22a1a2
1113888 1113889sin a2λ(xminus t)( 11138571113890 1113891dt xminus tgt 0
12
ks((tminusL)sin α)eminusa1λ(tminusx) cos a2λ(tminusx)( 1113857minus
a21 minus a2
2
2a1a21113888 1113889sin a2λ(tminusx)( 11138571113890 1113891dt xminus tle 0
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A5)
B1 Coefficients of the Matrix ki5
k15 minusbiqiminus1
4a1ia2ici4EiIi a1i
2 + a2i2( 1113857
minus2a1ia2i + 2a1ia2iF2i
+ a1i2 minus a2i
2( 1113857F4i
⎛⎜⎝ ⎞⎟⎠
+bi qi minus qiminus1( 1113857
4a1ia2ici5EiIiL3 a1i
2 + a2i2( 1113857
2
a2i minus3a21i + a2
2i( 1113857F3i
+a1i 2a2i a21i + a2
2i( 1113857cix + a1i2 minus 3a2i
2( 1113857F1i( 1113857
⎛⎜⎝ ⎞⎟⎠
(B1)
k25 biqiminus1 a2iF3i minus a1iF1i( 1113857
2a1a2 a21 + a 2
2( 1113857c3EiIi
+bi qi minus qiminus1( 1113857 2a1ia2i minus 2a1ia2iF2i minus a1i
2 minus a2i2( 1113857F4i( 1113857( 1113857
2a1ia2i a1i2 + a2i
2( 11138572ci
4EiIix1113872 1113873 (B2)
k35 minusbiqiminus1F4i
2a1ia2ici2 minus
bi qi minus qiminus1( 1113857 a2iF3i minus a1iF1i( 1113857
2a1ia2ici3 a1i
2 + a2i2( 1113857x
(B3)
k45 minusbiqiminus1
a1i2 + a2i
2( 1113857ci
1113888a1i 1 + 2a1i2a22i minus 2a2i
41113872 1113873F3i + a2i 1 + 2a1i
2a22i minus 2a1i
41113872 1113873F1i1113889
minusbi qi minus qiminus1( 1113857
a21i + a2i
2( 11138572c 2
i x
4a1i2a2i
2 minus 1( 1113857 a1i2 minus a2i
2( 1113857 + a2i2 minus 4a1i
4a2i2 + a1i
2 minus1 + 4a2i4( 1113857( 1113857F2i
minus2a1ia2i a 21i minus a 2
2i( 11138572 minus 11113872 1113873F4i
⎛⎜⎝ ⎞⎟⎠
(B4)
Conflicts of Interest
)e author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
)e author appreciates the help from the staff of Hangzhouand Hong Kong Tunnel Company Ltd during in-strumentation setup and data acquisition )is study issupported by the National Natural Science Foundation ofChina (no 41702313)
References
[1] S H Kim Model testing and analysis of interactions betweentunnels in clay [PhD 3esis] Department of EngineeringScience University of Oxford Oxford UK 1996
[2] H Y Liu J C Small J P Carter and D J Williams ldquoEffectsof tunnelling on existing support systems of perpendicularlypassing tunnelsrdquo Computers and Geotechnics vol 36 no 5pp 880ndash894 2009
[3] I Yamaguchi I Yamazaki and Y Kiritani ldquoStudy ofgroundndashtunnel interactions of four shield tunnels driven inclose proximity in relation to design and construction ofparallel shield tunnelsrdquo Tunnelling and Underground SpaceTechnology vol 13 no 3 pp 289ndash304 1998
[4] T I Addenbrooke and D M Potts ldquoTwin tunnel interactionsurface and subsurface effectsrdquo International Journal ofGeomechanics vol 1 no 2 pp 249ndash271 2001
[5] J P Kimmance S Lawrence O Hassan and N J PurchaseldquoObservations of deformations created in existing tunnels byadjacent and cross cutting excavationsrdquo in Proceedings ofGeotechnical Aspects of Underground Construction in SoftGround pp 707ndash712 London UK July 1996
[6] E Leca and B New ldquoSettlements induced by tunnelling in softgroundrdquo Tunnelling and Underground Space Technologyvol 22 no 2 pp 119ndash149 2007
Advances in Civil Engineering 15
[7] S M Liao J H Liu R L Wang and Z M Li ldquoShieldtunnelling and environment protection in Shanghai softgroundrdquo Tunnelling and Underground Space Technologyvol 24 no 4 pp 454ndash465 2009
[8] F H Chehade and I Shahrour ldquoNumerical analysis of theinteraction between twin-tunnels influence of the relativeposition and construction procedurerdquo Tunnelling and Un-derground Space Technology vol 23 no 2 pp 210ndash214 2008
[9] X G Li and D J Yuan ldquoResponse of a double-decked metrotunnel to shield driving of twin closely underpasses tunnelsrdquoTunnelling and Underground Space Technology vol 28 no 1pp 18ndash30 2012
[10] T E Vorster A Klar K Soga and R J Mair ldquoEstimating theeffects of tunnelling on existing pipelinesrdquo Journal of Geo-technical and Geoenvironmental Engineering vol 131 no 11pp 1399ndash1410 2005
[11] Z G Zhang M S Huang and W D Wang ldquoEvaluation ofdeformation response for adjacent tunnels due to soilunloading in excavation engineeringrdquo Tunnelling and Un-derground Space Technology vol 38 no 3 pp 244ndash253 2013
[12] A P S Selvadurai Elastic Analysis of Soil-Foundation InteractionElsevier Scientific Pub Co Amsterdam Netherlands 1979
[13] N Loganathan and H G Poulos ldquoAnalytical prediction fortunnelling-induced ground movements in claysrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124no 9 pp 846ndash856 1998
[14] K M Lee R K Rowe and K Y Lo ldquoSubsidence owing totunnelling I Estimating the gap parameterrdquo CanadianGeotechnical Journal vol 29 no 6 pp 929ndash940 1992
[15] R K Rowe and KM Lee ldquoSubsidence owing to tunnelling IIEvaluation of a prediction techniquerdquo Canadian GeotechnicalJournal vol 29 no 6 pp 941ndash954 1992
[16] P B Attewell J Yeates and A R Selby Soil MovementsInduced by Tunnelling and their Effects on Pipelines andStructures Methuen Inc New York NY USA 1986
[17] F Ye C He H H Zhu et al ldquoAnalysis of longitudinalequivalent stiffness of shield tunnel considering lateral per-formancerdquo Chinese Journal of Geotechnical Engineeringvol 33 no 12 pp 1870ndash1876 2011 in Chinese
[18] P R Lancaster and D Mitchell ldquoProblems in Bending of Barsand Plates [M]rdquo Advanced Solid Mechanics Macmillan UK1980
[19] L Xu ldquoStudy on vertical settlement of shield tunnel in soft soilrdquoPhD dissertation Tongji University Shanghai China 2005
[20] M Yang and X Zhao ldquoAn approach for a single pile in layeredsoilrdquo Tongji University Natural Science vol 20 no 4pp 421ndash428 1992 in Chinese
16 Advances in Civil Engineering
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leading to the reduction of the horizontal diameter of theexisting tunnel With the tunnelling of the southboundtunnel there are a reduction negative convergence dis-placement on the left side of point B and an addition positiveconvergence displacement on the right side of point B Ascan be seen from Figures 10 and 11 the shapes of theconvergence displacement curve and the horizontal dis-placement curve are similar because the causes of conver-gence displacement and horizontal displacement aretheoretically the same
5 Reinforcement Schemeof theExistingTunnel
Figure 12 shows the reinforcement in the existing tunnelRadial reinforcement by an arc-shaped supporting steelplate connected to the tunnel segment and longitudinalreinforcement by channel section steel to provide lon-gitudinal tensile stress are conducted in the existingtunnel Radial steels and longitudinal steels are connected
by welding and so the reinforcement becomes a wholeone 25 rings are reinforced at rst on either side of theintersection point A before the underpassing of thenorthbound tunnel and the whole reinforcement is
Northbound tunnelSouthbound tunnel
430 440 450 460 470 480 490 500 510 518
35
40
45
50
55
60
65
70
75
80
e corresponding ring number of the upline tunnel of Line 1
45 50 60 70 80 90 100 110 120 130 135e ring number of the northbound tunnel of Line 4
220 210 200 190 180 170 160 150 140e ring number of the southbound tunnel of Line 4
e g
rout
ing
volu
me (
m3 )
Figure 7 e grouting volume
e manual monitoring points ofconvergence displacement
Track bed
Take one point as the automatic monitoringpoint of the vertical and horizontal displacement
Centerline
e manual monitoring pointsof horizontal displacement
Figure 8 Arrangement of the monitoring points at the crosssection of the existing tunnel
400 420 440 460 480 500 520 540 560 580 600 620 640 660 670minus12
minus10
minus8
minus6
minus4
minus2
0
2
4
6 C
e construction ofnorthbound tunnel
e construction ofsouthbound tunnel
201453 Ring 135 2014515 Ring 226
Long-term monitoring201465
20146152014714
20141019
B
2013129 Ring 4820131227 Ring 72 2014112 Ring 1762014128 Ring 275
e shield is shut down2014227
Ver
tical
disp
lace
men
t (m
m)
e ring number of upline tunnel of Line 1
A
Figure 9 Time-varying vertical displacement of the existingtunnel
400 420 440 460 480 500 520 540 560 580 600 620 640 660 670
minus14minus12minus10minus8minus6minus4minus2
02468
10121416
C
e construction of thesouthbound tunnel
2014420 Ring 362014502 Ring 1242014528 Ring 270
Long-term monitoring20141019
e construction of thenorthbound tunnel
2013129 Ring 4820131227 Ring 72201415 Ring 1252014128 Rring 275
e ring number of upline tunnel of Line 1
Hor
izon
tal d
ispla
cem
ent (
mm
)
A B
Figure 10 Time-varying horizontal displacement of the existingtunnel
6 Advances in Civil Engineering
completed after the construction of the northboundtunnel e reinforcement range is from ring 424 to ring510 (ie 25 rings on either sides of the intersection pointsA and B)
6 Theoretical Analysis of the ReinforcementDesign
e deformed tunnel lining reinforced by inner bondingsteel plates has been partially or entirely used in manybuilt tunnels or currently under construction To better
understand the benets coming from the reinforcementmethod the behavior of the tunnel is investigated
Figure 13 shows a schematic view of the existingtunnel-soil tunnelling interaction e analysis methoddemonstrated in this paper can be divided into two stepsrstly estimating the greeneld displacement induced bythe tunnelling Secondly calculating the responses of theexisting tunnel subjected to the soil displacement emethod of analysis is based on three assumptions (1) theabove existing tunnel does not aect the displacement ofsoil due to tunnelling (2) the soil foundation is assumedas the Winkler or the Pasternak foundation and (3) thesoil displacement is calculated by superimposing theindependent settlement predicted for each individualtunnelling
In this paper the tunnels are considered as an innitebeam on the Winkler foundation an innite beam on thePasternak foundation and a nite beam on the Pasternakfoundation (the new method) Comparing the decentectionrotation angle normalized bendingmoment and shear forceof the existing tunnel with constant stiness based on thethree models the new method is veried en dierentstiness of the reinforcement is taken into considerationand the optimal reinforcement range of the existing tunnel isdiscussed
61 e Subsurface Soil Displacement due to Tunnelling Forthe theoretical analysis the alignment of the new tunnels andthe existing tunnel is assumed to be straight Figure 14 showsa schematic diagram of the new tunnels and the existingtunnel in this case e greeneld settlement s(x) due to thetunnelling can be replaced by the equivalent distributed loadq(x) acting on the beam on the third assumption as follows
q(x) ks(x) (2)
In this study subsurface settlement s(x) at the depth of zinduced by tunnelling is calculated based on closed-formanalytical solutions presented by Loganathan and Poulos[13] as follows
s(x) ε0R2 zminusHx2 +(zminusH)2
+(3minus 4v)z +H
x2 +(z +H)2minus2z x2 minus(z +H)2( )
x2 +(z +H)2( )2
middot e minus138x2(H+R)2minus069z2H2( ) (3)
Conv
erge
nce d
ispla
cem
ent (
mm
)
400 420 440 460 480 500 520 540 560 580 600 620 640 660minus7minus6minus5minus4minus3minus2minus1
012345
e construction ofsouthbound tunnel
201453 Ring 135201459 Ring 1912014528 Ring 270
e shield is shut down201461
e construction ofnorthbound tunnel
2013129 Ring 4820131222 Ring 6420131230 Ring 87
e shield is shut down201424
e ring number of the upline of Line 1
A B
Figure 11 Convergence displacement of the existing tunnel
Figure 12 Reinforcement in the existing tunnel
xk1
k2
e existing tunnel
Sz(x)
Figure 13 A schematic view of the existing tunnel-soil tunnellinginteraction
Advances in Civil Engineering 7
where s(x) subsurface settlement of the soils due to tun-nelling R tunnel radius z depth below ground surfaceH depth of the axis of the new tunnel vPoissonrsquos ratiox lateral distance from tunnel centerline and ε0 equivalentground loss ratio which is dened as
ε0 4Rg + g2
4R2 (4)
g Gp + ulowast3D + w asymp Gp (5)
where g is the gap parameter (Lee et al [14]) ulowast3D is thethree-dimensional elastoplastic deformation at the tunnelface w is the workmanship factor and Gp represents thephysical gap between the outer skin of the shield and thelining which is given as
Gp 2Δ + δ (6)
where Δ is the thickness of the tail piece and δ is the clearancerequired for erection of the tail piece e parameter w can beneglected due to considerable experience with the equipmentand good tunnelling technique of construction workers basedon the study of Rowe and Lee [15]
62 Analytical Methods Derived from Prior Studies Attewellet al [16] proposed a tunnel-soil interaction model andsolved the analytical solution of the longitudinal displace-ment of an innite tunnel
d4w(x)dx4
+ 4λ4p(x) 4λ4q(x) (7)
where q(x) is the concentrated load acting on the originpoint of the beam EI is the centexural stiness of a tunnel
which is recommended by Ye et al [17] and k is the subgrademodulus that represents the pipe-soil interaction
If the soil foundation is assumed as the Winklerfoundation the foundation counterforce p(x) can beexpressed as
p(x) kw(x) (8)
e two-parameter model of foundation can capture theshear resistance of soil Based on the assumption of theplane strain condition the displacement of ground in they-direction is neglected If the soil foundation is assumed asthe Pasternak foundation the foundation counterforce p(x)can be expressed as
p(x) minusGnabla2w(x) + kw(x) (9)
where G is the coelaquocient of the shear element in thePasternakrsquos model with the dimension of force per unitlength
Attewell et al [16] presented a solution for the con-centrated load q on an innite beam resting on the Winklerfoundation and computed the decentection as
w(x) 1
8EIλ3qeminusλx( cos λx + sin λx) (10)
It can be concluded that each of the concentrated loadks(τ) at point τ contributes the following amount to thedecentection of the existing tunnel on the Winkler foundation
w(x) 1
8EIλ3int+infin
minusinfin2Rks tminus L1( )sin α( ) eminusλ|xminusτ|
(cos λ|xminus τ| + sin λ|xminus τ|)dτ(11)
where L1 is the distance between the coordinate origin of theexisting tunnel and the intersection point between the newtunnel and the existing tunnel and α is the intersection anglebetween the new tunnel and existing tunnel e decentectionrotational angle bending moment and shear force of theexisting tunnel which is assumed as an innite beam on theWinkler foundation and the Pasternak foundation are il-lustrated in Appendix A
63 A Brief Description of the NewMethod In this study wewant to gure out the eects of the reinforcement on theexisting tunnel the stiness varies in this case and theexisting tunnel assumed as an innite beam is not suitableAssuming the existing tunnel with nite length and in-troducing (7) to (9) the basic dierential equation gov-erning the centexure of the beam resting on the Pasternakfoundation can be written in the form with an assumptionof Gλ2klt 1
EiIid4wi(x)dx4minusGibi
d2wi(x)dx4
+ kibiwi(x) biqi(x) (12)
where b1i bi(1 +(Gikiradic
bi)) subscript i represents the ithelement of the beam bi is the width of the beam and b1i isthe modied width of the beam Assuming a solution of (12)in the form of w(x) eλx the general solution of verticaldisplacement of the beam is as follows
y
e southbound tunnel
e northbound tunnel
e e
xistin
g tun
nel
x
L 1L 2
0x 1
x 2x 3
x ix i+
1
x nminus2
x nminus1
x n
O2
O1
xx
Figure 14 A schematic diagram of tunnels for the analytical methods
8 Advances in Civil Engineering
w(x) sum4
i1CiFi(x) (13)
where Ci (i 1 2 3 4) are undetermined parameters whichare dependent on the boundary conditions and Fi (x)(i 1 2 3 4) are the basic functions of the general solutionwhich are given as follows
F1i(x) cos a1icix( ) sinh a2icix( ) (14)
F2i(x) cos a1icix( ) cosh a2icix( ) (15)
F3i(x) sin a1icix( ) cosh a2icix( ) (16)
F4i(x) sin a1icix( ) sinh a2icix( ) (17)
where ci kib14EiIi4radic
a1i 1minusGic2i kiradic
a2i1 + Gic2i kiradic
e relationship of the decentection rotational angle
bending moment and shear force of the beam proposed byLancaster and Mitchell [18] is as follows
θi(x) wprimei(x)Mi(x) minusEiIiwPrime(x)
Qi(x) minusEiIiw(x) + Gpib1iwprimei(x)
(18)
where θi(x) Mi(x) and Qi(x) are the amplitudes of ro-tational angle bending moment and shear force of thesection area respectively
Figure 15 shows that dierent stiness is replaced byequivalent stepped stiness According to dierent stinessand distributed load the local coordinate system is establishedas illustrated in Figure 16 Selvadurai [12] suggests an initialparameter method to solve the parameters of the beam efour parameters at the origin of the coordinates arewi(0i) θi(0i)Mi(0i) and Qi(0i) and four parameters at theend of the element are wi(xi) θi(xi) Mi(xi) and Qi(xi)C1 C2 C3 and C4 can be expressed by wi(0i) θi(0i)Mi(0i) and Qi(0i) by introducing xi 0 to (13) and (18)
wi(xi) θi(xi)Mi(xi) and Qi(xi) can be rewritten in thematrix-array form as
wi xi( )θi xi( )Mi xi( )Vi xi( )1
k11 k12 k13 k14 k15
k21 k22 k23 k24 k25
k31 k32 k33 k34 k35
k41 k42 k43 k44 k45
0 0 0 0 1
middot
wi 0i( )θi 0i( )Mi 0i( )Vi 0i( )1
(19)
where [kij] is a stiness submatrix with 5 times 5 order ij 1 2 5M and V are respectively the bending momentand the generalized shear force Formula (19) can be sim-plied as
Ni xi( ) KijNi 0i( ) (20)
where Ni(xi) [Wi(xi) θi(xi)Mi(xi) Qi(xi) 1]T andNi(0i) [Wi(0i) θi(0i)Mi(0i) Qi(0i) 1]T
Figure 17 shows that curved distributed load is replacedby equivalent triangular and square distributed load whichcan be expressed as
qiminus1 +qi qiminus1 + ξqi minus qiminus1Li
( ) (21)
where qiminus1 and qi are respectively the load of the startingpoint and the ending point of the ith element and ξ is theposition of arbitrary load From (19) we can obtain
k15 minusbintx0 qiminus1 +qi( ) aminus12i F1i[xminus ξ]minus aminus11i F3i[xminus ξ]( )dξ
4c3i EiIi
(22)In order to avoid numerical errors in the calculation
process the integration of ki5 is shown in Appendix B erest of kij can be referred in the work by Selvadurai [12]
Figure 18 shows the tunnel matrix transfer diagramAssuming that the tunnel is composed of N elements thex-coordinate of beginning points of the ith element isxi(i 0 1 2 n) and the length of each element is Liwhere Li xi minus ximinus1
E0I0
0 x1 x2 x3 ximinus1 xi xnminus1 xn x
E1I1E2I2
Eiminus1Iiminus1Enminus1Inminus1
Figure 15 Sketch of equivalent stepped stiness
qi(x)
pi
Qi Qi+1Mi Mi+1
Oimi
xi
Figure 16 Forces and local coordinate system
q0q1 q2
qiminus1 qnminus1
x1 x2 x3 ximinus1 xi xnminus1 xn x
q (x)
0
Figure 17 Replacement of variable load with trapezoid load
Advances in Civil Engineering 9
For the rst element
N1(x) K1(x) middotN1 01( ) x isin 0 L1( ) (23)
For the end point of the rst element
N1 x1( ) K1 x1 minus 0( ) middotN1 01( ) (24)
For the end point of the ith element
Ni xi( ) Ki xi( ) middotKiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( ) middotN1 01( )(25)
For x at the ith element section where x is a globalcoordinate
Ni(x)Ki xminus ximinus1( ) middotKiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( ) middotN1 01( )(26)
Suppose
Ai(x) Ki xminusximinus1( ) middot Kiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( )(27)
en (20) can be expressed as
Ni(x) Ai(x) middotN1 01( ) (28)
us the relation between the vectorNn(xn) at the pointxn and the vector N1(01) at the start point x0 is
Nn xn( ) An xn( ) middotN1 01( ) (29)
where
An xn( ) prod1
inKi Li( ) (30)
e boundary condition of the existing tunnel assumedas an innite long tunnel considering the shear stress of soilcan be written as
M(0) 0
V(0) 2
k
G
radic
middot G middot R middot w(0)
V(L) 0
V(L) minus2
k
G
radic
middot G middot R middot w(L)
(31)
By applying the boundary conditions to (19) the de-centection rotational angle bending moment and shear forceof the overall beam can be obtained
64 Case Study
641 Parameter Selection To enable a direct compari-son assumed parameters of the soil and existing tunnelare selected in this study Poissonrsquos ratio of the soil is0388 e grouting process is regarded as the inverseprocess of soil loss due to tunnelling and g can beexpressed as g (2Δ + δ)(1minus η) where η is the injectionratio
For the Pasternak foundation as the thickness of theelastic layer is assumed to be 25 B the subgrade elasticmodulus k and shear stiness G are suggested by Xu [19] asfollows
k 5E0β
16 1minus v05pt2( )times 122
G 13E0B
2
32 1 + v0( )βtimes 085
(32)
where β is a modied parameter (β 129) and E0 and v0 arerespectively the elastic modulus and Poissonrsquos ratio forelastic foundation which are given as follows
E0 Es
1minus v2s
v0 vs
1minus vs
(33)
where the empirical relation between the Youngrsquos modulusEs and the compression modulus Es01minus02 of soft soils is builtup by Yang and Zhao [20]
E0 (25 sim 35)Es01minus02 (34)
As illustrated in Figure 14 points O1 and O2 are re-spectively the intersection points A and B in practicalcoordinate transformation expressions of the axis xprime and xPrimeare respectively
xprime xminusL1( )sin αxPrime xminusL2( )sin α
(35)
where L1 and L2 are the distance from the beginningpoint of the existing tunnel to points O1 and O2 re-spectively e length of the existing tunnel is assumedas 250 m and the ring spacing is 12 m So the tunnel isdivided into 209 sections and each section is 12 m inlength
N1(01) Ni(xi)N1(x1) = N2(02) N2(x2)L1
L1
E1I1
G1 k1 G2 k2 G3 k3 Gi ki Gn knGnminus1 knminus1
E2I2 E3I3 EiIi EnInEnminus1Inminus1
L2 L3 Li LnLnminus1
L2 Li
x x x
xx1 x2 x3 xi-1 xnminus2 xnminus1 xnxi0
Ni(0i)
Figure 18 e matrix transfer diagram
10 Advances in Civil Engineering
7 Discussions
e normalized bending moment and shear force are de-ned as
Mi MiL0EiIi
Qi QiL
20
EiIi
(36)
where Mi and Qi are respectively the normalized bendingmoment and shear force Mi and Qi are respectively thecalculated bending moment and shear force and L0 is thelength of the existing tunnel
Figure 19 shows the comparison of the decentection ro-tational angle bending moment and shear force of theexisting tunnel with a constant stiness which is based onthree kinds of models an innite tunnel on the Winklerfoundation an innite tunnel on the Pasternak foundationand a nite tunnel on the Winkler foundation Note that the
368 380 400 420 440 460 480 500 520 540 560 580 600 612
minus0007
minus0006
minus0005
minus0004
minus0003
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)
minus0002
minus0001
0000
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundatione observed data on 201465
e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
minus000015
minus000010
minus000005
000000
000005
000010
000015
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
(b)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
e b
endi
ng m
omen
t of t
he ex
istin
g tu
nnel
(kN
middotm)
387 400 420 440 460 480 500 520 540 560 580 595
0
minus6000
minus5000
minus4000
minus3000
minus2000
minus1000
1000
2000
3000
4000
e ring number of the existing tunnel
(c)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
387 400 420 440 460 480 500 520 540 560 580 595
minus600
minus400
minus200
0
200
400
600
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(kN
)
(d)
Figure 19 (a) Comparison of the decentection of the existing tunnel using dierent models (b) Comparison of the rotational angle ofthe existing tunnel using dierent models (c) Comparison of the normalized bending moment of the existing tunnel using dierentmodels (d) Comparison of the normalized shear force of the existing tunnel using dierent models
Advances in Civil Engineering 11
existing tunnel is only subject to soil deformation force inthis case e displacement and the decentection of beam areparticular similar concepts owing to the fact that the existingtunnel is securely attached to the foundation generallywithout rigid body motion Fairly good agreement can beobserved between the innite tunnel on the Pasternakfoundation and the nite tunnel on the Pasternak founda-tion which shows that the method proposed in this paper iscorrect and can be used to analyze similar problems Becauseof considering soil shear stress vertical stress and de-formation are aected and weakened the displacement of
the existing tunnel on the Pasternak foundation is smallerthan that on the Winkler foundation without consideringthe shear stress of soil Using the Winkler foundation modelis in good agreement with the observed data It can be seenfrom (3) that the displacement of the existing tunnel isproportional to the grouting elaquociency (ie η) Due to theuncertainty of η it cannot be decided which one is the bestanalytical method It is clear that the maximum normalizedbending moment occurs at the points O1 and O2 which arethe nearest points between the existing tunnel and the newtunnels
minus0003
minus0002
minus0001
0000387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
minus000010
minus000005
000000
000005
000010
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e ring number of the existing tunnel
(b)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10
minus0003
minus0002
minus0001
0000
0001
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
xL
(c)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10xL
minus008
minus006
minus004
minus002
000
002
004
006
008
The n
orm
alise
d sh
ear f
orce
of t
he ex
istin
g tu
nnel
(d)
Figure 20 (a) e eects of dierent reinforcement stiness on the decentection of the existing tunnel (b) e eects of dierent re-inforcement stiness on the slope of the existing tunnel (c)e eects of dierent reinforcement stiness on the normalized moment of theexisting tunnel (d) e eects of dierent reinforcement stiness on the normalized shear force of the existing tunnel
12 Advances in Civil Engineering
e reinforce range of the existing tunnel is assumed as25 rings on either side of the intersection point A and thevarious stiness is taken into consideration using themethod proposed in this paper Figure 20 shows the in-centuence of dierent stiness of the reinforcement on theexisting tunnel (ie 4 EI EI and 14 EI) It is evident thatthe reinforcement can reduce the displacement rotationalangle normalized bending moment and shear force of theexisting tunnel while tunnel damage (reduce the stinessof reinforcement) has the opposite eect It is evident thatthe change of reinforcement stiness has greater eect onthe normalized bending moment and the normalized shearforce of the existing tunnel particularly at the incentection
points of the normalized bending moment and normalizedshear force curve
e reinforce stiness of the reinforcement range isassumed as 4 EI and the various reinforcement ranges istaken into consideration using the method proposed in thispaper Figure 21 shows the incentuence of dierent re-inforcement ranges on the existing tunnel when thenorthbound tunnel of Line 4 underpasses e normalizedbending moment on either side of the intersection point istoo large when the reinforcement range is 69ndash892m andthe normalized shear force on either side of the intersectionpoint is relatively too large when the reinforcement range isless than 592ndash992m So it is necessary to reinforce larger
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00035
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
The d
eflec
tions
of t
he ex
istin
g tu
nnel
(m)
xL
(a)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus000010
minus000005
000000
000005
000010
The s
lope
of t
he ex
istin
g tu
nnel
(rad
)
xL
(b)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
00005
00010
00015
xL
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
(c)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus010
minus008
minus006
minus004
minus002
000
002
004
006
008
010
xLTh
e nor
mal
ised
shea
r for
ce o
f the
exist
ing
tunn
el
(d)
Figure 21 (a)e eects of dierent reinforcement ranges on the decentection of the existing tunnel (b)e eects of dierent reinforcementranges on the slope of the existing tunnel (c)e eects of dierent reinforcement ranges on the normalized moment of the existing tunnel(d) e eects of dierent reinforcement ranges on the normalized shear force of the existing tunnel
Advances in Civil Engineering 13
than the range of 492ndash1092m in the existing tunnel in thiscase that is larger than 30m (25 rings) on either side of theintersection point )ere is no big difference in the bendingmoment and shear force of the existing tunnel when thereinforcement ranges are 492ndash1092m 392ndash1192m and292ndash1292m In order to reduce the costs of reinforcementreinforcement in the 30m range (ie 25 rings) on eitherside of the intersection point is the best choice whichverifies the actual design
8 Conclusions
Based on the abovementioned statements the response of theexisting tunnel is analyzed and analytical methods are proposedfor verification and the reinforcement design Main conclusionsderived from the analysis are as follows
(1) )e displacement of the existing tunnel changes withthe relative position of the shield and the existingtunnel Shield parameters have a major impact on theexisting tunnel )e heaves of the existing tunnelmight be caused by the large applied face pressure andbulkhead addictive thrust When the shield tail isdriving beyond the selected monitoring rings theselected monitoring rings settle rapidly due to theclosure of the shield tail void
(2) )e southbound tunnel of Line 4 has less effect onthe left side of the intersection point A on theexisting tunnel because of the northbound tunnelrsquosbarrier effect A variety of measures of displacementprevention of Line 1 such as the control of shield
parameters reinforcement of the existing tunnel canensure the normal operation of metro Line 1
(3) )e analytical method proposed in the paper is ver-ified by the methods derived from prior studies It ispossible to take the different stiffness and curved loadsinto account by applying the stepped stiffness andtrapezoidal load while using a local coordinate systemin the derivation of the new method )e change ofreinforcement stiffness has greater effect on thenormalized bending moment and the normalizedshear force of the existing tunnel particularly at theinflection points of the normalized bending momentand normalized shear force curve )e analyticalmethod can be applied for the reinforcement designand safety check of the built tunnel
Appendices
A1 An Infinite Beam on the WinklerFoundation
θ(x) 1113946+infin
minusinfin
minusλ2s((tminus L)sin α)eminusλ(xminust) sin(λ(xminus t))
b
λ2s((tminus L)sin α)eλ(xminust)sin(λ(tminus x))
b
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎭
dt
(A1)
M(x) 14λ
1113946+infin
minusinfinks((tminusL)sin α)e
minusλ|xminust|( cos(λ|xminus t|)minus sin(λ|xminus t|))dt (A2)
Q(x) 1113946+infin
minusinfin
minus12
ks((tminusL)sin α)eminusλ(xminust)cos(λ(xminus t))dt xminus tgt 0
12
ks((tminusL)sin α)eminusλ(tminusx)cos(λ(tminusx))dt xminus tle 0
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
⎫⎪⎪⎪⎬
⎪⎪⎪⎭
dt (A3)
A2 An Infinite Beam on the PasternakFoundation
w(x) λ2bk
1113946+infin
minusinfinks((tminus L)sin α)De
minusa1λ|xminusτ| 1a1
cos a2λ|xminus τ|( 1113857 +1a2
sin a2λ|xminus τ|( 11138571113888 1113889dτ
θ(x) 1113946+infin
minusinfin
minusλ2s((tminusL))eminusa1λ(xminust)sin a2(xminus t)( 1113857
a1a2b
λ2s((tminus L))ea1λ(xminust)sin a2(tminus x)( 1113857
a1a2b
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A4)
14 Advances in Civil Engineering
M(x) 14λ
1113946+infin
minusinfinks((tminus L)sin α)e
minusa1λ|xminust| 1a1
cos a2λ|xminus t|( 1113857minus1a2
sin a2λ|xminus t|( 11138571113888 1113889dt
Q(x) 1113946+infin
minusinfin
minus12
ks((tminus L)sin α)eminusa1λ(xminust) cos a2λ(xminus t)( 1113857minus
a21 minus a 2
22a1a2
1113888 1113889sin a2λ(xminus t)( 11138571113890 1113891dt xminus tgt 0
12
ks((tminusL)sin α)eminusa1λ(tminusx) cos a2λ(tminusx)( 1113857minus
a21 minus a2
2
2a1a21113888 1113889sin a2λ(tminusx)( 11138571113890 1113891dt xminus tle 0
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A5)
B1 Coefficients of the Matrix ki5
k15 minusbiqiminus1
4a1ia2ici4EiIi a1i
2 + a2i2( 1113857
minus2a1ia2i + 2a1ia2iF2i
+ a1i2 minus a2i
2( 1113857F4i
⎛⎜⎝ ⎞⎟⎠
+bi qi minus qiminus1( 1113857
4a1ia2ici5EiIiL3 a1i
2 + a2i2( 1113857
2
a2i minus3a21i + a2
2i( 1113857F3i
+a1i 2a2i a21i + a2
2i( 1113857cix + a1i2 minus 3a2i
2( 1113857F1i( 1113857
⎛⎜⎝ ⎞⎟⎠
(B1)
k25 biqiminus1 a2iF3i minus a1iF1i( 1113857
2a1a2 a21 + a 2
2( 1113857c3EiIi
+bi qi minus qiminus1( 1113857 2a1ia2i minus 2a1ia2iF2i minus a1i
2 minus a2i2( 1113857F4i( 1113857( 1113857
2a1ia2i a1i2 + a2i
2( 11138572ci
4EiIix1113872 1113873 (B2)
k35 minusbiqiminus1F4i
2a1ia2ici2 minus
bi qi minus qiminus1( 1113857 a2iF3i minus a1iF1i( 1113857
2a1ia2ici3 a1i
2 + a2i2( 1113857x
(B3)
k45 minusbiqiminus1
a1i2 + a2i
2( 1113857ci
1113888a1i 1 + 2a1i2a22i minus 2a2i
41113872 1113873F3i + a2i 1 + 2a1i
2a22i minus 2a1i
41113872 1113873F1i1113889
minusbi qi minus qiminus1( 1113857
a21i + a2i
2( 11138572c 2
i x
4a1i2a2i
2 minus 1( 1113857 a1i2 minus a2i
2( 1113857 + a2i2 minus 4a1i
4a2i2 + a1i
2 minus1 + 4a2i4( 1113857( 1113857F2i
minus2a1ia2i a 21i minus a 2
2i( 11138572 minus 11113872 1113873F4i
⎛⎜⎝ ⎞⎟⎠
(B4)
Conflicts of Interest
)e author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
)e author appreciates the help from the staff of Hangzhouand Hong Kong Tunnel Company Ltd during in-strumentation setup and data acquisition )is study issupported by the National Natural Science Foundation ofChina (no 41702313)
References
[1] S H Kim Model testing and analysis of interactions betweentunnels in clay [PhD 3esis] Department of EngineeringScience University of Oxford Oxford UK 1996
[2] H Y Liu J C Small J P Carter and D J Williams ldquoEffectsof tunnelling on existing support systems of perpendicularlypassing tunnelsrdquo Computers and Geotechnics vol 36 no 5pp 880ndash894 2009
[3] I Yamaguchi I Yamazaki and Y Kiritani ldquoStudy ofgroundndashtunnel interactions of four shield tunnels driven inclose proximity in relation to design and construction ofparallel shield tunnelsrdquo Tunnelling and Underground SpaceTechnology vol 13 no 3 pp 289ndash304 1998
[4] T I Addenbrooke and D M Potts ldquoTwin tunnel interactionsurface and subsurface effectsrdquo International Journal ofGeomechanics vol 1 no 2 pp 249ndash271 2001
[5] J P Kimmance S Lawrence O Hassan and N J PurchaseldquoObservations of deformations created in existing tunnels byadjacent and cross cutting excavationsrdquo in Proceedings ofGeotechnical Aspects of Underground Construction in SoftGround pp 707ndash712 London UK July 1996
[6] E Leca and B New ldquoSettlements induced by tunnelling in softgroundrdquo Tunnelling and Underground Space Technologyvol 22 no 2 pp 119ndash149 2007
Advances in Civil Engineering 15
[7] S M Liao J H Liu R L Wang and Z M Li ldquoShieldtunnelling and environment protection in Shanghai softgroundrdquo Tunnelling and Underground Space Technologyvol 24 no 4 pp 454ndash465 2009
[8] F H Chehade and I Shahrour ldquoNumerical analysis of theinteraction between twin-tunnels influence of the relativeposition and construction procedurerdquo Tunnelling and Un-derground Space Technology vol 23 no 2 pp 210ndash214 2008
[9] X G Li and D J Yuan ldquoResponse of a double-decked metrotunnel to shield driving of twin closely underpasses tunnelsrdquoTunnelling and Underground Space Technology vol 28 no 1pp 18ndash30 2012
[10] T E Vorster A Klar K Soga and R J Mair ldquoEstimating theeffects of tunnelling on existing pipelinesrdquo Journal of Geo-technical and Geoenvironmental Engineering vol 131 no 11pp 1399ndash1410 2005
[11] Z G Zhang M S Huang and W D Wang ldquoEvaluation ofdeformation response for adjacent tunnels due to soilunloading in excavation engineeringrdquo Tunnelling and Un-derground Space Technology vol 38 no 3 pp 244ndash253 2013
[12] A P S Selvadurai Elastic Analysis of Soil-Foundation InteractionElsevier Scientific Pub Co Amsterdam Netherlands 1979
[13] N Loganathan and H G Poulos ldquoAnalytical prediction fortunnelling-induced ground movements in claysrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124no 9 pp 846ndash856 1998
[14] K M Lee R K Rowe and K Y Lo ldquoSubsidence owing totunnelling I Estimating the gap parameterrdquo CanadianGeotechnical Journal vol 29 no 6 pp 929ndash940 1992
[15] R K Rowe and KM Lee ldquoSubsidence owing to tunnelling IIEvaluation of a prediction techniquerdquo Canadian GeotechnicalJournal vol 29 no 6 pp 941ndash954 1992
[16] P B Attewell J Yeates and A R Selby Soil MovementsInduced by Tunnelling and their Effects on Pipelines andStructures Methuen Inc New York NY USA 1986
[17] F Ye C He H H Zhu et al ldquoAnalysis of longitudinalequivalent stiffness of shield tunnel considering lateral per-formancerdquo Chinese Journal of Geotechnical Engineeringvol 33 no 12 pp 1870ndash1876 2011 in Chinese
[18] P R Lancaster and D Mitchell ldquoProblems in Bending of Barsand Plates [M]rdquo Advanced Solid Mechanics Macmillan UK1980
[19] L Xu ldquoStudy on vertical settlement of shield tunnel in soft soilrdquoPhD dissertation Tongji University Shanghai China 2005
[20] M Yang and X Zhao ldquoAn approach for a single pile in layeredsoilrdquo Tongji University Natural Science vol 20 no 4pp 421ndash428 1992 in Chinese
16 Advances in Civil Engineering
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completed after the construction of the northboundtunnel e reinforcement range is from ring 424 to ring510 (ie 25 rings on either sides of the intersection pointsA and B)
6 Theoretical Analysis of the ReinforcementDesign
e deformed tunnel lining reinforced by inner bondingsteel plates has been partially or entirely used in manybuilt tunnels or currently under construction To better
understand the benets coming from the reinforcementmethod the behavior of the tunnel is investigated
Figure 13 shows a schematic view of the existingtunnel-soil tunnelling interaction e analysis methoddemonstrated in this paper can be divided into two stepsrstly estimating the greeneld displacement induced bythe tunnelling Secondly calculating the responses of theexisting tunnel subjected to the soil displacement emethod of analysis is based on three assumptions (1) theabove existing tunnel does not aect the displacement ofsoil due to tunnelling (2) the soil foundation is assumedas the Winkler or the Pasternak foundation and (3) thesoil displacement is calculated by superimposing theindependent settlement predicted for each individualtunnelling
In this paper the tunnels are considered as an innitebeam on the Winkler foundation an innite beam on thePasternak foundation and a nite beam on the Pasternakfoundation (the new method) Comparing the decentectionrotation angle normalized bendingmoment and shear forceof the existing tunnel with constant stiness based on thethree models the new method is veried en dierentstiness of the reinforcement is taken into considerationand the optimal reinforcement range of the existing tunnel isdiscussed
61 e Subsurface Soil Displacement due to Tunnelling Forthe theoretical analysis the alignment of the new tunnels andthe existing tunnel is assumed to be straight Figure 14 showsa schematic diagram of the new tunnels and the existingtunnel in this case e greeneld settlement s(x) due to thetunnelling can be replaced by the equivalent distributed loadq(x) acting on the beam on the third assumption as follows
q(x) ks(x) (2)
In this study subsurface settlement s(x) at the depth of zinduced by tunnelling is calculated based on closed-formanalytical solutions presented by Loganathan and Poulos[13] as follows
s(x) ε0R2 zminusHx2 +(zminusH)2
+(3minus 4v)z +H
x2 +(z +H)2minus2z x2 minus(z +H)2( )
x2 +(z +H)2( )2
middot e minus138x2(H+R)2minus069z2H2( ) (3)
Conv
erge
nce d
ispla
cem
ent (
mm
)
400 420 440 460 480 500 520 540 560 580 600 620 640 660minus7minus6minus5minus4minus3minus2minus1
012345
e construction ofsouthbound tunnel
201453 Ring 135201459 Ring 1912014528 Ring 270
e shield is shut down201461
e construction ofnorthbound tunnel
2013129 Ring 4820131222 Ring 6420131230 Ring 87
e shield is shut down201424
e ring number of the upline of Line 1
A B
Figure 11 Convergence displacement of the existing tunnel
Figure 12 Reinforcement in the existing tunnel
xk1
k2
e existing tunnel
Sz(x)
Figure 13 A schematic view of the existing tunnel-soil tunnellinginteraction
Advances in Civil Engineering 7
where s(x) subsurface settlement of the soils due to tun-nelling R tunnel radius z depth below ground surfaceH depth of the axis of the new tunnel vPoissonrsquos ratiox lateral distance from tunnel centerline and ε0 equivalentground loss ratio which is dened as
ε0 4Rg + g2
4R2 (4)
g Gp + ulowast3D + w asymp Gp (5)
where g is the gap parameter (Lee et al [14]) ulowast3D is thethree-dimensional elastoplastic deformation at the tunnelface w is the workmanship factor and Gp represents thephysical gap between the outer skin of the shield and thelining which is given as
Gp 2Δ + δ (6)
where Δ is the thickness of the tail piece and δ is the clearancerequired for erection of the tail piece e parameter w can beneglected due to considerable experience with the equipmentand good tunnelling technique of construction workers basedon the study of Rowe and Lee [15]
62 Analytical Methods Derived from Prior Studies Attewellet al [16] proposed a tunnel-soil interaction model andsolved the analytical solution of the longitudinal displace-ment of an innite tunnel
d4w(x)dx4
+ 4λ4p(x) 4λ4q(x) (7)
where q(x) is the concentrated load acting on the originpoint of the beam EI is the centexural stiness of a tunnel
which is recommended by Ye et al [17] and k is the subgrademodulus that represents the pipe-soil interaction
If the soil foundation is assumed as the Winklerfoundation the foundation counterforce p(x) can beexpressed as
p(x) kw(x) (8)
e two-parameter model of foundation can capture theshear resistance of soil Based on the assumption of theplane strain condition the displacement of ground in they-direction is neglected If the soil foundation is assumed asthe Pasternak foundation the foundation counterforce p(x)can be expressed as
p(x) minusGnabla2w(x) + kw(x) (9)
where G is the coelaquocient of the shear element in thePasternakrsquos model with the dimension of force per unitlength
Attewell et al [16] presented a solution for the con-centrated load q on an innite beam resting on the Winklerfoundation and computed the decentection as
w(x) 1
8EIλ3qeminusλx( cos λx + sin λx) (10)
It can be concluded that each of the concentrated loadks(τ) at point τ contributes the following amount to thedecentection of the existing tunnel on the Winkler foundation
w(x) 1
8EIλ3int+infin
minusinfin2Rks tminus L1( )sin α( ) eminusλ|xminusτ|
(cos λ|xminus τ| + sin λ|xminus τ|)dτ(11)
where L1 is the distance between the coordinate origin of theexisting tunnel and the intersection point between the newtunnel and the existing tunnel and α is the intersection anglebetween the new tunnel and existing tunnel e decentectionrotational angle bending moment and shear force of theexisting tunnel which is assumed as an innite beam on theWinkler foundation and the Pasternak foundation are il-lustrated in Appendix A
63 A Brief Description of the NewMethod In this study wewant to gure out the eects of the reinforcement on theexisting tunnel the stiness varies in this case and theexisting tunnel assumed as an innite beam is not suitableAssuming the existing tunnel with nite length and in-troducing (7) to (9) the basic dierential equation gov-erning the centexure of the beam resting on the Pasternakfoundation can be written in the form with an assumptionof Gλ2klt 1
EiIid4wi(x)dx4minusGibi
d2wi(x)dx4
+ kibiwi(x) biqi(x) (12)
where b1i bi(1 +(Gikiradic
bi)) subscript i represents the ithelement of the beam bi is the width of the beam and b1i isthe modied width of the beam Assuming a solution of (12)in the form of w(x) eλx the general solution of verticaldisplacement of the beam is as follows
y
e southbound tunnel
e northbound tunnel
e e
xistin
g tun
nel
x
L 1L 2
0x 1
x 2x 3
x ix i+
1
x nminus2
x nminus1
x n
O2
O1
xx
Figure 14 A schematic diagram of tunnels for the analytical methods
8 Advances in Civil Engineering
w(x) sum4
i1CiFi(x) (13)
where Ci (i 1 2 3 4) are undetermined parameters whichare dependent on the boundary conditions and Fi (x)(i 1 2 3 4) are the basic functions of the general solutionwhich are given as follows
F1i(x) cos a1icix( ) sinh a2icix( ) (14)
F2i(x) cos a1icix( ) cosh a2icix( ) (15)
F3i(x) sin a1icix( ) cosh a2icix( ) (16)
F4i(x) sin a1icix( ) sinh a2icix( ) (17)
where ci kib14EiIi4radic
a1i 1minusGic2i kiradic
a2i1 + Gic2i kiradic
e relationship of the decentection rotational angle
bending moment and shear force of the beam proposed byLancaster and Mitchell [18] is as follows
θi(x) wprimei(x)Mi(x) minusEiIiwPrime(x)
Qi(x) minusEiIiw(x) + Gpib1iwprimei(x)
(18)
where θi(x) Mi(x) and Qi(x) are the amplitudes of ro-tational angle bending moment and shear force of thesection area respectively
Figure 15 shows that dierent stiness is replaced byequivalent stepped stiness According to dierent stinessand distributed load the local coordinate system is establishedas illustrated in Figure 16 Selvadurai [12] suggests an initialparameter method to solve the parameters of the beam efour parameters at the origin of the coordinates arewi(0i) θi(0i)Mi(0i) and Qi(0i) and four parameters at theend of the element are wi(xi) θi(xi) Mi(xi) and Qi(xi)C1 C2 C3 and C4 can be expressed by wi(0i) θi(0i)Mi(0i) and Qi(0i) by introducing xi 0 to (13) and (18)
wi(xi) θi(xi)Mi(xi) and Qi(xi) can be rewritten in thematrix-array form as
wi xi( )θi xi( )Mi xi( )Vi xi( )1
k11 k12 k13 k14 k15
k21 k22 k23 k24 k25
k31 k32 k33 k34 k35
k41 k42 k43 k44 k45
0 0 0 0 1
middot
wi 0i( )θi 0i( )Mi 0i( )Vi 0i( )1
(19)
where [kij] is a stiness submatrix with 5 times 5 order ij 1 2 5M and V are respectively the bending momentand the generalized shear force Formula (19) can be sim-plied as
Ni xi( ) KijNi 0i( ) (20)
where Ni(xi) [Wi(xi) θi(xi)Mi(xi) Qi(xi) 1]T andNi(0i) [Wi(0i) θi(0i)Mi(0i) Qi(0i) 1]T
Figure 17 shows that curved distributed load is replacedby equivalent triangular and square distributed load whichcan be expressed as
qiminus1 +qi qiminus1 + ξqi minus qiminus1Li
( ) (21)
where qiminus1 and qi are respectively the load of the startingpoint and the ending point of the ith element and ξ is theposition of arbitrary load From (19) we can obtain
k15 minusbintx0 qiminus1 +qi( ) aminus12i F1i[xminus ξ]minus aminus11i F3i[xminus ξ]( )dξ
4c3i EiIi
(22)In order to avoid numerical errors in the calculation
process the integration of ki5 is shown in Appendix B erest of kij can be referred in the work by Selvadurai [12]
Figure 18 shows the tunnel matrix transfer diagramAssuming that the tunnel is composed of N elements thex-coordinate of beginning points of the ith element isxi(i 0 1 2 n) and the length of each element is Liwhere Li xi minus ximinus1
E0I0
0 x1 x2 x3 ximinus1 xi xnminus1 xn x
E1I1E2I2
Eiminus1Iiminus1Enminus1Inminus1
Figure 15 Sketch of equivalent stepped stiness
qi(x)
pi
Qi Qi+1Mi Mi+1
Oimi
xi
Figure 16 Forces and local coordinate system
q0q1 q2
qiminus1 qnminus1
x1 x2 x3 ximinus1 xi xnminus1 xn x
q (x)
0
Figure 17 Replacement of variable load with trapezoid load
Advances in Civil Engineering 9
For the rst element
N1(x) K1(x) middotN1 01( ) x isin 0 L1( ) (23)
For the end point of the rst element
N1 x1( ) K1 x1 minus 0( ) middotN1 01( ) (24)
For the end point of the ith element
Ni xi( ) Ki xi( ) middotKiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( ) middotN1 01( )(25)
For x at the ith element section where x is a globalcoordinate
Ni(x)Ki xminus ximinus1( ) middotKiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( ) middotN1 01( )(26)
Suppose
Ai(x) Ki xminusximinus1( ) middot Kiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( )(27)
en (20) can be expressed as
Ni(x) Ai(x) middotN1 01( ) (28)
us the relation between the vectorNn(xn) at the pointxn and the vector N1(01) at the start point x0 is
Nn xn( ) An xn( ) middotN1 01( ) (29)
where
An xn( ) prod1
inKi Li( ) (30)
e boundary condition of the existing tunnel assumedas an innite long tunnel considering the shear stress of soilcan be written as
M(0) 0
V(0) 2
k
G
radic
middot G middot R middot w(0)
V(L) 0
V(L) minus2
k
G
radic
middot G middot R middot w(L)
(31)
By applying the boundary conditions to (19) the de-centection rotational angle bending moment and shear forceof the overall beam can be obtained
64 Case Study
641 Parameter Selection To enable a direct compari-son assumed parameters of the soil and existing tunnelare selected in this study Poissonrsquos ratio of the soil is0388 e grouting process is regarded as the inverseprocess of soil loss due to tunnelling and g can beexpressed as g (2Δ + δ)(1minus η) where η is the injectionratio
For the Pasternak foundation as the thickness of theelastic layer is assumed to be 25 B the subgrade elasticmodulus k and shear stiness G are suggested by Xu [19] asfollows
k 5E0β
16 1minus v05pt2( )times 122
G 13E0B
2
32 1 + v0( )βtimes 085
(32)
where β is a modied parameter (β 129) and E0 and v0 arerespectively the elastic modulus and Poissonrsquos ratio forelastic foundation which are given as follows
E0 Es
1minus v2s
v0 vs
1minus vs
(33)
where the empirical relation between the Youngrsquos modulusEs and the compression modulus Es01minus02 of soft soils is builtup by Yang and Zhao [20]
E0 (25 sim 35)Es01minus02 (34)
As illustrated in Figure 14 points O1 and O2 are re-spectively the intersection points A and B in practicalcoordinate transformation expressions of the axis xprime and xPrimeare respectively
xprime xminusL1( )sin αxPrime xminusL2( )sin α
(35)
where L1 and L2 are the distance from the beginningpoint of the existing tunnel to points O1 and O2 re-spectively e length of the existing tunnel is assumedas 250 m and the ring spacing is 12 m So the tunnel isdivided into 209 sections and each section is 12 m inlength
N1(01) Ni(xi)N1(x1) = N2(02) N2(x2)L1
L1
E1I1
G1 k1 G2 k2 G3 k3 Gi ki Gn knGnminus1 knminus1
E2I2 E3I3 EiIi EnInEnminus1Inminus1
L2 L3 Li LnLnminus1
L2 Li
x x x
xx1 x2 x3 xi-1 xnminus2 xnminus1 xnxi0
Ni(0i)
Figure 18 e matrix transfer diagram
10 Advances in Civil Engineering
7 Discussions
e normalized bending moment and shear force are de-ned as
Mi MiL0EiIi
Qi QiL
20
EiIi
(36)
where Mi and Qi are respectively the normalized bendingmoment and shear force Mi and Qi are respectively thecalculated bending moment and shear force and L0 is thelength of the existing tunnel
Figure 19 shows the comparison of the decentection ro-tational angle bending moment and shear force of theexisting tunnel with a constant stiness which is based onthree kinds of models an innite tunnel on the Winklerfoundation an innite tunnel on the Pasternak foundationand a nite tunnel on the Winkler foundation Note that the
368 380 400 420 440 460 480 500 520 540 560 580 600 612
minus0007
minus0006
minus0005
minus0004
minus0003
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)
minus0002
minus0001
0000
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundatione observed data on 201465
e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
minus000015
minus000010
minus000005
000000
000005
000010
000015
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
(b)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
e b
endi
ng m
omen
t of t
he ex
istin
g tu
nnel
(kN
middotm)
387 400 420 440 460 480 500 520 540 560 580 595
0
minus6000
minus5000
minus4000
minus3000
minus2000
minus1000
1000
2000
3000
4000
e ring number of the existing tunnel
(c)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
387 400 420 440 460 480 500 520 540 560 580 595
minus600
minus400
minus200
0
200
400
600
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(kN
)
(d)
Figure 19 (a) Comparison of the decentection of the existing tunnel using dierent models (b) Comparison of the rotational angle ofthe existing tunnel using dierent models (c) Comparison of the normalized bending moment of the existing tunnel using dierentmodels (d) Comparison of the normalized shear force of the existing tunnel using dierent models
Advances in Civil Engineering 11
existing tunnel is only subject to soil deformation force inthis case e displacement and the decentection of beam areparticular similar concepts owing to the fact that the existingtunnel is securely attached to the foundation generallywithout rigid body motion Fairly good agreement can beobserved between the innite tunnel on the Pasternakfoundation and the nite tunnel on the Pasternak founda-tion which shows that the method proposed in this paper iscorrect and can be used to analyze similar problems Becauseof considering soil shear stress vertical stress and de-formation are aected and weakened the displacement of
the existing tunnel on the Pasternak foundation is smallerthan that on the Winkler foundation without consideringthe shear stress of soil Using the Winkler foundation modelis in good agreement with the observed data It can be seenfrom (3) that the displacement of the existing tunnel isproportional to the grouting elaquociency (ie η) Due to theuncertainty of η it cannot be decided which one is the bestanalytical method It is clear that the maximum normalizedbending moment occurs at the points O1 and O2 which arethe nearest points between the existing tunnel and the newtunnels
minus0003
minus0002
minus0001
0000387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
minus000010
minus000005
000000
000005
000010
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e ring number of the existing tunnel
(b)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10
minus0003
minus0002
minus0001
0000
0001
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
xL
(c)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10xL
minus008
minus006
minus004
minus002
000
002
004
006
008
The n
orm
alise
d sh
ear f
orce
of t
he ex
istin
g tu
nnel
(d)
Figure 20 (a) e eects of dierent reinforcement stiness on the decentection of the existing tunnel (b) e eects of dierent re-inforcement stiness on the slope of the existing tunnel (c)e eects of dierent reinforcement stiness on the normalized moment of theexisting tunnel (d) e eects of dierent reinforcement stiness on the normalized shear force of the existing tunnel
12 Advances in Civil Engineering
e reinforce range of the existing tunnel is assumed as25 rings on either side of the intersection point A and thevarious stiness is taken into consideration using themethod proposed in this paper Figure 20 shows the in-centuence of dierent stiness of the reinforcement on theexisting tunnel (ie 4 EI EI and 14 EI) It is evident thatthe reinforcement can reduce the displacement rotationalangle normalized bending moment and shear force of theexisting tunnel while tunnel damage (reduce the stinessof reinforcement) has the opposite eect It is evident thatthe change of reinforcement stiness has greater eect onthe normalized bending moment and the normalized shearforce of the existing tunnel particularly at the incentection
points of the normalized bending moment and normalizedshear force curve
e reinforce stiness of the reinforcement range isassumed as 4 EI and the various reinforcement ranges istaken into consideration using the method proposed in thispaper Figure 21 shows the incentuence of dierent re-inforcement ranges on the existing tunnel when thenorthbound tunnel of Line 4 underpasses e normalizedbending moment on either side of the intersection point istoo large when the reinforcement range is 69ndash892m andthe normalized shear force on either side of the intersectionpoint is relatively too large when the reinforcement range isless than 592ndash992m So it is necessary to reinforce larger
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00035
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
The d
eflec
tions
of t
he ex
istin
g tu
nnel
(m)
xL
(a)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus000010
minus000005
000000
000005
000010
The s
lope
of t
he ex
istin
g tu
nnel
(rad
)
xL
(b)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
00005
00010
00015
xL
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
(c)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus010
minus008
minus006
minus004
minus002
000
002
004
006
008
010
xLTh
e nor
mal
ised
shea
r for
ce o
f the
exist
ing
tunn
el
(d)
Figure 21 (a)e eects of dierent reinforcement ranges on the decentection of the existing tunnel (b)e eects of dierent reinforcementranges on the slope of the existing tunnel (c)e eects of dierent reinforcement ranges on the normalized moment of the existing tunnel(d) e eects of dierent reinforcement ranges on the normalized shear force of the existing tunnel
Advances in Civil Engineering 13
than the range of 492ndash1092m in the existing tunnel in thiscase that is larger than 30m (25 rings) on either side of theintersection point )ere is no big difference in the bendingmoment and shear force of the existing tunnel when thereinforcement ranges are 492ndash1092m 392ndash1192m and292ndash1292m In order to reduce the costs of reinforcementreinforcement in the 30m range (ie 25 rings) on eitherside of the intersection point is the best choice whichverifies the actual design
8 Conclusions
Based on the abovementioned statements the response of theexisting tunnel is analyzed and analytical methods are proposedfor verification and the reinforcement design Main conclusionsderived from the analysis are as follows
(1) )e displacement of the existing tunnel changes withthe relative position of the shield and the existingtunnel Shield parameters have a major impact on theexisting tunnel )e heaves of the existing tunnelmight be caused by the large applied face pressure andbulkhead addictive thrust When the shield tail isdriving beyond the selected monitoring rings theselected monitoring rings settle rapidly due to theclosure of the shield tail void
(2) )e southbound tunnel of Line 4 has less effect onthe left side of the intersection point A on theexisting tunnel because of the northbound tunnelrsquosbarrier effect A variety of measures of displacementprevention of Line 1 such as the control of shield
parameters reinforcement of the existing tunnel canensure the normal operation of metro Line 1
(3) )e analytical method proposed in the paper is ver-ified by the methods derived from prior studies It ispossible to take the different stiffness and curved loadsinto account by applying the stepped stiffness andtrapezoidal load while using a local coordinate systemin the derivation of the new method )e change ofreinforcement stiffness has greater effect on thenormalized bending moment and the normalizedshear force of the existing tunnel particularly at theinflection points of the normalized bending momentand normalized shear force curve )e analyticalmethod can be applied for the reinforcement designand safety check of the built tunnel
Appendices
A1 An Infinite Beam on the WinklerFoundation
θ(x) 1113946+infin
minusinfin
minusλ2s((tminus L)sin α)eminusλ(xminust) sin(λ(xminus t))
b
λ2s((tminus L)sin α)eλ(xminust)sin(λ(tminus x))
b
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎭
dt
(A1)
M(x) 14λ
1113946+infin
minusinfinks((tminusL)sin α)e
minusλ|xminust|( cos(λ|xminus t|)minus sin(λ|xminus t|))dt (A2)
Q(x) 1113946+infin
minusinfin
minus12
ks((tminusL)sin α)eminusλ(xminust)cos(λ(xminus t))dt xminus tgt 0
12
ks((tminusL)sin α)eminusλ(tminusx)cos(λ(tminusx))dt xminus tle 0
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
⎫⎪⎪⎪⎬
⎪⎪⎪⎭
dt (A3)
A2 An Infinite Beam on the PasternakFoundation
w(x) λ2bk
1113946+infin
minusinfinks((tminus L)sin α)De
minusa1λ|xminusτ| 1a1
cos a2λ|xminus τ|( 1113857 +1a2
sin a2λ|xminus τ|( 11138571113888 1113889dτ
θ(x) 1113946+infin
minusinfin
minusλ2s((tminusL))eminusa1λ(xminust)sin a2(xminus t)( 1113857
a1a2b
λ2s((tminus L))ea1λ(xminust)sin a2(tminus x)( 1113857
a1a2b
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A4)
14 Advances in Civil Engineering
M(x) 14λ
1113946+infin
minusinfinks((tminus L)sin α)e
minusa1λ|xminust| 1a1
cos a2λ|xminus t|( 1113857minus1a2
sin a2λ|xminus t|( 11138571113888 1113889dt
Q(x) 1113946+infin
minusinfin
minus12
ks((tminus L)sin α)eminusa1λ(xminust) cos a2λ(xminus t)( 1113857minus
a21 minus a 2
22a1a2
1113888 1113889sin a2λ(xminus t)( 11138571113890 1113891dt xminus tgt 0
12
ks((tminusL)sin α)eminusa1λ(tminusx) cos a2λ(tminusx)( 1113857minus
a21 minus a2
2
2a1a21113888 1113889sin a2λ(tminusx)( 11138571113890 1113891dt xminus tle 0
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A5)
B1 Coefficients of the Matrix ki5
k15 minusbiqiminus1
4a1ia2ici4EiIi a1i
2 + a2i2( 1113857
minus2a1ia2i + 2a1ia2iF2i
+ a1i2 minus a2i
2( 1113857F4i
⎛⎜⎝ ⎞⎟⎠
+bi qi minus qiminus1( 1113857
4a1ia2ici5EiIiL3 a1i
2 + a2i2( 1113857
2
a2i minus3a21i + a2
2i( 1113857F3i
+a1i 2a2i a21i + a2
2i( 1113857cix + a1i2 minus 3a2i
2( 1113857F1i( 1113857
⎛⎜⎝ ⎞⎟⎠
(B1)
k25 biqiminus1 a2iF3i minus a1iF1i( 1113857
2a1a2 a21 + a 2
2( 1113857c3EiIi
+bi qi minus qiminus1( 1113857 2a1ia2i minus 2a1ia2iF2i minus a1i
2 minus a2i2( 1113857F4i( 1113857( 1113857
2a1ia2i a1i2 + a2i
2( 11138572ci
4EiIix1113872 1113873 (B2)
k35 minusbiqiminus1F4i
2a1ia2ici2 minus
bi qi minus qiminus1( 1113857 a2iF3i minus a1iF1i( 1113857
2a1ia2ici3 a1i
2 + a2i2( 1113857x
(B3)
k45 minusbiqiminus1
a1i2 + a2i
2( 1113857ci
1113888a1i 1 + 2a1i2a22i minus 2a2i
41113872 1113873F3i + a2i 1 + 2a1i
2a22i minus 2a1i
41113872 1113873F1i1113889
minusbi qi minus qiminus1( 1113857
a21i + a2i
2( 11138572c 2
i x
4a1i2a2i
2 minus 1( 1113857 a1i2 minus a2i
2( 1113857 + a2i2 minus 4a1i
4a2i2 + a1i
2 minus1 + 4a2i4( 1113857( 1113857F2i
minus2a1ia2i a 21i minus a 2
2i( 11138572 minus 11113872 1113873F4i
⎛⎜⎝ ⎞⎟⎠
(B4)
Conflicts of Interest
)e author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
)e author appreciates the help from the staff of Hangzhouand Hong Kong Tunnel Company Ltd during in-strumentation setup and data acquisition )is study issupported by the National Natural Science Foundation ofChina (no 41702313)
References
[1] S H Kim Model testing and analysis of interactions betweentunnels in clay [PhD 3esis] Department of EngineeringScience University of Oxford Oxford UK 1996
[2] H Y Liu J C Small J P Carter and D J Williams ldquoEffectsof tunnelling on existing support systems of perpendicularlypassing tunnelsrdquo Computers and Geotechnics vol 36 no 5pp 880ndash894 2009
[3] I Yamaguchi I Yamazaki and Y Kiritani ldquoStudy ofgroundndashtunnel interactions of four shield tunnels driven inclose proximity in relation to design and construction ofparallel shield tunnelsrdquo Tunnelling and Underground SpaceTechnology vol 13 no 3 pp 289ndash304 1998
[4] T I Addenbrooke and D M Potts ldquoTwin tunnel interactionsurface and subsurface effectsrdquo International Journal ofGeomechanics vol 1 no 2 pp 249ndash271 2001
[5] J P Kimmance S Lawrence O Hassan and N J PurchaseldquoObservations of deformations created in existing tunnels byadjacent and cross cutting excavationsrdquo in Proceedings ofGeotechnical Aspects of Underground Construction in SoftGround pp 707ndash712 London UK July 1996
[6] E Leca and B New ldquoSettlements induced by tunnelling in softgroundrdquo Tunnelling and Underground Space Technologyvol 22 no 2 pp 119ndash149 2007
Advances in Civil Engineering 15
[7] S M Liao J H Liu R L Wang and Z M Li ldquoShieldtunnelling and environment protection in Shanghai softgroundrdquo Tunnelling and Underground Space Technologyvol 24 no 4 pp 454ndash465 2009
[8] F H Chehade and I Shahrour ldquoNumerical analysis of theinteraction between twin-tunnels influence of the relativeposition and construction procedurerdquo Tunnelling and Un-derground Space Technology vol 23 no 2 pp 210ndash214 2008
[9] X G Li and D J Yuan ldquoResponse of a double-decked metrotunnel to shield driving of twin closely underpasses tunnelsrdquoTunnelling and Underground Space Technology vol 28 no 1pp 18ndash30 2012
[10] T E Vorster A Klar K Soga and R J Mair ldquoEstimating theeffects of tunnelling on existing pipelinesrdquo Journal of Geo-technical and Geoenvironmental Engineering vol 131 no 11pp 1399ndash1410 2005
[11] Z G Zhang M S Huang and W D Wang ldquoEvaluation ofdeformation response for adjacent tunnels due to soilunloading in excavation engineeringrdquo Tunnelling and Un-derground Space Technology vol 38 no 3 pp 244ndash253 2013
[12] A P S Selvadurai Elastic Analysis of Soil-Foundation InteractionElsevier Scientific Pub Co Amsterdam Netherlands 1979
[13] N Loganathan and H G Poulos ldquoAnalytical prediction fortunnelling-induced ground movements in claysrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124no 9 pp 846ndash856 1998
[14] K M Lee R K Rowe and K Y Lo ldquoSubsidence owing totunnelling I Estimating the gap parameterrdquo CanadianGeotechnical Journal vol 29 no 6 pp 929ndash940 1992
[15] R K Rowe and KM Lee ldquoSubsidence owing to tunnelling IIEvaluation of a prediction techniquerdquo Canadian GeotechnicalJournal vol 29 no 6 pp 941ndash954 1992
[16] P B Attewell J Yeates and A R Selby Soil MovementsInduced by Tunnelling and their Effects on Pipelines andStructures Methuen Inc New York NY USA 1986
[17] F Ye C He H H Zhu et al ldquoAnalysis of longitudinalequivalent stiffness of shield tunnel considering lateral per-formancerdquo Chinese Journal of Geotechnical Engineeringvol 33 no 12 pp 1870ndash1876 2011 in Chinese
[18] P R Lancaster and D Mitchell ldquoProblems in Bending of Barsand Plates [M]rdquo Advanced Solid Mechanics Macmillan UK1980
[19] L Xu ldquoStudy on vertical settlement of shield tunnel in soft soilrdquoPhD dissertation Tongji University Shanghai China 2005
[20] M Yang and X Zhao ldquoAn approach for a single pile in layeredsoilrdquo Tongji University Natural Science vol 20 no 4pp 421ndash428 1992 in Chinese
16 Advances in Civil Engineering
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where s(x) subsurface settlement of the soils due to tun-nelling R tunnel radius z depth below ground surfaceH depth of the axis of the new tunnel vPoissonrsquos ratiox lateral distance from tunnel centerline and ε0 equivalentground loss ratio which is dened as
ε0 4Rg + g2
4R2 (4)
g Gp + ulowast3D + w asymp Gp (5)
where g is the gap parameter (Lee et al [14]) ulowast3D is thethree-dimensional elastoplastic deformation at the tunnelface w is the workmanship factor and Gp represents thephysical gap between the outer skin of the shield and thelining which is given as
Gp 2Δ + δ (6)
where Δ is the thickness of the tail piece and δ is the clearancerequired for erection of the tail piece e parameter w can beneglected due to considerable experience with the equipmentand good tunnelling technique of construction workers basedon the study of Rowe and Lee [15]
62 Analytical Methods Derived from Prior Studies Attewellet al [16] proposed a tunnel-soil interaction model andsolved the analytical solution of the longitudinal displace-ment of an innite tunnel
d4w(x)dx4
+ 4λ4p(x) 4λ4q(x) (7)
where q(x) is the concentrated load acting on the originpoint of the beam EI is the centexural stiness of a tunnel
which is recommended by Ye et al [17] and k is the subgrademodulus that represents the pipe-soil interaction
If the soil foundation is assumed as the Winklerfoundation the foundation counterforce p(x) can beexpressed as
p(x) kw(x) (8)
e two-parameter model of foundation can capture theshear resistance of soil Based on the assumption of theplane strain condition the displacement of ground in they-direction is neglected If the soil foundation is assumed asthe Pasternak foundation the foundation counterforce p(x)can be expressed as
p(x) minusGnabla2w(x) + kw(x) (9)
where G is the coelaquocient of the shear element in thePasternakrsquos model with the dimension of force per unitlength
Attewell et al [16] presented a solution for the con-centrated load q on an innite beam resting on the Winklerfoundation and computed the decentection as
w(x) 1
8EIλ3qeminusλx( cos λx + sin λx) (10)
It can be concluded that each of the concentrated loadks(τ) at point τ contributes the following amount to thedecentection of the existing tunnel on the Winkler foundation
w(x) 1
8EIλ3int+infin
minusinfin2Rks tminus L1( )sin α( ) eminusλ|xminusτ|
(cos λ|xminus τ| + sin λ|xminus τ|)dτ(11)
where L1 is the distance between the coordinate origin of theexisting tunnel and the intersection point between the newtunnel and the existing tunnel and α is the intersection anglebetween the new tunnel and existing tunnel e decentectionrotational angle bending moment and shear force of theexisting tunnel which is assumed as an innite beam on theWinkler foundation and the Pasternak foundation are il-lustrated in Appendix A
63 A Brief Description of the NewMethod In this study wewant to gure out the eects of the reinforcement on theexisting tunnel the stiness varies in this case and theexisting tunnel assumed as an innite beam is not suitableAssuming the existing tunnel with nite length and in-troducing (7) to (9) the basic dierential equation gov-erning the centexure of the beam resting on the Pasternakfoundation can be written in the form with an assumptionof Gλ2klt 1
EiIid4wi(x)dx4minusGibi
d2wi(x)dx4
+ kibiwi(x) biqi(x) (12)
where b1i bi(1 +(Gikiradic
bi)) subscript i represents the ithelement of the beam bi is the width of the beam and b1i isthe modied width of the beam Assuming a solution of (12)in the form of w(x) eλx the general solution of verticaldisplacement of the beam is as follows
y
e southbound tunnel
e northbound tunnel
e e
xistin
g tun
nel
x
L 1L 2
0x 1
x 2x 3
x ix i+
1
x nminus2
x nminus1
x n
O2
O1
xx
Figure 14 A schematic diagram of tunnels for the analytical methods
8 Advances in Civil Engineering
w(x) sum4
i1CiFi(x) (13)
where Ci (i 1 2 3 4) are undetermined parameters whichare dependent on the boundary conditions and Fi (x)(i 1 2 3 4) are the basic functions of the general solutionwhich are given as follows
F1i(x) cos a1icix( ) sinh a2icix( ) (14)
F2i(x) cos a1icix( ) cosh a2icix( ) (15)
F3i(x) sin a1icix( ) cosh a2icix( ) (16)
F4i(x) sin a1icix( ) sinh a2icix( ) (17)
where ci kib14EiIi4radic
a1i 1minusGic2i kiradic
a2i1 + Gic2i kiradic
e relationship of the decentection rotational angle
bending moment and shear force of the beam proposed byLancaster and Mitchell [18] is as follows
θi(x) wprimei(x)Mi(x) minusEiIiwPrime(x)
Qi(x) minusEiIiw(x) + Gpib1iwprimei(x)
(18)
where θi(x) Mi(x) and Qi(x) are the amplitudes of ro-tational angle bending moment and shear force of thesection area respectively
Figure 15 shows that dierent stiness is replaced byequivalent stepped stiness According to dierent stinessand distributed load the local coordinate system is establishedas illustrated in Figure 16 Selvadurai [12] suggests an initialparameter method to solve the parameters of the beam efour parameters at the origin of the coordinates arewi(0i) θi(0i)Mi(0i) and Qi(0i) and four parameters at theend of the element are wi(xi) θi(xi) Mi(xi) and Qi(xi)C1 C2 C3 and C4 can be expressed by wi(0i) θi(0i)Mi(0i) and Qi(0i) by introducing xi 0 to (13) and (18)
wi(xi) θi(xi)Mi(xi) and Qi(xi) can be rewritten in thematrix-array form as
wi xi( )θi xi( )Mi xi( )Vi xi( )1
k11 k12 k13 k14 k15
k21 k22 k23 k24 k25
k31 k32 k33 k34 k35
k41 k42 k43 k44 k45
0 0 0 0 1
middot
wi 0i( )θi 0i( )Mi 0i( )Vi 0i( )1
(19)
where [kij] is a stiness submatrix with 5 times 5 order ij 1 2 5M and V are respectively the bending momentand the generalized shear force Formula (19) can be sim-plied as
Ni xi( ) KijNi 0i( ) (20)
where Ni(xi) [Wi(xi) θi(xi)Mi(xi) Qi(xi) 1]T andNi(0i) [Wi(0i) θi(0i)Mi(0i) Qi(0i) 1]T
Figure 17 shows that curved distributed load is replacedby equivalent triangular and square distributed load whichcan be expressed as
qiminus1 +qi qiminus1 + ξqi minus qiminus1Li
( ) (21)
where qiminus1 and qi are respectively the load of the startingpoint and the ending point of the ith element and ξ is theposition of arbitrary load From (19) we can obtain
k15 minusbintx0 qiminus1 +qi( ) aminus12i F1i[xminus ξ]minus aminus11i F3i[xminus ξ]( )dξ
4c3i EiIi
(22)In order to avoid numerical errors in the calculation
process the integration of ki5 is shown in Appendix B erest of kij can be referred in the work by Selvadurai [12]
Figure 18 shows the tunnel matrix transfer diagramAssuming that the tunnel is composed of N elements thex-coordinate of beginning points of the ith element isxi(i 0 1 2 n) and the length of each element is Liwhere Li xi minus ximinus1
E0I0
0 x1 x2 x3 ximinus1 xi xnminus1 xn x
E1I1E2I2
Eiminus1Iiminus1Enminus1Inminus1
Figure 15 Sketch of equivalent stepped stiness
qi(x)
pi
Qi Qi+1Mi Mi+1
Oimi
xi
Figure 16 Forces and local coordinate system
q0q1 q2
qiminus1 qnminus1
x1 x2 x3 ximinus1 xi xnminus1 xn x
q (x)
0
Figure 17 Replacement of variable load with trapezoid load
Advances in Civil Engineering 9
For the rst element
N1(x) K1(x) middotN1 01( ) x isin 0 L1( ) (23)
For the end point of the rst element
N1 x1( ) K1 x1 minus 0( ) middotN1 01( ) (24)
For the end point of the ith element
Ni xi( ) Ki xi( ) middotKiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( ) middotN1 01( )(25)
For x at the ith element section where x is a globalcoordinate
Ni(x)Ki xminus ximinus1( ) middotKiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( ) middotN1 01( )(26)
Suppose
Ai(x) Ki xminusximinus1( ) middot Kiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( )(27)
en (20) can be expressed as
Ni(x) Ai(x) middotN1 01( ) (28)
us the relation between the vectorNn(xn) at the pointxn and the vector N1(01) at the start point x0 is
Nn xn( ) An xn( ) middotN1 01( ) (29)
where
An xn( ) prod1
inKi Li( ) (30)
e boundary condition of the existing tunnel assumedas an innite long tunnel considering the shear stress of soilcan be written as
M(0) 0
V(0) 2
k
G
radic
middot G middot R middot w(0)
V(L) 0
V(L) minus2
k
G
radic
middot G middot R middot w(L)
(31)
By applying the boundary conditions to (19) the de-centection rotational angle bending moment and shear forceof the overall beam can be obtained
64 Case Study
641 Parameter Selection To enable a direct compari-son assumed parameters of the soil and existing tunnelare selected in this study Poissonrsquos ratio of the soil is0388 e grouting process is regarded as the inverseprocess of soil loss due to tunnelling and g can beexpressed as g (2Δ + δ)(1minus η) where η is the injectionratio
For the Pasternak foundation as the thickness of theelastic layer is assumed to be 25 B the subgrade elasticmodulus k and shear stiness G are suggested by Xu [19] asfollows
k 5E0β
16 1minus v05pt2( )times 122
G 13E0B
2
32 1 + v0( )βtimes 085
(32)
where β is a modied parameter (β 129) and E0 and v0 arerespectively the elastic modulus and Poissonrsquos ratio forelastic foundation which are given as follows
E0 Es
1minus v2s
v0 vs
1minus vs
(33)
where the empirical relation between the Youngrsquos modulusEs and the compression modulus Es01minus02 of soft soils is builtup by Yang and Zhao [20]
E0 (25 sim 35)Es01minus02 (34)
As illustrated in Figure 14 points O1 and O2 are re-spectively the intersection points A and B in practicalcoordinate transformation expressions of the axis xprime and xPrimeare respectively
xprime xminusL1( )sin αxPrime xminusL2( )sin α
(35)
where L1 and L2 are the distance from the beginningpoint of the existing tunnel to points O1 and O2 re-spectively e length of the existing tunnel is assumedas 250 m and the ring spacing is 12 m So the tunnel isdivided into 209 sections and each section is 12 m inlength
N1(01) Ni(xi)N1(x1) = N2(02) N2(x2)L1
L1
E1I1
G1 k1 G2 k2 G3 k3 Gi ki Gn knGnminus1 knminus1
E2I2 E3I3 EiIi EnInEnminus1Inminus1
L2 L3 Li LnLnminus1
L2 Li
x x x
xx1 x2 x3 xi-1 xnminus2 xnminus1 xnxi0
Ni(0i)
Figure 18 e matrix transfer diagram
10 Advances in Civil Engineering
7 Discussions
e normalized bending moment and shear force are de-ned as
Mi MiL0EiIi
Qi QiL
20
EiIi
(36)
where Mi and Qi are respectively the normalized bendingmoment and shear force Mi and Qi are respectively thecalculated bending moment and shear force and L0 is thelength of the existing tunnel
Figure 19 shows the comparison of the decentection ro-tational angle bending moment and shear force of theexisting tunnel with a constant stiness which is based onthree kinds of models an innite tunnel on the Winklerfoundation an innite tunnel on the Pasternak foundationand a nite tunnel on the Winkler foundation Note that the
368 380 400 420 440 460 480 500 520 540 560 580 600 612
minus0007
minus0006
minus0005
minus0004
minus0003
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)
minus0002
minus0001
0000
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundatione observed data on 201465
e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
minus000015
minus000010
minus000005
000000
000005
000010
000015
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
(b)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
e b
endi
ng m
omen
t of t
he ex
istin
g tu
nnel
(kN
middotm)
387 400 420 440 460 480 500 520 540 560 580 595
0
minus6000
minus5000
minus4000
minus3000
minus2000
minus1000
1000
2000
3000
4000
e ring number of the existing tunnel
(c)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
387 400 420 440 460 480 500 520 540 560 580 595
minus600
minus400
minus200
0
200
400
600
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(kN
)
(d)
Figure 19 (a) Comparison of the decentection of the existing tunnel using dierent models (b) Comparison of the rotational angle ofthe existing tunnel using dierent models (c) Comparison of the normalized bending moment of the existing tunnel using dierentmodels (d) Comparison of the normalized shear force of the existing tunnel using dierent models
Advances in Civil Engineering 11
existing tunnel is only subject to soil deformation force inthis case e displacement and the decentection of beam areparticular similar concepts owing to the fact that the existingtunnel is securely attached to the foundation generallywithout rigid body motion Fairly good agreement can beobserved between the innite tunnel on the Pasternakfoundation and the nite tunnel on the Pasternak founda-tion which shows that the method proposed in this paper iscorrect and can be used to analyze similar problems Becauseof considering soil shear stress vertical stress and de-formation are aected and weakened the displacement of
the existing tunnel on the Pasternak foundation is smallerthan that on the Winkler foundation without consideringthe shear stress of soil Using the Winkler foundation modelis in good agreement with the observed data It can be seenfrom (3) that the displacement of the existing tunnel isproportional to the grouting elaquociency (ie η) Due to theuncertainty of η it cannot be decided which one is the bestanalytical method It is clear that the maximum normalizedbending moment occurs at the points O1 and O2 which arethe nearest points between the existing tunnel and the newtunnels
minus0003
minus0002
minus0001
0000387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
minus000010
minus000005
000000
000005
000010
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e ring number of the existing tunnel
(b)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10
minus0003
minus0002
minus0001
0000
0001
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
xL
(c)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10xL
minus008
minus006
minus004
minus002
000
002
004
006
008
The n
orm
alise
d sh
ear f
orce
of t
he ex
istin
g tu
nnel
(d)
Figure 20 (a) e eects of dierent reinforcement stiness on the decentection of the existing tunnel (b) e eects of dierent re-inforcement stiness on the slope of the existing tunnel (c)e eects of dierent reinforcement stiness on the normalized moment of theexisting tunnel (d) e eects of dierent reinforcement stiness on the normalized shear force of the existing tunnel
12 Advances in Civil Engineering
e reinforce range of the existing tunnel is assumed as25 rings on either side of the intersection point A and thevarious stiness is taken into consideration using themethod proposed in this paper Figure 20 shows the in-centuence of dierent stiness of the reinforcement on theexisting tunnel (ie 4 EI EI and 14 EI) It is evident thatthe reinforcement can reduce the displacement rotationalangle normalized bending moment and shear force of theexisting tunnel while tunnel damage (reduce the stinessof reinforcement) has the opposite eect It is evident thatthe change of reinforcement stiness has greater eect onthe normalized bending moment and the normalized shearforce of the existing tunnel particularly at the incentection
points of the normalized bending moment and normalizedshear force curve
e reinforce stiness of the reinforcement range isassumed as 4 EI and the various reinforcement ranges istaken into consideration using the method proposed in thispaper Figure 21 shows the incentuence of dierent re-inforcement ranges on the existing tunnel when thenorthbound tunnel of Line 4 underpasses e normalizedbending moment on either side of the intersection point istoo large when the reinforcement range is 69ndash892m andthe normalized shear force on either side of the intersectionpoint is relatively too large when the reinforcement range isless than 592ndash992m So it is necessary to reinforce larger
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00035
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
The d
eflec
tions
of t
he ex
istin
g tu
nnel
(m)
xL
(a)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus000010
minus000005
000000
000005
000010
The s
lope
of t
he ex
istin
g tu
nnel
(rad
)
xL
(b)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
00005
00010
00015
xL
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
(c)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus010
minus008
minus006
minus004
minus002
000
002
004
006
008
010
xLTh
e nor
mal
ised
shea
r for
ce o
f the
exist
ing
tunn
el
(d)
Figure 21 (a)e eects of dierent reinforcement ranges on the decentection of the existing tunnel (b)e eects of dierent reinforcementranges on the slope of the existing tunnel (c)e eects of dierent reinforcement ranges on the normalized moment of the existing tunnel(d) e eects of dierent reinforcement ranges on the normalized shear force of the existing tunnel
Advances in Civil Engineering 13
than the range of 492ndash1092m in the existing tunnel in thiscase that is larger than 30m (25 rings) on either side of theintersection point )ere is no big difference in the bendingmoment and shear force of the existing tunnel when thereinforcement ranges are 492ndash1092m 392ndash1192m and292ndash1292m In order to reduce the costs of reinforcementreinforcement in the 30m range (ie 25 rings) on eitherside of the intersection point is the best choice whichverifies the actual design
8 Conclusions
Based on the abovementioned statements the response of theexisting tunnel is analyzed and analytical methods are proposedfor verification and the reinforcement design Main conclusionsderived from the analysis are as follows
(1) )e displacement of the existing tunnel changes withthe relative position of the shield and the existingtunnel Shield parameters have a major impact on theexisting tunnel )e heaves of the existing tunnelmight be caused by the large applied face pressure andbulkhead addictive thrust When the shield tail isdriving beyond the selected monitoring rings theselected monitoring rings settle rapidly due to theclosure of the shield tail void
(2) )e southbound tunnel of Line 4 has less effect onthe left side of the intersection point A on theexisting tunnel because of the northbound tunnelrsquosbarrier effect A variety of measures of displacementprevention of Line 1 such as the control of shield
parameters reinforcement of the existing tunnel canensure the normal operation of metro Line 1
(3) )e analytical method proposed in the paper is ver-ified by the methods derived from prior studies It ispossible to take the different stiffness and curved loadsinto account by applying the stepped stiffness andtrapezoidal load while using a local coordinate systemin the derivation of the new method )e change ofreinforcement stiffness has greater effect on thenormalized bending moment and the normalizedshear force of the existing tunnel particularly at theinflection points of the normalized bending momentand normalized shear force curve )e analyticalmethod can be applied for the reinforcement designand safety check of the built tunnel
Appendices
A1 An Infinite Beam on the WinklerFoundation
θ(x) 1113946+infin
minusinfin
minusλ2s((tminus L)sin α)eminusλ(xminust) sin(λ(xminus t))
b
λ2s((tminus L)sin α)eλ(xminust)sin(λ(tminus x))
b
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎭
dt
(A1)
M(x) 14λ
1113946+infin
minusinfinks((tminusL)sin α)e
minusλ|xminust|( cos(λ|xminus t|)minus sin(λ|xminus t|))dt (A2)
Q(x) 1113946+infin
minusinfin
minus12
ks((tminusL)sin α)eminusλ(xminust)cos(λ(xminus t))dt xminus tgt 0
12
ks((tminusL)sin α)eminusλ(tminusx)cos(λ(tminusx))dt xminus tle 0
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
⎫⎪⎪⎪⎬
⎪⎪⎪⎭
dt (A3)
A2 An Infinite Beam on the PasternakFoundation
w(x) λ2bk
1113946+infin
minusinfinks((tminus L)sin α)De
minusa1λ|xminusτ| 1a1
cos a2λ|xminus τ|( 1113857 +1a2
sin a2λ|xminus τ|( 11138571113888 1113889dτ
θ(x) 1113946+infin
minusinfin
minusλ2s((tminusL))eminusa1λ(xminust)sin a2(xminus t)( 1113857
a1a2b
λ2s((tminus L))ea1λ(xminust)sin a2(tminus x)( 1113857
a1a2b
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A4)
14 Advances in Civil Engineering
M(x) 14λ
1113946+infin
minusinfinks((tminus L)sin α)e
minusa1λ|xminust| 1a1
cos a2λ|xminus t|( 1113857minus1a2
sin a2λ|xminus t|( 11138571113888 1113889dt
Q(x) 1113946+infin
minusinfin
minus12
ks((tminus L)sin α)eminusa1λ(xminust) cos a2λ(xminus t)( 1113857minus
a21 minus a 2
22a1a2
1113888 1113889sin a2λ(xminus t)( 11138571113890 1113891dt xminus tgt 0
12
ks((tminusL)sin α)eminusa1λ(tminusx) cos a2λ(tminusx)( 1113857minus
a21 minus a2
2
2a1a21113888 1113889sin a2λ(tminusx)( 11138571113890 1113891dt xminus tle 0
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A5)
B1 Coefficients of the Matrix ki5
k15 minusbiqiminus1
4a1ia2ici4EiIi a1i
2 + a2i2( 1113857
minus2a1ia2i + 2a1ia2iF2i
+ a1i2 minus a2i
2( 1113857F4i
⎛⎜⎝ ⎞⎟⎠
+bi qi minus qiminus1( 1113857
4a1ia2ici5EiIiL3 a1i
2 + a2i2( 1113857
2
a2i minus3a21i + a2
2i( 1113857F3i
+a1i 2a2i a21i + a2
2i( 1113857cix + a1i2 minus 3a2i
2( 1113857F1i( 1113857
⎛⎜⎝ ⎞⎟⎠
(B1)
k25 biqiminus1 a2iF3i minus a1iF1i( 1113857
2a1a2 a21 + a 2
2( 1113857c3EiIi
+bi qi minus qiminus1( 1113857 2a1ia2i minus 2a1ia2iF2i minus a1i
2 minus a2i2( 1113857F4i( 1113857( 1113857
2a1ia2i a1i2 + a2i
2( 11138572ci
4EiIix1113872 1113873 (B2)
k35 minusbiqiminus1F4i
2a1ia2ici2 minus
bi qi minus qiminus1( 1113857 a2iF3i minus a1iF1i( 1113857
2a1ia2ici3 a1i
2 + a2i2( 1113857x
(B3)
k45 minusbiqiminus1
a1i2 + a2i
2( 1113857ci
1113888a1i 1 + 2a1i2a22i minus 2a2i
41113872 1113873F3i + a2i 1 + 2a1i
2a22i minus 2a1i
41113872 1113873F1i1113889
minusbi qi minus qiminus1( 1113857
a21i + a2i
2( 11138572c 2
i x
4a1i2a2i
2 minus 1( 1113857 a1i2 minus a2i
2( 1113857 + a2i2 minus 4a1i
4a2i2 + a1i
2 minus1 + 4a2i4( 1113857( 1113857F2i
minus2a1ia2i a 21i minus a 2
2i( 11138572 minus 11113872 1113873F4i
⎛⎜⎝ ⎞⎟⎠
(B4)
Conflicts of Interest
)e author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
)e author appreciates the help from the staff of Hangzhouand Hong Kong Tunnel Company Ltd during in-strumentation setup and data acquisition )is study issupported by the National Natural Science Foundation ofChina (no 41702313)
References
[1] S H Kim Model testing and analysis of interactions betweentunnels in clay [PhD 3esis] Department of EngineeringScience University of Oxford Oxford UK 1996
[2] H Y Liu J C Small J P Carter and D J Williams ldquoEffectsof tunnelling on existing support systems of perpendicularlypassing tunnelsrdquo Computers and Geotechnics vol 36 no 5pp 880ndash894 2009
[3] I Yamaguchi I Yamazaki and Y Kiritani ldquoStudy ofgroundndashtunnel interactions of four shield tunnels driven inclose proximity in relation to design and construction ofparallel shield tunnelsrdquo Tunnelling and Underground SpaceTechnology vol 13 no 3 pp 289ndash304 1998
[4] T I Addenbrooke and D M Potts ldquoTwin tunnel interactionsurface and subsurface effectsrdquo International Journal ofGeomechanics vol 1 no 2 pp 249ndash271 2001
[5] J P Kimmance S Lawrence O Hassan and N J PurchaseldquoObservations of deformations created in existing tunnels byadjacent and cross cutting excavationsrdquo in Proceedings ofGeotechnical Aspects of Underground Construction in SoftGround pp 707ndash712 London UK July 1996
[6] E Leca and B New ldquoSettlements induced by tunnelling in softgroundrdquo Tunnelling and Underground Space Technologyvol 22 no 2 pp 119ndash149 2007
Advances in Civil Engineering 15
[7] S M Liao J H Liu R L Wang and Z M Li ldquoShieldtunnelling and environment protection in Shanghai softgroundrdquo Tunnelling and Underground Space Technologyvol 24 no 4 pp 454ndash465 2009
[8] F H Chehade and I Shahrour ldquoNumerical analysis of theinteraction between twin-tunnels influence of the relativeposition and construction procedurerdquo Tunnelling and Un-derground Space Technology vol 23 no 2 pp 210ndash214 2008
[9] X G Li and D J Yuan ldquoResponse of a double-decked metrotunnel to shield driving of twin closely underpasses tunnelsrdquoTunnelling and Underground Space Technology vol 28 no 1pp 18ndash30 2012
[10] T E Vorster A Klar K Soga and R J Mair ldquoEstimating theeffects of tunnelling on existing pipelinesrdquo Journal of Geo-technical and Geoenvironmental Engineering vol 131 no 11pp 1399ndash1410 2005
[11] Z G Zhang M S Huang and W D Wang ldquoEvaluation ofdeformation response for adjacent tunnels due to soilunloading in excavation engineeringrdquo Tunnelling and Un-derground Space Technology vol 38 no 3 pp 244ndash253 2013
[12] A P S Selvadurai Elastic Analysis of Soil-Foundation InteractionElsevier Scientific Pub Co Amsterdam Netherlands 1979
[13] N Loganathan and H G Poulos ldquoAnalytical prediction fortunnelling-induced ground movements in claysrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124no 9 pp 846ndash856 1998
[14] K M Lee R K Rowe and K Y Lo ldquoSubsidence owing totunnelling I Estimating the gap parameterrdquo CanadianGeotechnical Journal vol 29 no 6 pp 929ndash940 1992
[15] R K Rowe and KM Lee ldquoSubsidence owing to tunnelling IIEvaluation of a prediction techniquerdquo Canadian GeotechnicalJournal vol 29 no 6 pp 941ndash954 1992
[16] P B Attewell J Yeates and A R Selby Soil MovementsInduced by Tunnelling and their Effects on Pipelines andStructures Methuen Inc New York NY USA 1986
[17] F Ye C He H H Zhu et al ldquoAnalysis of longitudinalequivalent stiffness of shield tunnel considering lateral per-formancerdquo Chinese Journal of Geotechnical Engineeringvol 33 no 12 pp 1870ndash1876 2011 in Chinese
[18] P R Lancaster and D Mitchell ldquoProblems in Bending of Barsand Plates [M]rdquo Advanced Solid Mechanics Macmillan UK1980
[19] L Xu ldquoStudy on vertical settlement of shield tunnel in soft soilrdquoPhD dissertation Tongji University Shanghai China 2005
[20] M Yang and X Zhao ldquoAn approach for a single pile in layeredsoilrdquo Tongji University Natural Science vol 20 no 4pp 421ndash428 1992 in Chinese
16 Advances in Civil Engineering
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w(x) sum4
i1CiFi(x) (13)
where Ci (i 1 2 3 4) are undetermined parameters whichare dependent on the boundary conditions and Fi (x)(i 1 2 3 4) are the basic functions of the general solutionwhich are given as follows
F1i(x) cos a1icix( ) sinh a2icix( ) (14)
F2i(x) cos a1icix( ) cosh a2icix( ) (15)
F3i(x) sin a1icix( ) cosh a2icix( ) (16)
F4i(x) sin a1icix( ) sinh a2icix( ) (17)
where ci kib14EiIi4radic
a1i 1minusGic2i kiradic
a2i1 + Gic2i kiradic
e relationship of the decentection rotational angle
bending moment and shear force of the beam proposed byLancaster and Mitchell [18] is as follows
θi(x) wprimei(x)Mi(x) minusEiIiwPrime(x)
Qi(x) minusEiIiw(x) + Gpib1iwprimei(x)
(18)
where θi(x) Mi(x) and Qi(x) are the amplitudes of ro-tational angle bending moment and shear force of thesection area respectively
Figure 15 shows that dierent stiness is replaced byequivalent stepped stiness According to dierent stinessand distributed load the local coordinate system is establishedas illustrated in Figure 16 Selvadurai [12] suggests an initialparameter method to solve the parameters of the beam efour parameters at the origin of the coordinates arewi(0i) θi(0i)Mi(0i) and Qi(0i) and four parameters at theend of the element are wi(xi) θi(xi) Mi(xi) and Qi(xi)C1 C2 C3 and C4 can be expressed by wi(0i) θi(0i)Mi(0i) and Qi(0i) by introducing xi 0 to (13) and (18)
wi(xi) θi(xi)Mi(xi) and Qi(xi) can be rewritten in thematrix-array form as
wi xi( )θi xi( )Mi xi( )Vi xi( )1
k11 k12 k13 k14 k15
k21 k22 k23 k24 k25
k31 k32 k33 k34 k35
k41 k42 k43 k44 k45
0 0 0 0 1
middot
wi 0i( )θi 0i( )Mi 0i( )Vi 0i( )1
(19)
where [kij] is a stiness submatrix with 5 times 5 order ij 1 2 5M and V are respectively the bending momentand the generalized shear force Formula (19) can be sim-plied as
Ni xi( ) KijNi 0i( ) (20)
where Ni(xi) [Wi(xi) θi(xi)Mi(xi) Qi(xi) 1]T andNi(0i) [Wi(0i) θi(0i)Mi(0i) Qi(0i) 1]T
Figure 17 shows that curved distributed load is replacedby equivalent triangular and square distributed load whichcan be expressed as
qiminus1 +qi qiminus1 + ξqi minus qiminus1Li
( ) (21)
where qiminus1 and qi are respectively the load of the startingpoint and the ending point of the ith element and ξ is theposition of arbitrary load From (19) we can obtain
k15 minusbintx0 qiminus1 +qi( ) aminus12i F1i[xminus ξ]minus aminus11i F3i[xminus ξ]( )dξ
4c3i EiIi
(22)In order to avoid numerical errors in the calculation
process the integration of ki5 is shown in Appendix B erest of kij can be referred in the work by Selvadurai [12]
Figure 18 shows the tunnel matrix transfer diagramAssuming that the tunnel is composed of N elements thex-coordinate of beginning points of the ith element isxi(i 0 1 2 n) and the length of each element is Liwhere Li xi minus ximinus1
E0I0
0 x1 x2 x3 ximinus1 xi xnminus1 xn x
E1I1E2I2
Eiminus1Iiminus1Enminus1Inminus1
Figure 15 Sketch of equivalent stepped stiness
qi(x)
pi
Qi Qi+1Mi Mi+1
Oimi
xi
Figure 16 Forces and local coordinate system
q0q1 q2
qiminus1 qnminus1
x1 x2 x3 ximinus1 xi xnminus1 xn x
q (x)
0
Figure 17 Replacement of variable load with trapezoid load
Advances in Civil Engineering 9
For the rst element
N1(x) K1(x) middotN1 01( ) x isin 0 L1( ) (23)
For the end point of the rst element
N1 x1( ) K1 x1 minus 0( ) middotN1 01( ) (24)
For the end point of the ith element
Ni xi( ) Ki xi( ) middotKiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( ) middotN1 01( )(25)
For x at the ith element section where x is a globalcoordinate
Ni(x)Ki xminus ximinus1( ) middotKiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( ) middotN1 01( )(26)
Suppose
Ai(x) Ki xminusximinus1( ) middot Kiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( )(27)
en (20) can be expressed as
Ni(x) Ai(x) middotN1 01( ) (28)
us the relation between the vectorNn(xn) at the pointxn and the vector N1(01) at the start point x0 is
Nn xn( ) An xn( ) middotN1 01( ) (29)
where
An xn( ) prod1
inKi Li( ) (30)
e boundary condition of the existing tunnel assumedas an innite long tunnel considering the shear stress of soilcan be written as
M(0) 0
V(0) 2
k
G
radic
middot G middot R middot w(0)
V(L) 0
V(L) minus2
k
G
radic
middot G middot R middot w(L)
(31)
By applying the boundary conditions to (19) the de-centection rotational angle bending moment and shear forceof the overall beam can be obtained
64 Case Study
641 Parameter Selection To enable a direct compari-son assumed parameters of the soil and existing tunnelare selected in this study Poissonrsquos ratio of the soil is0388 e grouting process is regarded as the inverseprocess of soil loss due to tunnelling and g can beexpressed as g (2Δ + δ)(1minus η) where η is the injectionratio
For the Pasternak foundation as the thickness of theelastic layer is assumed to be 25 B the subgrade elasticmodulus k and shear stiness G are suggested by Xu [19] asfollows
k 5E0β
16 1minus v05pt2( )times 122
G 13E0B
2
32 1 + v0( )βtimes 085
(32)
where β is a modied parameter (β 129) and E0 and v0 arerespectively the elastic modulus and Poissonrsquos ratio forelastic foundation which are given as follows
E0 Es
1minus v2s
v0 vs
1minus vs
(33)
where the empirical relation between the Youngrsquos modulusEs and the compression modulus Es01minus02 of soft soils is builtup by Yang and Zhao [20]
E0 (25 sim 35)Es01minus02 (34)
As illustrated in Figure 14 points O1 and O2 are re-spectively the intersection points A and B in practicalcoordinate transformation expressions of the axis xprime and xPrimeare respectively
xprime xminusL1( )sin αxPrime xminusL2( )sin α
(35)
where L1 and L2 are the distance from the beginningpoint of the existing tunnel to points O1 and O2 re-spectively e length of the existing tunnel is assumedas 250 m and the ring spacing is 12 m So the tunnel isdivided into 209 sections and each section is 12 m inlength
N1(01) Ni(xi)N1(x1) = N2(02) N2(x2)L1
L1
E1I1
G1 k1 G2 k2 G3 k3 Gi ki Gn knGnminus1 knminus1
E2I2 E3I3 EiIi EnInEnminus1Inminus1
L2 L3 Li LnLnminus1
L2 Li
x x x
xx1 x2 x3 xi-1 xnminus2 xnminus1 xnxi0
Ni(0i)
Figure 18 e matrix transfer diagram
10 Advances in Civil Engineering
7 Discussions
e normalized bending moment and shear force are de-ned as
Mi MiL0EiIi
Qi QiL
20
EiIi
(36)
where Mi and Qi are respectively the normalized bendingmoment and shear force Mi and Qi are respectively thecalculated bending moment and shear force and L0 is thelength of the existing tunnel
Figure 19 shows the comparison of the decentection ro-tational angle bending moment and shear force of theexisting tunnel with a constant stiness which is based onthree kinds of models an innite tunnel on the Winklerfoundation an innite tunnel on the Pasternak foundationand a nite tunnel on the Winkler foundation Note that the
368 380 400 420 440 460 480 500 520 540 560 580 600 612
minus0007
minus0006
minus0005
minus0004
minus0003
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)
minus0002
minus0001
0000
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundatione observed data on 201465
e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
minus000015
minus000010
minus000005
000000
000005
000010
000015
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
(b)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
e b
endi
ng m
omen
t of t
he ex
istin
g tu
nnel
(kN
middotm)
387 400 420 440 460 480 500 520 540 560 580 595
0
minus6000
minus5000
minus4000
minus3000
minus2000
minus1000
1000
2000
3000
4000
e ring number of the existing tunnel
(c)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
387 400 420 440 460 480 500 520 540 560 580 595
minus600
minus400
minus200
0
200
400
600
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(kN
)
(d)
Figure 19 (a) Comparison of the decentection of the existing tunnel using dierent models (b) Comparison of the rotational angle ofthe existing tunnel using dierent models (c) Comparison of the normalized bending moment of the existing tunnel using dierentmodels (d) Comparison of the normalized shear force of the existing tunnel using dierent models
Advances in Civil Engineering 11
existing tunnel is only subject to soil deformation force inthis case e displacement and the decentection of beam areparticular similar concepts owing to the fact that the existingtunnel is securely attached to the foundation generallywithout rigid body motion Fairly good agreement can beobserved between the innite tunnel on the Pasternakfoundation and the nite tunnel on the Pasternak founda-tion which shows that the method proposed in this paper iscorrect and can be used to analyze similar problems Becauseof considering soil shear stress vertical stress and de-formation are aected and weakened the displacement of
the existing tunnel on the Pasternak foundation is smallerthan that on the Winkler foundation without consideringthe shear stress of soil Using the Winkler foundation modelis in good agreement with the observed data It can be seenfrom (3) that the displacement of the existing tunnel isproportional to the grouting elaquociency (ie η) Due to theuncertainty of η it cannot be decided which one is the bestanalytical method It is clear that the maximum normalizedbending moment occurs at the points O1 and O2 which arethe nearest points between the existing tunnel and the newtunnels
minus0003
minus0002
minus0001
0000387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
minus000010
minus000005
000000
000005
000010
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e ring number of the existing tunnel
(b)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10
minus0003
minus0002
minus0001
0000
0001
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
xL
(c)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10xL
minus008
minus006
minus004
minus002
000
002
004
006
008
The n
orm
alise
d sh
ear f
orce
of t
he ex
istin
g tu
nnel
(d)
Figure 20 (a) e eects of dierent reinforcement stiness on the decentection of the existing tunnel (b) e eects of dierent re-inforcement stiness on the slope of the existing tunnel (c)e eects of dierent reinforcement stiness on the normalized moment of theexisting tunnel (d) e eects of dierent reinforcement stiness on the normalized shear force of the existing tunnel
12 Advances in Civil Engineering
e reinforce range of the existing tunnel is assumed as25 rings on either side of the intersection point A and thevarious stiness is taken into consideration using themethod proposed in this paper Figure 20 shows the in-centuence of dierent stiness of the reinforcement on theexisting tunnel (ie 4 EI EI and 14 EI) It is evident thatthe reinforcement can reduce the displacement rotationalangle normalized bending moment and shear force of theexisting tunnel while tunnel damage (reduce the stinessof reinforcement) has the opposite eect It is evident thatthe change of reinforcement stiness has greater eect onthe normalized bending moment and the normalized shearforce of the existing tunnel particularly at the incentection
points of the normalized bending moment and normalizedshear force curve
e reinforce stiness of the reinforcement range isassumed as 4 EI and the various reinforcement ranges istaken into consideration using the method proposed in thispaper Figure 21 shows the incentuence of dierent re-inforcement ranges on the existing tunnel when thenorthbound tunnel of Line 4 underpasses e normalizedbending moment on either side of the intersection point istoo large when the reinforcement range is 69ndash892m andthe normalized shear force on either side of the intersectionpoint is relatively too large when the reinforcement range isless than 592ndash992m So it is necessary to reinforce larger
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00035
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
The d
eflec
tions
of t
he ex
istin
g tu
nnel
(m)
xL
(a)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus000010
minus000005
000000
000005
000010
The s
lope
of t
he ex
istin
g tu
nnel
(rad
)
xL
(b)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
00005
00010
00015
xL
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
(c)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus010
minus008
minus006
minus004
minus002
000
002
004
006
008
010
xLTh
e nor
mal
ised
shea
r for
ce o
f the
exist
ing
tunn
el
(d)
Figure 21 (a)e eects of dierent reinforcement ranges on the decentection of the existing tunnel (b)e eects of dierent reinforcementranges on the slope of the existing tunnel (c)e eects of dierent reinforcement ranges on the normalized moment of the existing tunnel(d) e eects of dierent reinforcement ranges on the normalized shear force of the existing tunnel
Advances in Civil Engineering 13
than the range of 492ndash1092m in the existing tunnel in thiscase that is larger than 30m (25 rings) on either side of theintersection point )ere is no big difference in the bendingmoment and shear force of the existing tunnel when thereinforcement ranges are 492ndash1092m 392ndash1192m and292ndash1292m In order to reduce the costs of reinforcementreinforcement in the 30m range (ie 25 rings) on eitherside of the intersection point is the best choice whichverifies the actual design
8 Conclusions
Based on the abovementioned statements the response of theexisting tunnel is analyzed and analytical methods are proposedfor verification and the reinforcement design Main conclusionsderived from the analysis are as follows
(1) )e displacement of the existing tunnel changes withthe relative position of the shield and the existingtunnel Shield parameters have a major impact on theexisting tunnel )e heaves of the existing tunnelmight be caused by the large applied face pressure andbulkhead addictive thrust When the shield tail isdriving beyond the selected monitoring rings theselected monitoring rings settle rapidly due to theclosure of the shield tail void
(2) )e southbound tunnel of Line 4 has less effect onthe left side of the intersection point A on theexisting tunnel because of the northbound tunnelrsquosbarrier effect A variety of measures of displacementprevention of Line 1 such as the control of shield
parameters reinforcement of the existing tunnel canensure the normal operation of metro Line 1
(3) )e analytical method proposed in the paper is ver-ified by the methods derived from prior studies It ispossible to take the different stiffness and curved loadsinto account by applying the stepped stiffness andtrapezoidal load while using a local coordinate systemin the derivation of the new method )e change ofreinforcement stiffness has greater effect on thenormalized bending moment and the normalizedshear force of the existing tunnel particularly at theinflection points of the normalized bending momentand normalized shear force curve )e analyticalmethod can be applied for the reinforcement designand safety check of the built tunnel
Appendices
A1 An Infinite Beam on the WinklerFoundation
θ(x) 1113946+infin
minusinfin
minusλ2s((tminus L)sin α)eminusλ(xminust) sin(λ(xminus t))
b
λ2s((tminus L)sin α)eλ(xminust)sin(λ(tminus x))
b
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎭
dt
(A1)
M(x) 14λ
1113946+infin
minusinfinks((tminusL)sin α)e
minusλ|xminust|( cos(λ|xminus t|)minus sin(λ|xminus t|))dt (A2)
Q(x) 1113946+infin
minusinfin
minus12
ks((tminusL)sin α)eminusλ(xminust)cos(λ(xminus t))dt xminus tgt 0
12
ks((tminusL)sin α)eminusλ(tminusx)cos(λ(tminusx))dt xminus tle 0
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
⎫⎪⎪⎪⎬
⎪⎪⎪⎭
dt (A3)
A2 An Infinite Beam on the PasternakFoundation
w(x) λ2bk
1113946+infin
minusinfinks((tminus L)sin α)De
minusa1λ|xminusτ| 1a1
cos a2λ|xminus τ|( 1113857 +1a2
sin a2λ|xminus τ|( 11138571113888 1113889dτ
θ(x) 1113946+infin
minusinfin
minusλ2s((tminusL))eminusa1λ(xminust)sin a2(xminus t)( 1113857
a1a2b
λ2s((tminus L))ea1λ(xminust)sin a2(tminus x)( 1113857
a1a2b
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A4)
14 Advances in Civil Engineering
M(x) 14λ
1113946+infin
minusinfinks((tminus L)sin α)e
minusa1λ|xminust| 1a1
cos a2λ|xminus t|( 1113857minus1a2
sin a2λ|xminus t|( 11138571113888 1113889dt
Q(x) 1113946+infin
minusinfin
minus12
ks((tminus L)sin α)eminusa1λ(xminust) cos a2λ(xminus t)( 1113857minus
a21 minus a 2
22a1a2
1113888 1113889sin a2λ(xminus t)( 11138571113890 1113891dt xminus tgt 0
12
ks((tminusL)sin α)eminusa1λ(tminusx) cos a2λ(tminusx)( 1113857minus
a21 minus a2
2
2a1a21113888 1113889sin a2λ(tminusx)( 11138571113890 1113891dt xminus tle 0
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A5)
B1 Coefficients of the Matrix ki5
k15 minusbiqiminus1
4a1ia2ici4EiIi a1i
2 + a2i2( 1113857
minus2a1ia2i + 2a1ia2iF2i
+ a1i2 minus a2i
2( 1113857F4i
⎛⎜⎝ ⎞⎟⎠
+bi qi minus qiminus1( 1113857
4a1ia2ici5EiIiL3 a1i
2 + a2i2( 1113857
2
a2i minus3a21i + a2
2i( 1113857F3i
+a1i 2a2i a21i + a2
2i( 1113857cix + a1i2 minus 3a2i
2( 1113857F1i( 1113857
⎛⎜⎝ ⎞⎟⎠
(B1)
k25 biqiminus1 a2iF3i minus a1iF1i( 1113857
2a1a2 a21 + a 2
2( 1113857c3EiIi
+bi qi minus qiminus1( 1113857 2a1ia2i minus 2a1ia2iF2i minus a1i
2 minus a2i2( 1113857F4i( 1113857( 1113857
2a1ia2i a1i2 + a2i
2( 11138572ci
4EiIix1113872 1113873 (B2)
k35 minusbiqiminus1F4i
2a1ia2ici2 minus
bi qi minus qiminus1( 1113857 a2iF3i minus a1iF1i( 1113857
2a1ia2ici3 a1i
2 + a2i2( 1113857x
(B3)
k45 minusbiqiminus1
a1i2 + a2i
2( 1113857ci
1113888a1i 1 + 2a1i2a22i minus 2a2i
41113872 1113873F3i + a2i 1 + 2a1i
2a22i minus 2a1i
41113872 1113873F1i1113889
minusbi qi minus qiminus1( 1113857
a21i + a2i
2( 11138572c 2
i x
4a1i2a2i
2 minus 1( 1113857 a1i2 minus a2i
2( 1113857 + a2i2 minus 4a1i
4a2i2 + a1i
2 minus1 + 4a2i4( 1113857( 1113857F2i
minus2a1ia2i a 21i minus a 2
2i( 11138572 minus 11113872 1113873F4i
⎛⎜⎝ ⎞⎟⎠
(B4)
Conflicts of Interest
)e author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
)e author appreciates the help from the staff of Hangzhouand Hong Kong Tunnel Company Ltd during in-strumentation setup and data acquisition )is study issupported by the National Natural Science Foundation ofChina (no 41702313)
References
[1] S H Kim Model testing and analysis of interactions betweentunnels in clay [PhD 3esis] Department of EngineeringScience University of Oxford Oxford UK 1996
[2] H Y Liu J C Small J P Carter and D J Williams ldquoEffectsof tunnelling on existing support systems of perpendicularlypassing tunnelsrdquo Computers and Geotechnics vol 36 no 5pp 880ndash894 2009
[3] I Yamaguchi I Yamazaki and Y Kiritani ldquoStudy ofgroundndashtunnel interactions of four shield tunnels driven inclose proximity in relation to design and construction ofparallel shield tunnelsrdquo Tunnelling and Underground SpaceTechnology vol 13 no 3 pp 289ndash304 1998
[4] T I Addenbrooke and D M Potts ldquoTwin tunnel interactionsurface and subsurface effectsrdquo International Journal ofGeomechanics vol 1 no 2 pp 249ndash271 2001
[5] J P Kimmance S Lawrence O Hassan and N J PurchaseldquoObservations of deformations created in existing tunnels byadjacent and cross cutting excavationsrdquo in Proceedings ofGeotechnical Aspects of Underground Construction in SoftGround pp 707ndash712 London UK July 1996
[6] E Leca and B New ldquoSettlements induced by tunnelling in softgroundrdquo Tunnelling and Underground Space Technologyvol 22 no 2 pp 119ndash149 2007
Advances in Civil Engineering 15
[7] S M Liao J H Liu R L Wang and Z M Li ldquoShieldtunnelling and environment protection in Shanghai softgroundrdquo Tunnelling and Underground Space Technologyvol 24 no 4 pp 454ndash465 2009
[8] F H Chehade and I Shahrour ldquoNumerical analysis of theinteraction between twin-tunnels influence of the relativeposition and construction procedurerdquo Tunnelling and Un-derground Space Technology vol 23 no 2 pp 210ndash214 2008
[9] X G Li and D J Yuan ldquoResponse of a double-decked metrotunnel to shield driving of twin closely underpasses tunnelsrdquoTunnelling and Underground Space Technology vol 28 no 1pp 18ndash30 2012
[10] T E Vorster A Klar K Soga and R J Mair ldquoEstimating theeffects of tunnelling on existing pipelinesrdquo Journal of Geo-technical and Geoenvironmental Engineering vol 131 no 11pp 1399ndash1410 2005
[11] Z G Zhang M S Huang and W D Wang ldquoEvaluation ofdeformation response for adjacent tunnels due to soilunloading in excavation engineeringrdquo Tunnelling and Un-derground Space Technology vol 38 no 3 pp 244ndash253 2013
[12] A P S Selvadurai Elastic Analysis of Soil-Foundation InteractionElsevier Scientific Pub Co Amsterdam Netherlands 1979
[13] N Loganathan and H G Poulos ldquoAnalytical prediction fortunnelling-induced ground movements in claysrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124no 9 pp 846ndash856 1998
[14] K M Lee R K Rowe and K Y Lo ldquoSubsidence owing totunnelling I Estimating the gap parameterrdquo CanadianGeotechnical Journal vol 29 no 6 pp 929ndash940 1992
[15] R K Rowe and KM Lee ldquoSubsidence owing to tunnelling IIEvaluation of a prediction techniquerdquo Canadian GeotechnicalJournal vol 29 no 6 pp 941ndash954 1992
[16] P B Attewell J Yeates and A R Selby Soil MovementsInduced by Tunnelling and their Effects on Pipelines andStructures Methuen Inc New York NY USA 1986
[17] F Ye C He H H Zhu et al ldquoAnalysis of longitudinalequivalent stiffness of shield tunnel considering lateral per-formancerdquo Chinese Journal of Geotechnical Engineeringvol 33 no 12 pp 1870ndash1876 2011 in Chinese
[18] P R Lancaster and D Mitchell ldquoProblems in Bending of Barsand Plates [M]rdquo Advanced Solid Mechanics Macmillan UK1980
[19] L Xu ldquoStudy on vertical settlement of shield tunnel in soft soilrdquoPhD dissertation Tongji University Shanghai China 2005
[20] M Yang and X Zhao ldquoAn approach for a single pile in layeredsoilrdquo Tongji University Natural Science vol 20 no 4pp 421ndash428 1992 in Chinese
16 Advances in Civil Engineering
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For the rst element
N1(x) K1(x) middotN1 01( ) x isin 0 L1( ) (23)
For the end point of the rst element
N1 x1( ) K1 x1 minus 0( ) middotN1 01( ) (24)
For the end point of the ith element
Ni xi( ) Ki xi( ) middotKiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( ) middotN1 01( )(25)
For x at the ith element section where x is a globalcoordinate
Ni(x)Ki xminus ximinus1( ) middotKiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( ) middotN1 01( )(26)
Suppose
Ai(x) Ki xminusximinus1( ) middot Kiminus1 Liminus1( ) middotKiminus2 Liminus2( ) middot middot middotK1 L1( )(27)
en (20) can be expressed as
Ni(x) Ai(x) middotN1 01( ) (28)
us the relation between the vectorNn(xn) at the pointxn and the vector N1(01) at the start point x0 is
Nn xn( ) An xn( ) middotN1 01( ) (29)
where
An xn( ) prod1
inKi Li( ) (30)
e boundary condition of the existing tunnel assumedas an innite long tunnel considering the shear stress of soilcan be written as
M(0) 0
V(0) 2
k
G
radic
middot G middot R middot w(0)
V(L) 0
V(L) minus2
k
G
radic
middot G middot R middot w(L)
(31)
By applying the boundary conditions to (19) the de-centection rotational angle bending moment and shear forceof the overall beam can be obtained
64 Case Study
641 Parameter Selection To enable a direct compari-son assumed parameters of the soil and existing tunnelare selected in this study Poissonrsquos ratio of the soil is0388 e grouting process is regarded as the inverseprocess of soil loss due to tunnelling and g can beexpressed as g (2Δ + δ)(1minus η) where η is the injectionratio
For the Pasternak foundation as the thickness of theelastic layer is assumed to be 25 B the subgrade elasticmodulus k and shear stiness G are suggested by Xu [19] asfollows
k 5E0β
16 1minus v05pt2( )times 122
G 13E0B
2
32 1 + v0( )βtimes 085
(32)
where β is a modied parameter (β 129) and E0 and v0 arerespectively the elastic modulus and Poissonrsquos ratio forelastic foundation which are given as follows
E0 Es
1minus v2s
v0 vs
1minus vs
(33)
where the empirical relation between the Youngrsquos modulusEs and the compression modulus Es01minus02 of soft soils is builtup by Yang and Zhao [20]
E0 (25 sim 35)Es01minus02 (34)
As illustrated in Figure 14 points O1 and O2 are re-spectively the intersection points A and B in practicalcoordinate transformation expressions of the axis xprime and xPrimeare respectively
xprime xminusL1( )sin αxPrime xminusL2( )sin α
(35)
where L1 and L2 are the distance from the beginningpoint of the existing tunnel to points O1 and O2 re-spectively e length of the existing tunnel is assumedas 250 m and the ring spacing is 12 m So the tunnel isdivided into 209 sections and each section is 12 m inlength
N1(01) Ni(xi)N1(x1) = N2(02) N2(x2)L1
L1
E1I1
G1 k1 G2 k2 G3 k3 Gi ki Gn knGnminus1 knminus1
E2I2 E3I3 EiIi EnInEnminus1Inminus1
L2 L3 Li LnLnminus1
L2 Li
x x x
xx1 x2 x3 xi-1 xnminus2 xnminus1 xnxi0
Ni(0i)
Figure 18 e matrix transfer diagram
10 Advances in Civil Engineering
7 Discussions
e normalized bending moment and shear force are de-ned as
Mi MiL0EiIi
Qi QiL
20
EiIi
(36)
where Mi and Qi are respectively the normalized bendingmoment and shear force Mi and Qi are respectively thecalculated bending moment and shear force and L0 is thelength of the existing tunnel
Figure 19 shows the comparison of the decentection ro-tational angle bending moment and shear force of theexisting tunnel with a constant stiness which is based onthree kinds of models an innite tunnel on the Winklerfoundation an innite tunnel on the Pasternak foundationand a nite tunnel on the Winkler foundation Note that the
368 380 400 420 440 460 480 500 520 540 560 580 600 612
minus0007
minus0006
minus0005
minus0004
minus0003
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)
minus0002
minus0001
0000
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundatione observed data on 201465
e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
minus000015
minus000010
minus000005
000000
000005
000010
000015
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
(b)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
e b
endi
ng m
omen
t of t
he ex
istin
g tu
nnel
(kN
middotm)
387 400 420 440 460 480 500 520 540 560 580 595
0
minus6000
minus5000
minus4000
minus3000
minus2000
minus1000
1000
2000
3000
4000
e ring number of the existing tunnel
(c)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
387 400 420 440 460 480 500 520 540 560 580 595
minus600
minus400
minus200
0
200
400
600
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(kN
)
(d)
Figure 19 (a) Comparison of the decentection of the existing tunnel using dierent models (b) Comparison of the rotational angle ofthe existing tunnel using dierent models (c) Comparison of the normalized bending moment of the existing tunnel using dierentmodels (d) Comparison of the normalized shear force of the existing tunnel using dierent models
Advances in Civil Engineering 11
existing tunnel is only subject to soil deformation force inthis case e displacement and the decentection of beam areparticular similar concepts owing to the fact that the existingtunnel is securely attached to the foundation generallywithout rigid body motion Fairly good agreement can beobserved between the innite tunnel on the Pasternakfoundation and the nite tunnel on the Pasternak founda-tion which shows that the method proposed in this paper iscorrect and can be used to analyze similar problems Becauseof considering soil shear stress vertical stress and de-formation are aected and weakened the displacement of
the existing tunnel on the Pasternak foundation is smallerthan that on the Winkler foundation without consideringthe shear stress of soil Using the Winkler foundation modelis in good agreement with the observed data It can be seenfrom (3) that the displacement of the existing tunnel isproportional to the grouting elaquociency (ie η) Due to theuncertainty of η it cannot be decided which one is the bestanalytical method It is clear that the maximum normalizedbending moment occurs at the points O1 and O2 which arethe nearest points between the existing tunnel and the newtunnels
minus0003
minus0002
minus0001
0000387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
minus000010
minus000005
000000
000005
000010
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e ring number of the existing tunnel
(b)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10
minus0003
minus0002
minus0001
0000
0001
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
xL
(c)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10xL
minus008
minus006
minus004
minus002
000
002
004
006
008
The n
orm
alise
d sh
ear f
orce
of t
he ex
istin
g tu
nnel
(d)
Figure 20 (a) e eects of dierent reinforcement stiness on the decentection of the existing tunnel (b) e eects of dierent re-inforcement stiness on the slope of the existing tunnel (c)e eects of dierent reinforcement stiness on the normalized moment of theexisting tunnel (d) e eects of dierent reinforcement stiness on the normalized shear force of the existing tunnel
12 Advances in Civil Engineering
e reinforce range of the existing tunnel is assumed as25 rings on either side of the intersection point A and thevarious stiness is taken into consideration using themethod proposed in this paper Figure 20 shows the in-centuence of dierent stiness of the reinforcement on theexisting tunnel (ie 4 EI EI and 14 EI) It is evident thatthe reinforcement can reduce the displacement rotationalangle normalized bending moment and shear force of theexisting tunnel while tunnel damage (reduce the stinessof reinforcement) has the opposite eect It is evident thatthe change of reinforcement stiness has greater eect onthe normalized bending moment and the normalized shearforce of the existing tunnel particularly at the incentection
points of the normalized bending moment and normalizedshear force curve
e reinforce stiness of the reinforcement range isassumed as 4 EI and the various reinforcement ranges istaken into consideration using the method proposed in thispaper Figure 21 shows the incentuence of dierent re-inforcement ranges on the existing tunnel when thenorthbound tunnel of Line 4 underpasses e normalizedbending moment on either side of the intersection point istoo large when the reinforcement range is 69ndash892m andthe normalized shear force on either side of the intersectionpoint is relatively too large when the reinforcement range isless than 592ndash992m So it is necessary to reinforce larger
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00035
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
The d
eflec
tions
of t
he ex
istin
g tu
nnel
(m)
xL
(a)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus000010
minus000005
000000
000005
000010
The s
lope
of t
he ex
istin
g tu
nnel
(rad
)
xL
(b)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
00005
00010
00015
xL
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
(c)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus010
minus008
minus006
minus004
minus002
000
002
004
006
008
010
xLTh
e nor
mal
ised
shea
r for
ce o
f the
exist
ing
tunn
el
(d)
Figure 21 (a)e eects of dierent reinforcement ranges on the decentection of the existing tunnel (b)e eects of dierent reinforcementranges on the slope of the existing tunnel (c)e eects of dierent reinforcement ranges on the normalized moment of the existing tunnel(d) e eects of dierent reinforcement ranges on the normalized shear force of the existing tunnel
Advances in Civil Engineering 13
than the range of 492ndash1092m in the existing tunnel in thiscase that is larger than 30m (25 rings) on either side of theintersection point )ere is no big difference in the bendingmoment and shear force of the existing tunnel when thereinforcement ranges are 492ndash1092m 392ndash1192m and292ndash1292m In order to reduce the costs of reinforcementreinforcement in the 30m range (ie 25 rings) on eitherside of the intersection point is the best choice whichverifies the actual design
8 Conclusions
Based on the abovementioned statements the response of theexisting tunnel is analyzed and analytical methods are proposedfor verification and the reinforcement design Main conclusionsderived from the analysis are as follows
(1) )e displacement of the existing tunnel changes withthe relative position of the shield and the existingtunnel Shield parameters have a major impact on theexisting tunnel )e heaves of the existing tunnelmight be caused by the large applied face pressure andbulkhead addictive thrust When the shield tail isdriving beyond the selected monitoring rings theselected monitoring rings settle rapidly due to theclosure of the shield tail void
(2) )e southbound tunnel of Line 4 has less effect onthe left side of the intersection point A on theexisting tunnel because of the northbound tunnelrsquosbarrier effect A variety of measures of displacementprevention of Line 1 such as the control of shield
parameters reinforcement of the existing tunnel canensure the normal operation of metro Line 1
(3) )e analytical method proposed in the paper is ver-ified by the methods derived from prior studies It ispossible to take the different stiffness and curved loadsinto account by applying the stepped stiffness andtrapezoidal load while using a local coordinate systemin the derivation of the new method )e change ofreinforcement stiffness has greater effect on thenormalized bending moment and the normalizedshear force of the existing tunnel particularly at theinflection points of the normalized bending momentand normalized shear force curve )e analyticalmethod can be applied for the reinforcement designand safety check of the built tunnel
Appendices
A1 An Infinite Beam on the WinklerFoundation
θ(x) 1113946+infin
minusinfin
minusλ2s((tminus L)sin α)eminusλ(xminust) sin(λ(xminus t))
b
λ2s((tminus L)sin α)eλ(xminust)sin(λ(tminus x))
b
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎭
dt
(A1)
M(x) 14λ
1113946+infin
minusinfinks((tminusL)sin α)e
minusλ|xminust|( cos(λ|xminus t|)minus sin(λ|xminus t|))dt (A2)
Q(x) 1113946+infin
minusinfin
minus12
ks((tminusL)sin α)eminusλ(xminust)cos(λ(xminus t))dt xminus tgt 0
12
ks((tminusL)sin α)eminusλ(tminusx)cos(λ(tminusx))dt xminus tle 0
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
⎫⎪⎪⎪⎬
⎪⎪⎪⎭
dt (A3)
A2 An Infinite Beam on the PasternakFoundation
w(x) λ2bk
1113946+infin
minusinfinks((tminus L)sin α)De
minusa1λ|xminusτ| 1a1
cos a2λ|xminus τ|( 1113857 +1a2
sin a2λ|xminus τ|( 11138571113888 1113889dτ
θ(x) 1113946+infin
minusinfin
minusλ2s((tminusL))eminusa1λ(xminust)sin a2(xminus t)( 1113857
a1a2b
λ2s((tminus L))ea1λ(xminust)sin a2(tminus x)( 1113857
a1a2b
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A4)
14 Advances in Civil Engineering
M(x) 14λ
1113946+infin
minusinfinks((tminus L)sin α)e
minusa1λ|xminust| 1a1
cos a2λ|xminus t|( 1113857minus1a2
sin a2λ|xminus t|( 11138571113888 1113889dt
Q(x) 1113946+infin
minusinfin
minus12
ks((tminus L)sin α)eminusa1λ(xminust) cos a2λ(xminus t)( 1113857minus
a21 minus a 2
22a1a2
1113888 1113889sin a2λ(xminus t)( 11138571113890 1113891dt xminus tgt 0
12
ks((tminusL)sin α)eminusa1λ(tminusx) cos a2λ(tminusx)( 1113857minus
a21 minus a2
2
2a1a21113888 1113889sin a2λ(tminusx)( 11138571113890 1113891dt xminus tle 0
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A5)
B1 Coefficients of the Matrix ki5
k15 minusbiqiminus1
4a1ia2ici4EiIi a1i
2 + a2i2( 1113857
minus2a1ia2i + 2a1ia2iF2i
+ a1i2 minus a2i
2( 1113857F4i
⎛⎜⎝ ⎞⎟⎠
+bi qi minus qiminus1( 1113857
4a1ia2ici5EiIiL3 a1i
2 + a2i2( 1113857
2
a2i minus3a21i + a2
2i( 1113857F3i
+a1i 2a2i a21i + a2
2i( 1113857cix + a1i2 minus 3a2i
2( 1113857F1i( 1113857
⎛⎜⎝ ⎞⎟⎠
(B1)
k25 biqiminus1 a2iF3i minus a1iF1i( 1113857
2a1a2 a21 + a 2
2( 1113857c3EiIi
+bi qi minus qiminus1( 1113857 2a1ia2i minus 2a1ia2iF2i minus a1i
2 minus a2i2( 1113857F4i( 1113857( 1113857
2a1ia2i a1i2 + a2i
2( 11138572ci
4EiIix1113872 1113873 (B2)
k35 minusbiqiminus1F4i
2a1ia2ici2 minus
bi qi minus qiminus1( 1113857 a2iF3i minus a1iF1i( 1113857
2a1ia2ici3 a1i
2 + a2i2( 1113857x
(B3)
k45 minusbiqiminus1
a1i2 + a2i
2( 1113857ci
1113888a1i 1 + 2a1i2a22i minus 2a2i
41113872 1113873F3i + a2i 1 + 2a1i
2a22i minus 2a1i
41113872 1113873F1i1113889
minusbi qi minus qiminus1( 1113857
a21i + a2i
2( 11138572c 2
i x
4a1i2a2i
2 minus 1( 1113857 a1i2 minus a2i
2( 1113857 + a2i2 minus 4a1i
4a2i2 + a1i
2 minus1 + 4a2i4( 1113857( 1113857F2i
minus2a1ia2i a 21i minus a 2
2i( 11138572 minus 11113872 1113873F4i
⎛⎜⎝ ⎞⎟⎠
(B4)
Conflicts of Interest
)e author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
)e author appreciates the help from the staff of Hangzhouand Hong Kong Tunnel Company Ltd during in-strumentation setup and data acquisition )is study issupported by the National Natural Science Foundation ofChina (no 41702313)
References
[1] S H Kim Model testing and analysis of interactions betweentunnels in clay [PhD 3esis] Department of EngineeringScience University of Oxford Oxford UK 1996
[2] H Y Liu J C Small J P Carter and D J Williams ldquoEffectsof tunnelling on existing support systems of perpendicularlypassing tunnelsrdquo Computers and Geotechnics vol 36 no 5pp 880ndash894 2009
[3] I Yamaguchi I Yamazaki and Y Kiritani ldquoStudy ofgroundndashtunnel interactions of four shield tunnels driven inclose proximity in relation to design and construction ofparallel shield tunnelsrdquo Tunnelling and Underground SpaceTechnology vol 13 no 3 pp 289ndash304 1998
[4] T I Addenbrooke and D M Potts ldquoTwin tunnel interactionsurface and subsurface effectsrdquo International Journal ofGeomechanics vol 1 no 2 pp 249ndash271 2001
[5] J P Kimmance S Lawrence O Hassan and N J PurchaseldquoObservations of deformations created in existing tunnels byadjacent and cross cutting excavationsrdquo in Proceedings ofGeotechnical Aspects of Underground Construction in SoftGround pp 707ndash712 London UK July 1996
[6] E Leca and B New ldquoSettlements induced by tunnelling in softgroundrdquo Tunnelling and Underground Space Technologyvol 22 no 2 pp 119ndash149 2007
Advances in Civil Engineering 15
[7] S M Liao J H Liu R L Wang and Z M Li ldquoShieldtunnelling and environment protection in Shanghai softgroundrdquo Tunnelling and Underground Space Technologyvol 24 no 4 pp 454ndash465 2009
[8] F H Chehade and I Shahrour ldquoNumerical analysis of theinteraction between twin-tunnels influence of the relativeposition and construction procedurerdquo Tunnelling and Un-derground Space Technology vol 23 no 2 pp 210ndash214 2008
[9] X G Li and D J Yuan ldquoResponse of a double-decked metrotunnel to shield driving of twin closely underpasses tunnelsrdquoTunnelling and Underground Space Technology vol 28 no 1pp 18ndash30 2012
[10] T E Vorster A Klar K Soga and R J Mair ldquoEstimating theeffects of tunnelling on existing pipelinesrdquo Journal of Geo-technical and Geoenvironmental Engineering vol 131 no 11pp 1399ndash1410 2005
[11] Z G Zhang M S Huang and W D Wang ldquoEvaluation ofdeformation response for adjacent tunnels due to soilunloading in excavation engineeringrdquo Tunnelling and Un-derground Space Technology vol 38 no 3 pp 244ndash253 2013
[12] A P S Selvadurai Elastic Analysis of Soil-Foundation InteractionElsevier Scientific Pub Co Amsterdam Netherlands 1979
[13] N Loganathan and H G Poulos ldquoAnalytical prediction fortunnelling-induced ground movements in claysrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124no 9 pp 846ndash856 1998
[14] K M Lee R K Rowe and K Y Lo ldquoSubsidence owing totunnelling I Estimating the gap parameterrdquo CanadianGeotechnical Journal vol 29 no 6 pp 929ndash940 1992
[15] R K Rowe and KM Lee ldquoSubsidence owing to tunnelling IIEvaluation of a prediction techniquerdquo Canadian GeotechnicalJournal vol 29 no 6 pp 941ndash954 1992
[16] P B Attewell J Yeates and A R Selby Soil MovementsInduced by Tunnelling and their Effects on Pipelines andStructures Methuen Inc New York NY USA 1986
[17] F Ye C He H H Zhu et al ldquoAnalysis of longitudinalequivalent stiffness of shield tunnel considering lateral per-formancerdquo Chinese Journal of Geotechnical Engineeringvol 33 no 12 pp 1870ndash1876 2011 in Chinese
[18] P R Lancaster and D Mitchell ldquoProblems in Bending of Barsand Plates [M]rdquo Advanced Solid Mechanics Macmillan UK1980
[19] L Xu ldquoStudy on vertical settlement of shield tunnel in soft soilrdquoPhD dissertation Tongji University Shanghai China 2005
[20] M Yang and X Zhao ldquoAn approach for a single pile in layeredsoilrdquo Tongji University Natural Science vol 20 no 4pp 421ndash428 1992 in Chinese
16 Advances in Civil Engineering
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7 Discussions
e normalized bending moment and shear force are de-ned as
Mi MiL0EiIi
Qi QiL
20
EiIi
(36)
where Mi and Qi are respectively the normalized bendingmoment and shear force Mi and Qi are respectively thecalculated bending moment and shear force and L0 is thelength of the existing tunnel
Figure 19 shows the comparison of the decentection ro-tational angle bending moment and shear force of theexisting tunnel with a constant stiness which is based onthree kinds of models an innite tunnel on the Winklerfoundation an innite tunnel on the Pasternak foundationand a nite tunnel on the Winkler foundation Note that the
368 380 400 420 440 460 480 500 520 540 560 580 600 612
minus0007
minus0006
minus0005
minus0004
minus0003
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)
minus0002
minus0001
0000
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundatione observed data on 201465
e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
minus000015
minus000010
minus000005
000000
000005
000010
000015
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
(b)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
e b
endi
ng m
omen
t of t
he ex
istin
g tu
nnel
(kN
middotm)
387 400 420 440 460 480 500 520 540 560 580 595
0
minus6000
minus5000
minus4000
minus3000
minus2000
minus1000
1000
2000
3000
4000
e ring number of the existing tunnel
(c)
e infinite exisiting tunnel on the winkler foundatione infinite exisiting tunnel on the pasternak foundatione finite existing tunnel on the pasternak foundation
387 400 420 440 460 480 500 520 540 560 580 595
minus600
minus400
minus200
0
200
400
600
e ring number of the existing tunnel
e s
lope
of t
he ex
istin
g tu
nnel
(kN
)
(d)
Figure 19 (a) Comparison of the decentection of the existing tunnel using dierent models (b) Comparison of the rotational angle ofthe existing tunnel using dierent models (c) Comparison of the normalized bending moment of the existing tunnel using dierentmodels (d) Comparison of the normalized shear force of the existing tunnel using dierent models
Advances in Civil Engineering 11
existing tunnel is only subject to soil deformation force inthis case e displacement and the decentection of beam areparticular similar concepts owing to the fact that the existingtunnel is securely attached to the foundation generallywithout rigid body motion Fairly good agreement can beobserved between the innite tunnel on the Pasternakfoundation and the nite tunnel on the Pasternak founda-tion which shows that the method proposed in this paper iscorrect and can be used to analyze similar problems Becauseof considering soil shear stress vertical stress and de-formation are aected and weakened the displacement of
the existing tunnel on the Pasternak foundation is smallerthan that on the Winkler foundation without consideringthe shear stress of soil Using the Winkler foundation modelis in good agreement with the observed data It can be seenfrom (3) that the displacement of the existing tunnel isproportional to the grouting elaquociency (ie η) Due to theuncertainty of η it cannot be decided which one is the bestanalytical method It is clear that the maximum normalizedbending moment occurs at the points O1 and O2 which arethe nearest points between the existing tunnel and the newtunnels
minus0003
minus0002
minus0001
0000387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
minus000010
minus000005
000000
000005
000010
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e ring number of the existing tunnel
(b)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10
minus0003
minus0002
minus0001
0000
0001
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
xL
(c)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10xL
minus008
minus006
minus004
minus002
000
002
004
006
008
The n
orm
alise
d sh
ear f
orce
of t
he ex
istin
g tu
nnel
(d)
Figure 20 (a) e eects of dierent reinforcement stiness on the decentection of the existing tunnel (b) e eects of dierent re-inforcement stiness on the slope of the existing tunnel (c)e eects of dierent reinforcement stiness on the normalized moment of theexisting tunnel (d) e eects of dierent reinforcement stiness on the normalized shear force of the existing tunnel
12 Advances in Civil Engineering
e reinforce range of the existing tunnel is assumed as25 rings on either side of the intersection point A and thevarious stiness is taken into consideration using themethod proposed in this paper Figure 20 shows the in-centuence of dierent stiness of the reinforcement on theexisting tunnel (ie 4 EI EI and 14 EI) It is evident thatthe reinforcement can reduce the displacement rotationalangle normalized bending moment and shear force of theexisting tunnel while tunnel damage (reduce the stinessof reinforcement) has the opposite eect It is evident thatthe change of reinforcement stiness has greater eect onthe normalized bending moment and the normalized shearforce of the existing tunnel particularly at the incentection
points of the normalized bending moment and normalizedshear force curve
e reinforce stiness of the reinforcement range isassumed as 4 EI and the various reinforcement ranges istaken into consideration using the method proposed in thispaper Figure 21 shows the incentuence of dierent re-inforcement ranges on the existing tunnel when thenorthbound tunnel of Line 4 underpasses e normalizedbending moment on either side of the intersection point istoo large when the reinforcement range is 69ndash892m andthe normalized shear force on either side of the intersectionpoint is relatively too large when the reinforcement range isless than 592ndash992m So it is necessary to reinforce larger
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00035
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
The d
eflec
tions
of t
he ex
istin
g tu
nnel
(m)
xL
(a)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus000010
minus000005
000000
000005
000010
The s
lope
of t
he ex
istin
g tu
nnel
(rad
)
xL
(b)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
00005
00010
00015
xL
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
(c)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus010
minus008
minus006
minus004
minus002
000
002
004
006
008
010
xLTh
e nor
mal
ised
shea
r for
ce o
f the
exist
ing
tunn
el
(d)
Figure 21 (a)e eects of dierent reinforcement ranges on the decentection of the existing tunnel (b)e eects of dierent reinforcementranges on the slope of the existing tunnel (c)e eects of dierent reinforcement ranges on the normalized moment of the existing tunnel(d) e eects of dierent reinforcement ranges on the normalized shear force of the existing tunnel
Advances in Civil Engineering 13
than the range of 492ndash1092m in the existing tunnel in thiscase that is larger than 30m (25 rings) on either side of theintersection point )ere is no big difference in the bendingmoment and shear force of the existing tunnel when thereinforcement ranges are 492ndash1092m 392ndash1192m and292ndash1292m In order to reduce the costs of reinforcementreinforcement in the 30m range (ie 25 rings) on eitherside of the intersection point is the best choice whichverifies the actual design
8 Conclusions
Based on the abovementioned statements the response of theexisting tunnel is analyzed and analytical methods are proposedfor verification and the reinforcement design Main conclusionsderived from the analysis are as follows
(1) )e displacement of the existing tunnel changes withthe relative position of the shield and the existingtunnel Shield parameters have a major impact on theexisting tunnel )e heaves of the existing tunnelmight be caused by the large applied face pressure andbulkhead addictive thrust When the shield tail isdriving beyond the selected monitoring rings theselected monitoring rings settle rapidly due to theclosure of the shield tail void
(2) )e southbound tunnel of Line 4 has less effect onthe left side of the intersection point A on theexisting tunnel because of the northbound tunnelrsquosbarrier effect A variety of measures of displacementprevention of Line 1 such as the control of shield
parameters reinforcement of the existing tunnel canensure the normal operation of metro Line 1
(3) )e analytical method proposed in the paper is ver-ified by the methods derived from prior studies It ispossible to take the different stiffness and curved loadsinto account by applying the stepped stiffness andtrapezoidal load while using a local coordinate systemin the derivation of the new method )e change ofreinforcement stiffness has greater effect on thenormalized bending moment and the normalizedshear force of the existing tunnel particularly at theinflection points of the normalized bending momentand normalized shear force curve )e analyticalmethod can be applied for the reinforcement designand safety check of the built tunnel
Appendices
A1 An Infinite Beam on the WinklerFoundation
θ(x) 1113946+infin
minusinfin
minusλ2s((tminus L)sin α)eminusλ(xminust) sin(λ(xminus t))
b
λ2s((tminus L)sin α)eλ(xminust)sin(λ(tminus x))
b
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎭
dt
(A1)
M(x) 14λ
1113946+infin
minusinfinks((tminusL)sin α)e
minusλ|xminust|( cos(λ|xminus t|)minus sin(λ|xminus t|))dt (A2)
Q(x) 1113946+infin
minusinfin
minus12
ks((tminusL)sin α)eminusλ(xminust)cos(λ(xminus t))dt xminus tgt 0
12
ks((tminusL)sin α)eminusλ(tminusx)cos(λ(tminusx))dt xminus tle 0
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
⎫⎪⎪⎪⎬
⎪⎪⎪⎭
dt (A3)
A2 An Infinite Beam on the PasternakFoundation
w(x) λ2bk
1113946+infin
minusinfinks((tminus L)sin α)De
minusa1λ|xminusτ| 1a1
cos a2λ|xminus τ|( 1113857 +1a2
sin a2λ|xminus τ|( 11138571113888 1113889dτ
θ(x) 1113946+infin
minusinfin
minusλ2s((tminusL))eminusa1λ(xminust)sin a2(xminus t)( 1113857
a1a2b
λ2s((tminus L))ea1λ(xminust)sin a2(tminus x)( 1113857
a1a2b
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A4)
14 Advances in Civil Engineering
M(x) 14λ
1113946+infin
minusinfinks((tminus L)sin α)e
minusa1λ|xminust| 1a1
cos a2λ|xminus t|( 1113857minus1a2
sin a2λ|xminus t|( 11138571113888 1113889dt
Q(x) 1113946+infin
minusinfin
minus12
ks((tminus L)sin α)eminusa1λ(xminust) cos a2λ(xminus t)( 1113857minus
a21 minus a 2
22a1a2
1113888 1113889sin a2λ(xminus t)( 11138571113890 1113891dt xminus tgt 0
12
ks((tminusL)sin α)eminusa1λ(tminusx) cos a2λ(tminusx)( 1113857minus
a21 minus a2
2
2a1a21113888 1113889sin a2λ(tminusx)( 11138571113890 1113891dt xminus tle 0
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A5)
B1 Coefficients of the Matrix ki5
k15 minusbiqiminus1
4a1ia2ici4EiIi a1i
2 + a2i2( 1113857
minus2a1ia2i + 2a1ia2iF2i
+ a1i2 minus a2i
2( 1113857F4i
⎛⎜⎝ ⎞⎟⎠
+bi qi minus qiminus1( 1113857
4a1ia2ici5EiIiL3 a1i
2 + a2i2( 1113857
2
a2i minus3a21i + a2
2i( 1113857F3i
+a1i 2a2i a21i + a2
2i( 1113857cix + a1i2 minus 3a2i
2( 1113857F1i( 1113857
⎛⎜⎝ ⎞⎟⎠
(B1)
k25 biqiminus1 a2iF3i minus a1iF1i( 1113857
2a1a2 a21 + a 2
2( 1113857c3EiIi
+bi qi minus qiminus1( 1113857 2a1ia2i minus 2a1ia2iF2i minus a1i
2 minus a2i2( 1113857F4i( 1113857( 1113857
2a1ia2i a1i2 + a2i
2( 11138572ci
4EiIix1113872 1113873 (B2)
k35 minusbiqiminus1F4i
2a1ia2ici2 minus
bi qi minus qiminus1( 1113857 a2iF3i minus a1iF1i( 1113857
2a1ia2ici3 a1i
2 + a2i2( 1113857x
(B3)
k45 minusbiqiminus1
a1i2 + a2i
2( 1113857ci
1113888a1i 1 + 2a1i2a22i minus 2a2i
41113872 1113873F3i + a2i 1 + 2a1i
2a22i minus 2a1i
41113872 1113873F1i1113889
minusbi qi minus qiminus1( 1113857
a21i + a2i
2( 11138572c 2
i x
4a1i2a2i
2 minus 1( 1113857 a1i2 minus a2i
2( 1113857 + a2i2 minus 4a1i
4a2i2 + a1i
2 minus1 + 4a2i4( 1113857( 1113857F2i
minus2a1ia2i a 21i minus a 2
2i( 11138572 minus 11113872 1113873F4i
⎛⎜⎝ ⎞⎟⎠
(B4)
Conflicts of Interest
)e author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
)e author appreciates the help from the staff of Hangzhouand Hong Kong Tunnel Company Ltd during in-strumentation setup and data acquisition )is study issupported by the National Natural Science Foundation ofChina (no 41702313)
References
[1] S H Kim Model testing and analysis of interactions betweentunnels in clay [PhD 3esis] Department of EngineeringScience University of Oxford Oxford UK 1996
[2] H Y Liu J C Small J P Carter and D J Williams ldquoEffectsof tunnelling on existing support systems of perpendicularlypassing tunnelsrdquo Computers and Geotechnics vol 36 no 5pp 880ndash894 2009
[3] I Yamaguchi I Yamazaki and Y Kiritani ldquoStudy ofgroundndashtunnel interactions of four shield tunnels driven inclose proximity in relation to design and construction ofparallel shield tunnelsrdquo Tunnelling and Underground SpaceTechnology vol 13 no 3 pp 289ndash304 1998
[4] T I Addenbrooke and D M Potts ldquoTwin tunnel interactionsurface and subsurface effectsrdquo International Journal ofGeomechanics vol 1 no 2 pp 249ndash271 2001
[5] J P Kimmance S Lawrence O Hassan and N J PurchaseldquoObservations of deformations created in existing tunnels byadjacent and cross cutting excavationsrdquo in Proceedings ofGeotechnical Aspects of Underground Construction in SoftGround pp 707ndash712 London UK July 1996
[6] E Leca and B New ldquoSettlements induced by tunnelling in softgroundrdquo Tunnelling and Underground Space Technologyvol 22 no 2 pp 119ndash149 2007
Advances in Civil Engineering 15
[7] S M Liao J H Liu R L Wang and Z M Li ldquoShieldtunnelling and environment protection in Shanghai softgroundrdquo Tunnelling and Underground Space Technologyvol 24 no 4 pp 454ndash465 2009
[8] F H Chehade and I Shahrour ldquoNumerical analysis of theinteraction between twin-tunnels influence of the relativeposition and construction procedurerdquo Tunnelling and Un-derground Space Technology vol 23 no 2 pp 210ndash214 2008
[9] X G Li and D J Yuan ldquoResponse of a double-decked metrotunnel to shield driving of twin closely underpasses tunnelsrdquoTunnelling and Underground Space Technology vol 28 no 1pp 18ndash30 2012
[10] T E Vorster A Klar K Soga and R J Mair ldquoEstimating theeffects of tunnelling on existing pipelinesrdquo Journal of Geo-technical and Geoenvironmental Engineering vol 131 no 11pp 1399ndash1410 2005
[11] Z G Zhang M S Huang and W D Wang ldquoEvaluation ofdeformation response for adjacent tunnels due to soilunloading in excavation engineeringrdquo Tunnelling and Un-derground Space Technology vol 38 no 3 pp 244ndash253 2013
[12] A P S Selvadurai Elastic Analysis of Soil-Foundation InteractionElsevier Scientific Pub Co Amsterdam Netherlands 1979
[13] N Loganathan and H G Poulos ldquoAnalytical prediction fortunnelling-induced ground movements in claysrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124no 9 pp 846ndash856 1998
[14] K M Lee R K Rowe and K Y Lo ldquoSubsidence owing totunnelling I Estimating the gap parameterrdquo CanadianGeotechnical Journal vol 29 no 6 pp 929ndash940 1992
[15] R K Rowe and KM Lee ldquoSubsidence owing to tunnelling IIEvaluation of a prediction techniquerdquo Canadian GeotechnicalJournal vol 29 no 6 pp 941ndash954 1992
[16] P B Attewell J Yeates and A R Selby Soil MovementsInduced by Tunnelling and their Effects on Pipelines andStructures Methuen Inc New York NY USA 1986
[17] F Ye C He H H Zhu et al ldquoAnalysis of longitudinalequivalent stiffness of shield tunnel considering lateral per-formancerdquo Chinese Journal of Geotechnical Engineeringvol 33 no 12 pp 1870ndash1876 2011 in Chinese
[18] P R Lancaster and D Mitchell ldquoProblems in Bending of Barsand Plates [M]rdquo Advanced Solid Mechanics Macmillan UK1980
[19] L Xu ldquoStudy on vertical settlement of shield tunnel in soft soilrdquoPhD dissertation Tongji University Shanghai China 2005
[20] M Yang and X Zhao ldquoAn approach for a single pile in layeredsoilrdquo Tongji University Natural Science vol 20 no 4pp 421ndash428 1992 in Chinese
16 Advances in Civil Engineering
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
existing tunnel is only subject to soil deformation force inthis case e displacement and the decentection of beam areparticular similar concepts owing to the fact that the existingtunnel is securely attached to the foundation generallywithout rigid body motion Fairly good agreement can beobserved between the innite tunnel on the Pasternakfoundation and the nite tunnel on the Pasternak founda-tion which shows that the method proposed in this paper iscorrect and can be used to analyze similar problems Becauseof considering soil shear stress vertical stress and de-formation are aected and weakened the displacement of
the existing tunnel on the Pasternak foundation is smallerthan that on the Winkler foundation without consideringthe shear stress of soil Using the Winkler foundation modelis in good agreement with the observed data It can be seenfrom (3) that the displacement of the existing tunnel isproportional to the grouting elaquociency (ie η) Due to theuncertainty of η it cannot be decided which one is the bestanalytical method It is clear that the maximum normalizedbending moment occurs at the points O1 and O2 which arethe nearest points between the existing tunnel and the newtunnels
minus0003
minus0002
minus0001
0000387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
e d
eflec
tion
of th
e exi
sting
tunn
el (m
)e ring number of the existing tunnel
(a)
387 400 420 440 460 480 500 520 540 560 580 595
4EI14EI
EI
minus000010
minus000005
000000
000005
000010
e s
lope
of t
he ex
istin
g tu
nnel
(rad
)
e ring number of the existing tunnel
(b)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10
minus0003
minus0002
minus0001
0000
0001
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
xL
(c)
4EI14EI
EI
00 01 02 03 04 05 06 07 08 09 10xL
minus008
minus006
minus004
minus002
000
002
004
006
008
The n
orm
alise
d sh
ear f
orce
of t
he ex
istin
g tu
nnel
(d)
Figure 20 (a) e eects of dierent reinforcement stiness on the decentection of the existing tunnel (b) e eects of dierent re-inforcement stiness on the slope of the existing tunnel (c)e eects of dierent reinforcement stiness on the normalized moment of theexisting tunnel (d) e eects of dierent reinforcement stiness on the normalized shear force of the existing tunnel
12 Advances in Civil Engineering
e reinforce range of the existing tunnel is assumed as25 rings on either side of the intersection point A and thevarious stiness is taken into consideration using themethod proposed in this paper Figure 20 shows the in-centuence of dierent stiness of the reinforcement on theexisting tunnel (ie 4 EI EI and 14 EI) It is evident thatthe reinforcement can reduce the displacement rotationalangle normalized bending moment and shear force of theexisting tunnel while tunnel damage (reduce the stinessof reinforcement) has the opposite eect It is evident thatthe change of reinforcement stiness has greater eect onthe normalized bending moment and the normalized shearforce of the existing tunnel particularly at the incentection
points of the normalized bending moment and normalizedshear force curve
e reinforce stiness of the reinforcement range isassumed as 4 EI and the various reinforcement ranges istaken into consideration using the method proposed in thispaper Figure 21 shows the incentuence of dierent re-inforcement ranges on the existing tunnel when thenorthbound tunnel of Line 4 underpasses e normalizedbending moment on either side of the intersection point istoo large when the reinforcement range is 69ndash892m andthe normalized shear force on either side of the intersectionpoint is relatively too large when the reinforcement range isless than 592ndash992m So it is necessary to reinforce larger
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00035
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
The d
eflec
tions
of t
he ex
istin
g tu
nnel
(m)
xL
(a)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus000010
minus000005
000000
000005
000010
The s
lope
of t
he ex
istin
g tu
nnel
(rad
)
xL
(b)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
00005
00010
00015
xL
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
(c)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus010
minus008
minus006
minus004
minus002
000
002
004
006
008
010
xLTh
e nor
mal
ised
shea
r for
ce o
f the
exist
ing
tunn
el
(d)
Figure 21 (a)e eects of dierent reinforcement ranges on the decentection of the existing tunnel (b)e eects of dierent reinforcementranges on the slope of the existing tunnel (c)e eects of dierent reinforcement ranges on the normalized moment of the existing tunnel(d) e eects of dierent reinforcement ranges on the normalized shear force of the existing tunnel
Advances in Civil Engineering 13
than the range of 492ndash1092m in the existing tunnel in thiscase that is larger than 30m (25 rings) on either side of theintersection point )ere is no big difference in the bendingmoment and shear force of the existing tunnel when thereinforcement ranges are 492ndash1092m 392ndash1192m and292ndash1292m In order to reduce the costs of reinforcementreinforcement in the 30m range (ie 25 rings) on eitherside of the intersection point is the best choice whichverifies the actual design
8 Conclusions
Based on the abovementioned statements the response of theexisting tunnel is analyzed and analytical methods are proposedfor verification and the reinforcement design Main conclusionsderived from the analysis are as follows
(1) )e displacement of the existing tunnel changes withthe relative position of the shield and the existingtunnel Shield parameters have a major impact on theexisting tunnel )e heaves of the existing tunnelmight be caused by the large applied face pressure andbulkhead addictive thrust When the shield tail isdriving beyond the selected monitoring rings theselected monitoring rings settle rapidly due to theclosure of the shield tail void
(2) )e southbound tunnel of Line 4 has less effect onthe left side of the intersection point A on theexisting tunnel because of the northbound tunnelrsquosbarrier effect A variety of measures of displacementprevention of Line 1 such as the control of shield
parameters reinforcement of the existing tunnel canensure the normal operation of metro Line 1
(3) )e analytical method proposed in the paper is ver-ified by the methods derived from prior studies It ispossible to take the different stiffness and curved loadsinto account by applying the stepped stiffness andtrapezoidal load while using a local coordinate systemin the derivation of the new method )e change ofreinforcement stiffness has greater effect on thenormalized bending moment and the normalizedshear force of the existing tunnel particularly at theinflection points of the normalized bending momentand normalized shear force curve )e analyticalmethod can be applied for the reinforcement designand safety check of the built tunnel
Appendices
A1 An Infinite Beam on the WinklerFoundation
θ(x) 1113946+infin
minusinfin
minusλ2s((tminus L)sin α)eminusλ(xminust) sin(λ(xminus t))
b
λ2s((tminus L)sin α)eλ(xminust)sin(λ(tminus x))
b
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎭
dt
(A1)
M(x) 14λ
1113946+infin
minusinfinks((tminusL)sin α)e
minusλ|xminust|( cos(λ|xminus t|)minus sin(λ|xminus t|))dt (A2)
Q(x) 1113946+infin
minusinfin
minus12
ks((tminusL)sin α)eminusλ(xminust)cos(λ(xminus t))dt xminus tgt 0
12
ks((tminusL)sin α)eminusλ(tminusx)cos(λ(tminusx))dt xminus tle 0
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
⎫⎪⎪⎪⎬
⎪⎪⎪⎭
dt (A3)
A2 An Infinite Beam on the PasternakFoundation
w(x) λ2bk
1113946+infin
minusinfinks((tminus L)sin α)De
minusa1λ|xminusτ| 1a1
cos a2λ|xminus τ|( 1113857 +1a2
sin a2λ|xminus τ|( 11138571113888 1113889dτ
θ(x) 1113946+infin
minusinfin
minusλ2s((tminusL))eminusa1λ(xminust)sin a2(xminus t)( 1113857
a1a2b
λ2s((tminus L))ea1λ(xminust)sin a2(tminus x)( 1113857
a1a2b
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A4)
14 Advances in Civil Engineering
M(x) 14λ
1113946+infin
minusinfinks((tminus L)sin α)e
minusa1λ|xminust| 1a1
cos a2λ|xminus t|( 1113857minus1a2
sin a2λ|xminus t|( 11138571113888 1113889dt
Q(x) 1113946+infin
minusinfin
minus12
ks((tminus L)sin α)eminusa1λ(xminust) cos a2λ(xminus t)( 1113857minus
a21 minus a 2
22a1a2
1113888 1113889sin a2λ(xminus t)( 11138571113890 1113891dt xminus tgt 0
12
ks((tminusL)sin α)eminusa1λ(tminusx) cos a2λ(tminusx)( 1113857minus
a21 minus a2
2
2a1a21113888 1113889sin a2λ(tminusx)( 11138571113890 1113891dt xminus tle 0
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A5)
B1 Coefficients of the Matrix ki5
k15 minusbiqiminus1
4a1ia2ici4EiIi a1i
2 + a2i2( 1113857
minus2a1ia2i + 2a1ia2iF2i
+ a1i2 minus a2i
2( 1113857F4i
⎛⎜⎝ ⎞⎟⎠
+bi qi minus qiminus1( 1113857
4a1ia2ici5EiIiL3 a1i
2 + a2i2( 1113857
2
a2i minus3a21i + a2
2i( 1113857F3i
+a1i 2a2i a21i + a2
2i( 1113857cix + a1i2 minus 3a2i
2( 1113857F1i( 1113857
⎛⎜⎝ ⎞⎟⎠
(B1)
k25 biqiminus1 a2iF3i minus a1iF1i( 1113857
2a1a2 a21 + a 2
2( 1113857c3EiIi
+bi qi minus qiminus1( 1113857 2a1ia2i minus 2a1ia2iF2i minus a1i
2 minus a2i2( 1113857F4i( 1113857( 1113857
2a1ia2i a1i2 + a2i
2( 11138572ci
4EiIix1113872 1113873 (B2)
k35 minusbiqiminus1F4i
2a1ia2ici2 minus
bi qi minus qiminus1( 1113857 a2iF3i minus a1iF1i( 1113857
2a1ia2ici3 a1i
2 + a2i2( 1113857x
(B3)
k45 minusbiqiminus1
a1i2 + a2i
2( 1113857ci
1113888a1i 1 + 2a1i2a22i minus 2a2i
41113872 1113873F3i + a2i 1 + 2a1i
2a22i minus 2a1i
41113872 1113873F1i1113889
minusbi qi minus qiminus1( 1113857
a21i + a2i
2( 11138572c 2
i x
4a1i2a2i
2 minus 1( 1113857 a1i2 minus a2i
2( 1113857 + a2i2 minus 4a1i
4a2i2 + a1i
2 minus1 + 4a2i4( 1113857( 1113857F2i
minus2a1ia2i a 21i minus a 2
2i( 11138572 minus 11113872 1113873F4i
⎛⎜⎝ ⎞⎟⎠
(B4)
Conflicts of Interest
)e author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
)e author appreciates the help from the staff of Hangzhouand Hong Kong Tunnel Company Ltd during in-strumentation setup and data acquisition )is study issupported by the National Natural Science Foundation ofChina (no 41702313)
References
[1] S H Kim Model testing and analysis of interactions betweentunnels in clay [PhD 3esis] Department of EngineeringScience University of Oxford Oxford UK 1996
[2] H Y Liu J C Small J P Carter and D J Williams ldquoEffectsof tunnelling on existing support systems of perpendicularlypassing tunnelsrdquo Computers and Geotechnics vol 36 no 5pp 880ndash894 2009
[3] I Yamaguchi I Yamazaki and Y Kiritani ldquoStudy ofgroundndashtunnel interactions of four shield tunnels driven inclose proximity in relation to design and construction ofparallel shield tunnelsrdquo Tunnelling and Underground SpaceTechnology vol 13 no 3 pp 289ndash304 1998
[4] T I Addenbrooke and D M Potts ldquoTwin tunnel interactionsurface and subsurface effectsrdquo International Journal ofGeomechanics vol 1 no 2 pp 249ndash271 2001
[5] J P Kimmance S Lawrence O Hassan and N J PurchaseldquoObservations of deformations created in existing tunnels byadjacent and cross cutting excavationsrdquo in Proceedings ofGeotechnical Aspects of Underground Construction in SoftGround pp 707ndash712 London UK July 1996
[6] E Leca and B New ldquoSettlements induced by tunnelling in softgroundrdquo Tunnelling and Underground Space Technologyvol 22 no 2 pp 119ndash149 2007
Advances in Civil Engineering 15
[7] S M Liao J H Liu R L Wang and Z M Li ldquoShieldtunnelling and environment protection in Shanghai softgroundrdquo Tunnelling and Underground Space Technologyvol 24 no 4 pp 454ndash465 2009
[8] F H Chehade and I Shahrour ldquoNumerical analysis of theinteraction between twin-tunnels influence of the relativeposition and construction procedurerdquo Tunnelling and Un-derground Space Technology vol 23 no 2 pp 210ndash214 2008
[9] X G Li and D J Yuan ldquoResponse of a double-decked metrotunnel to shield driving of twin closely underpasses tunnelsrdquoTunnelling and Underground Space Technology vol 28 no 1pp 18ndash30 2012
[10] T E Vorster A Klar K Soga and R J Mair ldquoEstimating theeffects of tunnelling on existing pipelinesrdquo Journal of Geo-technical and Geoenvironmental Engineering vol 131 no 11pp 1399ndash1410 2005
[11] Z G Zhang M S Huang and W D Wang ldquoEvaluation ofdeformation response for adjacent tunnels due to soilunloading in excavation engineeringrdquo Tunnelling and Un-derground Space Technology vol 38 no 3 pp 244ndash253 2013
[12] A P S Selvadurai Elastic Analysis of Soil-Foundation InteractionElsevier Scientific Pub Co Amsterdam Netherlands 1979
[13] N Loganathan and H G Poulos ldquoAnalytical prediction fortunnelling-induced ground movements in claysrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124no 9 pp 846ndash856 1998
[14] K M Lee R K Rowe and K Y Lo ldquoSubsidence owing totunnelling I Estimating the gap parameterrdquo CanadianGeotechnical Journal vol 29 no 6 pp 929ndash940 1992
[15] R K Rowe and KM Lee ldquoSubsidence owing to tunnelling IIEvaluation of a prediction techniquerdquo Canadian GeotechnicalJournal vol 29 no 6 pp 941ndash954 1992
[16] P B Attewell J Yeates and A R Selby Soil MovementsInduced by Tunnelling and their Effects on Pipelines andStructures Methuen Inc New York NY USA 1986
[17] F Ye C He H H Zhu et al ldquoAnalysis of longitudinalequivalent stiffness of shield tunnel considering lateral per-formancerdquo Chinese Journal of Geotechnical Engineeringvol 33 no 12 pp 1870ndash1876 2011 in Chinese
[18] P R Lancaster and D Mitchell ldquoProblems in Bending of Barsand Plates [M]rdquo Advanced Solid Mechanics Macmillan UK1980
[19] L Xu ldquoStudy on vertical settlement of shield tunnel in soft soilrdquoPhD dissertation Tongji University Shanghai China 2005
[20] M Yang and X Zhao ldquoAn approach for a single pile in layeredsoilrdquo Tongji University Natural Science vol 20 no 4pp 421ndash428 1992 in Chinese
16 Advances in Civil Engineering
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
e reinforce range of the existing tunnel is assumed as25 rings on either side of the intersection point A and thevarious stiness is taken into consideration using themethod proposed in this paper Figure 20 shows the in-centuence of dierent stiness of the reinforcement on theexisting tunnel (ie 4 EI EI and 14 EI) It is evident thatthe reinforcement can reduce the displacement rotationalangle normalized bending moment and shear force of theexisting tunnel while tunnel damage (reduce the stinessof reinforcement) has the opposite eect It is evident thatthe change of reinforcement stiness has greater eect onthe normalized bending moment and the normalized shearforce of the existing tunnel particularly at the incentection
points of the normalized bending moment and normalizedshear force curve
e reinforce stiness of the reinforcement range isassumed as 4 EI and the various reinforcement ranges istaken into consideration using the method proposed in thispaper Figure 21 shows the incentuence of dierent re-inforcement ranges on the existing tunnel when thenorthbound tunnel of Line 4 underpasses e normalizedbending moment on either side of the intersection point istoo large when the reinforcement range is 69ndash892m andthe normalized shear force on either side of the intersectionpoint is relatively too large when the reinforcement range isless than 592ndash992m So it is necessary to reinforce larger
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00035
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
The d
eflec
tions
of t
he ex
istin
g tu
nnel
(m)
xL
(a)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus000010
minus000005
000000
000005
000010
The s
lope
of t
he ex
istin
g tu
nnel
(rad
)
xL
(b)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus00030
minus00025
minus00020
minus00015
minus00010
minus00005
00000
00005
00010
00015
xL
The n
orm
alise
d be
ndin
g m
omen
t of t
he ex
istin
g tu
nnel
(c)
Reinforcement range (m)(692 892)(592 992)(492 1092)
(392 1192)(292 1292)
00 01 02 03 04 05 06 07 08 09 10
minus010
minus008
minus006
minus004
minus002
000
002
004
006
008
010
xLTh
e nor
mal
ised
shea
r for
ce o
f the
exist
ing
tunn
el
(d)
Figure 21 (a)e eects of dierent reinforcement ranges on the decentection of the existing tunnel (b)e eects of dierent reinforcementranges on the slope of the existing tunnel (c)e eects of dierent reinforcement ranges on the normalized moment of the existing tunnel(d) e eects of dierent reinforcement ranges on the normalized shear force of the existing tunnel
Advances in Civil Engineering 13
than the range of 492ndash1092m in the existing tunnel in thiscase that is larger than 30m (25 rings) on either side of theintersection point )ere is no big difference in the bendingmoment and shear force of the existing tunnel when thereinforcement ranges are 492ndash1092m 392ndash1192m and292ndash1292m In order to reduce the costs of reinforcementreinforcement in the 30m range (ie 25 rings) on eitherside of the intersection point is the best choice whichverifies the actual design
8 Conclusions
Based on the abovementioned statements the response of theexisting tunnel is analyzed and analytical methods are proposedfor verification and the reinforcement design Main conclusionsderived from the analysis are as follows
(1) )e displacement of the existing tunnel changes withthe relative position of the shield and the existingtunnel Shield parameters have a major impact on theexisting tunnel )e heaves of the existing tunnelmight be caused by the large applied face pressure andbulkhead addictive thrust When the shield tail isdriving beyond the selected monitoring rings theselected monitoring rings settle rapidly due to theclosure of the shield tail void
(2) )e southbound tunnel of Line 4 has less effect onthe left side of the intersection point A on theexisting tunnel because of the northbound tunnelrsquosbarrier effect A variety of measures of displacementprevention of Line 1 such as the control of shield
parameters reinforcement of the existing tunnel canensure the normal operation of metro Line 1
(3) )e analytical method proposed in the paper is ver-ified by the methods derived from prior studies It ispossible to take the different stiffness and curved loadsinto account by applying the stepped stiffness andtrapezoidal load while using a local coordinate systemin the derivation of the new method )e change ofreinforcement stiffness has greater effect on thenormalized bending moment and the normalizedshear force of the existing tunnel particularly at theinflection points of the normalized bending momentand normalized shear force curve )e analyticalmethod can be applied for the reinforcement designand safety check of the built tunnel
Appendices
A1 An Infinite Beam on the WinklerFoundation
θ(x) 1113946+infin
minusinfin
minusλ2s((tminus L)sin α)eminusλ(xminust) sin(λ(xminus t))
b
λ2s((tminus L)sin α)eλ(xminust)sin(λ(tminus x))
b
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎭
dt
(A1)
M(x) 14λ
1113946+infin
minusinfinks((tminusL)sin α)e
minusλ|xminust|( cos(λ|xminus t|)minus sin(λ|xminus t|))dt (A2)
Q(x) 1113946+infin
minusinfin
minus12
ks((tminusL)sin α)eminusλ(xminust)cos(λ(xminus t))dt xminus tgt 0
12
ks((tminusL)sin α)eminusλ(tminusx)cos(λ(tminusx))dt xminus tle 0
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
⎫⎪⎪⎪⎬
⎪⎪⎪⎭
dt (A3)
A2 An Infinite Beam on the PasternakFoundation
w(x) λ2bk
1113946+infin
minusinfinks((tminus L)sin α)De
minusa1λ|xminusτ| 1a1
cos a2λ|xminus τ|( 1113857 +1a2
sin a2λ|xminus τ|( 11138571113888 1113889dτ
θ(x) 1113946+infin
minusinfin
minusλ2s((tminusL))eminusa1λ(xminust)sin a2(xminus t)( 1113857
a1a2b
λ2s((tminus L))ea1λ(xminust)sin a2(tminus x)( 1113857
a1a2b
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A4)
14 Advances in Civil Engineering
M(x) 14λ
1113946+infin
minusinfinks((tminus L)sin α)e
minusa1λ|xminust| 1a1
cos a2λ|xminus t|( 1113857minus1a2
sin a2λ|xminus t|( 11138571113888 1113889dt
Q(x) 1113946+infin
minusinfin
minus12
ks((tminus L)sin α)eminusa1λ(xminust) cos a2λ(xminus t)( 1113857minus
a21 minus a 2
22a1a2
1113888 1113889sin a2λ(xminus t)( 11138571113890 1113891dt xminus tgt 0
12
ks((tminusL)sin α)eminusa1λ(tminusx) cos a2λ(tminusx)( 1113857minus
a21 minus a2
2
2a1a21113888 1113889sin a2λ(tminusx)( 11138571113890 1113891dt xminus tle 0
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A5)
B1 Coefficients of the Matrix ki5
k15 minusbiqiminus1
4a1ia2ici4EiIi a1i
2 + a2i2( 1113857
minus2a1ia2i + 2a1ia2iF2i
+ a1i2 minus a2i
2( 1113857F4i
⎛⎜⎝ ⎞⎟⎠
+bi qi minus qiminus1( 1113857
4a1ia2ici5EiIiL3 a1i
2 + a2i2( 1113857
2
a2i minus3a21i + a2
2i( 1113857F3i
+a1i 2a2i a21i + a2
2i( 1113857cix + a1i2 minus 3a2i
2( 1113857F1i( 1113857
⎛⎜⎝ ⎞⎟⎠
(B1)
k25 biqiminus1 a2iF3i minus a1iF1i( 1113857
2a1a2 a21 + a 2
2( 1113857c3EiIi
+bi qi minus qiminus1( 1113857 2a1ia2i minus 2a1ia2iF2i minus a1i
2 minus a2i2( 1113857F4i( 1113857( 1113857
2a1ia2i a1i2 + a2i
2( 11138572ci
4EiIix1113872 1113873 (B2)
k35 minusbiqiminus1F4i
2a1ia2ici2 minus
bi qi minus qiminus1( 1113857 a2iF3i minus a1iF1i( 1113857
2a1ia2ici3 a1i
2 + a2i2( 1113857x
(B3)
k45 minusbiqiminus1
a1i2 + a2i
2( 1113857ci
1113888a1i 1 + 2a1i2a22i minus 2a2i
41113872 1113873F3i + a2i 1 + 2a1i
2a22i minus 2a1i
41113872 1113873F1i1113889
minusbi qi minus qiminus1( 1113857
a21i + a2i
2( 11138572c 2
i x
4a1i2a2i
2 minus 1( 1113857 a1i2 minus a2i
2( 1113857 + a2i2 minus 4a1i
4a2i2 + a1i
2 minus1 + 4a2i4( 1113857( 1113857F2i
minus2a1ia2i a 21i minus a 2
2i( 11138572 minus 11113872 1113873F4i
⎛⎜⎝ ⎞⎟⎠
(B4)
Conflicts of Interest
)e author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
)e author appreciates the help from the staff of Hangzhouand Hong Kong Tunnel Company Ltd during in-strumentation setup and data acquisition )is study issupported by the National Natural Science Foundation ofChina (no 41702313)
References
[1] S H Kim Model testing and analysis of interactions betweentunnels in clay [PhD 3esis] Department of EngineeringScience University of Oxford Oxford UK 1996
[2] H Y Liu J C Small J P Carter and D J Williams ldquoEffectsof tunnelling on existing support systems of perpendicularlypassing tunnelsrdquo Computers and Geotechnics vol 36 no 5pp 880ndash894 2009
[3] I Yamaguchi I Yamazaki and Y Kiritani ldquoStudy ofgroundndashtunnel interactions of four shield tunnels driven inclose proximity in relation to design and construction ofparallel shield tunnelsrdquo Tunnelling and Underground SpaceTechnology vol 13 no 3 pp 289ndash304 1998
[4] T I Addenbrooke and D M Potts ldquoTwin tunnel interactionsurface and subsurface effectsrdquo International Journal ofGeomechanics vol 1 no 2 pp 249ndash271 2001
[5] J P Kimmance S Lawrence O Hassan and N J PurchaseldquoObservations of deformations created in existing tunnels byadjacent and cross cutting excavationsrdquo in Proceedings ofGeotechnical Aspects of Underground Construction in SoftGround pp 707ndash712 London UK July 1996
[6] E Leca and B New ldquoSettlements induced by tunnelling in softgroundrdquo Tunnelling and Underground Space Technologyvol 22 no 2 pp 119ndash149 2007
Advances in Civil Engineering 15
[7] S M Liao J H Liu R L Wang and Z M Li ldquoShieldtunnelling and environment protection in Shanghai softgroundrdquo Tunnelling and Underground Space Technologyvol 24 no 4 pp 454ndash465 2009
[8] F H Chehade and I Shahrour ldquoNumerical analysis of theinteraction between twin-tunnels influence of the relativeposition and construction procedurerdquo Tunnelling and Un-derground Space Technology vol 23 no 2 pp 210ndash214 2008
[9] X G Li and D J Yuan ldquoResponse of a double-decked metrotunnel to shield driving of twin closely underpasses tunnelsrdquoTunnelling and Underground Space Technology vol 28 no 1pp 18ndash30 2012
[10] T E Vorster A Klar K Soga and R J Mair ldquoEstimating theeffects of tunnelling on existing pipelinesrdquo Journal of Geo-technical and Geoenvironmental Engineering vol 131 no 11pp 1399ndash1410 2005
[11] Z G Zhang M S Huang and W D Wang ldquoEvaluation ofdeformation response for adjacent tunnels due to soilunloading in excavation engineeringrdquo Tunnelling and Un-derground Space Technology vol 38 no 3 pp 244ndash253 2013
[12] A P S Selvadurai Elastic Analysis of Soil-Foundation InteractionElsevier Scientific Pub Co Amsterdam Netherlands 1979
[13] N Loganathan and H G Poulos ldquoAnalytical prediction fortunnelling-induced ground movements in claysrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124no 9 pp 846ndash856 1998
[14] K M Lee R K Rowe and K Y Lo ldquoSubsidence owing totunnelling I Estimating the gap parameterrdquo CanadianGeotechnical Journal vol 29 no 6 pp 929ndash940 1992
[15] R K Rowe and KM Lee ldquoSubsidence owing to tunnelling IIEvaluation of a prediction techniquerdquo Canadian GeotechnicalJournal vol 29 no 6 pp 941ndash954 1992
[16] P B Attewell J Yeates and A R Selby Soil MovementsInduced by Tunnelling and their Effects on Pipelines andStructures Methuen Inc New York NY USA 1986
[17] F Ye C He H H Zhu et al ldquoAnalysis of longitudinalequivalent stiffness of shield tunnel considering lateral per-formancerdquo Chinese Journal of Geotechnical Engineeringvol 33 no 12 pp 1870ndash1876 2011 in Chinese
[18] P R Lancaster and D Mitchell ldquoProblems in Bending of Barsand Plates [M]rdquo Advanced Solid Mechanics Macmillan UK1980
[19] L Xu ldquoStudy on vertical settlement of shield tunnel in soft soilrdquoPhD dissertation Tongji University Shanghai China 2005
[20] M Yang and X Zhao ldquoAn approach for a single pile in layeredsoilrdquo Tongji University Natural Science vol 20 no 4pp 421ndash428 1992 in Chinese
16 Advances in Civil Engineering
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
than the range of 492ndash1092m in the existing tunnel in thiscase that is larger than 30m (25 rings) on either side of theintersection point )ere is no big difference in the bendingmoment and shear force of the existing tunnel when thereinforcement ranges are 492ndash1092m 392ndash1192m and292ndash1292m In order to reduce the costs of reinforcementreinforcement in the 30m range (ie 25 rings) on eitherside of the intersection point is the best choice whichverifies the actual design
8 Conclusions
Based on the abovementioned statements the response of theexisting tunnel is analyzed and analytical methods are proposedfor verification and the reinforcement design Main conclusionsderived from the analysis are as follows
(1) )e displacement of the existing tunnel changes withthe relative position of the shield and the existingtunnel Shield parameters have a major impact on theexisting tunnel )e heaves of the existing tunnelmight be caused by the large applied face pressure andbulkhead addictive thrust When the shield tail isdriving beyond the selected monitoring rings theselected monitoring rings settle rapidly due to theclosure of the shield tail void
(2) )e southbound tunnel of Line 4 has less effect onthe left side of the intersection point A on theexisting tunnel because of the northbound tunnelrsquosbarrier effect A variety of measures of displacementprevention of Line 1 such as the control of shield
parameters reinforcement of the existing tunnel canensure the normal operation of metro Line 1
(3) )e analytical method proposed in the paper is ver-ified by the methods derived from prior studies It ispossible to take the different stiffness and curved loadsinto account by applying the stepped stiffness andtrapezoidal load while using a local coordinate systemin the derivation of the new method )e change ofreinforcement stiffness has greater effect on thenormalized bending moment and the normalizedshear force of the existing tunnel particularly at theinflection points of the normalized bending momentand normalized shear force curve )e analyticalmethod can be applied for the reinforcement designand safety check of the built tunnel
Appendices
A1 An Infinite Beam on the WinklerFoundation
θ(x) 1113946+infin
minusinfin
minusλ2s((tminus L)sin α)eminusλ(xminust) sin(λ(xminus t))
b
λ2s((tminus L)sin α)eλ(xminust)sin(λ(tminus x))
b
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎭
dt
(A1)
M(x) 14λ
1113946+infin
minusinfinks((tminusL)sin α)e
minusλ|xminust|( cos(λ|xminus t|)minus sin(λ|xminus t|))dt (A2)
Q(x) 1113946+infin
minusinfin
minus12
ks((tminusL)sin α)eminusλ(xminust)cos(λ(xminus t))dt xminus tgt 0
12
ks((tminusL)sin α)eminusλ(tminusx)cos(λ(tminusx))dt xminus tle 0
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
⎫⎪⎪⎪⎬
⎪⎪⎪⎭
dt (A3)
A2 An Infinite Beam on the PasternakFoundation
w(x) λ2bk
1113946+infin
minusinfinks((tminus L)sin α)De
minusa1λ|xminusτ| 1a1
cos a2λ|xminus τ|( 1113857 +1a2
sin a2λ|xminus τ|( 11138571113888 1113889dτ
θ(x) 1113946+infin
minusinfin
minusλ2s((tminusL))eminusa1λ(xminust)sin a2(xminus t)( 1113857
a1a2b
λ2s((tminus L))ea1λ(xminust)sin a2(tminus x)( 1113857
a1a2b
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A4)
14 Advances in Civil Engineering
M(x) 14λ
1113946+infin
minusinfinks((tminus L)sin α)e
minusa1λ|xminust| 1a1
cos a2λ|xminus t|( 1113857minus1a2
sin a2λ|xminus t|( 11138571113888 1113889dt
Q(x) 1113946+infin
minusinfin
minus12
ks((tminus L)sin α)eminusa1λ(xminust) cos a2λ(xminus t)( 1113857minus
a21 minus a 2
22a1a2
1113888 1113889sin a2λ(xminus t)( 11138571113890 1113891dt xminus tgt 0
12
ks((tminusL)sin α)eminusa1λ(tminusx) cos a2λ(tminusx)( 1113857minus
a21 minus a2
2
2a1a21113888 1113889sin a2λ(tminusx)( 11138571113890 1113891dt xminus tle 0
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A5)
B1 Coefficients of the Matrix ki5
k15 minusbiqiminus1
4a1ia2ici4EiIi a1i
2 + a2i2( 1113857
minus2a1ia2i + 2a1ia2iF2i
+ a1i2 minus a2i
2( 1113857F4i
⎛⎜⎝ ⎞⎟⎠
+bi qi minus qiminus1( 1113857
4a1ia2ici5EiIiL3 a1i
2 + a2i2( 1113857
2
a2i minus3a21i + a2
2i( 1113857F3i
+a1i 2a2i a21i + a2
2i( 1113857cix + a1i2 minus 3a2i
2( 1113857F1i( 1113857
⎛⎜⎝ ⎞⎟⎠
(B1)
k25 biqiminus1 a2iF3i minus a1iF1i( 1113857
2a1a2 a21 + a 2
2( 1113857c3EiIi
+bi qi minus qiminus1( 1113857 2a1ia2i minus 2a1ia2iF2i minus a1i
2 minus a2i2( 1113857F4i( 1113857( 1113857
2a1ia2i a1i2 + a2i
2( 11138572ci
4EiIix1113872 1113873 (B2)
k35 minusbiqiminus1F4i
2a1ia2ici2 minus
bi qi minus qiminus1( 1113857 a2iF3i minus a1iF1i( 1113857
2a1ia2ici3 a1i
2 + a2i2( 1113857x
(B3)
k45 minusbiqiminus1
a1i2 + a2i
2( 1113857ci
1113888a1i 1 + 2a1i2a22i minus 2a2i
41113872 1113873F3i + a2i 1 + 2a1i
2a22i minus 2a1i
41113872 1113873F1i1113889
minusbi qi minus qiminus1( 1113857
a21i + a2i
2( 11138572c 2
i x
4a1i2a2i
2 minus 1( 1113857 a1i2 minus a2i
2( 1113857 + a2i2 minus 4a1i
4a2i2 + a1i
2 minus1 + 4a2i4( 1113857( 1113857F2i
minus2a1ia2i a 21i minus a 2
2i( 11138572 minus 11113872 1113873F4i
⎛⎜⎝ ⎞⎟⎠
(B4)
Conflicts of Interest
)e author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
)e author appreciates the help from the staff of Hangzhouand Hong Kong Tunnel Company Ltd during in-strumentation setup and data acquisition )is study issupported by the National Natural Science Foundation ofChina (no 41702313)
References
[1] S H Kim Model testing and analysis of interactions betweentunnels in clay [PhD 3esis] Department of EngineeringScience University of Oxford Oxford UK 1996
[2] H Y Liu J C Small J P Carter and D J Williams ldquoEffectsof tunnelling on existing support systems of perpendicularlypassing tunnelsrdquo Computers and Geotechnics vol 36 no 5pp 880ndash894 2009
[3] I Yamaguchi I Yamazaki and Y Kiritani ldquoStudy ofgroundndashtunnel interactions of four shield tunnels driven inclose proximity in relation to design and construction ofparallel shield tunnelsrdquo Tunnelling and Underground SpaceTechnology vol 13 no 3 pp 289ndash304 1998
[4] T I Addenbrooke and D M Potts ldquoTwin tunnel interactionsurface and subsurface effectsrdquo International Journal ofGeomechanics vol 1 no 2 pp 249ndash271 2001
[5] J P Kimmance S Lawrence O Hassan and N J PurchaseldquoObservations of deformations created in existing tunnels byadjacent and cross cutting excavationsrdquo in Proceedings ofGeotechnical Aspects of Underground Construction in SoftGround pp 707ndash712 London UK July 1996
[6] E Leca and B New ldquoSettlements induced by tunnelling in softgroundrdquo Tunnelling and Underground Space Technologyvol 22 no 2 pp 119ndash149 2007
Advances in Civil Engineering 15
[7] S M Liao J H Liu R L Wang and Z M Li ldquoShieldtunnelling and environment protection in Shanghai softgroundrdquo Tunnelling and Underground Space Technologyvol 24 no 4 pp 454ndash465 2009
[8] F H Chehade and I Shahrour ldquoNumerical analysis of theinteraction between twin-tunnels influence of the relativeposition and construction procedurerdquo Tunnelling and Un-derground Space Technology vol 23 no 2 pp 210ndash214 2008
[9] X G Li and D J Yuan ldquoResponse of a double-decked metrotunnel to shield driving of twin closely underpasses tunnelsrdquoTunnelling and Underground Space Technology vol 28 no 1pp 18ndash30 2012
[10] T E Vorster A Klar K Soga and R J Mair ldquoEstimating theeffects of tunnelling on existing pipelinesrdquo Journal of Geo-technical and Geoenvironmental Engineering vol 131 no 11pp 1399ndash1410 2005
[11] Z G Zhang M S Huang and W D Wang ldquoEvaluation ofdeformation response for adjacent tunnels due to soilunloading in excavation engineeringrdquo Tunnelling and Un-derground Space Technology vol 38 no 3 pp 244ndash253 2013
[12] A P S Selvadurai Elastic Analysis of Soil-Foundation InteractionElsevier Scientific Pub Co Amsterdam Netherlands 1979
[13] N Loganathan and H G Poulos ldquoAnalytical prediction fortunnelling-induced ground movements in claysrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124no 9 pp 846ndash856 1998
[14] K M Lee R K Rowe and K Y Lo ldquoSubsidence owing totunnelling I Estimating the gap parameterrdquo CanadianGeotechnical Journal vol 29 no 6 pp 929ndash940 1992
[15] R K Rowe and KM Lee ldquoSubsidence owing to tunnelling IIEvaluation of a prediction techniquerdquo Canadian GeotechnicalJournal vol 29 no 6 pp 941ndash954 1992
[16] P B Attewell J Yeates and A R Selby Soil MovementsInduced by Tunnelling and their Effects on Pipelines andStructures Methuen Inc New York NY USA 1986
[17] F Ye C He H H Zhu et al ldquoAnalysis of longitudinalequivalent stiffness of shield tunnel considering lateral per-formancerdquo Chinese Journal of Geotechnical Engineeringvol 33 no 12 pp 1870ndash1876 2011 in Chinese
[18] P R Lancaster and D Mitchell ldquoProblems in Bending of Barsand Plates [M]rdquo Advanced Solid Mechanics Macmillan UK1980
[19] L Xu ldquoStudy on vertical settlement of shield tunnel in soft soilrdquoPhD dissertation Tongji University Shanghai China 2005
[20] M Yang and X Zhao ldquoAn approach for a single pile in layeredsoilrdquo Tongji University Natural Science vol 20 no 4pp 421ndash428 1992 in Chinese
16 Advances in Civil Engineering
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
M(x) 14λ
1113946+infin
minusinfinks((tminus L)sin α)e
minusa1λ|xminust| 1a1
cos a2λ|xminus t|( 1113857minus1a2
sin a2λ|xminus t|( 11138571113888 1113889dt
Q(x) 1113946+infin
minusinfin
minus12
ks((tminus L)sin α)eminusa1λ(xminust) cos a2λ(xminus t)( 1113857minus
a21 minus a 2
22a1a2
1113888 1113889sin a2λ(xminus t)( 11138571113890 1113891dt xminus tgt 0
12
ks((tminusL)sin α)eminusa1λ(tminusx) cos a2λ(tminusx)( 1113857minus
a21 minus a2
2
2a1a21113888 1113889sin a2λ(tminusx)( 11138571113890 1113891dt xminus tle 0
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
dt
(A5)
B1 Coefficients of the Matrix ki5
k15 minusbiqiminus1
4a1ia2ici4EiIi a1i
2 + a2i2( 1113857
minus2a1ia2i + 2a1ia2iF2i
+ a1i2 minus a2i
2( 1113857F4i
⎛⎜⎝ ⎞⎟⎠
+bi qi minus qiminus1( 1113857
4a1ia2ici5EiIiL3 a1i
2 + a2i2( 1113857
2
a2i minus3a21i + a2
2i( 1113857F3i
+a1i 2a2i a21i + a2
2i( 1113857cix + a1i2 minus 3a2i
2( 1113857F1i( 1113857
⎛⎜⎝ ⎞⎟⎠
(B1)
k25 biqiminus1 a2iF3i minus a1iF1i( 1113857
2a1a2 a21 + a 2
2( 1113857c3EiIi
+bi qi minus qiminus1( 1113857 2a1ia2i minus 2a1ia2iF2i minus a1i
2 minus a2i2( 1113857F4i( 1113857( 1113857
2a1ia2i a1i2 + a2i
2( 11138572ci
4EiIix1113872 1113873 (B2)
k35 minusbiqiminus1F4i
2a1ia2ici2 minus
bi qi minus qiminus1( 1113857 a2iF3i minus a1iF1i( 1113857
2a1ia2ici3 a1i
2 + a2i2( 1113857x
(B3)
k45 minusbiqiminus1
a1i2 + a2i
2( 1113857ci
1113888a1i 1 + 2a1i2a22i minus 2a2i
41113872 1113873F3i + a2i 1 + 2a1i
2a22i minus 2a1i
41113872 1113873F1i1113889
minusbi qi minus qiminus1( 1113857
a21i + a2i
2( 11138572c 2
i x
4a1i2a2i
2 minus 1( 1113857 a1i2 minus a2i
2( 1113857 + a2i2 minus 4a1i
4a2i2 + a1i
2 minus1 + 4a2i4( 1113857( 1113857F2i
minus2a1ia2i a 21i minus a 2
2i( 11138572 minus 11113872 1113873F4i
⎛⎜⎝ ⎞⎟⎠
(B4)
Conflicts of Interest
)e author declares that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
)e author appreciates the help from the staff of Hangzhouand Hong Kong Tunnel Company Ltd during in-strumentation setup and data acquisition )is study issupported by the National Natural Science Foundation ofChina (no 41702313)
References
[1] S H Kim Model testing and analysis of interactions betweentunnels in clay [PhD 3esis] Department of EngineeringScience University of Oxford Oxford UK 1996
[2] H Y Liu J C Small J P Carter and D J Williams ldquoEffectsof tunnelling on existing support systems of perpendicularlypassing tunnelsrdquo Computers and Geotechnics vol 36 no 5pp 880ndash894 2009
[3] I Yamaguchi I Yamazaki and Y Kiritani ldquoStudy ofgroundndashtunnel interactions of four shield tunnels driven inclose proximity in relation to design and construction ofparallel shield tunnelsrdquo Tunnelling and Underground SpaceTechnology vol 13 no 3 pp 289ndash304 1998
[4] T I Addenbrooke and D M Potts ldquoTwin tunnel interactionsurface and subsurface effectsrdquo International Journal ofGeomechanics vol 1 no 2 pp 249ndash271 2001
[5] J P Kimmance S Lawrence O Hassan and N J PurchaseldquoObservations of deformations created in existing tunnels byadjacent and cross cutting excavationsrdquo in Proceedings ofGeotechnical Aspects of Underground Construction in SoftGround pp 707ndash712 London UK July 1996
[6] E Leca and B New ldquoSettlements induced by tunnelling in softgroundrdquo Tunnelling and Underground Space Technologyvol 22 no 2 pp 119ndash149 2007
Advances in Civil Engineering 15
[7] S M Liao J H Liu R L Wang and Z M Li ldquoShieldtunnelling and environment protection in Shanghai softgroundrdquo Tunnelling and Underground Space Technologyvol 24 no 4 pp 454ndash465 2009
[8] F H Chehade and I Shahrour ldquoNumerical analysis of theinteraction between twin-tunnels influence of the relativeposition and construction procedurerdquo Tunnelling and Un-derground Space Technology vol 23 no 2 pp 210ndash214 2008
[9] X G Li and D J Yuan ldquoResponse of a double-decked metrotunnel to shield driving of twin closely underpasses tunnelsrdquoTunnelling and Underground Space Technology vol 28 no 1pp 18ndash30 2012
[10] T E Vorster A Klar K Soga and R J Mair ldquoEstimating theeffects of tunnelling on existing pipelinesrdquo Journal of Geo-technical and Geoenvironmental Engineering vol 131 no 11pp 1399ndash1410 2005
[11] Z G Zhang M S Huang and W D Wang ldquoEvaluation ofdeformation response for adjacent tunnels due to soilunloading in excavation engineeringrdquo Tunnelling and Un-derground Space Technology vol 38 no 3 pp 244ndash253 2013
[12] A P S Selvadurai Elastic Analysis of Soil-Foundation InteractionElsevier Scientific Pub Co Amsterdam Netherlands 1979
[13] N Loganathan and H G Poulos ldquoAnalytical prediction fortunnelling-induced ground movements in claysrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124no 9 pp 846ndash856 1998
[14] K M Lee R K Rowe and K Y Lo ldquoSubsidence owing totunnelling I Estimating the gap parameterrdquo CanadianGeotechnical Journal vol 29 no 6 pp 929ndash940 1992
[15] R K Rowe and KM Lee ldquoSubsidence owing to tunnelling IIEvaluation of a prediction techniquerdquo Canadian GeotechnicalJournal vol 29 no 6 pp 941ndash954 1992
[16] P B Attewell J Yeates and A R Selby Soil MovementsInduced by Tunnelling and their Effects on Pipelines andStructures Methuen Inc New York NY USA 1986
[17] F Ye C He H H Zhu et al ldquoAnalysis of longitudinalequivalent stiffness of shield tunnel considering lateral per-formancerdquo Chinese Journal of Geotechnical Engineeringvol 33 no 12 pp 1870ndash1876 2011 in Chinese
[18] P R Lancaster and D Mitchell ldquoProblems in Bending of Barsand Plates [M]rdquo Advanced Solid Mechanics Macmillan UK1980
[19] L Xu ldquoStudy on vertical settlement of shield tunnel in soft soilrdquoPhD dissertation Tongji University Shanghai China 2005
[20] M Yang and X Zhao ldquoAn approach for a single pile in layeredsoilrdquo Tongji University Natural Science vol 20 no 4pp 421ndash428 1992 in Chinese
16 Advances in Civil Engineering
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
[7] S M Liao J H Liu R L Wang and Z M Li ldquoShieldtunnelling and environment protection in Shanghai softgroundrdquo Tunnelling and Underground Space Technologyvol 24 no 4 pp 454ndash465 2009
[8] F H Chehade and I Shahrour ldquoNumerical analysis of theinteraction between twin-tunnels influence of the relativeposition and construction procedurerdquo Tunnelling and Un-derground Space Technology vol 23 no 2 pp 210ndash214 2008
[9] X G Li and D J Yuan ldquoResponse of a double-decked metrotunnel to shield driving of twin closely underpasses tunnelsrdquoTunnelling and Underground Space Technology vol 28 no 1pp 18ndash30 2012
[10] T E Vorster A Klar K Soga and R J Mair ldquoEstimating theeffects of tunnelling on existing pipelinesrdquo Journal of Geo-technical and Geoenvironmental Engineering vol 131 no 11pp 1399ndash1410 2005
[11] Z G Zhang M S Huang and W D Wang ldquoEvaluation ofdeformation response for adjacent tunnels due to soilunloading in excavation engineeringrdquo Tunnelling and Un-derground Space Technology vol 38 no 3 pp 244ndash253 2013
[12] A P S Selvadurai Elastic Analysis of Soil-Foundation InteractionElsevier Scientific Pub Co Amsterdam Netherlands 1979
[13] N Loganathan and H G Poulos ldquoAnalytical prediction fortunnelling-induced ground movements in claysrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 124no 9 pp 846ndash856 1998
[14] K M Lee R K Rowe and K Y Lo ldquoSubsidence owing totunnelling I Estimating the gap parameterrdquo CanadianGeotechnical Journal vol 29 no 6 pp 929ndash940 1992
[15] R K Rowe and KM Lee ldquoSubsidence owing to tunnelling IIEvaluation of a prediction techniquerdquo Canadian GeotechnicalJournal vol 29 no 6 pp 941ndash954 1992
[16] P B Attewell J Yeates and A R Selby Soil MovementsInduced by Tunnelling and their Effects on Pipelines andStructures Methuen Inc New York NY USA 1986
[17] F Ye C He H H Zhu et al ldquoAnalysis of longitudinalequivalent stiffness of shield tunnel considering lateral per-formancerdquo Chinese Journal of Geotechnical Engineeringvol 33 no 12 pp 1870ndash1876 2011 in Chinese
[18] P R Lancaster and D Mitchell ldquoProblems in Bending of Barsand Plates [M]rdquo Advanced Solid Mechanics Macmillan UK1980
[19] L Xu ldquoStudy on vertical settlement of shield tunnel in soft soilrdquoPhD dissertation Tongji University Shanghai China 2005
[20] M Yang and X Zhao ldquoAn approach for a single pile in layeredsoilrdquo Tongji University Natural Science vol 20 no 4pp 421ndash428 1992 in Chinese
16 Advances in Civil Engineering
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom