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

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


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


3ndash5 963times10minus5 174times10minus4 mdash3ndash6 329times10minus4 301times10minus4 mdash4ndash3 239times10minus7 784times10minus7 6ndash1 206times10minus7 303times10minus7 mdash6ndash2 262times10minus7 128times10minus7 mdash8ndash1 188times10minus7 511times10minus7 mdash


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


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


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


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


e construction of thenorthbound tunnel

2013129 Ring 4820131227 Ring 72201415 Ring 1252014128 Rring 275


Hor

izon

tal d

ispla

cem

ent (

mm

)

A B

Figure 10 Time-varying horizontal displacement of the existingtunnel


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


201453 Ring 135201459 Ring 1912014528 Ring 270



2013129 Ring 4820131222 Ring 6420131230 Ring 87


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


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


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


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


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 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 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


(c)


387 400 420 440 460 480 500 520 540 560 580 595

minus600

minus400

minus200

0

200

400

600


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


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

)


(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


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)


(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)


(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)


(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


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)


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


[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


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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






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


Guanhe Station

786 m

966 m835 m

35302520

674

669

664

659

654205

160

504

240245250255






Da-ju-yuan Station

A B

C






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




208 212

K20+

898

754

K21+

053

342

K21+

094

361

K21+

608

099




K21+

015

389

Water pipeline

K21+

144

111

Groundwater table

Chainage (m)


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






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




gravel

r (kNm)

Dep

th (m

)

wpwL

w

Soil layer

uucucu


uucucu

c (kPa) φordm





Kv Kh K









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


Tunn

el ch

aina

ge (m

)

Date

46 days


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


430 440 450 460 470 480 490 500 510 515

020

022

024

026

028

030

032

034

036

038

040


225 220 210 200 190 180 170 160 150 143




50 60 70 80 90 100 110 120 130

The a

pplie

d tu

nnel

face

pre

ssur

e (M

Pa)








430 440 450 460 470 480 490 500 510 518

35

40

45

50

55

60

65

70

75

80




e g

rout

ing

volu

me (

m3 )



Track bed


Centerline



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



201453 Ring 135 2014515 Ring 226


20146152014714

20141019

B



Ver

tical

disp

lace

men

t (m

m)


A


400 420 440 460 480 500 520 540 560 580 600 620 640 660 670


02468

10121416

C


2014420 Ring 362014502 Ring 1242014528 Ring 270





Hor

izon

tal d

ispla

cem

ent (

mm

)

A B










q(x) ks(x) (2)



+(3minus 4v)z +H


x2 +(z +H)2( )2


Conv

erge

nce d

ispla

cem

ent (

mm

)


012345


201453 Ring 135201459 Ring 1912014528 Ring 270



2013129 Ring 4820131222 Ring 6420131230 Ring 87



A B



xk1

k2

e existing tunnel

Sz(x)




ε0 4Rg + g2

4R2 (4)



Gp 2Δ + δ (6)



d4w(x)dx4

+ 4λ4p(x) 4λ4q(x) (7)




p(x) kw(x) (8)





w(x) 1



w(x) 1

8EIλ3int+infin






d2wi(x)dx4




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



w(x) sum4

i1CiFi(x) (13)













(18)





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)






( ) (21)



4c3i EiIi




E0I0


E1I1E2I2



qi(x)

pi

Qi Qi+1Mi Mi+1

Oimi

xi


q0q1 q2

qiminus1 qnminus1


q (x)

0



For the rst element








Suppose






where

An xn( ) prod1

inKi Li( ) (30)


M(0) 0

V(0) 2

k

G

radic


V(L) 0

V(L) minus2

k

G

radic


(31)


64 Case Study



k 5E0β


G 13E0B

2

32 1 + v0( )βtimes 085

(32)


E0 Es

1minus v2s

v0 vs

1minus vs

(33)


E0 (25 sim 35)Es01minus02 (34)



(35)


N1(01) Ni(xi)N1(x1) = N2(02) N2(x2)L1

L1

E1I1



L2 L3 Li LnLnminus1

L2 Li

x x x


Ni(0i)



7 Discussions


Mi MiL0EiIi

Qi QiL

20

EiIi

(36)



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



(a)

387 400 420 440 460 480 500 520 540 560 580 595

minus000015

minus000010

minus000005

000000

000005

000010

000015


e s

lope

of t

he ex

istin

g tu

nnel

(rad

)


(b)


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


(c)


387 400 420 440 460 480 500 520 540 560 580 595

minus600

minus400

minus200

0

200

400

600


e s

lope

of t

he ex

istin

g tu

nnel

(kN

)

(d)





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


(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

)


(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)







(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)


(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)


(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)


(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)




8 Conclusions






Appendices


θ(x) 1113946+infin

minusinfin


b


b

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

dt

(A1)

M(x) 14λ

1113946+infin



Q(x) 1113946+infin

minusinfin

minus12


12


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

dt (A3)


w(x) λ2bk

1113946+infin




sin a2λ|xminus τ|( 11138571113888 1113889dτ

θ(x) 1113946+infin

minusinfin


a1a2b


a1a2b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A4)


M(x) 14λ

1113946+infin




sin a2λ|xminus t|( 11138571113888 1113889dt

Q(x) 1113946+infin

minusinfin

minus12


a21 minus a 2

22a1a2


12


a21 minus a2

2


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A5)


k15 minusbiqiminus1

4a1ia2ici4EiIi a1i

2 + a2i2( 1113857


+ a1i2 minus a2i

2( 1113857F4i

⎛⎜⎝ ⎞⎟⎠



2 + a2i2( 1113857

2

a2i minus3a21i + a2

2i( 1113857F3i

+a1i 2a2i a21i + a2


2( 1113857F1i( 1113857

⎛⎜⎝ ⎞⎟⎠

(B1)


2a1a2 a21 + a 2

2( 1113857c3EiIi


2 minus a2i2( 1113857F4i( 1113857( 1113857

2a1ia2i a1i2 + a2i

2( 11138572ci

4EiIix1113872 1113873 (B2)


2a1ia2ici2 minus


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


a21i + a2i

2( 11138572c 2

i x

4a1i2a2i


2( 1113857 + a2i2 minus 4a1i

4a2i2 + a1i

2 minus1 + 4a2i4( 1113857( 1113857F2i


2i( 11138572 minus 11113872 1113873F4i

⎛⎜⎝ ⎞⎟⎠

(B4)



Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






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


Guanhe Station

786 m

966 m835 m

35302520

674

669

664

659

654205

160

504

240245250255






Da-ju-yuan Station

A B

C






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




208 212

K20+

898

754

K21+

053

342

K21+

094

361

K21+

608

099




K21+

015

389

Water pipeline

K21+

144

111

Groundwater table

Chainage (m)


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






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




gravel

r (kNm)

Dep

th (m

)

wpwL

w

Soil layer

uucucu


uucucu

c (kPa) φordm





Kv Kh K









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


Tunn

el ch

aina

ge (m

)

Date

46 days


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


430 440 450 460 470 480 490 500 510 515

020

022

024

026

028

030

032

034

036

038

040


225 220 210 200 190 180 170 160 150 143




50 60 70 80 90 100 110 120 130

The a

pplie

d tu

nnel

face

pre

ssur

e (M

Pa)








430 440 450 460 470 480 490 500 510 518

35

40

45

50

55

60

65

70

75

80




e g

rout

ing

volu

me (

m3 )



Track bed


Centerline



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



201453 Ring 135 2014515 Ring 226


20146152014714

20141019

B



Ver

tical

disp

lace

men

t (m

m)


A


400 420 440 460 480 500 520 540 560 580 600 620 640 660 670


02468

10121416

C


2014420 Ring 362014502 Ring 1242014528 Ring 270





Hor

izon

tal d

ispla

cem

ent (

mm

)

A B










q(x) ks(x) (2)



+(3minus 4v)z +H


x2 +(z +H)2( )2


Conv

erge

nce d

ispla

cem

ent (

mm

)


012345


201453 Ring 135201459 Ring 1912014528 Ring 270



2013129 Ring 4820131222 Ring 6420131230 Ring 87



A B



xk1

k2

e existing tunnel

Sz(x)




ε0 4Rg + g2

4R2 (4)



Gp 2Δ + δ (6)



d4w(x)dx4

+ 4λ4p(x) 4λ4q(x) (7)




p(x) kw(x) (8)





w(x) 1



w(x) 1

8EIλ3int+infin






d2wi(x)dx4




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



w(x) sum4

i1CiFi(x) (13)













(18)





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)






( ) (21)



4c3i EiIi




E0I0


E1I1E2I2



qi(x)

pi

Qi Qi+1Mi Mi+1

Oimi

xi


q0q1 q2

qiminus1 qnminus1


q (x)

0



For the rst element








Suppose






where

An xn( ) prod1

inKi Li( ) (30)


M(0) 0

V(0) 2

k

G

radic


V(L) 0

V(L) minus2

k

G

radic


(31)


64 Case Study



k 5E0β


G 13E0B

2

32 1 + v0( )βtimes 085

(32)


E0 Es

1minus v2s

v0 vs

1minus vs

(33)


E0 (25 sim 35)Es01minus02 (34)



(35)


N1(01) Ni(xi)N1(x1) = N2(02) N2(x2)L1

L1

E1I1



L2 L3 Li LnLnminus1

L2 Li

x x x


Ni(0i)



7 Discussions


Mi MiL0EiIi

Qi QiL

20

EiIi

(36)



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



(a)

387 400 420 440 460 480 500 520 540 560 580 595

minus000015

minus000010

minus000005

000000

000005

000010

000015


e s

lope

of t

he ex

istin

g tu

nnel

(rad

)


(b)


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


(c)


387 400 420 440 460 480 500 520 540 560 580 595

minus600

minus400

minus200

0

200

400

600


e s

lope

of t

he ex

istin

g tu

nnel

(kN

)

(d)





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


(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

)


(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)







(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)


(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)


(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)


(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)




8 Conclusions






Appendices


θ(x) 1113946+infin

minusinfin


b


b

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

dt

(A1)

M(x) 14λ

1113946+infin



Q(x) 1113946+infin

minusinfin

minus12


12


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

dt (A3)


w(x) λ2bk

1113946+infin




sin a2λ|xminus τ|( 11138571113888 1113889dτ

θ(x) 1113946+infin

minusinfin


a1a2b


a1a2b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A4)


M(x) 14λ

1113946+infin




sin a2λ|xminus t|( 11138571113888 1113889dt

Q(x) 1113946+infin

minusinfin

minus12


a21 minus a 2

22a1a2


12


a21 minus a2

2


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A5)


k15 minusbiqiminus1

4a1ia2ici4EiIi a1i

2 + a2i2( 1113857


+ a1i2 minus a2i

2( 1113857F4i

⎛⎜⎝ ⎞⎟⎠



2 + a2i2( 1113857

2

a2i minus3a21i + a2

2i( 1113857F3i

+a1i 2a2i a21i + a2


2( 1113857F1i( 1113857

⎛⎜⎝ ⎞⎟⎠

(B1)


2a1a2 a21 + a 2

2( 1113857c3EiIi


2 minus a2i2( 1113857F4i( 1113857( 1113857

2a1ia2i a1i2 + a2i

2( 11138572ci

4EiIix1113872 1113873 (B2)


2a1ia2ici2 minus


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


a21i + a2i

2( 11138572c 2

i x

4a1i2a2i


2( 1113857 + a2i2 minus 4a1i

4a2i2 + a1i

2 minus1 + 4a2i4( 1113857( 1113857F2i


2i( 11138572 minus 11113872 1113873F4i

⎛⎜⎝ ⎞⎟⎠

(B4)



Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





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




gravel

r (kNm)

Dep

th (m

)

wpwL

w

Soil layer

uucucu


uucucu

c (kPa) φordm





Kv Kh K









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


Tunn

el ch

aina

ge (m

)

Date

46 days


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


430 440 450 460 470 480 490 500 510 515

020

022

024

026

028

030

032

034

036

038

040


225 220 210 200 190 180 170 160 150 143




50 60 70 80 90 100 110 120 130

The a

pplie

d tu

nnel

face

pre

ssur

e (M

Pa)








430 440 450 460 470 480 490 500 510 518

35

40

45

50

55

60

65

70

75

80




e g

rout

ing

volu

me (

m3 )



Track bed


Centerline



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



201453 Ring 135 2014515 Ring 226


20146152014714

20141019

B



Ver

tical

disp

lace

men

t (m

m)


A


400 420 440 460 480 500 520 540 560 580 600 620 640 660 670


02468

10121416

C


2014420 Ring 362014502 Ring 1242014528 Ring 270





Hor

izon

tal d

ispla

cem

ent (

mm

)

A B










q(x) ks(x) (2)



+(3minus 4v)z +H


x2 +(z +H)2( )2


Conv

erge

nce d

ispla

cem

ent (

mm

)


012345


201453 Ring 135201459 Ring 1912014528 Ring 270



2013129 Ring 4820131222 Ring 6420131230 Ring 87



A B



xk1

k2

e existing tunnel

Sz(x)




ε0 4Rg + g2

4R2 (4)



Gp 2Δ + δ (6)



d4w(x)dx4

+ 4λ4p(x) 4λ4q(x) (7)




p(x) kw(x) (8)





w(x) 1



w(x) 1

8EIλ3int+infin






d2wi(x)dx4




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



w(x) sum4

i1CiFi(x) (13)













(18)





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)






( ) (21)



4c3i EiIi




E0I0


E1I1E2I2



qi(x)

pi

Qi Qi+1Mi Mi+1

Oimi

xi


q0q1 q2

qiminus1 qnminus1


q (x)

0



For the rst element








Suppose






where

An xn( ) prod1

inKi Li( ) (30)


M(0) 0

V(0) 2

k

G

radic


V(L) 0

V(L) minus2

k

G

radic


(31)


64 Case Study



k 5E0β


G 13E0B

2

32 1 + v0( )βtimes 085

(32)


E0 Es

1minus v2s

v0 vs

1minus vs

(33)


E0 (25 sim 35)Es01minus02 (34)



(35)


N1(01) Ni(xi)N1(x1) = N2(02) N2(x2)L1

L1

E1I1



L2 L3 Li LnLnminus1

L2 Li

x x x


Ni(0i)



7 Discussions


Mi MiL0EiIi

Qi QiL

20

EiIi

(36)



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



(a)

387 400 420 440 460 480 500 520 540 560 580 595

minus000015

minus000010

minus000005

000000

000005

000010

000015


e s

lope

of t

he ex

istin

g tu

nnel

(rad

)


(b)


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


(c)


387 400 420 440 460 480 500 520 540 560 580 595

minus600

minus400

minus200

0

200

400

600


e s

lope

of t

he ex

istin

g tu

nnel

(kN

)

(d)





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


(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

)


(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)







(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)


(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)


(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)


(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)




8 Conclusions






Appendices


θ(x) 1113946+infin

minusinfin


b


b

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

dt

(A1)

M(x) 14λ

1113946+infin



Q(x) 1113946+infin

minusinfin

minus12


12


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

dt (A3)


w(x) λ2bk

1113946+infin




sin a2λ|xminus τ|( 11138571113888 1113889dτ

θ(x) 1113946+infin

minusinfin


a1a2b


a1a2b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A4)


M(x) 14λ

1113946+infin




sin a2λ|xminus t|( 11138571113888 1113889dt

Q(x) 1113946+infin

minusinfin

minus12


a21 minus a 2

22a1a2


12


a21 minus a2

2


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A5)


k15 minusbiqiminus1

4a1ia2ici4EiIi a1i

2 + a2i2( 1113857


+ a1i2 minus a2i

2( 1113857F4i

⎛⎜⎝ ⎞⎟⎠



2 + a2i2( 1113857

2

a2i minus3a21i + a2

2i( 1113857F3i

+a1i 2a2i a21i + a2


2( 1113857F1i( 1113857

⎛⎜⎝ ⎞⎟⎠

(B1)


2a1a2 a21 + a 2

2( 1113857c3EiIi


2 minus a2i2( 1113857F4i( 1113857( 1113857

2a1ia2i a1i2 + a2i

2( 11138572ci

4EiIix1113872 1113873 (B2)


2a1ia2ici2 minus


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


a21i + a2i

2( 11138572c 2

i x

4a1i2a2i


2( 1113857 + a2i2 minus 4a1i

4a2i2 + a1i

2 minus1 + 4a2i4( 1113857( 1113857F2i


2i( 11138572 minus 11113872 1113873F4i

⎛⎜⎝ ⎞⎟⎠

(B4)



Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






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


Tunn

el ch

aina

ge (m

)

Date

46 days


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


430 440 450 460 470 480 490 500 510 515

020

022

024

026

028

030

032

034

036

038

040


225 220 210 200 190 180 170 160 150 143




50 60 70 80 90 100 110 120 130

The a

pplie

d tu

nnel

face

pre

ssur

e (M

Pa)








430 440 450 460 470 480 490 500 510 518

35

40

45

50

55

60

65

70

75

80




e g

rout

ing

volu

me (

m3 )



Track bed


Centerline



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



201453 Ring 135 2014515 Ring 226


20146152014714

20141019

B



Ver

tical

disp

lace

men

t (m

m)


A


400 420 440 460 480 500 520 540 560 580 600 620 640 660 670


02468

10121416

C


2014420 Ring 362014502 Ring 1242014528 Ring 270





Hor

izon

tal d

ispla

cem

ent (

mm

)

A B










q(x) ks(x) (2)



+(3minus 4v)z +H


x2 +(z +H)2( )2


Conv

erge

nce d

ispla

cem

ent (

mm

)


012345


201453 Ring 135201459 Ring 1912014528 Ring 270



2013129 Ring 4820131222 Ring 6420131230 Ring 87



A B



xk1

k2

e existing tunnel

Sz(x)




ε0 4Rg + g2

4R2 (4)



Gp 2Δ + δ (6)



d4w(x)dx4

+ 4λ4p(x) 4λ4q(x) (7)




p(x) kw(x) (8)





w(x) 1



w(x) 1

8EIλ3int+infin






d2wi(x)dx4




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



w(x) sum4

i1CiFi(x) (13)













(18)





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)






( ) (21)



4c3i EiIi




E0I0


E1I1E2I2



qi(x)

pi

Qi Qi+1Mi Mi+1

Oimi

xi


q0q1 q2

qiminus1 qnminus1


q (x)

0



For the rst element








Suppose






where

An xn( ) prod1

inKi Li( ) (30)


M(0) 0

V(0) 2

k

G

radic


V(L) 0

V(L) minus2

k

G

radic


(31)


64 Case Study



k 5E0β


G 13E0B

2

32 1 + v0( )βtimes 085

(32)


E0 Es

1minus v2s

v0 vs

1minus vs

(33)


E0 (25 sim 35)Es01minus02 (34)



(35)


N1(01) Ni(xi)N1(x1) = N2(02) N2(x2)L1

L1

E1I1



L2 L3 Li LnLnminus1

L2 Li

x x x


Ni(0i)



7 Discussions


Mi MiL0EiIi

Qi QiL

20

EiIi

(36)



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



(a)

387 400 420 440 460 480 500 520 540 560 580 595

minus000015

minus000010

minus000005

000000

000005

000010

000015


e s

lope

of t

he ex

istin

g tu

nnel

(rad

)


(b)


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


(c)


387 400 420 440 460 480 500 520 540 560 580 595

minus600

minus400

minus200

0

200

400

600


e s

lope

of t

he ex

istin

g tu

nnel

(kN

)

(d)





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


(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

)


(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)







(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)


(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)


(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)


(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)




8 Conclusions






Appendices


θ(x) 1113946+infin

minusinfin


b


b

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

dt

(A1)

M(x) 14λ

1113946+infin



Q(x) 1113946+infin

minusinfin

minus12


12


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

dt (A3)


w(x) λ2bk

1113946+infin




sin a2λ|xminus τ|( 11138571113888 1113889dτ

θ(x) 1113946+infin

minusinfin


a1a2b


a1a2b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A4)


M(x) 14λ

1113946+infin




sin a2λ|xminus t|( 11138571113888 1113889dt

Q(x) 1113946+infin

minusinfin

minus12


a21 minus a 2

22a1a2


12


a21 minus a2

2


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A5)


k15 minusbiqiminus1

4a1ia2ici4EiIi a1i

2 + a2i2( 1113857


+ a1i2 minus a2i

2( 1113857F4i

⎛⎜⎝ ⎞⎟⎠



2 + a2i2( 1113857

2

a2i minus3a21i + a2

2i( 1113857F3i

+a1i 2a2i a21i + a2


2( 1113857F1i( 1113857

⎛⎜⎝ ⎞⎟⎠

(B1)


2a1a2 a21 + a 2

2( 1113857c3EiIi


2 minus a2i2( 1113857F4i( 1113857( 1113857

2a1ia2i a1i2 + a2i

2( 11138572ci

4EiIix1113872 1113873 (B2)


2a1ia2ici2 minus


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


a21i + a2i

2( 11138572c 2

i x

4a1i2a2i


2( 1113857 + a2i2 minus 4a1i

4a2i2 + a1i

2 minus1 + 4a2i4( 1113857( 1113857F2i


2i( 11138572 minus 11113872 1113873F4i

⎛⎜⎝ ⎞⎟⎠

(B4)



Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







430 440 450 460 470 480 490 500 510 518

35

40

45

50

55

60

65

70

75

80




e g

rout

ing

volu

me (

m3 )



Track bed


Centerline



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



201453 Ring 135 2014515 Ring 226


20146152014714

20141019

B



Ver

tical

disp

lace

men

t (m

m)


A


400 420 440 460 480 500 520 540 560 580 600 620 640 660 670


02468

10121416

C


2014420 Ring 362014502 Ring 1242014528 Ring 270





Hor

izon

tal d

ispla

cem

ent (

mm

)

A B










q(x) ks(x) (2)



+(3minus 4v)z +H


x2 +(z +H)2( )2


Conv

erge

nce d

ispla

cem

ent (

mm

)


012345


201453 Ring 135201459 Ring 1912014528 Ring 270



2013129 Ring 4820131222 Ring 6420131230 Ring 87



A B



xk1

k2

e existing tunnel

Sz(x)




ε0 4Rg + g2

4R2 (4)



Gp 2Δ + δ (6)



d4w(x)dx4

+ 4λ4p(x) 4λ4q(x) (7)




p(x) kw(x) (8)





w(x) 1



w(x) 1

8EIλ3int+infin






d2wi(x)dx4




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



w(x) sum4

i1CiFi(x) (13)













(18)





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)






( ) (21)



4c3i EiIi




E0I0


E1I1E2I2



qi(x)

pi

Qi Qi+1Mi Mi+1

Oimi

xi


q0q1 q2

qiminus1 qnminus1


q (x)

0



For the rst element








Suppose






where

An xn( ) prod1

inKi Li( ) (30)


M(0) 0

V(0) 2

k

G

radic


V(L) 0

V(L) minus2

k

G

radic


(31)


64 Case Study



k 5E0β


G 13E0B

2

32 1 + v0( )βtimes 085

(32)


E0 Es

1minus v2s

v0 vs

1minus vs

(33)


E0 (25 sim 35)Es01minus02 (34)



(35)


N1(01) Ni(xi)N1(x1) = N2(02) N2(x2)L1

L1

E1I1



L2 L3 Li LnLnminus1

L2 Li

x x x


Ni(0i)



7 Discussions


Mi MiL0EiIi

Qi QiL

20

EiIi

(36)



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



(a)

387 400 420 440 460 480 500 520 540 560 580 595

minus000015

minus000010

minus000005

000000

000005

000010

000015


e s

lope

of t

he ex

istin

g tu

nnel

(rad

)


(b)


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


(c)


387 400 420 440 460 480 500 520 540 560 580 595

minus600

minus400

minus200

0

200

400

600


e s

lope

of t

he ex

istin

g tu

nnel

(kN

)

(d)





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


(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

)


(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)







(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)


(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)


(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)


(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)




8 Conclusions






Appendices


θ(x) 1113946+infin

minusinfin


b


b

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

dt

(A1)

M(x) 14λ

1113946+infin



Q(x) 1113946+infin

minusinfin

minus12


12


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

dt (A3)


w(x) λ2bk

1113946+infin




sin a2λ|xminus τ|( 11138571113888 1113889dτ

θ(x) 1113946+infin

minusinfin


a1a2b


a1a2b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A4)


M(x) 14λ

1113946+infin




sin a2λ|xminus t|( 11138571113888 1113889dt

Q(x) 1113946+infin

minusinfin

minus12


a21 minus a 2

22a1a2


12


a21 minus a2

2


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A5)


k15 minusbiqiminus1

4a1ia2ici4EiIi a1i

2 + a2i2( 1113857


+ a1i2 minus a2i

2( 1113857F4i

⎛⎜⎝ ⎞⎟⎠



2 + a2i2( 1113857

2

a2i minus3a21i + a2

2i( 1113857F3i

+a1i 2a2i a21i + a2


2( 1113857F1i( 1113857

⎛⎜⎝ ⎞⎟⎠

(B1)


2a1a2 a21 + a 2

2( 1113857c3EiIi


2 minus a2i2( 1113857F4i( 1113857( 1113857

2a1ia2i a1i2 + a2i

2( 11138572ci

4EiIix1113872 1113873 (B2)


2a1ia2ici2 minus


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


a21i + a2i

2( 11138572c 2

i x

4a1i2a2i


2( 1113857 + a2i2 minus 4a1i

4a2i2 + a1i

2 minus1 + 4a2i4( 1113857( 1113857F2i


2i( 11138572 minus 11113872 1113873F4i

⎛⎜⎝ ⎞⎟⎠

(B4)



Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









q(x) ks(x) (2)



+(3minus 4v)z +H


x2 +(z +H)2( )2


Conv

erge

nce d

ispla

cem

ent (

mm

)


012345


201453 Ring 135201459 Ring 1912014528 Ring 270



2013129 Ring 4820131222 Ring 6420131230 Ring 87



A B



xk1

k2

e existing tunnel

Sz(x)




ε0 4Rg + g2

4R2 (4)



Gp 2Δ + δ (6)



d4w(x)dx4

+ 4λ4p(x) 4λ4q(x) (7)




p(x) kw(x) (8)





w(x) 1



w(x) 1

8EIλ3int+infin






d2wi(x)dx4




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



w(x) sum4

i1CiFi(x) (13)













(18)





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)






( ) (21)



4c3i EiIi




E0I0


E1I1E2I2



qi(x)

pi

Qi Qi+1Mi Mi+1

Oimi

xi


q0q1 q2

qiminus1 qnminus1


q (x)

0



For the rst element








Suppose






where

An xn( ) prod1

inKi Li( ) (30)


M(0) 0

V(0) 2

k

G

radic


V(L) 0

V(L) minus2

k

G

radic


(31)


64 Case Study



k 5E0β


G 13E0B

2

32 1 + v0( )βtimes 085

(32)


E0 Es

1minus v2s

v0 vs

1minus vs

(33)


E0 (25 sim 35)Es01minus02 (34)



(35)


N1(01) Ni(xi)N1(x1) = N2(02) N2(x2)L1

L1

E1I1



L2 L3 Li LnLnminus1

L2 Li

x x x


Ni(0i)



7 Discussions


Mi MiL0EiIi

Qi QiL

20

EiIi

(36)



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



(a)

387 400 420 440 460 480 500 520 540 560 580 595

minus000015

minus000010

minus000005

000000

000005

000010

000015


e s

lope

of t

he ex

istin

g tu

nnel

(rad

)


(b)


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


(c)


387 400 420 440 460 480 500 520 540 560 580 595

minus600

minus400

minus200

0

200

400

600


e s

lope

of t

he ex

istin

g tu

nnel

(kN

)

(d)





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


(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

)


(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)







(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)


(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)


(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)


(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)




8 Conclusions






Appendices


θ(x) 1113946+infin

minusinfin


b


b

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

dt

(A1)

M(x) 14λ

1113946+infin



Q(x) 1113946+infin

minusinfin

minus12


12


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

dt (A3)


w(x) λ2bk

1113946+infin




sin a2λ|xminus τ|( 11138571113888 1113889dτ

θ(x) 1113946+infin

minusinfin


a1a2b


a1a2b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A4)


M(x) 14λ

1113946+infin




sin a2λ|xminus t|( 11138571113888 1113889dt

Q(x) 1113946+infin

minusinfin

minus12


a21 minus a 2

22a1a2


12


a21 minus a2

2


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A5)


k15 minusbiqiminus1

4a1ia2ici4EiIi a1i

2 + a2i2( 1113857


+ a1i2 minus a2i

2( 1113857F4i

⎛⎜⎝ ⎞⎟⎠



2 + a2i2( 1113857

2

a2i minus3a21i + a2

2i( 1113857F3i

+a1i 2a2i a21i + a2


2( 1113857F1i( 1113857

⎛⎜⎝ ⎞⎟⎠

(B1)


2a1a2 a21 + a 2

2( 1113857c3EiIi


2 minus a2i2( 1113857F4i( 1113857( 1113857

2a1ia2i a1i2 + a2i

2( 11138572ci

4EiIix1113872 1113873 (B2)


2a1ia2ici2 minus


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


a21i + a2i

2( 11138572c 2

i x

4a1i2a2i


2( 1113857 + a2i2 minus 4a1i

4a2i2 + a1i

2 minus1 + 4a2i4( 1113857( 1113857F2i


2i( 11138572 minus 11113872 1113873F4i

⎛⎜⎝ ⎞⎟⎠

(B4)



Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



ε0 4Rg + g2

4R2 (4)



Gp 2Δ + δ (6)



d4w(x)dx4

+ 4λ4p(x) 4λ4q(x) (7)




p(x) kw(x) (8)





w(x) 1



w(x) 1

8EIλ3int+infin






d2wi(x)dx4




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



w(x) sum4

i1CiFi(x) (13)













(18)





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)






( ) (21)



4c3i EiIi




E0I0


E1I1E2I2



qi(x)

pi

Qi Qi+1Mi Mi+1

Oimi

xi


q0q1 q2

qiminus1 qnminus1


q (x)

0



For the rst element








Suppose






where

An xn( ) prod1

inKi Li( ) (30)


M(0) 0

V(0) 2

k

G

radic


V(L) 0

V(L) minus2

k

G

radic


(31)


64 Case Study



k 5E0β


G 13E0B

2

32 1 + v0( )βtimes 085

(32)


E0 Es

1minus v2s

v0 vs

1minus vs

(33)


E0 (25 sim 35)Es01minus02 (34)



(35)


N1(01) Ni(xi)N1(x1) = N2(02) N2(x2)L1

L1

E1I1



L2 L3 Li LnLnminus1

L2 Li

x x x


Ni(0i)



7 Discussions


Mi MiL0EiIi

Qi QiL

20

EiIi

(36)



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



(a)

387 400 420 440 460 480 500 520 540 560 580 595

minus000015

minus000010

minus000005

000000

000005

000010

000015


e s

lope

of t

he ex

istin

g tu

nnel

(rad

)


(b)


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


(c)


387 400 420 440 460 480 500 520 540 560 580 595

minus600

minus400

minus200

0

200

400

600


e s

lope

of t

he ex

istin

g tu

nnel

(kN

)

(d)





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


(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

)


(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)







(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)


(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)


(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)


(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)




8 Conclusions






Appendices


θ(x) 1113946+infin

minusinfin


b


b

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

dt

(A1)

M(x) 14λ

1113946+infin



Q(x) 1113946+infin

minusinfin

minus12


12


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

dt (A3)


w(x) λ2bk

1113946+infin




sin a2λ|xminus τ|( 11138571113888 1113889dτ

θ(x) 1113946+infin

minusinfin


a1a2b


a1a2b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A4)


M(x) 14λ

1113946+infin




sin a2λ|xminus t|( 11138571113888 1113889dt

Q(x) 1113946+infin

minusinfin

minus12


a21 minus a 2

22a1a2


12


a21 minus a2

2


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A5)


k15 minusbiqiminus1

4a1ia2ici4EiIi a1i

2 + a2i2( 1113857


+ a1i2 minus a2i

2( 1113857F4i

⎛⎜⎝ ⎞⎟⎠



2 + a2i2( 1113857

2

a2i minus3a21i + a2

2i( 1113857F3i

+a1i 2a2i a21i + a2


2( 1113857F1i( 1113857

⎛⎜⎝ ⎞⎟⎠

(B1)


2a1a2 a21 + a 2

2( 1113857c3EiIi


2 minus a2i2( 1113857F4i( 1113857( 1113857

2a1ia2i a1i2 + a2i

2( 11138572ci

4EiIix1113872 1113873 (B2)


2a1ia2ici2 minus


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


a21i + a2i

2( 11138572c 2

i x

4a1i2a2i


2( 1113857 + a2i2 minus 4a1i

4a2i2 + a1i

2 minus1 + 4a2i4( 1113857( 1113857F2i


2i( 11138572 minus 11113872 1113873F4i

⎛⎜⎝ ⎞⎟⎠

(B4)



Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


w(x) sum4

i1CiFi(x) (13)













(18)





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)






( ) (21)



4c3i EiIi




E0I0


E1I1E2I2



qi(x)

pi

Qi Qi+1Mi Mi+1

Oimi

xi


q0q1 q2

qiminus1 qnminus1


q (x)

0



For the rst element








Suppose






where

An xn( ) prod1

inKi Li( ) (30)


M(0) 0

V(0) 2

k

G

radic


V(L) 0

V(L) minus2

k

G

radic


(31)


64 Case Study



k 5E0β


G 13E0B

2

32 1 + v0( )βtimes 085

(32)


E0 Es

1minus v2s

v0 vs

1minus vs

(33)


E0 (25 sim 35)Es01minus02 (34)



(35)


N1(01) Ni(xi)N1(x1) = N2(02) N2(x2)L1

L1

E1I1



L2 L3 Li LnLnminus1

L2 Li

x x x


Ni(0i)



7 Discussions


Mi MiL0EiIi

Qi QiL

20

EiIi

(36)



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



(a)

387 400 420 440 460 480 500 520 540 560 580 595

minus000015

minus000010

minus000005

000000

000005

000010

000015


e s

lope

of t

he ex

istin

g tu

nnel

(rad

)


(b)


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


(c)


387 400 420 440 460 480 500 520 540 560 580 595

minus600

minus400

minus200

0

200

400

600


e s

lope

of t

he ex

istin

g tu

nnel

(kN

)

(d)





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


(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

)


(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)







(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)


(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)


(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)


(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)




8 Conclusions






Appendices


θ(x) 1113946+infin

minusinfin


b


b

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

dt

(A1)

M(x) 14λ

1113946+infin



Q(x) 1113946+infin

minusinfin

minus12


12


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

dt (A3)


w(x) λ2bk

1113946+infin




sin a2λ|xminus τ|( 11138571113888 1113889dτ

θ(x) 1113946+infin

minusinfin


a1a2b


a1a2b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A4)


M(x) 14λ

1113946+infin




sin a2λ|xminus t|( 11138571113888 1113889dt

Q(x) 1113946+infin

minusinfin

minus12


a21 minus a 2

22a1a2


12


a21 minus a2

2


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A5)


k15 minusbiqiminus1

4a1ia2ici4EiIi a1i

2 + a2i2( 1113857


+ a1i2 minus a2i

2( 1113857F4i

⎛⎜⎝ ⎞⎟⎠



2 + a2i2( 1113857

2

a2i minus3a21i + a2

2i( 1113857F3i

+a1i 2a2i a21i + a2


2( 1113857F1i( 1113857

⎛⎜⎝ ⎞⎟⎠

(B1)


2a1a2 a21 + a 2

2( 1113857c3EiIi


2 minus a2i2( 1113857F4i( 1113857( 1113857

2a1ia2i a1i2 + a2i

2( 11138572ci

4EiIix1113872 1113873 (B2)


2a1ia2ici2 minus


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


a21i + a2i

2( 11138572c 2

i x

4a1i2a2i


2( 1113857 + a2i2 minus 4a1i

4a2i2 + a1i

2 minus1 + 4a2i4( 1113857( 1113857F2i


2i( 11138572 minus 11113872 1113873F4i

⎛⎜⎝ ⎞⎟⎠

(B4)



Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


For the rst element








Suppose






where

An xn( ) prod1

inKi Li( ) (30)


M(0) 0

V(0) 2

k

G

radic


V(L) 0

V(L) minus2

k

G

radic


(31)


64 Case Study



k 5E0β


G 13E0B

2

32 1 + v0( )βtimes 085

(32)


E0 Es

1minus v2s

v0 vs

1minus vs

(33)


E0 (25 sim 35)Es01minus02 (34)



(35)


N1(01) Ni(xi)N1(x1) = N2(02) N2(x2)L1

L1

E1I1



L2 L3 Li LnLnminus1

L2 Li

x x x


Ni(0i)



7 Discussions


Mi MiL0EiIi

Qi QiL

20

EiIi

(36)



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



(a)

387 400 420 440 460 480 500 520 540 560 580 595

minus000015

minus000010

minus000005

000000

000005

000010

000015


e s

lope

of t

he ex

istin

g tu

nnel

(rad

)


(b)


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


(c)


387 400 420 440 460 480 500 520 540 560 580 595

minus600

minus400

minus200

0

200

400

600


e s

lope

of t

he ex

istin

g tu

nnel

(kN

)

(d)





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


(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

)


(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)







(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)


(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)


(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)


(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)




8 Conclusions






Appendices


θ(x) 1113946+infin

minusinfin


b


b

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

dt

(A1)

M(x) 14λ

1113946+infin



Q(x) 1113946+infin

minusinfin

minus12


12


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

dt (A3)


w(x) λ2bk

1113946+infin




sin a2λ|xminus τ|( 11138571113888 1113889dτ

θ(x) 1113946+infin

minusinfin


a1a2b


a1a2b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A4)


M(x) 14λ

1113946+infin




sin a2λ|xminus t|( 11138571113888 1113889dt

Q(x) 1113946+infin

minusinfin

minus12


a21 minus a 2

22a1a2


12


a21 minus a2

2


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A5)


k15 minusbiqiminus1

4a1ia2ici4EiIi a1i

2 + a2i2( 1113857


+ a1i2 minus a2i

2( 1113857F4i

⎛⎜⎝ ⎞⎟⎠



2 + a2i2( 1113857

2

a2i minus3a21i + a2

2i( 1113857F3i

+a1i 2a2i a21i + a2


2( 1113857F1i( 1113857

⎛⎜⎝ ⎞⎟⎠

(B1)


2a1a2 a21 + a 2

2( 1113857c3EiIi


2 minus a2i2( 1113857F4i( 1113857( 1113857

2a1ia2i a1i2 + a2i

2( 11138572ci

4EiIix1113872 1113873 (B2)


2a1ia2ici2 minus


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


a21i + a2i

2( 11138572c 2

i x

4a1i2a2i


2( 1113857 + a2i2 minus 4a1i

4a2i2 + a1i

2 minus1 + 4a2i4( 1113857( 1113857F2i


2i( 11138572 minus 11113872 1113873F4i

⎛⎜⎝ ⎞⎟⎠

(B4)



Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


7 Discussions


Mi MiL0EiIi

Qi QiL

20

EiIi

(36)



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



(a)

387 400 420 440 460 480 500 520 540 560 580 595

minus000015

minus000010

minus000005

000000

000005

000010

000015


e s

lope

of t

he ex

istin

g tu

nnel

(rad

)


(b)


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


(c)


387 400 420 440 460 480 500 520 540 560 580 595

minus600

minus400

minus200

0

200

400

600


e s

lope

of t

he ex

istin

g tu

nnel

(kN

)

(d)





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


(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

)


(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)







(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)


(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)


(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)


(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)




8 Conclusions






Appendices


θ(x) 1113946+infin

minusinfin


b


b

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

dt

(A1)

M(x) 14λ

1113946+infin



Q(x) 1113946+infin

minusinfin

minus12


12


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

dt (A3)


w(x) λ2bk

1113946+infin




sin a2λ|xminus τ|( 11138571113888 1113889dτ

θ(x) 1113946+infin

minusinfin


a1a2b


a1a2b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A4)


M(x) 14λ

1113946+infin




sin a2λ|xminus t|( 11138571113888 1113889dt

Q(x) 1113946+infin

minusinfin

minus12


a21 minus a 2

22a1a2


12


a21 minus a2

2


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A5)


k15 minusbiqiminus1

4a1ia2ici4EiIi a1i

2 + a2i2( 1113857


+ a1i2 minus a2i

2( 1113857F4i

⎛⎜⎝ ⎞⎟⎠



2 + a2i2( 1113857

2

a2i minus3a21i + a2

2i( 1113857F3i

+a1i 2a2i a21i + a2


2( 1113857F1i( 1113857

⎛⎜⎝ ⎞⎟⎠

(B1)


2a1a2 a21 + a 2

2( 1113857c3EiIi


2 minus a2i2( 1113857F4i( 1113857( 1113857

2a1ia2i a1i2 + a2i

2( 11138572ci

4EiIix1113872 1113873 (B2)


2a1ia2ici2 minus


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


a21i + a2i

2( 11138572c 2

i x

4a1i2a2i


2( 1113857 + a2i2 minus 4a1i

4a2i2 + a1i

2 minus1 + 4a2i4( 1113857( 1113857F2i


2i( 11138572 minus 11113872 1113873F4i

⎛⎜⎝ ⎞⎟⎠

(B4)



Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




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


(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

)


(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)







(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)


(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)


(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)


(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)




8 Conclusions






Appendices


θ(x) 1113946+infin

minusinfin


b


b

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

dt

(A1)

M(x) 14λ

1113946+infin



Q(x) 1113946+infin

minusinfin

minus12


12


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

dt (A3)


w(x) λ2bk

1113946+infin




sin a2λ|xminus τ|( 11138571113888 1113889dτ

θ(x) 1113946+infin

minusinfin


a1a2b


a1a2b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A4)


M(x) 14λ

1113946+infin




sin a2λ|xminus t|( 11138571113888 1113889dt

Q(x) 1113946+infin

minusinfin

minus12


a21 minus a 2

22a1a2


12


a21 minus a2

2


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A5)


k15 minusbiqiminus1

4a1ia2ici4EiIi a1i

2 + a2i2( 1113857


+ a1i2 minus a2i

2( 1113857F4i

⎛⎜⎝ ⎞⎟⎠



2 + a2i2( 1113857

2

a2i minus3a21i + a2

2i( 1113857F3i

+a1i 2a2i a21i + a2


2( 1113857F1i( 1113857

⎛⎜⎝ ⎞⎟⎠

(B1)


2a1a2 a21 + a 2

2( 1113857c3EiIi


2 minus a2i2( 1113857F4i( 1113857( 1113857

2a1ia2i a1i2 + a2i

2( 11138572ci

4EiIix1113872 1113873 (B2)


2a1ia2ici2 minus


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


a21i + a2i

2( 11138572c 2

i x

4a1i2a2i


2( 1113857 + a2i2 minus 4a1i

4a2i2 + a1i

2 minus1 + 4a2i4( 1113857( 1113857F2i


2i( 11138572 minus 11113872 1113873F4i

⎛⎜⎝ ⎞⎟⎠

(B4)



Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






(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)


(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)


(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)


(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)




8 Conclusions






Appendices


θ(x) 1113946+infin

minusinfin


b


b

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

dt

(A1)

M(x) 14λ

1113946+infin



Q(x) 1113946+infin

minusinfin

minus12


12


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

dt (A3)


w(x) λ2bk

1113946+infin




sin a2λ|xminus τ|( 11138571113888 1113889dτ

θ(x) 1113946+infin

minusinfin


a1a2b


a1a2b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A4)


M(x) 14λ

1113946+infin




sin a2λ|xminus t|( 11138571113888 1113889dt

Q(x) 1113946+infin

minusinfin

minus12


a21 minus a 2

22a1a2


12


a21 minus a2

2


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A5)


k15 minusbiqiminus1

4a1ia2ici4EiIi a1i

2 + a2i2( 1113857


+ a1i2 minus a2i

2( 1113857F4i

⎛⎜⎝ ⎞⎟⎠



2 + a2i2( 1113857

2

a2i minus3a21i + a2

2i( 1113857F3i

+a1i 2a2i a21i + a2


2( 1113857F1i( 1113857

⎛⎜⎝ ⎞⎟⎠

(B1)


2a1a2 a21 + a 2

2( 1113857c3EiIi


2 minus a2i2( 1113857F4i( 1113857( 1113857

2a1ia2i a1i2 + a2i

2( 11138572ci

4EiIix1113872 1113873 (B2)


2a1ia2ici2 minus


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


a21i + a2i

2( 11138572c 2

i x

4a1i2a2i


2( 1113857 + a2i2 minus 4a1i

4a2i2 + a1i

2 minus1 + 4a2i4( 1113857( 1113857F2i


2i( 11138572 minus 11113872 1113873F4i

⎛⎜⎝ ⎞⎟⎠

(B4)



Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



8 Conclusions






Appendices


θ(x) 1113946+infin

minusinfin


b


b

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

dt

(A1)

M(x) 14λ

1113946+infin



Q(x) 1113946+infin

minusinfin

minus12


12


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

dt (A3)


w(x) λ2bk

1113946+infin




sin a2λ|xminus τ|( 11138571113888 1113889dτ

θ(x) 1113946+infin

minusinfin


a1a2b


a1a2b

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A4)


M(x) 14λ

1113946+infin




sin a2λ|xminus t|( 11138571113888 1113889dt

Q(x) 1113946+infin

minusinfin

minus12


a21 minus a 2

22a1a2


12


a21 minus a2

2


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A5)


k15 minusbiqiminus1

4a1ia2ici4EiIi a1i

2 + a2i2( 1113857


+ a1i2 minus a2i

2( 1113857F4i

⎛⎜⎝ ⎞⎟⎠



2 + a2i2( 1113857

2

a2i minus3a21i + a2

2i( 1113857F3i

+a1i 2a2i a21i + a2


2( 1113857F1i( 1113857

⎛⎜⎝ ⎞⎟⎠

(B1)


2a1a2 a21 + a 2

2( 1113857c3EiIi


2 minus a2i2( 1113857F4i( 1113857( 1113857

2a1ia2i a1i2 + a2i

2( 11138572ci

4EiIix1113872 1113873 (B2)


2a1ia2ici2 minus


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


a21i + a2i

2( 11138572c 2

i x

4a1i2a2i


2( 1113857 + a2i2 minus 4a1i

4a2i2 + a1i

2 minus1 + 4a2i4( 1113857( 1113857F2i


2i( 11138572 minus 11113872 1113873F4i

⎛⎜⎝ ⎞⎟⎠

(B4)



Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


M(x) 14λ

1113946+infin




sin a2λ|xminus t|( 11138571113888 1113889dt

Q(x) 1113946+infin

minusinfin

minus12


a21 minus a 2

22a1a2


12


a21 minus a2

2


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

dt

(A5)


k15 minusbiqiminus1

4a1ia2ici4EiIi a1i

2 + a2i2( 1113857


+ a1i2 minus a2i

2( 1113857F4i

⎛⎜⎝ ⎞⎟⎠



2 + a2i2( 1113857

2

a2i minus3a21i + a2

2i( 1113857F3i

+a1i 2a2i a21i + a2


2( 1113857F1i( 1113857

⎛⎜⎝ ⎞⎟⎠

(B1)


2a1a2 a21 + a 2

2( 1113857c3EiIi


2 minus a2i2( 1113857F4i( 1113857( 1113857

2a1ia2i a1i2 + a2i

2( 11138572ci

4EiIix1113872 1113873 (B2)


2a1ia2ici2 minus


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


a21i + a2i

2( 11138572c 2

i x

4a1i2a2i


2( 1113857 + a2i2 minus 4a1i

4a2i2 + a1i

2 minus1 + 4a2i4( 1113857( 1113857F2i


2i( 11138572 minus 11113872 1113873F4i

⎛⎜⎝ ⎞⎟⎠

(B4)



Acknowledgments


References

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


field observation and theoretical study on an existing...

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