Research ArticleDynamic Responses of a Twin-Tunnel Subjected to MovingLoads in a Saturated Half-Space
Hong-lei Sun 1 An-hua Chen1 Li Shi 2 Xue-yu Geng3 and YuWang4
1 Institute of Disaster Prevention Engineering Zhejiang University Hangzhou Zhejiang 310058 China2Institute of Geotechnical Engineering Zhejiang University of Technology Hangzhou Zhejiang 310000 China3School of Engineering University of Warwick Coventry CV4 7AL UK4China Railway Design Corporation Beijing China
Correspondence should be addressed to Li Shi lishizjuteducn
Received 25 July 2018 Accepted 2 December 2018 Published 12 December 2018
Academic Editor Nicolae Herisanu
Copyright copy 2018 Hong-lei Sun et al This 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
With the fast development of rail transit the environmental vibration problems caused by subways have received increasingattention A 3D finite element model was built in this study to investigate the ground vibrations induced by the moving loadoperating in the parallel twin tunnels Compared to the model consisting of a single tunnel that was commonly adopted in thepast studies a pair of tunnels is considered and the surrounding medium of the tunnels is taken as a saturated porous mediumThe governing equations of the 3D finite element method modeling of the saturated poroelastic soil have been derived accordingto Biotrsquos theory Computed results showed that the dynamic response of the twin-tunnel model is greater than that of the singletunnel model And the spacing between two tunnels tunnel buried depth and load moving speed are the essential parameters todetermine the dynamic response of the tunnel and soil
1 Introduction
Recently concerns about the environmental vibrationsinduced by the rail transit have increased substantiallyWhen the subways run close to the existing infrastructuresvibrations are transmitted to the infrastructure through theground which can cause annoyance to inhabitants or resultin malfunction interruption to sensitive equipment Due tothe high groundwater table in southeastern area of Chinaeg Shanghai Zhejiang Jiangsu and Guangdong subwaysin these areas are running in the saturated soil Besides it isa common practice to construct underground railway linesin pairs The vibrations caused by twin tunnels cannot besimplified to be the sumof those caused by two single tunnelsTherefore an investigation on ground vibrations inducedby subways in the context of twin tunnels embedding in asaturated ground is necessary and desirable
The vibrations caused by subways have been investigatedby many scholars using both analytical and computationalmethods Metrikine et al [1] presented an analytical method
to simplify the metro to Euler beams embedded in vis-coelastic medium Their study focused on the effect of loadvelocity on the structural response Forrest and Hunt [2 3]built a three-dimensional analytical model for a circularsubway tunnel buried deep underground in order to obtainits dynamic responses Liu et al [4 5] studied the transientresponse of partially sealed spherical cavity embedded inviscoelastic saturated soil The partial permeability of theboundary and the influence of the relative stiffness betweenlining and soil on the transient response were investigatedYuan et al [6] analyzed the influences of the load velocity andthe oscillating frequencies on the structural displacement ofthe track the displacement and pore pressure responses ofthe ground Considering the train load excitation as a randomprocess Hunt et al [7] studied the ground vibrations dueto the subway trains using a track-building model of infinitelength
Compared with the analytical method the computa-tional method eg finite element method (FEM) standsout for its easy handling of the material heterogeneity and
HindawiMathematical Problems in EngineeringVolume 2018 Article ID 6949507 12 pageshttpsdoiorg10115520186949507
2 Mathematical Problems in Engineering
complexirregular geometry Balendra et al [8] investigatedthe coupled vibrations between the subway the ground andthe building using a 2D plane-strain finite element modelA viscous absorbing boundary condition was applied tothe truncation boundary to suppress unwanted reflectionsA 2D finite element model has been established by Wanget al [9] including the subway tunnel the ground andthe adjacent building The impact of the tunnel depthson the vibrations of different floors and positions in thebuildings was investigated Yang et al [10] studied thedynamic interaction between the ground and tunnel sys-tem using the finite element method At the truncationboundary the infinite element was applied to damp out theoutgoing waves Similarly Zhang and Pan [11] establisheda coupled finite and infinite element model to study thevibration of metro tunnel structure and surrounding stra-tum A coupled finite element (FE)ndashboundary element (BE)approach was proposed by Jones et al [12] to study thewave transmissions and the ground vibrations induced by thesubways
However the wave field generated by the moving load isthree-dimensional in nature which restricts the applicabilityof 2D computational models Gardien and Stuit et al [13]analyzed the ground vibrations generated by the subwayusing the finite element code LS-DYNAWolf et al [14] builta 3D prediction model for low-frequency ground vibrationinduced by the subway using the finite difference methodwhich is calibrated using the empirical relationship estab-lished on the measured field data G Degrande et al [15]proposed a coupled finite element (FE)ndashboundary element(BE) formulation to study the dynamic interaction betweenthe tunnel and the ground in which the tunnel was modeledby FEM and the ground by BEM The dynamic responsesof a shallow cut-and-cover masonry tunnel were comparedto those of a deep bored one A 3D periodic FE-BE modelwas established by Liu et al [16] to obtain the train-inducedvibrations for both the tunnel and the free field The floatingslab track and the general slab track are compared for theireffectiveness in vibration reduction
As mentioned above the current research focuses onthe environmental vibrations problems generated by subwaysrunning in a single tunnel The effect of the neighboringtunnel is not considered although it is a common practicethat underground railway lines are constructed in pairs Thevibration caused by two metro tunnels is different to thesum of two single tunnels since the neighboring tunnel canimpede and screen the vibration waves generated by theoperating tunnel To the best of the authorsrsquo knowledge Kuoet al [17] presented the investigation on the dynamics of atwin tunnel embedded in elastic full-space adopting a cou-pled finite elementndashboundary element approach Parametricstudies were carried out for the load excitation frequencythe tunnel orientations and the tunnel geometry whichdemonstrated that the dynamic interactions between thetwo tunnels were highly significant However the full spacemodel used by Kuo et al is not applicable when the groundvibrations are of concern since the ground is semi-infinitewith a free surface Moreover the possible influences of theload speed the tunnel burying depth and the tunnel spacing
on the dynamics responses of the twin-tunnel and the groundremain untouched in the literature
The ground in the above-referred papers using numer-ical approach is generally modeled as single-phase elasticmedium in which the ground water is not consideredExisting research by Yuan et al [18] showed that the dynamicresponses of the subway tunnel in the saturated groundare different from those in the elastic ground Therefore itis necessary to model the ground as saturated poroelasticmedium such that the ground water can be taken intoaccount
This paper focuses on the coupled vibrations of the linerthe twin-parallel tunnels and the saturated ground using thefinite element method A moving point load is applied to theinvert to represent the subway excitationThe dynamics of thesurrounding saturated soil are governed by Biotrsquos poroelastictheory To suppress spurious reflections at the truncationboundaries an artificial boundary condition named themulti-transmitting formula (MTF) that is proposed by theLiao and Wong [19] and extended by Shi et al [20] isapplied to transmit the outgoing waves in the saturatedsoil medium The effectiveness and stability of MTF havebeen demonstrated in the paper by Shi et al [20] Usingthe established 3D finite element model parametric studieshave been performed for investigating the effects of theload velocity the burying depth of tunnel and the tunnelspacing on the coupled vibrations of the liner-tunnel-groundsystem Because the effect of the permeability coefficient onthe dynamic response of soil has been fully discussed bymany researchers andwehave not found any new conclusionsusing the presented model therefore the influence of thepermeability coefficient on vibration response has not beendiscussed in this paper
2 Governing Equations for Saturated Ground
According to Biotrsquos theory [21 22] the governing equationsfor the dynamics of the saturated poroelastic medium read asfollows
120590119894119895119895 + 120588119887119894 minus 120588119894 minus 120588f119894 = 0 (1)
minus119901119894 + 120588f119887119894 minus 120588f 119894 minus 119898119894 minus 119887119894 = 0 (2)
120590119894119895 = 12059010158401015840119894119895 minus 120572120575119894119895119901 (4)
where 120588119887119894 denotes the gravity force acting on the soil skeleton120588 and 120588f denote the density of the soil and the density ofthe pore fluid respectively 120588 = (1 minus 119899)120588s + 119899120588f 120588s is thedensity of the solid skeleton and 119899 is the porosity 119906119894 and 119908119894denote the displacement of soil skeleton and the infiltrationdisplacement of the pore fluid relative to the soil skeletonrespectively ldquosdotrdquo and ldquosdotsdotrdquo denote the first and second derivativeof time respectively 119898 = 120588f119899 119887 = 120588fg119896D 119892 is theacceleration of gravity 119896D is the Darcy permeability 120590119894119895 and12059011989411989510158401015840 denote the total and effective stress tensors respectively120575119894119895 is the Kronecker delta 119901 accounts for the pore pressure 120572
Mathematical Problems in Engineering 3
and119872 are Biot parameters and subscripts i and j denote thetensor operation and the summation convention is applied
Aftermanipulation the pore pressure119901 can be eliminatedfrom the governing equations leaving 119906119894 and 119908119894 the fieldvariables
There are two reasons for keeping 119906119894 and 119908119894 as theprimary unknowns the first reason is that the u minus wformulation is a complete description of Biotrsquos theory Forthe u minus 119901 formulation the relative pore fluid acceleration needs to be neglected [23] which would bring in certaininaccuracy especially for a high-frequency loading situation[23 24] the second reason is that the u minus w formulation isconsistent with the proposed artificial boundary conditionmulti-transmitting formula (MTF given in Section 3) sinceit is vector-based andmanipulates the displacement vectors uand w directly while the pore pressure p is a scalar and thusnot a suitable variable for applying the MTF
The same interpolation function is used for both the soil-skeleton displacement u and the relative pore-fluid displace-ment w which are given as
ue (x 119905) = 119899e
sum119894=1
119873119894 (x) Iue119894 (119905)
we (x 119905) = 119899e
sum119894=1
119873119894 (x) Iwe119894 (119905)
(7)
where x = (119909 119910 119911) is the coordinate 119905 represents time119873119894(x)is the shape function at node 119894 119899e is the number of nodes ofeach element I is the unit matrix of 2 times 2 or 3 times 3 and ue119894 (119905)and we
119894 (119905) are the nodal displacement vectorsFollowing the standard Galerkin procedure the govern-
ing equations of the finite element formulation representingthe saturated poroelastic soil medium can be derived as
[Mss Msw
Mws Mww] ue
we + [0 00 Cww
] uewe
+ [Kss + K1015840ss Ksw
Kws Kww]ue
we = fsfw
(8)
The expressions of the elements in the mass matrix thedamping matrix the stiffness matrix and the force vector aregiven in the appendix
3 The Multi-Transmitting Formula
Due to the semi-infinite nature of the ground the groundvibration analysis using the finite element method requiresartificial boundary conditions to make the computationaldomain finite Many artificial boundary conditions have beenproposed for absorbingdamping the outgoing waves for boththe elastic medium and the saturated poroelastic medium
For a detailed review one is referred to the paper by Shi etal [20] Among the existing artificial boundary conditionsthe multi-transmitting formula that is proposed by Liao andWong [19] for the elastic medium and then extended byShi et al [20] for the saturated poroelastic medium standsout for being local in both time and space and for the easyimplementation in the finite element analysis both for 2D and3D grids
The MTF extrapolates displacement on the artificialboundary at time 119905 = (119899 + 1)Δ119905 as a linear combination ofthe displacements at previous time steps along a straight linenormal to the boundary which is given as
where 119906119894(119895Δ119904 minus119896Δ119904 119899Δ119905) denotes the soil skeleton dis-placements of the grid point at (119909 = 119895Δ119904 119911 = minus119896Δ119904) andat time 119905 = 119899Δ119905 Similarly 119908119894(119895Δ119904 minus119896Δ119904 119899Δ119905) denotes therelative pore-fluid displacement 119888a represents the apparentvelocities of waves propagating along the normal direction ofartificial boundary C119873119896 is the binomial coefficient and N isthe order of the boundary condition
Since the computational points (119911 = minus119896119888aΔ119905) in (9) do notgenerally coincide with the grid points a quadratic interpo-lation scheme is employed to correlate the displacements atthe computational points to those at the grid points as
where the operator A is the backward space shift of 119888aΔ119905119860120601 (119909 119911 119905) = (11987911 + 11987912119885minus1119885 + 11987913119885minus2119885 ) 120601 (119909 119911 119905)
asymp 120601 (119909 119911 minus 119888aΔ119905 119905) (11)
in which the interpolation coefficients are given as
11987911 = (2 minus 119904) (1 minus 119904)211987912 = 119904 (2 minus 119904)11987913 = 119904 (119904 minus 1)2
119904 = 119888aΔ119905Δ119909
(12)
4 Mathematical Problems in Engineering
MTF
MTF
Sym
met
ric
Symmetric
5
4
2
3Y
X
1
0 543210
(a)
Pres
sure
0 1 2
f0
tlowast
(b)
Figure 1 A cavity embedded in an infinite saturated poroelastic medium (a) model setup and mesh grid (b) edge pressure
The operators 119885119905 and 119885119911 are the forward time and spaceshifts respectively
Correspondingly the operators119885minus1119905 and119885minus1119911 represent thebackward time and space shifts respectively
According to Shi et al [20] a second-order MTF (119873 = 2)and the artificial wave velocity 119888119886 = 119888s (119888s is the shear wavevelocity of the saturated medium) are efficient and accurateenough for configuring the MTF Moreover the numericalstability of the MTF can be guaranteed by (1) meeting theCourant-Friedrichs-Lewy (CFL) condition 119904 = 119888aΔ119905Δ119904 le1radic2 and (2) perturbing the interpolation coefficient 11987911 byadding a small negative value (= minus004119904 sim minus03119904)4 Verification of the Present Model
To verify the accuracy of the proposed finite element formu-lation and the effectiveness of the MTF boundary conditionthe transient response due to an edge pressure applied radiallyat a cavity surface is computed and compared to the analyticalsolution given by T Senjuntichai [25] which is developed fora cavity in an infinite saturated soil medium under the plane-strain condition
Due to the symmetry a 14 model is established for thecavity and the surrounding saturated soil medium Symmet-ric boundary conditions are imposed for the left and bottomboundaries while MTF boundary conditions are applied tothe top and right boundaries to account for the infiniteextension of the soil medium The mesh in the x-y plane isshown in Figure 1(a) while themovement in the z direction isfixed tomodel the plain strain conditionThe applied pressureis a triangular pulse as given in Figure 1(b) in which 119905lowast is the
normalized time 119905lowast = (119905119886)radic120583120588 119886 is the radius of tunnel 119891denotes the edge pressure and 1198910 is its peak value
The soil properties and the load parameters are thesame as those given by Senjuntichai [25] and summarizedin Table 1 Figure 2 shows the comparison on the radialdisplacement at the cavity surface where the vertical axis isthe normalized radial displacements 119880119903lowast = 120583119880119903(1198910119886) andthe horizontal axis is the normalized time 119905lowast From Figure 2certain reflections can be observed for the MTF conditionwhen the reflected waves reached the observation point at thefirst time (ie 119905lowast asymp 5) However the following impinging ofthe reflected waves is transmitted effectively by MTF rendinga very close comparison to the analytical solution for 119905lowast gt 8Also it can be seen from Figure 2 that significant reflectionscan happen during the entire time if the fixed boundarycondition is used
5 Numerical Results
In this section coupled vibrations of the liner the twintunnels and the saturated ground are investigated usingthe developed finite element along with the developed MTFboundary condition Parametric studies have been per-formed for the burying depth of tunnel the tunnel spacingthe load velocity and the soil permeability
51 Model Description The computation model consisting oftwo identical circular tunnels that are parallel to each otherand embedded in a three-dimensional saturated ground isshown in Figure 3 The model size is 119886 (width) times 119887 (height)times 119897 (length)The center-to-center spacing between the tunnelsis represented by 119889 and the burying depth measured fromthe ground surface to the tunnel center is denoted by ℎas shown in Figure 3(b) 119903 denotes the outer radius of theliner and the thickness of liner is represented by 119905 The sideand cross-sectional views with dimensions are presented in
Mathematical Problems in Engineering 5
Table 1 Parameters of the edge pressure and the saturated poroelastic medium
Parameter Symbol ValueYoungrsquos modulus 119864 533 times 107 PaPoissonrsquos ratio ] 033Density of the bulk material 120588 2000 kgm3
Density of the pore fluid 120588f 1000 kgm3
Porosity 119899 04Biot parameters 120572 098Biot parameters 119872 40 times 108 PaAdditional mass density 120588a 0 kgm3
Darcy permeability coefficient 119896119863 491 times 10minus3msTriangular pulse load 1198910 200 times 107 PaRadius of tunnel 119886 10m
present methodSenjundichai (1993)fixed boundary
2 4 6 8 10 12 14 160tlowast
minus02
minus01
00
01
02
03
UＬlowast
Figure 2 Comparison of present results with the solution proposedby Senjundichai [25]
Figures 3(a) and 3(b) respectively A nonoscillating pointload of magnitude 119865119899 that moves along the positive 119909 axle actsvertically to the invert of the left tunnel liner
Four observation points are chosen to record theresponses caused by the moving load points A and B arelocated at the liner bottom of the tunnel with the loadapplied and the other tunnel respectively points C and D arelocated on the ground surface right between the twin tunnelsand above the center of the tunnel loaded respectively Allobservation points are located at the middle of the modelalong the tunnel axial direction (ie 119909 = 1198972)
The bottom surface of the model is fixed to consider theunderlying hard stratum The top surface is free and set aspermeable The MTF boundary condition is applied to theremaining 4 side surfaces to account for the infinite extensionof the saturated ground For a reliable modeling of wavepropagations the element sizes of the ground shouldmeet therequirement Δ119904 le 120582s6 where 120582s = 2120587119888s120596max is the shearwave length and 120596max is the highest frequency of the shear
wave [26] The implicit Newmark method is employed forthe timemarching of the dynamic analysis with theNewmarkparameter being set to 06 to remove disturbances caused bysudden application of the point load Although the Newmarkintegration scheme is unconditionally stable the time stepshould meet the CFL condition as mentioned in Section 3since the MTF is essentially a forward Euler scheme
Parameters of the liner the saturated ground and themoving load are summarized in Table 2 In the subsequentanalysis the dynamic displacement and velocity are pre-sented in decibels (dB) ie 119902lowast = 20 lg(1199021199020) where 119902denotes an arbitrary variable and 1199020 is the reference valueThe reference values for displacement and velocity are 20 times10minus5m and 20 times 10minus5ms respectively
52 Models Comparisons In this section a single tunnelmodel and a model with both tunnels loaded are comparedwith the present twin-tunnel model The cross sections ofthe two models are presented in Figures 4(a) and 4(b)respectively The results of single phased model are alsocompared with those of the saturated poroelastic model inthis section
To investigate the influence of interactions between thetwo tunnels on the vibration-prediction results the compar-ison of dynamic responses between single tunnel model andthe present model is shown in Figures 5(a) and 5(b) Theburying depth and the spacing of the tunnels are respectivelyfixed at ℎ119903 = 4 and 119889119903 = 4 and the load velocity is 03 119888sIn Figure 5(a) the tunnel spacing for the single tunnel modeldenotes two times the distance between tunnel center and themodel center The dynamic responses between two modelson point A are very close while the results of the presentmodel are greater than those of the single tunnel model onthe ground observation points (point C and point D) Thedisplacements at point C for the single tunnel model decreasefaster than the present model between 119889119903 which is 4 to5 That is because the dynamic response at point C in thepresent model is the sum of the waves generated from thetunnel with the load applied and reflected wave from theother tunnel lining However the dynamic responses of thesingle-tunnel model only come from the tunnel with the loadapplied As the tunnel spacing increases the differences of the
6 Mathematical Problems in Engineering
D(C)
A(B)MTFMTF
Free and permeable
Fixed
z
y-Plane
l2
l
b
x
(a) Side view
D C
A B
z
MTFMTF
Fixed
Free and permeable
bＨ
d
tr
ℎ
a x-Plane y
(b) Cross section view
Figure 3 Schematic of the 3D computation model
D C
A
z
MTFMTF
Fixed
Free and permeable
b Ｈt r
ℎ
a x-Plane y
d2
(a) Single tunnel model
D C
A B
a
z
MTFMTF
Fixed
Free and permeable
b Ｈ Ｈ
d
tr
ℎ
x-Plane y
(b) Model with both tunnels loaded
Figure 4 Cross sections of single tunnel model and the model with both tunnels loaded
displacement at point C and D for both models decrease InFigure 5(b) it is observed that the maximum displacementsof single tunnelmodel are all smaller than those of the presentmodel when the burying depth changes from 3 to 5 Thedecrease rates for both models are almost the same
The comparison between the single tunnel model andthe present model is shown in Figure 6 under different loadvelocities It is observed from Figure 6 that the dynamicresponses of the twin-tunnel model are larger than those ofthe single tunnel model which is inconsistent with the resultsshown in Figure 5 And the difference of the two models atpoint C is greater than that at point D The differences reachpeak value when 119888119888s = 09 Since the dynamic response atpoint C is jointly affected by two tunnels while the dynamicresponse at point D is dominated by the tunnel loadedtherefore it is essential to consider the twin-tunnel model toobtain more accurate prediction results
Practically both metro tunnels are always loaded atthe same time due to the high operation frequency of thesubways The dynamic responses of the model with bothtunnel inverts loaded are studied in this section
In contrast to the result of the presentmodel the dynamicresponses of point C and point D due to two moving pointloads are larger in terms of the maximum displacement InFigure 7 the maximum displacements of the two models atpoint A are almost the same The dynamic response of thetwomodels at point C decreases a little with increasing tunnelspacing However as tunnel spacing increases the maximum
displacement of the two models at point A remains the sameand the results of two models become closer at point DThis is because as the tunnel spacing increases the effect ofthe other tunnel on point D decreases gradually And withthe increasing of the burying depth the dynamic responsesof points C and D have obvious downtrend Besides thedisplacement at the point C is larger than that at the point Din the model with two loads while the situation of the presentmodel is the opposite That is because the load in the othertunnel has a larger contribution to the dynamic response atpoint C than at point D
The comparison of the displacements and the vibrationvelocity between the present model and the model with bothtunnels loaded is shown in Figures 8 and 9 for differentload velocities For points C and D the dynamic responsesof the model with both tunnels loaded are obviously largerthan the model with only one tunnel loaded However theprediction results of the two models at point A are very closeAnd the maximum displacement at point C is smaller thanthat at point D in the model with only one tunnel loadedwhile the situation of the two-load model is the oppositewhich is explained in the previous section The differencebetween the maximum displacements of the two models isalmost unchanged at different load velocity However thedifference in the vibration velocity results of the two modelsat point D decreases as the velocity increases and reaches aminimum at 119888119888119904 = 09 From Figure 9 it can also be observedthat the maximum vibration velocities at points C and D are
Mathematical Problems in Engineering 7
Table 2 Parameters of the liner the saturated ground and the moving load
Saturated Soil Ground
Width a (m) Height b (m) Length l (m) Biot parameters 12057242 24 120 099
Spacing 119889 (m) Burying depth h (m) Porosity 119899 Poissonrsquos ratio ]9ndash24 6ndash18 0286 04
Youngrsquos modulus Darcy permeability coefficient Shear wave velocity 119888s Biot parameters119864 (Pa) 119896119863 (ms) (ms) 119872 (Nm)223 times 108 10 times 10minus5ndash10 times 10minus2 1911 574 times 109
Density of the bulk material 120588 (kgm3) Density of the pore fluid 120588f (kgm3)2178 times 103 10 times 103
Concrete LinerYoungrsquos modulus Poissonrsquos ratio Density of the bulk material outer radius Thickness119864 (Pa) ] 120588 (kgm3) r(m) t(m)
250 times 1010 02 2400 times 103 30 03
Moving Load Point load value (N) Load velocity (ms)20 times 104 1911ndash1911
0
4 45 5 55 6Tunnel spacing dr
point A(twin-tunnel)point C(twin-tunnel)point D(twin-tunnel)point A(single tunnel)point C(single tunnel)point D(single tunnel)
(b) Maximum displacement at the observation points for differentburying depth
Figure 5 Comparison of dynamic responses between a single-tunnel model and a twin-tunnel model
almost the same in the present model with only one tunnelloaded while the value at point C in the two-load model issignificantly larger than that at point D Because the load inthe other tunnel is closer to point C than point D it has agreater impact on the dynamic response of point C
To illustrate the necessity in introducing the saturatedground model the dynamic responses of the twin tunnelembedded in a single-phase ground are compared to thoseof the saturated ground model and presented in Figure 10The single-phase ground model is obtained by reducing thesaturated ground model after fixing the pore-fluid relativedisplacement and setting Biotrsquos parameters to zeros It isobserved that the dynamic responses of the elastic ground arelarger than those in the saturated ground The discrepancybetween the two ground models widens when the buryingdepth increases And as the load velocity increases the
difference of dynamic response between the two models atpoint D also becomes larger From the comparison it can beconcluded that the ground water has an obvious influence onthe vibration responses of the tunnel and the ground Theirvibration levels would be overestimated if the ground wateris not considered
53 e Influences of Load Velocity The influences of loadvelocity on the dynamic responses of the liner-tunnel-groundsystem are investigated in this section Six different velocities119888119888s = 01 03 05 07 09 and 10 are considered along withthe tunnel spacing 119889119903 = 4 and the burying depth ℎ119903 = 4
It is observed from Figure 11 that the displacements atpoints A and B change slightly with the increasing loadvelocity while the displacements at points C and D reacha peak at around 09 119888s Since C and D are on the ground
8 Mathematical Problems in Engineering
point C(twin-tunnel)point D(twin-tunnel)point C(single tunnel)point D(single tunnel)
Max
imum
disp
lace
men
t0 02 04 06 08 1
Load velocity s
minus365
minus360
minus355
minus350
minus345
minus340
minus335
minus330
UZlowast
(dB)
Figure 6 Comparison on maximum displacement at observation points between a single-tunnel model and a twin-tunnel model
(a) Maximum displacement at the observation points for differenttunnel spacing
3 35 4 45 5Burying depth hr
point C(one load)point D(one load)point C(two loads)point D(two loads)
Max
imum
disp
lace
men
t
minus45
minus40
minus35
minus30
minus25
minus20
minus15
minus10
UZlowast
(dB)
(b) Maximum displacement at the observation points for differentburying depth
Figure 7 Comparison of dynamic responses between the present model and a model with both tunnels loaded
surface their responses will be amplified when the loadvelocity approaches the Rayleigh wave velocity of the groundwhich is around 09 119888s The same results were reported inprevious studies on the critical velocity for a train runningon the ground surface [27 28] However points A and B areset on the concrete liner whose shear wave velocity is muchhigher than that of the soil Thus the load velocity is very lowcompared to the shear wave velocity of the concrete whichmakes the displacement of points A and B nonsensitive to thechange of load velocity
54 e Influence of Tunnel Spacing In this section theinfluence of the tunnel spacing on the dynamic responses isinvestigated by considering 6 different twin tunnel distancesie 119889119903 = 3 4 5 6 7 8 The load velocity is 03 119888119904 The
maximum vertical displacements at the observation pointswhen ℎ119903 = 4 are compared in Figure 12
The displacement at point A is significantly larger thanthose at other points since point A is traversed directlyby the point load When the tunnel spacing increases thedisplacements at points B and C decrease gradually Howeverthe displacements at points A and D remain almost the sameThis observation is within expectation because the positionsof points A and D remain the same while points B and C areshifted away from the load when the spacing increases Thedisplacements at points B and C vary greatly when the tunnelspacing 119889119903 increases from 4 to 6
55e Influence of Burying Depth In this section paramet-ric study is conducted for 5 different burying depths of the
Figure 9 Comparison of maximum vibration velocity at obser-vation points between a model containing one load and a modelcontaining two loads
twin tunnel ie ℎ119903 = 2 3 4 5 6 And the load velocity is setto 03 119888s
The variation of the maximum vertical displacements atthe observation points with respect to the burying depth ispresented in Figure 13 In this case the tunnel spacing is fixedas 119889119903 = 4 It is observed that the displacements at points AC and D decrease when ℎ119903 increases For the displacementat point B it increases slightly when ℎ119903 lt 4 however itdrops substantially when ℎ119903 increases further to 6 This
phenomenon is explained by the fact that when the tunnelis shallowly buried the waves generated by the source tunnel(ie the tunnel where the moving load is applied) impinge onthe ground surface at a large incident angle (resembling theglancing incident) correspondingly the reflected waves leavethe surface at a large angle and thus the surface reflectionregion at nonsourced tunnels is small resulting in a fractionof surface reflections contributing to the dynamic response atpoint B However the surface reflection region will expandwhen hr increases gradually but when the burying depthis large enough (ie ℎ119889 gt 5) the material and geometricaldamping are dominant which results in the decreasing of thedisplacement at point B
6 Conclusions
In this paper a 3D twin-tunnel model is presented to evaluatethe dynamic responses of saturated half-space induced bysubways The effects of spacing between two tunnels tunnelburied depth load moving speed and soil permeabilityon the vibration response are investigated Based on thederivation and numerical examples presented above thefollowing conclusions can be drawn
(1) The existence of the other tunnel has influences onthe vibration caused by the moving subway Andwhen the twin tunnels are both loaded the dynamicresponses cannot be calculated by simply summingup the results of a single tunnel model Therefore itis necessary to consider a twin-tunnel model whenpredicting the dynamic responses induced by sub-ways for all load velocities The dynamic responsesof the single-phase ground are larger than those inthe saturated ground which denotes that using asingle phase model to predict the vibrations is moreconservative
(2) The influence of the spacing between two tunnels onthe dynamic response of the tunnel without the loadapplied is greater than that on the ground surfaceWith the increase of the spacing 119889119903 the dynamicresponse of the adjacent tunnel decreases Tunnelburied depth has a great influence on the dynamicresponse of adjacent tunnel and the ground sur-face For points on the ground surface the dynamicresponse decreases with the increase of the tunnelburied depthDue to the presence of surface reflectionregion the dynamic response of adjacent tunnelincreases at first and then decreases as the burialdepth increases So more attention should be paid tothe spacing between tunnels and the tunnel burieddepth in the construction of subway
(3) The subways moving at a low speed will generate asmaller dynamic response in the saturated half-spacethan moving at a higher speed And when the loadvelocity approaches the Rayleigh wave velocity of theground which is around 09 119888119904 the responses of thesoil and liner will be amplified Therefore the subwaymoving speed is also an important parameter thataffects the environmental vibration
10 Mathematical Problems in Engineering
0
2 3 4 5 6Burying depth hr
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
Max
imum
vib
ratio
n ve
loci
ty
minus40
minus30
minus20
minus10
10
20
30V
Zlowast(d
B)
(a) Maximum vibration velocity at observation points for differentburying depth
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
Max
imum
disp
lace
men
t
01 03 05 07 09Load velocity s
minus40
minus35
minus30
minus25
minus20
minus15
minus10
minus5
0
UZlowast
(dB)
(b) Maximum displacement at observation points for different loadvelocity
minus30
minus20
minus10
0
10
20
30
Max
imum
vib
ratio
n ve
loci
ty
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
01 03 05 07 09Load velocity s
VZlowast
(dB)
(c) Maximum vibration velocity at observation points for differentload velocity
Figure 10 Comparisons of dynamic response at observation points between a saturated ground and an elastic ground
Appendix
The expressions of the elements in the mass matrix thedamping matrix the stiffness matrix and the force vector arepresented as follows
Figure 13 Vertical maximum displacements for different buryingdepth of tunnels
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the National Key RampD Programof China (Grant No 2016YFC0800200) the Projects ofInternational Cooperation and Exchanges NSFC (Grant No51620105008) National Science Foundation for Young Scien-tists of China (Grant No 51608482) and National NaturalScience Foundation of China (Grant No 51478424)
References
[1] A V Metrikine and A C W M Vrouwenvelder ldquoSurfaceground vibration due to a moving train in a tunnel two-dimensional modelrdquo Journal of Sound and Vibration vol 234no 1 pp 43ndash66 2000
[2] J A Forrest and H E M Hunt ldquoA three-dimensional tunnelmodel for calculation of train-induced ground vibrationrdquo Jour-nal of Sound and Vibration vol 294 no 4 pp 678ndash705 2006
[3] J A Forrest andH EMHunt ldquoGround vibration generated bytrains in underground tunnelsrdquo Journal of Sound and Vibrationvol 294 no 4 pp 706ndash736 2006
[4] K-H Xie G-B Liu and Z-Y Shi ldquoDynamic response ofpartially sealed circular tunnel in viscoelastic saturated soilrdquoSoil Dynamics and Earthquake Engineering vol 24 no 12 pp1003ndash1011 2004
[5] K-H Xie and G-B Liu ldquoTransient response of a sphericalcavity with a partially sealed shell embedded in viscoelasticsaturated soilrdquo Journal of Zhejiang University Science A vol 6no 3 pp 194ndash201 2005
[6] Z-H Yuan Y-Q Cai L Shi H-L Sun and Z-G CaoldquoResponse of rail structure and circular tunnel in saturatedsoil subjected to harmonic moving loadrdquo Chinese Journal ofGeotechnical Engineering vol 38 no 2 pp 311ndash322 2016
12 Mathematical Problems in Engineering
[7] H E M Hunt ldquoModelling of rail vehicles and track forcalculation of ground-vibration transmission into buildingsrdquoJournal of Sound and Vibration vol 193 no 1 pp 185ndash194 1996
[8] T Balendra K H Chua K W Lo and S L Lee ldquoSteady-statevibration of subway-soilbuilding systemrdquo Journal of EngineeringMechanics vol 115 no 1 pp 145ndash162 1989
[9] F-C Wang H Xia and H-R Zhang ldquoVibration effects ofsubway trains on surrounding buildingsrdquo Journal of NorthernJiaotong University vol 23 no 5 pp 45ndash48 1999
[10] Y Yang HHung and L Hsu ldquoGround vibrations due to under-ground trains considering soil-tunnel interactionrdquo Interactionand Multiscale Mechanics vol 1 no 1 pp 157ndash175 2007
[11] Y-E Zhang and Changshi-Pan ldquoTests and analyses of thedynamic response of tunnels subjected to passing train loadrdquoJournal of Shijiazhuang Railway Institute vol 6 no 2 pp 7ndash141993
[12] C J C Jones D J Thompson and M Petyt ldquoStudies usinga combined finite element and boundary element model forvibration propagation from railway tunnelsrdquo International Insti-tute of Acoustics and Vibration pp 2703ndash2710 2000
[13] W Gardien and H G Stuit ldquoModelling of soil vibrations fromrailway tunnelsrdquo Journal of Sound and Vibration vol 267 no 3pp 605ndash619 2003
[14] S Wolf ldquoPotential low frequency ground vibration (lt63 Hz)impacts from underground LRT operationsrdquo Journal of Soundand Vibration vol 267 no 3 pp 651ndash661 2003
[15] G Degrande D Clouteau R Othman et al ldquoA numericalmodel for ground-borne vibrations from underground railwaytraffic based on a periodic finite element-boundary elementformulationrdquo Journal of Sound and Vibration vol 293 no 3-5pp 645ndash666 2006
[16] W-F Liu W-N Liu S Gupta and G Degrande ldquoPrediction ofvibrations in the tunnel and free field due to passage of metrotrainsrdquo Engineering Mechanics vol 27 no 1 pp 250ndash256 2010
[17] K A Kuo H E M Hunt and M F M Hussein ldquoThe effectof a twin tunnel on the propagation of ground-borne vibrationfrom an underground railwayrdquo Journal of Sound and Vibrationvol 330 no 25 pp 6203ndash6222 2011
[18] Z-H Yuan Y-Q Cai W Yuan Y-L Xu and Z-G CaoldquoDynamic response of circular railway tunnel and track systemin saturated soil under moving train loadingrdquo Rock and SoilMechanics vol 38 no 4 pp 1003ndash1014 2017
[19] Z P Liao and H L Wong ldquoA transmitting boundary for thenumerical simulation of elasticwave propagationrdquo InternationalJournal of Soil Dynamics and Earthquake Engineering vol 3 pp133ndash144 1984
[20] L Shi P Wang Y Cai and Z Cao ldquoMulti-transmittingformula for finite element modeling of wave propagation in asaturated poroelastic mediumrdquo Soil Dynamics and EarthquakeEngineering vol 80 pp 11ndash24 2016
[21] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid I Low-frequency rangerdquoe Journal ofthe Acoustical Society of America vol 28 pp 168ndash178 1956
[22] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956
[23] O C Zienkiewicz C T Chang and P Bettess ldquoDrainedundrained consolidating and dynamic behaviour assumptionsin soilsrdquo Geotechnique vol 30 no 4 pp 385ndash395 1980
[24] L Shi H Sun Y Cai C Xu and P Wang ldquoValidity of fullydrained fully undrained and u-p formulations for modeling a
poroelastic half-space under a moving harmonic point loadrdquoSoil Dynamics and Earthquake Engineering vol 42 pp 292ndash3012012
[25] T Senjuntichai and R K N D Rajapakse ldquoTransient responseof a circular cavity in a poroelastic mediumrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 17 no 6 pp 357ndash383 1993
[26] Y-B Yang S-R Kuo and H-H Hung ldquoFrequency-independent infinite elements for analysing semi-infiniteproblemsrdquo International Journal for Numerical Methods inEngineering vol 39 no 20 pp 3553ndash3569 1996
[27] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic response of tracksystem and saturated soil in half space subjected to a movingtrain loadrdquo Chinese Journal of Rock Mechanics and Engineeringvol 26 no 8 pp 1705ndash1712 2007
[28] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic responses oftrack system and poroelastic soil under high-speed train loadrdquoJournal of Zhejiang University (Engineering Science) vol 42 no11 pp 2002ndash2008 2008
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2 Mathematical Problems in Engineering
complexirregular geometry Balendra et al [8] investigatedthe coupled vibrations between the subway the ground andthe building using a 2D plane-strain finite element modelA viscous absorbing boundary condition was applied tothe truncation boundary to suppress unwanted reflectionsA 2D finite element model has been established by Wanget al [9] including the subway tunnel the ground andthe adjacent building The impact of the tunnel depthson the vibrations of different floors and positions in thebuildings was investigated Yang et al [10] studied thedynamic interaction between the ground and tunnel sys-tem using the finite element method At the truncationboundary the infinite element was applied to damp out theoutgoing waves Similarly Zhang and Pan [11] establisheda coupled finite and infinite element model to study thevibration of metro tunnel structure and surrounding stra-tum A coupled finite element (FE)ndashboundary element (BE)approach was proposed by Jones et al [12] to study thewave transmissions and the ground vibrations induced by thesubways
However the wave field generated by the moving load isthree-dimensional in nature which restricts the applicabilityof 2D computational models Gardien and Stuit et al [13]analyzed the ground vibrations generated by the subwayusing the finite element code LS-DYNAWolf et al [14] builta 3D prediction model for low-frequency ground vibrationinduced by the subway using the finite difference methodwhich is calibrated using the empirical relationship estab-lished on the measured field data G Degrande et al [15]proposed a coupled finite element (FE)ndashboundary element(BE) formulation to study the dynamic interaction betweenthe tunnel and the ground in which the tunnel was modeledby FEM and the ground by BEM The dynamic responsesof a shallow cut-and-cover masonry tunnel were comparedto those of a deep bored one A 3D periodic FE-BE modelwas established by Liu et al [16] to obtain the train-inducedvibrations for both the tunnel and the free field The floatingslab track and the general slab track are compared for theireffectiveness in vibration reduction
As mentioned above the current research focuses onthe environmental vibrations problems generated by subwaysrunning in a single tunnel The effect of the neighboringtunnel is not considered although it is a common practicethat underground railway lines are constructed in pairs Thevibration caused by two metro tunnels is different to thesum of two single tunnels since the neighboring tunnel canimpede and screen the vibration waves generated by theoperating tunnel To the best of the authorsrsquo knowledge Kuoet al [17] presented the investigation on the dynamics of atwin tunnel embedded in elastic full-space adopting a cou-pled finite elementndashboundary element approach Parametricstudies were carried out for the load excitation frequencythe tunnel orientations and the tunnel geometry whichdemonstrated that the dynamic interactions between thetwo tunnels were highly significant However the full spacemodel used by Kuo et al is not applicable when the groundvibrations are of concern since the ground is semi-infinitewith a free surface Moreover the possible influences of theload speed the tunnel burying depth and the tunnel spacing
on the dynamics responses of the twin-tunnel and the groundremain untouched in the literature
The ground in the above-referred papers using numer-ical approach is generally modeled as single-phase elasticmedium in which the ground water is not consideredExisting research by Yuan et al [18] showed that the dynamicresponses of the subway tunnel in the saturated groundare different from those in the elastic ground Therefore itis necessary to model the ground as saturated poroelasticmedium such that the ground water can be taken intoaccount
This paper focuses on the coupled vibrations of the linerthe twin-parallel tunnels and the saturated ground using thefinite element method A moving point load is applied to theinvert to represent the subway excitationThe dynamics of thesurrounding saturated soil are governed by Biotrsquos poroelastictheory To suppress spurious reflections at the truncationboundaries an artificial boundary condition named themulti-transmitting formula (MTF) that is proposed by theLiao and Wong [19] and extended by Shi et al [20] isapplied to transmit the outgoing waves in the saturatedsoil medium The effectiveness and stability of MTF havebeen demonstrated in the paper by Shi et al [20] Usingthe established 3D finite element model parametric studieshave been performed for investigating the effects of theload velocity the burying depth of tunnel and the tunnelspacing on the coupled vibrations of the liner-tunnel-groundsystem Because the effect of the permeability coefficient onthe dynamic response of soil has been fully discussed bymany researchers andwehave not found any new conclusionsusing the presented model therefore the influence of thepermeability coefficient on vibration response has not beendiscussed in this paper
2 Governing Equations for Saturated Ground
According to Biotrsquos theory [21 22] the governing equationsfor the dynamics of the saturated poroelastic medium read asfollows
120590119894119895119895 + 120588119887119894 minus 120588119894 minus 120588f119894 = 0 (1)
minus119901119894 + 120588f119887119894 minus 120588f 119894 minus 119898119894 minus 119887119894 = 0 (2)
120590119894119895 = 12059010158401015840119894119895 minus 120572120575119894119895119901 (4)
where 120588119887119894 denotes the gravity force acting on the soil skeleton120588 and 120588f denote the density of the soil and the density ofthe pore fluid respectively 120588 = (1 minus 119899)120588s + 119899120588f 120588s is thedensity of the solid skeleton and 119899 is the porosity 119906119894 and 119908119894denote the displacement of soil skeleton and the infiltrationdisplacement of the pore fluid relative to the soil skeletonrespectively ldquosdotrdquo and ldquosdotsdotrdquo denote the first and second derivativeof time respectively 119898 = 120588f119899 119887 = 120588fg119896D 119892 is theacceleration of gravity 119896D is the Darcy permeability 120590119894119895 and12059011989411989510158401015840 denote the total and effective stress tensors respectively120575119894119895 is the Kronecker delta 119901 accounts for the pore pressure 120572
Mathematical Problems in Engineering 3
and119872 are Biot parameters and subscripts i and j denote thetensor operation and the summation convention is applied
Aftermanipulation the pore pressure119901 can be eliminatedfrom the governing equations leaving 119906119894 and 119908119894 the fieldvariables
There are two reasons for keeping 119906119894 and 119908119894 as theprimary unknowns the first reason is that the u minus wformulation is a complete description of Biotrsquos theory Forthe u minus 119901 formulation the relative pore fluid acceleration needs to be neglected [23] which would bring in certaininaccuracy especially for a high-frequency loading situation[23 24] the second reason is that the u minus w formulation isconsistent with the proposed artificial boundary conditionmulti-transmitting formula (MTF given in Section 3) sinceit is vector-based andmanipulates the displacement vectors uand w directly while the pore pressure p is a scalar and thusnot a suitable variable for applying the MTF
The same interpolation function is used for both the soil-skeleton displacement u and the relative pore-fluid displace-ment w which are given as
ue (x 119905) = 119899e
sum119894=1
119873119894 (x) Iue119894 (119905)
we (x 119905) = 119899e
sum119894=1
119873119894 (x) Iwe119894 (119905)
(7)
where x = (119909 119910 119911) is the coordinate 119905 represents time119873119894(x)is the shape function at node 119894 119899e is the number of nodes ofeach element I is the unit matrix of 2 times 2 or 3 times 3 and ue119894 (119905)and we
119894 (119905) are the nodal displacement vectorsFollowing the standard Galerkin procedure the govern-
ing equations of the finite element formulation representingthe saturated poroelastic soil medium can be derived as
[Mss Msw
Mws Mww] ue
we + [0 00 Cww
] uewe
+ [Kss + K1015840ss Ksw
Kws Kww]ue
we = fsfw
(8)
The expressions of the elements in the mass matrix thedamping matrix the stiffness matrix and the force vector aregiven in the appendix
3 The Multi-Transmitting Formula
Due to the semi-infinite nature of the ground the groundvibration analysis using the finite element method requiresartificial boundary conditions to make the computationaldomain finite Many artificial boundary conditions have beenproposed for absorbingdamping the outgoing waves for boththe elastic medium and the saturated poroelastic medium
For a detailed review one is referred to the paper by Shi etal [20] Among the existing artificial boundary conditionsthe multi-transmitting formula that is proposed by Liao andWong [19] for the elastic medium and then extended byShi et al [20] for the saturated poroelastic medium standsout for being local in both time and space and for the easyimplementation in the finite element analysis both for 2D and3D grids
The MTF extrapolates displacement on the artificialboundary at time 119905 = (119899 + 1)Δ119905 as a linear combination ofthe displacements at previous time steps along a straight linenormal to the boundary which is given as
where 119906119894(119895Δ119904 minus119896Δ119904 119899Δ119905) denotes the soil skeleton dis-placements of the grid point at (119909 = 119895Δ119904 119911 = minus119896Δ119904) andat time 119905 = 119899Δ119905 Similarly 119908119894(119895Δ119904 minus119896Δ119904 119899Δ119905) denotes therelative pore-fluid displacement 119888a represents the apparentvelocities of waves propagating along the normal direction ofartificial boundary C119873119896 is the binomial coefficient and N isthe order of the boundary condition
Since the computational points (119911 = minus119896119888aΔ119905) in (9) do notgenerally coincide with the grid points a quadratic interpo-lation scheme is employed to correlate the displacements atthe computational points to those at the grid points as
where the operator A is the backward space shift of 119888aΔ119905119860120601 (119909 119911 119905) = (11987911 + 11987912119885minus1119885 + 11987913119885minus2119885 ) 120601 (119909 119911 119905)
asymp 120601 (119909 119911 minus 119888aΔ119905 119905) (11)
in which the interpolation coefficients are given as
11987911 = (2 minus 119904) (1 minus 119904)211987912 = 119904 (2 minus 119904)11987913 = 119904 (119904 minus 1)2
119904 = 119888aΔ119905Δ119909
(12)
4 Mathematical Problems in Engineering
MTF
MTF
Sym
met
ric
Symmetric
5
4
2
3Y
X
1
0 543210
(a)
Pres
sure
0 1 2
f0
tlowast
(b)
Figure 1 A cavity embedded in an infinite saturated poroelastic medium (a) model setup and mesh grid (b) edge pressure
The operators 119885119905 and 119885119911 are the forward time and spaceshifts respectively
Correspondingly the operators119885minus1119905 and119885minus1119911 represent thebackward time and space shifts respectively
According to Shi et al [20] a second-order MTF (119873 = 2)and the artificial wave velocity 119888119886 = 119888s (119888s is the shear wavevelocity of the saturated medium) are efficient and accurateenough for configuring the MTF Moreover the numericalstability of the MTF can be guaranteed by (1) meeting theCourant-Friedrichs-Lewy (CFL) condition 119904 = 119888aΔ119905Δ119904 le1radic2 and (2) perturbing the interpolation coefficient 11987911 byadding a small negative value (= minus004119904 sim minus03119904)4 Verification of the Present Model
To verify the accuracy of the proposed finite element formu-lation and the effectiveness of the MTF boundary conditionthe transient response due to an edge pressure applied radiallyat a cavity surface is computed and compared to the analyticalsolution given by T Senjuntichai [25] which is developed fora cavity in an infinite saturated soil medium under the plane-strain condition
Due to the symmetry a 14 model is established for thecavity and the surrounding saturated soil medium Symmet-ric boundary conditions are imposed for the left and bottomboundaries while MTF boundary conditions are applied tothe top and right boundaries to account for the infiniteextension of the soil medium The mesh in the x-y plane isshown in Figure 1(a) while themovement in the z direction isfixed tomodel the plain strain conditionThe applied pressureis a triangular pulse as given in Figure 1(b) in which 119905lowast is the
normalized time 119905lowast = (119905119886)radic120583120588 119886 is the radius of tunnel 119891denotes the edge pressure and 1198910 is its peak value
The soil properties and the load parameters are thesame as those given by Senjuntichai [25] and summarizedin Table 1 Figure 2 shows the comparison on the radialdisplacement at the cavity surface where the vertical axis isthe normalized radial displacements 119880119903lowast = 120583119880119903(1198910119886) andthe horizontal axis is the normalized time 119905lowast From Figure 2certain reflections can be observed for the MTF conditionwhen the reflected waves reached the observation point at thefirst time (ie 119905lowast asymp 5) However the following impinging ofthe reflected waves is transmitted effectively by MTF rendinga very close comparison to the analytical solution for 119905lowast gt 8Also it can be seen from Figure 2 that significant reflectionscan happen during the entire time if the fixed boundarycondition is used
5 Numerical Results
In this section coupled vibrations of the liner the twintunnels and the saturated ground are investigated usingthe developed finite element along with the developed MTFboundary condition Parametric studies have been per-formed for the burying depth of tunnel the tunnel spacingthe load velocity and the soil permeability
51 Model Description The computation model consisting oftwo identical circular tunnels that are parallel to each otherand embedded in a three-dimensional saturated ground isshown in Figure 3 The model size is 119886 (width) times 119887 (height)times 119897 (length)The center-to-center spacing between the tunnelsis represented by 119889 and the burying depth measured fromthe ground surface to the tunnel center is denoted by ℎas shown in Figure 3(b) 119903 denotes the outer radius of theliner and the thickness of liner is represented by 119905 The sideand cross-sectional views with dimensions are presented in
Mathematical Problems in Engineering 5
Table 1 Parameters of the edge pressure and the saturated poroelastic medium
Parameter Symbol ValueYoungrsquos modulus 119864 533 times 107 PaPoissonrsquos ratio ] 033Density of the bulk material 120588 2000 kgm3
Density of the pore fluid 120588f 1000 kgm3
Porosity 119899 04Biot parameters 120572 098Biot parameters 119872 40 times 108 PaAdditional mass density 120588a 0 kgm3
Darcy permeability coefficient 119896119863 491 times 10minus3msTriangular pulse load 1198910 200 times 107 PaRadius of tunnel 119886 10m
present methodSenjundichai (1993)fixed boundary
2 4 6 8 10 12 14 160tlowast
minus02
minus01
00
01
02
03
UＬlowast
Figure 2 Comparison of present results with the solution proposedby Senjundichai [25]
Figures 3(a) and 3(b) respectively A nonoscillating pointload of magnitude 119865119899 that moves along the positive 119909 axle actsvertically to the invert of the left tunnel liner
Four observation points are chosen to record theresponses caused by the moving load points A and B arelocated at the liner bottom of the tunnel with the loadapplied and the other tunnel respectively points C and D arelocated on the ground surface right between the twin tunnelsand above the center of the tunnel loaded respectively Allobservation points are located at the middle of the modelalong the tunnel axial direction (ie 119909 = 1198972)
The bottom surface of the model is fixed to consider theunderlying hard stratum The top surface is free and set aspermeable The MTF boundary condition is applied to theremaining 4 side surfaces to account for the infinite extensionof the saturated ground For a reliable modeling of wavepropagations the element sizes of the ground shouldmeet therequirement Δ119904 le 120582s6 where 120582s = 2120587119888s120596max is the shearwave length and 120596max is the highest frequency of the shear
wave [26] The implicit Newmark method is employed forthe timemarching of the dynamic analysis with theNewmarkparameter being set to 06 to remove disturbances caused bysudden application of the point load Although the Newmarkintegration scheme is unconditionally stable the time stepshould meet the CFL condition as mentioned in Section 3since the MTF is essentially a forward Euler scheme
Parameters of the liner the saturated ground and themoving load are summarized in Table 2 In the subsequentanalysis the dynamic displacement and velocity are pre-sented in decibels (dB) ie 119902lowast = 20 lg(1199021199020) where 119902denotes an arbitrary variable and 1199020 is the reference valueThe reference values for displacement and velocity are 20 times10minus5m and 20 times 10minus5ms respectively
52 Models Comparisons In this section a single tunnelmodel and a model with both tunnels loaded are comparedwith the present twin-tunnel model The cross sections ofthe two models are presented in Figures 4(a) and 4(b)respectively The results of single phased model are alsocompared with those of the saturated poroelastic model inthis section
To investigate the influence of interactions between thetwo tunnels on the vibration-prediction results the compar-ison of dynamic responses between single tunnel model andthe present model is shown in Figures 5(a) and 5(b) Theburying depth and the spacing of the tunnels are respectivelyfixed at ℎ119903 = 4 and 119889119903 = 4 and the load velocity is 03 119888sIn Figure 5(a) the tunnel spacing for the single tunnel modeldenotes two times the distance between tunnel center and themodel center The dynamic responses between two modelson point A are very close while the results of the presentmodel are greater than those of the single tunnel model onthe ground observation points (point C and point D) Thedisplacements at point C for the single tunnel model decreasefaster than the present model between 119889119903 which is 4 to5 That is because the dynamic response at point C in thepresent model is the sum of the waves generated from thetunnel with the load applied and reflected wave from theother tunnel lining However the dynamic responses of thesingle-tunnel model only come from the tunnel with the loadapplied As the tunnel spacing increases the differences of the
6 Mathematical Problems in Engineering
D(C)
A(B)MTFMTF
Free and permeable
Fixed
z
y-Plane
l2
l
b
x
(a) Side view
D C
A B
z
MTFMTF
Fixed
Free and permeable
bＨ
d
tr
ℎ
a x-Plane y
(b) Cross section view
Figure 3 Schematic of the 3D computation model
D C
A
z
MTFMTF
Fixed
Free and permeable
b Ｈt r
ℎ
a x-Plane y
d2
(a) Single tunnel model
D C
A B
a
z
MTFMTF
Fixed
Free and permeable
b Ｈ Ｈ
d
tr
ℎ
x-Plane y
(b) Model with both tunnels loaded
Figure 4 Cross sections of single tunnel model and the model with both tunnels loaded
displacement at point C and D for both models decrease InFigure 5(b) it is observed that the maximum displacementsof single tunnelmodel are all smaller than those of the presentmodel when the burying depth changes from 3 to 5 Thedecrease rates for both models are almost the same
The comparison between the single tunnel model andthe present model is shown in Figure 6 under different loadvelocities It is observed from Figure 6 that the dynamicresponses of the twin-tunnel model are larger than those ofthe single tunnel model which is inconsistent with the resultsshown in Figure 5 And the difference of the two models atpoint C is greater than that at point D The differences reachpeak value when 119888119888s = 09 Since the dynamic response atpoint C is jointly affected by two tunnels while the dynamicresponse at point D is dominated by the tunnel loadedtherefore it is essential to consider the twin-tunnel model toobtain more accurate prediction results
Practically both metro tunnels are always loaded atthe same time due to the high operation frequency of thesubways The dynamic responses of the model with bothtunnel inverts loaded are studied in this section
In contrast to the result of the presentmodel the dynamicresponses of point C and point D due to two moving pointloads are larger in terms of the maximum displacement InFigure 7 the maximum displacements of the two models atpoint A are almost the same The dynamic response of thetwomodels at point C decreases a little with increasing tunnelspacing However as tunnel spacing increases the maximum
displacement of the two models at point A remains the sameand the results of two models become closer at point DThis is because as the tunnel spacing increases the effect ofthe other tunnel on point D decreases gradually And withthe increasing of the burying depth the dynamic responsesof points C and D have obvious downtrend Besides thedisplacement at the point C is larger than that at the point Din the model with two loads while the situation of the presentmodel is the opposite That is because the load in the othertunnel has a larger contribution to the dynamic response atpoint C than at point D
The comparison of the displacements and the vibrationvelocity between the present model and the model with bothtunnels loaded is shown in Figures 8 and 9 for differentload velocities For points C and D the dynamic responsesof the model with both tunnels loaded are obviously largerthan the model with only one tunnel loaded However theprediction results of the two models at point A are very closeAnd the maximum displacement at point C is smaller thanthat at point D in the model with only one tunnel loadedwhile the situation of the two-load model is the oppositewhich is explained in the previous section The differencebetween the maximum displacements of the two models isalmost unchanged at different load velocity However thedifference in the vibration velocity results of the two modelsat point D decreases as the velocity increases and reaches aminimum at 119888119888119904 = 09 From Figure 9 it can also be observedthat the maximum vibration velocities at points C and D are
Mathematical Problems in Engineering 7
Table 2 Parameters of the liner the saturated ground and the moving load
Saturated Soil Ground
Width a (m) Height b (m) Length l (m) Biot parameters 12057242 24 120 099
Spacing 119889 (m) Burying depth h (m) Porosity 119899 Poissonrsquos ratio ]9ndash24 6ndash18 0286 04
Youngrsquos modulus Darcy permeability coefficient Shear wave velocity 119888s Biot parameters119864 (Pa) 119896119863 (ms) (ms) 119872 (Nm)223 times 108 10 times 10minus5ndash10 times 10minus2 1911 574 times 109
Density of the bulk material 120588 (kgm3) Density of the pore fluid 120588f (kgm3)2178 times 103 10 times 103
Concrete LinerYoungrsquos modulus Poissonrsquos ratio Density of the bulk material outer radius Thickness119864 (Pa) ] 120588 (kgm3) r(m) t(m)
250 times 1010 02 2400 times 103 30 03
Moving Load Point load value (N) Load velocity (ms)20 times 104 1911ndash1911
0
4 45 5 55 6Tunnel spacing dr
point A(twin-tunnel)point C(twin-tunnel)point D(twin-tunnel)point A(single tunnel)point C(single tunnel)point D(single tunnel)
(b) Maximum displacement at the observation points for differentburying depth
Figure 5 Comparison of dynamic responses between a single-tunnel model and a twin-tunnel model
almost the same in the present model with only one tunnelloaded while the value at point C in the two-load model issignificantly larger than that at point D Because the load inthe other tunnel is closer to point C than point D it has agreater impact on the dynamic response of point C
To illustrate the necessity in introducing the saturatedground model the dynamic responses of the twin tunnelembedded in a single-phase ground are compared to thoseof the saturated ground model and presented in Figure 10The single-phase ground model is obtained by reducing thesaturated ground model after fixing the pore-fluid relativedisplacement and setting Biotrsquos parameters to zeros It isobserved that the dynamic responses of the elastic ground arelarger than those in the saturated ground The discrepancybetween the two ground models widens when the buryingdepth increases And as the load velocity increases the
difference of dynamic response between the two models atpoint D also becomes larger From the comparison it can beconcluded that the ground water has an obvious influence onthe vibration responses of the tunnel and the ground Theirvibration levels would be overestimated if the ground wateris not considered
53 e Influences of Load Velocity The influences of loadvelocity on the dynamic responses of the liner-tunnel-groundsystem are investigated in this section Six different velocities119888119888s = 01 03 05 07 09 and 10 are considered along withthe tunnel spacing 119889119903 = 4 and the burying depth ℎ119903 = 4
It is observed from Figure 11 that the displacements atpoints A and B change slightly with the increasing loadvelocity while the displacements at points C and D reacha peak at around 09 119888s Since C and D are on the ground
8 Mathematical Problems in Engineering
point C(twin-tunnel)point D(twin-tunnel)point C(single tunnel)point D(single tunnel)
Max
imum
disp
lace
men
t0 02 04 06 08 1
Load velocity s
minus365
minus360
minus355
minus350
minus345
minus340
minus335
minus330
UZlowast
(dB)
Figure 6 Comparison on maximum displacement at observation points between a single-tunnel model and a twin-tunnel model
(a) Maximum displacement at the observation points for differenttunnel spacing
3 35 4 45 5Burying depth hr
point C(one load)point D(one load)point C(two loads)point D(two loads)
Max
imum
disp
lace
men
t
minus45
minus40
minus35
minus30
minus25
minus20
minus15
minus10
UZlowast
(dB)
(b) Maximum displacement at the observation points for differentburying depth
Figure 7 Comparison of dynamic responses between the present model and a model with both tunnels loaded
surface their responses will be amplified when the loadvelocity approaches the Rayleigh wave velocity of the groundwhich is around 09 119888s The same results were reported inprevious studies on the critical velocity for a train runningon the ground surface [27 28] However points A and B areset on the concrete liner whose shear wave velocity is muchhigher than that of the soil Thus the load velocity is very lowcompared to the shear wave velocity of the concrete whichmakes the displacement of points A and B nonsensitive to thechange of load velocity
54 e Influence of Tunnel Spacing In this section theinfluence of the tunnel spacing on the dynamic responses isinvestigated by considering 6 different twin tunnel distancesie 119889119903 = 3 4 5 6 7 8 The load velocity is 03 119888119904 The
maximum vertical displacements at the observation pointswhen ℎ119903 = 4 are compared in Figure 12
The displacement at point A is significantly larger thanthose at other points since point A is traversed directlyby the point load When the tunnel spacing increases thedisplacements at points B and C decrease gradually Howeverthe displacements at points A and D remain almost the sameThis observation is within expectation because the positionsof points A and D remain the same while points B and C areshifted away from the load when the spacing increases Thedisplacements at points B and C vary greatly when the tunnelspacing 119889119903 increases from 4 to 6
55e Influence of Burying Depth In this section paramet-ric study is conducted for 5 different burying depths of the
Figure 9 Comparison of maximum vibration velocity at obser-vation points between a model containing one load and a modelcontaining two loads
twin tunnel ie ℎ119903 = 2 3 4 5 6 And the load velocity is setto 03 119888s
The variation of the maximum vertical displacements atthe observation points with respect to the burying depth ispresented in Figure 13 In this case the tunnel spacing is fixedas 119889119903 = 4 It is observed that the displacements at points AC and D decrease when ℎ119903 increases For the displacementat point B it increases slightly when ℎ119903 lt 4 however itdrops substantially when ℎ119903 increases further to 6 This
phenomenon is explained by the fact that when the tunnelis shallowly buried the waves generated by the source tunnel(ie the tunnel where the moving load is applied) impinge onthe ground surface at a large incident angle (resembling theglancing incident) correspondingly the reflected waves leavethe surface at a large angle and thus the surface reflectionregion at nonsourced tunnels is small resulting in a fractionof surface reflections contributing to the dynamic response atpoint B However the surface reflection region will expandwhen hr increases gradually but when the burying depthis large enough (ie ℎ119889 gt 5) the material and geometricaldamping are dominant which results in the decreasing of thedisplacement at point B
6 Conclusions
In this paper a 3D twin-tunnel model is presented to evaluatethe dynamic responses of saturated half-space induced bysubways The effects of spacing between two tunnels tunnelburied depth load moving speed and soil permeabilityon the vibration response are investigated Based on thederivation and numerical examples presented above thefollowing conclusions can be drawn
(1) The existence of the other tunnel has influences onthe vibration caused by the moving subway Andwhen the twin tunnels are both loaded the dynamicresponses cannot be calculated by simply summingup the results of a single tunnel model Therefore itis necessary to consider a twin-tunnel model whenpredicting the dynamic responses induced by sub-ways for all load velocities The dynamic responsesof the single-phase ground are larger than those inthe saturated ground which denotes that using asingle phase model to predict the vibrations is moreconservative
(2) The influence of the spacing between two tunnels onthe dynamic response of the tunnel without the loadapplied is greater than that on the ground surfaceWith the increase of the spacing 119889119903 the dynamicresponse of the adjacent tunnel decreases Tunnelburied depth has a great influence on the dynamicresponse of adjacent tunnel and the ground sur-face For points on the ground surface the dynamicresponse decreases with the increase of the tunnelburied depthDue to the presence of surface reflectionregion the dynamic response of adjacent tunnelincreases at first and then decreases as the burialdepth increases So more attention should be paid tothe spacing between tunnels and the tunnel burieddepth in the construction of subway
(3) The subways moving at a low speed will generate asmaller dynamic response in the saturated half-spacethan moving at a higher speed And when the loadvelocity approaches the Rayleigh wave velocity of theground which is around 09 119888119904 the responses of thesoil and liner will be amplified Therefore the subwaymoving speed is also an important parameter thataffects the environmental vibration
10 Mathematical Problems in Engineering
0
2 3 4 5 6Burying depth hr
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
Max
imum
vib
ratio
n ve
loci
ty
minus40
minus30
minus20
minus10
10
20
30V
Zlowast(d
B)
(a) Maximum vibration velocity at observation points for differentburying depth
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
Max
imum
disp
lace
men
t
01 03 05 07 09Load velocity s
minus40
minus35
minus30
minus25
minus20
minus15
minus10
minus5
0
UZlowast
(dB)
(b) Maximum displacement at observation points for different loadvelocity
minus30
minus20
minus10
0
10
20
30
Max
imum
vib
ratio
n ve
loci
ty
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
01 03 05 07 09Load velocity s
VZlowast
(dB)
(c) Maximum vibration velocity at observation points for differentload velocity
Figure 10 Comparisons of dynamic response at observation points between a saturated ground and an elastic ground
Appendix
The expressions of the elements in the mass matrix thedamping matrix the stiffness matrix and the force vector arepresented as follows
Figure 13 Vertical maximum displacements for different buryingdepth of tunnels
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the National Key RampD Programof China (Grant No 2016YFC0800200) the Projects ofInternational Cooperation and Exchanges NSFC (Grant No51620105008) National Science Foundation for Young Scien-tists of China (Grant No 51608482) and National NaturalScience Foundation of China (Grant No 51478424)
References
[1] A V Metrikine and A C W M Vrouwenvelder ldquoSurfaceground vibration due to a moving train in a tunnel two-dimensional modelrdquo Journal of Sound and Vibration vol 234no 1 pp 43ndash66 2000
[2] J A Forrest and H E M Hunt ldquoA three-dimensional tunnelmodel for calculation of train-induced ground vibrationrdquo Jour-nal of Sound and Vibration vol 294 no 4 pp 678ndash705 2006
[3] J A Forrest andH EMHunt ldquoGround vibration generated bytrains in underground tunnelsrdquo Journal of Sound and Vibrationvol 294 no 4 pp 706ndash736 2006
[4] K-H Xie G-B Liu and Z-Y Shi ldquoDynamic response ofpartially sealed circular tunnel in viscoelastic saturated soilrdquoSoil Dynamics and Earthquake Engineering vol 24 no 12 pp1003ndash1011 2004
[5] K-H Xie and G-B Liu ldquoTransient response of a sphericalcavity with a partially sealed shell embedded in viscoelasticsaturated soilrdquo Journal of Zhejiang University Science A vol 6no 3 pp 194ndash201 2005
[6] Z-H Yuan Y-Q Cai L Shi H-L Sun and Z-G CaoldquoResponse of rail structure and circular tunnel in saturatedsoil subjected to harmonic moving loadrdquo Chinese Journal ofGeotechnical Engineering vol 38 no 2 pp 311ndash322 2016
12 Mathematical Problems in Engineering
[7] H E M Hunt ldquoModelling of rail vehicles and track forcalculation of ground-vibration transmission into buildingsrdquoJournal of Sound and Vibration vol 193 no 1 pp 185ndash194 1996
[8] T Balendra K H Chua K W Lo and S L Lee ldquoSteady-statevibration of subway-soilbuilding systemrdquo Journal of EngineeringMechanics vol 115 no 1 pp 145ndash162 1989
[9] F-C Wang H Xia and H-R Zhang ldquoVibration effects ofsubway trains on surrounding buildingsrdquo Journal of NorthernJiaotong University vol 23 no 5 pp 45ndash48 1999
[10] Y Yang HHung and L Hsu ldquoGround vibrations due to under-ground trains considering soil-tunnel interactionrdquo Interactionand Multiscale Mechanics vol 1 no 1 pp 157ndash175 2007
[11] Y-E Zhang and Changshi-Pan ldquoTests and analyses of thedynamic response of tunnels subjected to passing train loadrdquoJournal of Shijiazhuang Railway Institute vol 6 no 2 pp 7ndash141993
[12] C J C Jones D J Thompson and M Petyt ldquoStudies usinga combined finite element and boundary element model forvibration propagation from railway tunnelsrdquo International Insti-tute of Acoustics and Vibration pp 2703ndash2710 2000
[13] W Gardien and H G Stuit ldquoModelling of soil vibrations fromrailway tunnelsrdquo Journal of Sound and Vibration vol 267 no 3pp 605ndash619 2003
[14] S Wolf ldquoPotential low frequency ground vibration (lt63 Hz)impacts from underground LRT operationsrdquo Journal of Soundand Vibration vol 267 no 3 pp 651ndash661 2003
[15] G Degrande D Clouteau R Othman et al ldquoA numericalmodel for ground-borne vibrations from underground railwaytraffic based on a periodic finite element-boundary elementformulationrdquo Journal of Sound and Vibration vol 293 no 3-5pp 645ndash666 2006
[16] W-F Liu W-N Liu S Gupta and G Degrande ldquoPrediction ofvibrations in the tunnel and free field due to passage of metrotrainsrdquo Engineering Mechanics vol 27 no 1 pp 250ndash256 2010
[17] K A Kuo H E M Hunt and M F M Hussein ldquoThe effectof a twin tunnel on the propagation of ground-borne vibrationfrom an underground railwayrdquo Journal of Sound and Vibrationvol 330 no 25 pp 6203ndash6222 2011
[18] Z-H Yuan Y-Q Cai W Yuan Y-L Xu and Z-G CaoldquoDynamic response of circular railway tunnel and track systemin saturated soil under moving train loadingrdquo Rock and SoilMechanics vol 38 no 4 pp 1003ndash1014 2017
[19] Z P Liao and H L Wong ldquoA transmitting boundary for thenumerical simulation of elasticwave propagationrdquo InternationalJournal of Soil Dynamics and Earthquake Engineering vol 3 pp133ndash144 1984
[20] L Shi P Wang Y Cai and Z Cao ldquoMulti-transmittingformula for finite element modeling of wave propagation in asaturated poroelastic mediumrdquo Soil Dynamics and EarthquakeEngineering vol 80 pp 11ndash24 2016
[21] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid I Low-frequency rangerdquoe Journal ofthe Acoustical Society of America vol 28 pp 168ndash178 1956
[22] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956
[23] O C Zienkiewicz C T Chang and P Bettess ldquoDrainedundrained consolidating and dynamic behaviour assumptionsin soilsrdquo Geotechnique vol 30 no 4 pp 385ndash395 1980
[24] L Shi H Sun Y Cai C Xu and P Wang ldquoValidity of fullydrained fully undrained and u-p formulations for modeling a
poroelastic half-space under a moving harmonic point loadrdquoSoil Dynamics and Earthquake Engineering vol 42 pp 292ndash3012012
[25] T Senjuntichai and R K N D Rajapakse ldquoTransient responseof a circular cavity in a poroelastic mediumrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 17 no 6 pp 357ndash383 1993
[26] Y-B Yang S-R Kuo and H-H Hung ldquoFrequency-independent infinite elements for analysing semi-infiniteproblemsrdquo International Journal for Numerical Methods inEngineering vol 39 no 20 pp 3553ndash3569 1996
[27] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic response of tracksystem and saturated soil in half space subjected to a movingtrain loadrdquo Chinese Journal of Rock Mechanics and Engineeringvol 26 no 8 pp 1705ndash1712 2007
[28] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic responses oftrack system and poroelastic soil under high-speed train loadrdquoJournal of Zhejiang University (Engineering Science) vol 42 no11 pp 2002ndash2008 2008
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
There are two reasons for keeping 119906119894 and 119908119894 as theprimary unknowns the first reason is that the u minus wformulation is a complete description of Biotrsquos theory Forthe u minus 119901 formulation the relative pore fluid acceleration needs to be neglected [23] which would bring in certaininaccuracy especially for a high-frequency loading situation[23 24] the second reason is that the u minus w formulation isconsistent with the proposed artificial boundary conditionmulti-transmitting formula (MTF given in Section 3) sinceit is vector-based andmanipulates the displacement vectors uand w directly while the pore pressure p is a scalar and thusnot a suitable variable for applying the MTF
The same interpolation function is used for both the soil-skeleton displacement u and the relative pore-fluid displace-ment w which are given as
ue (x 119905) = 119899e
sum119894=1
119873119894 (x) Iue119894 (119905)
we (x 119905) = 119899e
sum119894=1
119873119894 (x) Iwe119894 (119905)
(7)
where x = (119909 119910 119911) is the coordinate 119905 represents time119873119894(x)is the shape function at node 119894 119899e is the number of nodes ofeach element I is the unit matrix of 2 times 2 or 3 times 3 and ue119894 (119905)and we
119894 (119905) are the nodal displacement vectorsFollowing the standard Galerkin procedure the govern-
ing equations of the finite element formulation representingthe saturated poroelastic soil medium can be derived as
[Mss Msw
Mws Mww] ue
we + [0 00 Cww
] uewe
+ [Kss + K1015840ss Ksw
Kws Kww]ue
we = fsfw
(8)
The expressions of the elements in the mass matrix thedamping matrix the stiffness matrix and the force vector aregiven in the appendix
3 The Multi-Transmitting Formula
Due to the semi-infinite nature of the ground the groundvibration analysis using the finite element method requiresartificial boundary conditions to make the computationaldomain finite Many artificial boundary conditions have beenproposed for absorbingdamping the outgoing waves for boththe elastic medium and the saturated poroelastic medium
For a detailed review one is referred to the paper by Shi etal [20] Among the existing artificial boundary conditionsthe multi-transmitting formula that is proposed by Liao andWong [19] for the elastic medium and then extended byShi et al [20] for the saturated poroelastic medium standsout for being local in both time and space and for the easyimplementation in the finite element analysis both for 2D and3D grids
The MTF extrapolates displacement on the artificialboundary at time 119905 = (119899 + 1)Δ119905 as a linear combination ofthe displacements at previous time steps along a straight linenormal to the boundary which is given as
where 119906119894(119895Δ119904 minus119896Δ119904 119899Δ119905) denotes the soil skeleton dis-placements of the grid point at (119909 = 119895Δ119904 119911 = minus119896Δ119904) andat time 119905 = 119899Δ119905 Similarly 119908119894(119895Δ119904 minus119896Δ119904 119899Δ119905) denotes therelative pore-fluid displacement 119888a represents the apparentvelocities of waves propagating along the normal direction ofartificial boundary C119873119896 is the binomial coefficient and N isthe order of the boundary condition
Since the computational points (119911 = minus119896119888aΔ119905) in (9) do notgenerally coincide with the grid points a quadratic interpo-lation scheme is employed to correlate the displacements atthe computational points to those at the grid points as
where the operator A is the backward space shift of 119888aΔ119905119860120601 (119909 119911 119905) = (11987911 + 11987912119885minus1119885 + 11987913119885minus2119885 ) 120601 (119909 119911 119905)
asymp 120601 (119909 119911 minus 119888aΔ119905 119905) (11)
in which the interpolation coefficients are given as
11987911 = (2 minus 119904) (1 minus 119904)211987912 = 119904 (2 minus 119904)11987913 = 119904 (119904 minus 1)2
119904 = 119888aΔ119905Δ119909
(12)
4 Mathematical Problems in Engineering
MTF
MTF
Sym
met
ric
Symmetric
5
4
2
3Y
X
1
0 543210
(a)
Pres
sure
0 1 2
f0
tlowast
(b)
Figure 1 A cavity embedded in an infinite saturated poroelastic medium (a) model setup and mesh grid (b) edge pressure
The operators 119885119905 and 119885119911 are the forward time and spaceshifts respectively
Correspondingly the operators119885minus1119905 and119885minus1119911 represent thebackward time and space shifts respectively
According to Shi et al [20] a second-order MTF (119873 = 2)and the artificial wave velocity 119888119886 = 119888s (119888s is the shear wavevelocity of the saturated medium) are efficient and accurateenough for configuring the MTF Moreover the numericalstability of the MTF can be guaranteed by (1) meeting theCourant-Friedrichs-Lewy (CFL) condition 119904 = 119888aΔ119905Δ119904 le1radic2 and (2) perturbing the interpolation coefficient 11987911 byadding a small negative value (= minus004119904 sim minus03119904)4 Verification of the Present Model
To verify the accuracy of the proposed finite element formu-lation and the effectiveness of the MTF boundary conditionthe transient response due to an edge pressure applied radiallyat a cavity surface is computed and compared to the analyticalsolution given by T Senjuntichai [25] which is developed fora cavity in an infinite saturated soil medium under the plane-strain condition
Due to the symmetry a 14 model is established for thecavity and the surrounding saturated soil medium Symmet-ric boundary conditions are imposed for the left and bottomboundaries while MTF boundary conditions are applied tothe top and right boundaries to account for the infiniteextension of the soil medium The mesh in the x-y plane isshown in Figure 1(a) while themovement in the z direction isfixed tomodel the plain strain conditionThe applied pressureis a triangular pulse as given in Figure 1(b) in which 119905lowast is the
normalized time 119905lowast = (119905119886)radic120583120588 119886 is the radius of tunnel 119891denotes the edge pressure and 1198910 is its peak value
The soil properties and the load parameters are thesame as those given by Senjuntichai [25] and summarizedin Table 1 Figure 2 shows the comparison on the radialdisplacement at the cavity surface where the vertical axis isthe normalized radial displacements 119880119903lowast = 120583119880119903(1198910119886) andthe horizontal axis is the normalized time 119905lowast From Figure 2certain reflections can be observed for the MTF conditionwhen the reflected waves reached the observation point at thefirst time (ie 119905lowast asymp 5) However the following impinging ofthe reflected waves is transmitted effectively by MTF rendinga very close comparison to the analytical solution for 119905lowast gt 8Also it can be seen from Figure 2 that significant reflectionscan happen during the entire time if the fixed boundarycondition is used
5 Numerical Results
In this section coupled vibrations of the liner the twintunnels and the saturated ground are investigated usingthe developed finite element along with the developed MTFboundary condition Parametric studies have been per-formed for the burying depth of tunnel the tunnel spacingthe load velocity and the soil permeability
51 Model Description The computation model consisting oftwo identical circular tunnels that are parallel to each otherand embedded in a three-dimensional saturated ground isshown in Figure 3 The model size is 119886 (width) times 119887 (height)times 119897 (length)The center-to-center spacing between the tunnelsis represented by 119889 and the burying depth measured fromthe ground surface to the tunnel center is denoted by ℎas shown in Figure 3(b) 119903 denotes the outer radius of theliner and the thickness of liner is represented by 119905 The sideand cross-sectional views with dimensions are presented in
Mathematical Problems in Engineering 5
Table 1 Parameters of the edge pressure and the saturated poroelastic medium
Parameter Symbol ValueYoungrsquos modulus 119864 533 times 107 PaPoissonrsquos ratio ] 033Density of the bulk material 120588 2000 kgm3
Density of the pore fluid 120588f 1000 kgm3
Porosity 119899 04Biot parameters 120572 098Biot parameters 119872 40 times 108 PaAdditional mass density 120588a 0 kgm3
Darcy permeability coefficient 119896119863 491 times 10minus3msTriangular pulse load 1198910 200 times 107 PaRadius of tunnel 119886 10m
present methodSenjundichai (1993)fixed boundary
2 4 6 8 10 12 14 160tlowast
minus02
minus01
00
01
02
03
UＬlowast
Figure 2 Comparison of present results with the solution proposedby Senjundichai [25]
Figures 3(a) and 3(b) respectively A nonoscillating pointload of magnitude 119865119899 that moves along the positive 119909 axle actsvertically to the invert of the left tunnel liner
Four observation points are chosen to record theresponses caused by the moving load points A and B arelocated at the liner bottom of the tunnel with the loadapplied and the other tunnel respectively points C and D arelocated on the ground surface right between the twin tunnelsand above the center of the tunnel loaded respectively Allobservation points are located at the middle of the modelalong the tunnel axial direction (ie 119909 = 1198972)
The bottom surface of the model is fixed to consider theunderlying hard stratum The top surface is free and set aspermeable The MTF boundary condition is applied to theremaining 4 side surfaces to account for the infinite extensionof the saturated ground For a reliable modeling of wavepropagations the element sizes of the ground shouldmeet therequirement Δ119904 le 120582s6 where 120582s = 2120587119888s120596max is the shearwave length and 120596max is the highest frequency of the shear
wave [26] The implicit Newmark method is employed forthe timemarching of the dynamic analysis with theNewmarkparameter being set to 06 to remove disturbances caused bysudden application of the point load Although the Newmarkintegration scheme is unconditionally stable the time stepshould meet the CFL condition as mentioned in Section 3since the MTF is essentially a forward Euler scheme
Parameters of the liner the saturated ground and themoving load are summarized in Table 2 In the subsequentanalysis the dynamic displacement and velocity are pre-sented in decibels (dB) ie 119902lowast = 20 lg(1199021199020) where 119902denotes an arbitrary variable and 1199020 is the reference valueThe reference values for displacement and velocity are 20 times10minus5m and 20 times 10minus5ms respectively
52 Models Comparisons In this section a single tunnelmodel and a model with both tunnels loaded are comparedwith the present twin-tunnel model The cross sections ofthe two models are presented in Figures 4(a) and 4(b)respectively The results of single phased model are alsocompared with those of the saturated poroelastic model inthis section
To investigate the influence of interactions between thetwo tunnels on the vibration-prediction results the compar-ison of dynamic responses between single tunnel model andthe present model is shown in Figures 5(a) and 5(b) Theburying depth and the spacing of the tunnels are respectivelyfixed at ℎ119903 = 4 and 119889119903 = 4 and the load velocity is 03 119888sIn Figure 5(a) the tunnel spacing for the single tunnel modeldenotes two times the distance between tunnel center and themodel center The dynamic responses between two modelson point A are very close while the results of the presentmodel are greater than those of the single tunnel model onthe ground observation points (point C and point D) Thedisplacements at point C for the single tunnel model decreasefaster than the present model between 119889119903 which is 4 to5 That is because the dynamic response at point C in thepresent model is the sum of the waves generated from thetunnel with the load applied and reflected wave from theother tunnel lining However the dynamic responses of thesingle-tunnel model only come from the tunnel with the loadapplied As the tunnel spacing increases the differences of the
6 Mathematical Problems in Engineering
D(C)
A(B)MTFMTF
Free and permeable
Fixed
z
y-Plane
l2
l
b
x
(a) Side view
D C
A B
z
MTFMTF
Fixed
Free and permeable
bＨ
d
tr
ℎ
a x-Plane y
(b) Cross section view
Figure 3 Schematic of the 3D computation model
D C
A
z
MTFMTF
Fixed
Free and permeable
b Ｈt r
ℎ
a x-Plane y
d2
(a) Single tunnel model
D C
A B
a
z
MTFMTF
Fixed
Free and permeable
b Ｈ Ｈ
d
tr
ℎ
x-Plane y
(b) Model with both tunnels loaded
Figure 4 Cross sections of single tunnel model and the model with both tunnels loaded
displacement at point C and D for both models decrease InFigure 5(b) it is observed that the maximum displacementsof single tunnelmodel are all smaller than those of the presentmodel when the burying depth changes from 3 to 5 Thedecrease rates for both models are almost the same
The comparison between the single tunnel model andthe present model is shown in Figure 6 under different loadvelocities It is observed from Figure 6 that the dynamicresponses of the twin-tunnel model are larger than those ofthe single tunnel model which is inconsistent with the resultsshown in Figure 5 And the difference of the two models atpoint C is greater than that at point D The differences reachpeak value when 119888119888s = 09 Since the dynamic response atpoint C is jointly affected by two tunnels while the dynamicresponse at point D is dominated by the tunnel loadedtherefore it is essential to consider the twin-tunnel model toobtain more accurate prediction results
Practically both metro tunnels are always loaded atthe same time due to the high operation frequency of thesubways The dynamic responses of the model with bothtunnel inverts loaded are studied in this section
In contrast to the result of the presentmodel the dynamicresponses of point C and point D due to two moving pointloads are larger in terms of the maximum displacement InFigure 7 the maximum displacements of the two models atpoint A are almost the same The dynamic response of thetwomodels at point C decreases a little with increasing tunnelspacing However as tunnel spacing increases the maximum
displacement of the two models at point A remains the sameand the results of two models become closer at point DThis is because as the tunnel spacing increases the effect ofthe other tunnel on point D decreases gradually And withthe increasing of the burying depth the dynamic responsesof points C and D have obvious downtrend Besides thedisplacement at the point C is larger than that at the point Din the model with two loads while the situation of the presentmodel is the opposite That is because the load in the othertunnel has a larger contribution to the dynamic response atpoint C than at point D
The comparison of the displacements and the vibrationvelocity between the present model and the model with bothtunnels loaded is shown in Figures 8 and 9 for differentload velocities For points C and D the dynamic responsesof the model with both tunnels loaded are obviously largerthan the model with only one tunnel loaded However theprediction results of the two models at point A are very closeAnd the maximum displacement at point C is smaller thanthat at point D in the model with only one tunnel loadedwhile the situation of the two-load model is the oppositewhich is explained in the previous section The differencebetween the maximum displacements of the two models isalmost unchanged at different load velocity However thedifference in the vibration velocity results of the two modelsat point D decreases as the velocity increases and reaches aminimum at 119888119888119904 = 09 From Figure 9 it can also be observedthat the maximum vibration velocities at points C and D are
Mathematical Problems in Engineering 7
Table 2 Parameters of the liner the saturated ground and the moving load
Saturated Soil Ground
Width a (m) Height b (m) Length l (m) Biot parameters 12057242 24 120 099
Spacing 119889 (m) Burying depth h (m) Porosity 119899 Poissonrsquos ratio ]9ndash24 6ndash18 0286 04
Youngrsquos modulus Darcy permeability coefficient Shear wave velocity 119888s Biot parameters119864 (Pa) 119896119863 (ms) (ms) 119872 (Nm)223 times 108 10 times 10minus5ndash10 times 10minus2 1911 574 times 109
Density of the bulk material 120588 (kgm3) Density of the pore fluid 120588f (kgm3)2178 times 103 10 times 103
Concrete LinerYoungrsquos modulus Poissonrsquos ratio Density of the bulk material outer radius Thickness119864 (Pa) ] 120588 (kgm3) r(m) t(m)
250 times 1010 02 2400 times 103 30 03
Moving Load Point load value (N) Load velocity (ms)20 times 104 1911ndash1911
0
4 45 5 55 6Tunnel spacing dr
point A(twin-tunnel)point C(twin-tunnel)point D(twin-tunnel)point A(single tunnel)point C(single tunnel)point D(single tunnel)
(b) Maximum displacement at the observation points for differentburying depth
Figure 5 Comparison of dynamic responses between a single-tunnel model and a twin-tunnel model
almost the same in the present model with only one tunnelloaded while the value at point C in the two-load model issignificantly larger than that at point D Because the load inthe other tunnel is closer to point C than point D it has agreater impact on the dynamic response of point C
To illustrate the necessity in introducing the saturatedground model the dynamic responses of the twin tunnelembedded in a single-phase ground are compared to thoseof the saturated ground model and presented in Figure 10The single-phase ground model is obtained by reducing thesaturated ground model after fixing the pore-fluid relativedisplacement and setting Biotrsquos parameters to zeros It isobserved that the dynamic responses of the elastic ground arelarger than those in the saturated ground The discrepancybetween the two ground models widens when the buryingdepth increases And as the load velocity increases the
difference of dynamic response between the two models atpoint D also becomes larger From the comparison it can beconcluded that the ground water has an obvious influence onthe vibration responses of the tunnel and the ground Theirvibration levels would be overestimated if the ground wateris not considered
53 e Influences of Load Velocity The influences of loadvelocity on the dynamic responses of the liner-tunnel-groundsystem are investigated in this section Six different velocities119888119888s = 01 03 05 07 09 and 10 are considered along withthe tunnel spacing 119889119903 = 4 and the burying depth ℎ119903 = 4
It is observed from Figure 11 that the displacements atpoints A and B change slightly with the increasing loadvelocity while the displacements at points C and D reacha peak at around 09 119888s Since C and D are on the ground
8 Mathematical Problems in Engineering
point C(twin-tunnel)point D(twin-tunnel)point C(single tunnel)point D(single tunnel)
Max
imum
disp
lace
men
t0 02 04 06 08 1
Load velocity s
minus365
minus360
minus355
minus350
minus345
minus340
minus335
minus330
UZlowast
(dB)
Figure 6 Comparison on maximum displacement at observation points between a single-tunnel model and a twin-tunnel model
(a) Maximum displacement at the observation points for differenttunnel spacing
3 35 4 45 5Burying depth hr
point C(one load)point D(one load)point C(two loads)point D(two loads)
Max
imum
disp
lace
men
t
minus45
minus40
minus35
minus30
minus25
minus20
minus15
minus10
UZlowast
(dB)
(b) Maximum displacement at the observation points for differentburying depth
Figure 7 Comparison of dynamic responses between the present model and a model with both tunnels loaded
surface their responses will be amplified when the loadvelocity approaches the Rayleigh wave velocity of the groundwhich is around 09 119888s The same results were reported inprevious studies on the critical velocity for a train runningon the ground surface [27 28] However points A and B areset on the concrete liner whose shear wave velocity is muchhigher than that of the soil Thus the load velocity is very lowcompared to the shear wave velocity of the concrete whichmakes the displacement of points A and B nonsensitive to thechange of load velocity
54 e Influence of Tunnel Spacing In this section theinfluence of the tunnel spacing on the dynamic responses isinvestigated by considering 6 different twin tunnel distancesie 119889119903 = 3 4 5 6 7 8 The load velocity is 03 119888119904 The
maximum vertical displacements at the observation pointswhen ℎ119903 = 4 are compared in Figure 12
The displacement at point A is significantly larger thanthose at other points since point A is traversed directlyby the point load When the tunnel spacing increases thedisplacements at points B and C decrease gradually Howeverthe displacements at points A and D remain almost the sameThis observation is within expectation because the positionsof points A and D remain the same while points B and C areshifted away from the load when the spacing increases Thedisplacements at points B and C vary greatly when the tunnelspacing 119889119903 increases from 4 to 6
55e Influence of Burying Depth In this section paramet-ric study is conducted for 5 different burying depths of the
Figure 9 Comparison of maximum vibration velocity at obser-vation points between a model containing one load and a modelcontaining two loads
twin tunnel ie ℎ119903 = 2 3 4 5 6 And the load velocity is setto 03 119888s
The variation of the maximum vertical displacements atthe observation points with respect to the burying depth ispresented in Figure 13 In this case the tunnel spacing is fixedas 119889119903 = 4 It is observed that the displacements at points AC and D decrease when ℎ119903 increases For the displacementat point B it increases slightly when ℎ119903 lt 4 however itdrops substantially when ℎ119903 increases further to 6 This
phenomenon is explained by the fact that when the tunnelis shallowly buried the waves generated by the source tunnel(ie the tunnel where the moving load is applied) impinge onthe ground surface at a large incident angle (resembling theglancing incident) correspondingly the reflected waves leavethe surface at a large angle and thus the surface reflectionregion at nonsourced tunnels is small resulting in a fractionof surface reflections contributing to the dynamic response atpoint B However the surface reflection region will expandwhen hr increases gradually but when the burying depthis large enough (ie ℎ119889 gt 5) the material and geometricaldamping are dominant which results in the decreasing of thedisplacement at point B
6 Conclusions
In this paper a 3D twin-tunnel model is presented to evaluatethe dynamic responses of saturated half-space induced bysubways The effects of spacing between two tunnels tunnelburied depth load moving speed and soil permeabilityon the vibration response are investigated Based on thederivation and numerical examples presented above thefollowing conclusions can be drawn
(1) The existence of the other tunnel has influences onthe vibration caused by the moving subway Andwhen the twin tunnels are both loaded the dynamicresponses cannot be calculated by simply summingup the results of a single tunnel model Therefore itis necessary to consider a twin-tunnel model whenpredicting the dynamic responses induced by sub-ways for all load velocities The dynamic responsesof the single-phase ground are larger than those inthe saturated ground which denotes that using asingle phase model to predict the vibrations is moreconservative
(2) The influence of the spacing between two tunnels onthe dynamic response of the tunnel without the loadapplied is greater than that on the ground surfaceWith the increase of the spacing 119889119903 the dynamicresponse of the adjacent tunnel decreases Tunnelburied depth has a great influence on the dynamicresponse of adjacent tunnel and the ground sur-face For points on the ground surface the dynamicresponse decreases with the increase of the tunnelburied depthDue to the presence of surface reflectionregion the dynamic response of adjacent tunnelincreases at first and then decreases as the burialdepth increases So more attention should be paid tothe spacing between tunnels and the tunnel burieddepth in the construction of subway
(3) The subways moving at a low speed will generate asmaller dynamic response in the saturated half-spacethan moving at a higher speed And when the loadvelocity approaches the Rayleigh wave velocity of theground which is around 09 119888119904 the responses of thesoil and liner will be amplified Therefore the subwaymoving speed is also an important parameter thataffects the environmental vibration
10 Mathematical Problems in Engineering
0
2 3 4 5 6Burying depth hr
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
Max
imum
vib
ratio
n ve
loci
ty
minus40
minus30
minus20
minus10
10
20
30V
Zlowast(d
B)
(a) Maximum vibration velocity at observation points for differentburying depth
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
Max
imum
disp
lace
men
t
01 03 05 07 09Load velocity s
minus40
minus35
minus30
minus25
minus20
minus15
minus10
minus5
0
UZlowast
(dB)
(b) Maximum displacement at observation points for different loadvelocity
minus30
minus20
minus10
0
10
20
30
Max
imum
vib
ratio
n ve
loci
ty
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
01 03 05 07 09Load velocity s
VZlowast
(dB)
(c) Maximum vibration velocity at observation points for differentload velocity
Figure 10 Comparisons of dynamic response at observation points between a saturated ground and an elastic ground
Appendix
The expressions of the elements in the mass matrix thedamping matrix the stiffness matrix and the force vector arepresented as follows
Figure 13 Vertical maximum displacements for different buryingdepth of tunnels
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the National Key RampD Programof China (Grant No 2016YFC0800200) the Projects ofInternational Cooperation and Exchanges NSFC (Grant No51620105008) National Science Foundation for Young Scien-tists of China (Grant No 51608482) and National NaturalScience Foundation of China (Grant No 51478424)
References
[1] A V Metrikine and A C W M Vrouwenvelder ldquoSurfaceground vibration due to a moving train in a tunnel two-dimensional modelrdquo Journal of Sound and Vibration vol 234no 1 pp 43ndash66 2000
[2] J A Forrest and H E M Hunt ldquoA three-dimensional tunnelmodel for calculation of train-induced ground vibrationrdquo Jour-nal of Sound and Vibration vol 294 no 4 pp 678ndash705 2006
[3] J A Forrest andH EMHunt ldquoGround vibration generated bytrains in underground tunnelsrdquo Journal of Sound and Vibrationvol 294 no 4 pp 706ndash736 2006
[4] K-H Xie G-B Liu and Z-Y Shi ldquoDynamic response ofpartially sealed circular tunnel in viscoelastic saturated soilrdquoSoil Dynamics and Earthquake Engineering vol 24 no 12 pp1003ndash1011 2004
[5] K-H Xie and G-B Liu ldquoTransient response of a sphericalcavity with a partially sealed shell embedded in viscoelasticsaturated soilrdquo Journal of Zhejiang University Science A vol 6no 3 pp 194ndash201 2005
[6] Z-H Yuan Y-Q Cai L Shi H-L Sun and Z-G CaoldquoResponse of rail structure and circular tunnel in saturatedsoil subjected to harmonic moving loadrdquo Chinese Journal ofGeotechnical Engineering vol 38 no 2 pp 311ndash322 2016
12 Mathematical Problems in Engineering
[7] H E M Hunt ldquoModelling of rail vehicles and track forcalculation of ground-vibration transmission into buildingsrdquoJournal of Sound and Vibration vol 193 no 1 pp 185ndash194 1996
[8] T Balendra K H Chua K W Lo and S L Lee ldquoSteady-statevibration of subway-soilbuilding systemrdquo Journal of EngineeringMechanics vol 115 no 1 pp 145ndash162 1989
[9] F-C Wang H Xia and H-R Zhang ldquoVibration effects ofsubway trains on surrounding buildingsrdquo Journal of NorthernJiaotong University vol 23 no 5 pp 45ndash48 1999
[10] Y Yang HHung and L Hsu ldquoGround vibrations due to under-ground trains considering soil-tunnel interactionrdquo Interactionand Multiscale Mechanics vol 1 no 1 pp 157ndash175 2007
[11] Y-E Zhang and Changshi-Pan ldquoTests and analyses of thedynamic response of tunnels subjected to passing train loadrdquoJournal of Shijiazhuang Railway Institute vol 6 no 2 pp 7ndash141993
[12] C J C Jones D J Thompson and M Petyt ldquoStudies usinga combined finite element and boundary element model forvibration propagation from railway tunnelsrdquo International Insti-tute of Acoustics and Vibration pp 2703ndash2710 2000
[13] W Gardien and H G Stuit ldquoModelling of soil vibrations fromrailway tunnelsrdquo Journal of Sound and Vibration vol 267 no 3pp 605ndash619 2003
[14] S Wolf ldquoPotential low frequency ground vibration (lt63 Hz)impacts from underground LRT operationsrdquo Journal of Soundand Vibration vol 267 no 3 pp 651ndash661 2003
[15] G Degrande D Clouteau R Othman et al ldquoA numericalmodel for ground-borne vibrations from underground railwaytraffic based on a periodic finite element-boundary elementformulationrdquo Journal of Sound and Vibration vol 293 no 3-5pp 645ndash666 2006
[16] W-F Liu W-N Liu S Gupta and G Degrande ldquoPrediction ofvibrations in the tunnel and free field due to passage of metrotrainsrdquo Engineering Mechanics vol 27 no 1 pp 250ndash256 2010
[17] K A Kuo H E M Hunt and M F M Hussein ldquoThe effectof a twin tunnel on the propagation of ground-borne vibrationfrom an underground railwayrdquo Journal of Sound and Vibrationvol 330 no 25 pp 6203ndash6222 2011
[18] Z-H Yuan Y-Q Cai W Yuan Y-L Xu and Z-G CaoldquoDynamic response of circular railway tunnel and track systemin saturated soil under moving train loadingrdquo Rock and SoilMechanics vol 38 no 4 pp 1003ndash1014 2017
[19] Z P Liao and H L Wong ldquoA transmitting boundary for thenumerical simulation of elasticwave propagationrdquo InternationalJournal of Soil Dynamics and Earthquake Engineering vol 3 pp133ndash144 1984
[20] L Shi P Wang Y Cai and Z Cao ldquoMulti-transmittingformula for finite element modeling of wave propagation in asaturated poroelastic mediumrdquo Soil Dynamics and EarthquakeEngineering vol 80 pp 11ndash24 2016
[21] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid I Low-frequency rangerdquoe Journal ofthe Acoustical Society of America vol 28 pp 168ndash178 1956
[22] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956
[23] O C Zienkiewicz C T Chang and P Bettess ldquoDrainedundrained consolidating and dynamic behaviour assumptionsin soilsrdquo Geotechnique vol 30 no 4 pp 385ndash395 1980
[24] L Shi H Sun Y Cai C Xu and P Wang ldquoValidity of fullydrained fully undrained and u-p formulations for modeling a
poroelastic half-space under a moving harmonic point loadrdquoSoil Dynamics and Earthquake Engineering vol 42 pp 292ndash3012012
[25] T Senjuntichai and R K N D Rajapakse ldquoTransient responseof a circular cavity in a poroelastic mediumrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 17 no 6 pp 357ndash383 1993
[26] Y-B Yang S-R Kuo and H-H Hung ldquoFrequency-independent infinite elements for analysing semi-infiniteproblemsrdquo International Journal for Numerical Methods inEngineering vol 39 no 20 pp 3553ndash3569 1996
[27] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic response of tracksystem and saturated soil in half space subjected to a movingtrain loadrdquo Chinese Journal of Rock Mechanics and Engineeringvol 26 no 8 pp 1705ndash1712 2007
[28] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic responses oftrack system and poroelastic soil under high-speed train loadrdquoJournal of Zhejiang University (Engineering Science) vol 42 no11 pp 2002ndash2008 2008
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Correspondingly the operators119885minus1119905 and119885minus1119911 represent thebackward time and space shifts respectively
According to Shi et al [20] a second-order MTF (119873 = 2)and the artificial wave velocity 119888119886 = 119888s (119888s is the shear wavevelocity of the saturated medium) are efficient and accurateenough for configuring the MTF Moreover the numericalstability of the MTF can be guaranteed by (1) meeting theCourant-Friedrichs-Lewy (CFL) condition 119904 = 119888aΔ119905Δ119904 le1radic2 and (2) perturbing the interpolation coefficient 11987911 byadding a small negative value (= minus004119904 sim minus03119904)4 Verification of the Present Model
To verify the accuracy of the proposed finite element formu-lation and the effectiveness of the MTF boundary conditionthe transient response due to an edge pressure applied radiallyat a cavity surface is computed and compared to the analyticalsolution given by T Senjuntichai [25] which is developed fora cavity in an infinite saturated soil medium under the plane-strain condition
Due to the symmetry a 14 model is established for thecavity and the surrounding saturated soil medium Symmet-ric boundary conditions are imposed for the left and bottomboundaries while MTF boundary conditions are applied tothe top and right boundaries to account for the infiniteextension of the soil medium The mesh in the x-y plane isshown in Figure 1(a) while themovement in the z direction isfixed tomodel the plain strain conditionThe applied pressureis a triangular pulse as given in Figure 1(b) in which 119905lowast is the
normalized time 119905lowast = (119905119886)radic120583120588 119886 is the radius of tunnel 119891denotes the edge pressure and 1198910 is its peak value
The soil properties and the load parameters are thesame as those given by Senjuntichai [25] and summarizedin Table 1 Figure 2 shows the comparison on the radialdisplacement at the cavity surface where the vertical axis isthe normalized radial displacements 119880119903lowast = 120583119880119903(1198910119886) andthe horizontal axis is the normalized time 119905lowast From Figure 2certain reflections can be observed for the MTF conditionwhen the reflected waves reached the observation point at thefirst time (ie 119905lowast asymp 5) However the following impinging ofthe reflected waves is transmitted effectively by MTF rendinga very close comparison to the analytical solution for 119905lowast gt 8Also it can be seen from Figure 2 that significant reflectionscan happen during the entire time if the fixed boundarycondition is used
5 Numerical Results
In this section coupled vibrations of the liner the twintunnels and the saturated ground are investigated usingthe developed finite element along with the developed MTFboundary condition Parametric studies have been per-formed for the burying depth of tunnel the tunnel spacingthe load velocity and the soil permeability
51 Model Description The computation model consisting oftwo identical circular tunnels that are parallel to each otherand embedded in a three-dimensional saturated ground isshown in Figure 3 The model size is 119886 (width) times 119887 (height)times 119897 (length)The center-to-center spacing between the tunnelsis represented by 119889 and the burying depth measured fromthe ground surface to the tunnel center is denoted by ℎas shown in Figure 3(b) 119903 denotes the outer radius of theliner and the thickness of liner is represented by 119905 The sideand cross-sectional views with dimensions are presented in
Mathematical Problems in Engineering 5
Table 1 Parameters of the edge pressure and the saturated poroelastic medium
Parameter Symbol ValueYoungrsquos modulus 119864 533 times 107 PaPoissonrsquos ratio ] 033Density of the bulk material 120588 2000 kgm3
Density of the pore fluid 120588f 1000 kgm3
Porosity 119899 04Biot parameters 120572 098Biot parameters 119872 40 times 108 PaAdditional mass density 120588a 0 kgm3
Darcy permeability coefficient 119896119863 491 times 10minus3msTriangular pulse load 1198910 200 times 107 PaRadius of tunnel 119886 10m
present methodSenjundichai (1993)fixed boundary
2 4 6 8 10 12 14 160tlowast
minus02
minus01
00
01
02
03
UＬlowast
Figure 2 Comparison of present results with the solution proposedby Senjundichai [25]
Figures 3(a) and 3(b) respectively A nonoscillating pointload of magnitude 119865119899 that moves along the positive 119909 axle actsvertically to the invert of the left tunnel liner
Four observation points are chosen to record theresponses caused by the moving load points A and B arelocated at the liner bottom of the tunnel with the loadapplied and the other tunnel respectively points C and D arelocated on the ground surface right between the twin tunnelsand above the center of the tunnel loaded respectively Allobservation points are located at the middle of the modelalong the tunnel axial direction (ie 119909 = 1198972)
The bottom surface of the model is fixed to consider theunderlying hard stratum The top surface is free and set aspermeable The MTF boundary condition is applied to theremaining 4 side surfaces to account for the infinite extensionof the saturated ground For a reliable modeling of wavepropagations the element sizes of the ground shouldmeet therequirement Δ119904 le 120582s6 where 120582s = 2120587119888s120596max is the shearwave length and 120596max is the highest frequency of the shear
wave [26] The implicit Newmark method is employed forthe timemarching of the dynamic analysis with theNewmarkparameter being set to 06 to remove disturbances caused bysudden application of the point load Although the Newmarkintegration scheme is unconditionally stable the time stepshould meet the CFL condition as mentioned in Section 3since the MTF is essentially a forward Euler scheme
Parameters of the liner the saturated ground and themoving load are summarized in Table 2 In the subsequentanalysis the dynamic displacement and velocity are pre-sented in decibels (dB) ie 119902lowast = 20 lg(1199021199020) where 119902denotes an arbitrary variable and 1199020 is the reference valueThe reference values for displacement and velocity are 20 times10minus5m and 20 times 10minus5ms respectively
52 Models Comparisons In this section a single tunnelmodel and a model with both tunnels loaded are comparedwith the present twin-tunnel model The cross sections ofthe two models are presented in Figures 4(a) and 4(b)respectively The results of single phased model are alsocompared with those of the saturated poroelastic model inthis section
To investigate the influence of interactions between thetwo tunnels on the vibration-prediction results the compar-ison of dynamic responses between single tunnel model andthe present model is shown in Figures 5(a) and 5(b) Theburying depth and the spacing of the tunnels are respectivelyfixed at ℎ119903 = 4 and 119889119903 = 4 and the load velocity is 03 119888sIn Figure 5(a) the tunnel spacing for the single tunnel modeldenotes two times the distance between tunnel center and themodel center The dynamic responses between two modelson point A are very close while the results of the presentmodel are greater than those of the single tunnel model onthe ground observation points (point C and point D) Thedisplacements at point C for the single tunnel model decreasefaster than the present model between 119889119903 which is 4 to5 That is because the dynamic response at point C in thepresent model is the sum of the waves generated from thetunnel with the load applied and reflected wave from theother tunnel lining However the dynamic responses of thesingle-tunnel model only come from the tunnel with the loadapplied As the tunnel spacing increases the differences of the
6 Mathematical Problems in Engineering
D(C)
A(B)MTFMTF
Free and permeable
Fixed
z
y-Plane
l2
l
b
x
(a) Side view
D C
A B
z
MTFMTF
Fixed
Free and permeable
bＨ
d
tr
ℎ
a x-Plane y
(b) Cross section view
Figure 3 Schematic of the 3D computation model
D C
A
z
MTFMTF
Fixed
Free and permeable
b Ｈt r
ℎ
a x-Plane y
d2
(a) Single tunnel model
D C
A B
a
z
MTFMTF
Fixed
Free and permeable
b Ｈ Ｈ
d
tr
ℎ
x-Plane y
(b) Model with both tunnels loaded
Figure 4 Cross sections of single tunnel model and the model with both tunnels loaded
displacement at point C and D for both models decrease InFigure 5(b) it is observed that the maximum displacementsof single tunnelmodel are all smaller than those of the presentmodel when the burying depth changes from 3 to 5 Thedecrease rates for both models are almost the same
The comparison between the single tunnel model andthe present model is shown in Figure 6 under different loadvelocities It is observed from Figure 6 that the dynamicresponses of the twin-tunnel model are larger than those ofthe single tunnel model which is inconsistent with the resultsshown in Figure 5 And the difference of the two models atpoint C is greater than that at point D The differences reachpeak value when 119888119888s = 09 Since the dynamic response atpoint C is jointly affected by two tunnels while the dynamicresponse at point D is dominated by the tunnel loadedtherefore it is essential to consider the twin-tunnel model toobtain more accurate prediction results
Practically both metro tunnels are always loaded atthe same time due to the high operation frequency of thesubways The dynamic responses of the model with bothtunnel inverts loaded are studied in this section
In contrast to the result of the presentmodel the dynamicresponses of point C and point D due to two moving pointloads are larger in terms of the maximum displacement InFigure 7 the maximum displacements of the two models atpoint A are almost the same The dynamic response of thetwomodels at point C decreases a little with increasing tunnelspacing However as tunnel spacing increases the maximum
displacement of the two models at point A remains the sameand the results of two models become closer at point DThis is because as the tunnel spacing increases the effect ofthe other tunnel on point D decreases gradually And withthe increasing of the burying depth the dynamic responsesof points C and D have obvious downtrend Besides thedisplacement at the point C is larger than that at the point Din the model with two loads while the situation of the presentmodel is the opposite That is because the load in the othertunnel has a larger contribution to the dynamic response atpoint C than at point D
The comparison of the displacements and the vibrationvelocity between the present model and the model with bothtunnels loaded is shown in Figures 8 and 9 for differentload velocities For points C and D the dynamic responsesof the model with both tunnels loaded are obviously largerthan the model with only one tunnel loaded However theprediction results of the two models at point A are very closeAnd the maximum displacement at point C is smaller thanthat at point D in the model with only one tunnel loadedwhile the situation of the two-load model is the oppositewhich is explained in the previous section The differencebetween the maximum displacements of the two models isalmost unchanged at different load velocity However thedifference in the vibration velocity results of the two modelsat point D decreases as the velocity increases and reaches aminimum at 119888119888119904 = 09 From Figure 9 it can also be observedthat the maximum vibration velocities at points C and D are
Mathematical Problems in Engineering 7
Table 2 Parameters of the liner the saturated ground and the moving load
Saturated Soil Ground
Width a (m) Height b (m) Length l (m) Biot parameters 12057242 24 120 099
Spacing 119889 (m) Burying depth h (m) Porosity 119899 Poissonrsquos ratio ]9ndash24 6ndash18 0286 04
Youngrsquos modulus Darcy permeability coefficient Shear wave velocity 119888s Biot parameters119864 (Pa) 119896119863 (ms) (ms) 119872 (Nm)223 times 108 10 times 10minus5ndash10 times 10minus2 1911 574 times 109
Density of the bulk material 120588 (kgm3) Density of the pore fluid 120588f (kgm3)2178 times 103 10 times 103
Concrete LinerYoungrsquos modulus Poissonrsquos ratio Density of the bulk material outer radius Thickness119864 (Pa) ] 120588 (kgm3) r(m) t(m)
250 times 1010 02 2400 times 103 30 03
Moving Load Point load value (N) Load velocity (ms)20 times 104 1911ndash1911
0
4 45 5 55 6Tunnel spacing dr
point A(twin-tunnel)point C(twin-tunnel)point D(twin-tunnel)point A(single tunnel)point C(single tunnel)point D(single tunnel)
(b) Maximum displacement at the observation points for differentburying depth
Figure 5 Comparison of dynamic responses between a single-tunnel model and a twin-tunnel model
almost the same in the present model with only one tunnelloaded while the value at point C in the two-load model issignificantly larger than that at point D Because the load inthe other tunnel is closer to point C than point D it has agreater impact on the dynamic response of point C
To illustrate the necessity in introducing the saturatedground model the dynamic responses of the twin tunnelembedded in a single-phase ground are compared to thoseof the saturated ground model and presented in Figure 10The single-phase ground model is obtained by reducing thesaturated ground model after fixing the pore-fluid relativedisplacement and setting Biotrsquos parameters to zeros It isobserved that the dynamic responses of the elastic ground arelarger than those in the saturated ground The discrepancybetween the two ground models widens when the buryingdepth increases And as the load velocity increases the
difference of dynamic response between the two models atpoint D also becomes larger From the comparison it can beconcluded that the ground water has an obvious influence onthe vibration responses of the tunnel and the ground Theirvibration levels would be overestimated if the ground wateris not considered
53 e Influences of Load Velocity The influences of loadvelocity on the dynamic responses of the liner-tunnel-groundsystem are investigated in this section Six different velocities119888119888s = 01 03 05 07 09 and 10 are considered along withthe tunnel spacing 119889119903 = 4 and the burying depth ℎ119903 = 4
It is observed from Figure 11 that the displacements atpoints A and B change slightly with the increasing loadvelocity while the displacements at points C and D reacha peak at around 09 119888s Since C and D are on the ground
8 Mathematical Problems in Engineering
point C(twin-tunnel)point D(twin-tunnel)point C(single tunnel)point D(single tunnel)
Max
imum
disp
lace
men
t0 02 04 06 08 1
Load velocity s
minus365
minus360
minus355
minus350
minus345
minus340
minus335
minus330
UZlowast
(dB)
Figure 6 Comparison on maximum displacement at observation points between a single-tunnel model and a twin-tunnel model
(a) Maximum displacement at the observation points for differenttunnel spacing
3 35 4 45 5Burying depth hr
point C(one load)point D(one load)point C(two loads)point D(two loads)
Max
imum
disp
lace
men
t
minus45
minus40
minus35
minus30
minus25
minus20
minus15
minus10
UZlowast
(dB)
(b) Maximum displacement at the observation points for differentburying depth
Figure 7 Comparison of dynamic responses between the present model and a model with both tunnels loaded
surface their responses will be amplified when the loadvelocity approaches the Rayleigh wave velocity of the groundwhich is around 09 119888s The same results were reported inprevious studies on the critical velocity for a train runningon the ground surface [27 28] However points A and B areset on the concrete liner whose shear wave velocity is muchhigher than that of the soil Thus the load velocity is very lowcompared to the shear wave velocity of the concrete whichmakes the displacement of points A and B nonsensitive to thechange of load velocity
54 e Influence of Tunnel Spacing In this section theinfluence of the tunnel spacing on the dynamic responses isinvestigated by considering 6 different twin tunnel distancesie 119889119903 = 3 4 5 6 7 8 The load velocity is 03 119888119904 The
maximum vertical displacements at the observation pointswhen ℎ119903 = 4 are compared in Figure 12
The displacement at point A is significantly larger thanthose at other points since point A is traversed directlyby the point load When the tunnel spacing increases thedisplacements at points B and C decrease gradually Howeverthe displacements at points A and D remain almost the sameThis observation is within expectation because the positionsof points A and D remain the same while points B and C areshifted away from the load when the spacing increases Thedisplacements at points B and C vary greatly when the tunnelspacing 119889119903 increases from 4 to 6
55e Influence of Burying Depth In this section paramet-ric study is conducted for 5 different burying depths of the
Figure 9 Comparison of maximum vibration velocity at obser-vation points between a model containing one load and a modelcontaining two loads
twin tunnel ie ℎ119903 = 2 3 4 5 6 And the load velocity is setto 03 119888s
The variation of the maximum vertical displacements atthe observation points with respect to the burying depth ispresented in Figure 13 In this case the tunnel spacing is fixedas 119889119903 = 4 It is observed that the displacements at points AC and D decrease when ℎ119903 increases For the displacementat point B it increases slightly when ℎ119903 lt 4 however itdrops substantially when ℎ119903 increases further to 6 This
phenomenon is explained by the fact that when the tunnelis shallowly buried the waves generated by the source tunnel(ie the tunnel where the moving load is applied) impinge onthe ground surface at a large incident angle (resembling theglancing incident) correspondingly the reflected waves leavethe surface at a large angle and thus the surface reflectionregion at nonsourced tunnels is small resulting in a fractionof surface reflections contributing to the dynamic response atpoint B However the surface reflection region will expandwhen hr increases gradually but when the burying depthis large enough (ie ℎ119889 gt 5) the material and geometricaldamping are dominant which results in the decreasing of thedisplacement at point B
6 Conclusions
In this paper a 3D twin-tunnel model is presented to evaluatethe dynamic responses of saturated half-space induced bysubways The effects of spacing between two tunnels tunnelburied depth load moving speed and soil permeabilityon the vibration response are investigated Based on thederivation and numerical examples presented above thefollowing conclusions can be drawn
(1) The existence of the other tunnel has influences onthe vibration caused by the moving subway Andwhen the twin tunnels are both loaded the dynamicresponses cannot be calculated by simply summingup the results of a single tunnel model Therefore itis necessary to consider a twin-tunnel model whenpredicting the dynamic responses induced by sub-ways for all load velocities The dynamic responsesof the single-phase ground are larger than those inthe saturated ground which denotes that using asingle phase model to predict the vibrations is moreconservative
(2) The influence of the spacing between two tunnels onthe dynamic response of the tunnel without the loadapplied is greater than that on the ground surfaceWith the increase of the spacing 119889119903 the dynamicresponse of the adjacent tunnel decreases Tunnelburied depth has a great influence on the dynamicresponse of adjacent tunnel and the ground sur-face For points on the ground surface the dynamicresponse decreases with the increase of the tunnelburied depthDue to the presence of surface reflectionregion the dynamic response of adjacent tunnelincreases at first and then decreases as the burialdepth increases So more attention should be paid tothe spacing between tunnels and the tunnel burieddepth in the construction of subway
(3) The subways moving at a low speed will generate asmaller dynamic response in the saturated half-spacethan moving at a higher speed And when the loadvelocity approaches the Rayleigh wave velocity of theground which is around 09 119888119904 the responses of thesoil and liner will be amplified Therefore the subwaymoving speed is also an important parameter thataffects the environmental vibration
10 Mathematical Problems in Engineering
0
2 3 4 5 6Burying depth hr
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
Max
imum
vib
ratio
n ve
loci
ty
minus40
minus30
minus20
minus10
10
20
30V
Zlowast(d
B)
(a) Maximum vibration velocity at observation points for differentburying depth
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
Max
imum
disp
lace
men
t
01 03 05 07 09Load velocity s
minus40
minus35
minus30
minus25
minus20
minus15
minus10
minus5
0
UZlowast
(dB)
(b) Maximum displacement at observation points for different loadvelocity
minus30
minus20
minus10
0
10
20
30
Max
imum
vib
ratio
n ve
loci
ty
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
01 03 05 07 09Load velocity s
VZlowast
(dB)
(c) Maximum vibration velocity at observation points for differentload velocity
Figure 10 Comparisons of dynamic response at observation points between a saturated ground and an elastic ground
Appendix
The expressions of the elements in the mass matrix thedamping matrix the stiffness matrix and the force vector arepresented as follows
Figure 13 Vertical maximum displacements for different buryingdepth of tunnels
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the National Key RampD Programof China (Grant No 2016YFC0800200) the Projects ofInternational Cooperation and Exchanges NSFC (Grant No51620105008) National Science Foundation for Young Scien-tists of China (Grant No 51608482) and National NaturalScience Foundation of China (Grant No 51478424)
References
[1] A V Metrikine and A C W M Vrouwenvelder ldquoSurfaceground vibration due to a moving train in a tunnel two-dimensional modelrdquo Journal of Sound and Vibration vol 234no 1 pp 43ndash66 2000
[2] J A Forrest and H E M Hunt ldquoA three-dimensional tunnelmodel for calculation of train-induced ground vibrationrdquo Jour-nal of Sound and Vibration vol 294 no 4 pp 678ndash705 2006
[3] J A Forrest andH EMHunt ldquoGround vibration generated bytrains in underground tunnelsrdquo Journal of Sound and Vibrationvol 294 no 4 pp 706ndash736 2006
[4] K-H Xie G-B Liu and Z-Y Shi ldquoDynamic response ofpartially sealed circular tunnel in viscoelastic saturated soilrdquoSoil Dynamics and Earthquake Engineering vol 24 no 12 pp1003ndash1011 2004
[5] K-H Xie and G-B Liu ldquoTransient response of a sphericalcavity with a partially sealed shell embedded in viscoelasticsaturated soilrdquo Journal of Zhejiang University Science A vol 6no 3 pp 194ndash201 2005
[6] Z-H Yuan Y-Q Cai L Shi H-L Sun and Z-G CaoldquoResponse of rail structure and circular tunnel in saturatedsoil subjected to harmonic moving loadrdquo Chinese Journal ofGeotechnical Engineering vol 38 no 2 pp 311ndash322 2016
12 Mathematical Problems in Engineering
[7] H E M Hunt ldquoModelling of rail vehicles and track forcalculation of ground-vibration transmission into buildingsrdquoJournal of Sound and Vibration vol 193 no 1 pp 185ndash194 1996
[8] T Balendra K H Chua K W Lo and S L Lee ldquoSteady-statevibration of subway-soilbuilding systemrdquo Journal of EngineeringMechanics vol 115 no 1 pp 145ndash162 1989
[9] F-C Wang H Xia and H-R Zhang ldquoVibration effects ofsubway trains on surrounding buildingsrdquo Journal of NorthernJiaotong University vol 23 no 5 pp 45ndash48 1999
[10] Y Yang HHung and L Hsu ldquoGround vibrations due to under-ground trains considering soil-tunnel interactionrdquo Interactionand Multiscale Mechanics vol 1 no 1 pp 157ndash175 2007
[11] Y-E Zhang and Changshi-Pan ldquoTests and analyses of thedynamic response of tunnels subjected to passing train loadrdquoJournal of Shijiazhuang Railway Institute vol 6 no 2 pp 7ndash141993
[12] C J C Jones D J Thompson and M Petyt ldquoStudies usinga combined finite element and boundary element model forvibration propagation from railway tunnelsrdquo International Insti-tute of Acoustics and Vibration pp 2703ndash2710 2000
[13] W Gardien and H G Stuit ldquoModelling of soil vibrations fromrailway tunnelsrdquo Journal of Sound and Vibration vol 267 no 3pp 605ndash619 2003
[14] S Wolf ldquoPotential low frequency ground vibration (lt63 Hz)impacts from underground LRT operationsrdquo Journal of Soundand Vibration vol 267 no 3 pp 651ndash661 2003
[15] G Degrande D Clouteau R Othman et al ldquoA numericalmodel for ground-borne vibrations from underground railwaytraffic based on a periodic finite element-boundary elementformulationrdquo Journal of Sound and Vibration vol 293 no 3-5pp 645ndash666 2006
[16] W-F Liu W-N Liu S Gupta and G Degrande ldquoPrediction ofvibrations in the tunnel and free field due to passage of metrotrainsrdquo Engineering Mechanics vol 27 no 1 pp 250ndash256 2010
[17] K A Kuo H E M Hunt and M F M Hussein ldquoThe effectof a twin tunnel on the propagation of ground-borne vibrationfrom an underground railwayrdquo Journal of Sound and Vibrationvol 330 no 25 pp 6203ndash6222 2011
[18] Z-H Yuan Y-Q Cai W Yuan Y-L Xu and Z-G CaoldquoDynamic response of circular railway tunnel and track systemin saturated soil under moving train loadingrdquo Rock and SoilMechanics vol 38 no 4 pp 1003ndash1014 2017
[19] Z P Liao and H L Wong ldquoA transmitting boundary for thenumerical simulation of elasticwave propagationrdquo InternationalJournal of Soil Dynamics and Earthquake Engineering vol 3 pp133ndash144 1984
[20] L Shi P Wang Y Cai and Z Cao ldquoMulti-transmittingformula for finite element modeling of wave propagation in asaturated poroelastic mediumrdquo Soil Dynamics and EarthquakeEngineering vol 80 pp 11ndash24 2016
[21] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid I Low-frequency rangerdquoe Journal ofthe Acoustical Society of America vol 28 pp 168ndash178 1956
[22] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956
[23] O C Zienkiewicz C T Chang and P Bettess ldquoDrainedundrained consolidating and dynamic behaviour assumptionsin soilsrdquo Geotechnique vol 30 no 4 pp 385ndash395 1980
[24] L Shi H Sun Y Cai C Xu and P Wang ldquoValidity of fullydrained fully undrained and u-p formulations for modeling a
poroelastic half-space under a moving harmonic point loadrdquoSoil Dynamics and Earthquake Engineering vol 42 pp 292ndash3012012
[25] T Senjuntichai and R K N D Rajapakse ldquoTransient responseof a circular cavity in a poroelastic mediumrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 17 no 6 pp 357ndash383 1993
[26] Y-B Yang S-R Kuo and H-H Hung ldquoFrequency-independent infinite elements for analysing semi-infiniteproblemsrdquo International Journal for Numerical Methods inEngineering vol 39 no 20 pp 3553ndash3569 1996
[27] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic response of tracksystem and saturated soil in half space subjected to a movingtrain loadrdquo Chinese Journal of Rock Mechanics and Engineeringvol 26 no 8 pp 1705ndash1712 2007
[28] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic responses oftrack system and poroelastic soil under high-speed train loadrdquoJournal of Zhejiang University (Engineering Science) vol 42 no11 pp 2002ndash2008 2008
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Mathematical Problems in Engineering 5
Table 1 Parameters of the edge pressure and the saturated poroelastic medium
Parameter Symbol ValueYoungrsquos modulus 119864 533 times 107 PaPoissonrsquos ratio ] 033Density of the bulk material 120588 2000 kgm3
Density of the pore fluid 120588f 1000 kgm3
Porosity 119899 04Biot parameters 120572 098Biot parameters 119872 40 times 108 PaAdditional mass density 120588a 0 kgm3
Darcy permeability coefficient 119896119863 491 times 10minus3msTriangular pulse load 1198910 200 times 107 PaRadius of tunnel 119886 10m
present methodSenjundichai (1993)fixed boundary
2 4 6 8 10 12 14 160tlowast
minus02
minus01
00
01
02
03
UＬlowast
Figure 2 Comparison of present results with the solution proposedby Senjundichai [25]
Figures 3(a) and 3(b) respectively A nonoscillating pointload of magnitude 119865119899 that moves along the positive 119909 axle actsvertically to the invert of the left tunnel liner
Four observation points are chosen to record theresponses caused by the moving load points A and B arelocated at the liner bottom of the tunnel with the loadapplied and the other tunnel respectively points C and D arelocated on the ground surface right between the twin tunnelsand above the center of the tunnel loaded respectively Allobservation points are located at the middle of the modelalong the tunnel axial direction (ie 119909 = 1198972)
The bottom surface of the model is fixed to consider theunderlying hard stratum The top surface is free and set aspermeable The MTF boundary condition is applied to theremaining 4 side surfaces to account for the infinite extensionof the saturated ground For a reliable modeling of wavepropagations the element sizes of the ground shouldmeet therequirement Δ119904 le 120582s6 where 120582s = 2120587119888s120596max is the shearwave length and 120596max is the highest frequency of the shear
wave [26] The implicit Newmark method is employed forthe timemarching of the dynamic analysis with theNewmarkparameter being set to 06 to remove disturbances caused bysudden application of the point load Although the Newmarkintegration scheme is unconditionally stable the time stepshould meet the CFL condition as mentioned in Section 3since the MTF is essentially a forward Euler scheme
Parameters of the liner the saturated ground and themoving load are summarized in Table 2 In the subsequentanalysis the dynamic displacement and velocity are pre-sented in decibels (dB) ie 119902lowast = 20 lg(1199021199020) where 119902denotes an arbitrary variable and 1199020 is the reference valueThe reference values for displacement and velocity are 20 times10minus5m and 20 times 10minus5ms respectively
52 Models Comparisons In this section a single tunnelmodel and a model with both tunnels loaded are comparedwith the present twin-tunnel model The cross sections ofthe two models are presented in Figures 4(a) and 4(b)respectively The results of single phased model are alsocompared with those of the saturated poroelastic model inthis section
To investigate the influence of interactions between thetwo tunnels on the vibration-prediction results the compar-ison of dynamic responses between single tunnel model andthe present model is shown in Figures 5(a) and 5(b) Theburying depth and the spacing of the tunnels are respectivelyfixed at ℎ119903 = 4 and 119889119903 = 4 and the load velocity is 03 119888sIn Figure 5(a) the tunnel spacing for the single tunnel modeldenotes two times the distance between tunnel center and themodel center The dynamic responses between two modelson point A are very close while the results of the presentmodel are greater than those of the single tunnel model onthe ground observation points (point C and point D) Thedisplacements at point C for the single tunnel model decreasefaster than the present model between 119889119903 which is 4 to5 That is because the dynamic response at point C in thepresent model is the sum of the waves generated from thetunnel with the load applied and reflected wave from theother tunnel lining However the dynamic responses of thesingle-tunnel model only come from the tunnel with the loadapplied As the tunnel spacing increases the differences of the
6 Mathematical Problems in Engineering
D(C)
A(B)MTFMTF
Free and permeable
Fixed
z
y-Plane
l2
l
b
x
(a) Side view
D C
A B
z
MTFMTF
Fixed
Free and permeable
bＨ
d
tr
ℎ
a x-Plane y
(b) Cross section view
Figure 3 Schematic of the 3D computation model
D C
A
z
MTFMTF
Fixed
Free and permeable
b Ｈt r
ℎ
a x-Plane y
d2
(a) Single tunnel model
D C
A B
a
z
MTFMTF
Fixed
Free and permeable
b Ｈ Ｈ
d
tr
ℎ
x-Plane y
(b) Model with both tunnels loaded
Figure 4 Cross sections of single tunnel model and the model with both tunnels loaded
displacement at point C and D for both models decrease InFigure 5(b) it is observed that the maximum displacementsof single tunnelmodel are all smaller than those of the presentmodel when the burying depth changes from 3 to 5 Thedecrease rates for both models are almost the same
The comparison between the single tunnel model andthe present model is shown in Figure 6 under different loadvelocities It is observed from Figure 6 that the dynamicresponses of the twin-tunnel model are larger than those ofthe single tunnel model which is inconsistent with the resultsshown in Figure 5 And the difference of the two models atpoint C is greater than that at point D The differences reachpeak value when 119888119888s = 09 Since the dynamic response atpoint C is jointly affected by two tunnels while the dynamicresponse at point D is dominated by the tunnel loadedtherefore it is essential to consider the twin-tunnel model toobtain more accurate prediction results
Practically both metro tunnels are always loaded atthe same time due to the high operation frequency of thesubways The dynamic responses of the model with bothtunnel inverts loaded are studied in this section
In contrast to the result of the presentmodel the dynamicresponses of point C and point D due to two moving pointloads are larger in terms of the maximum displacement InFigure 7 the maximum displacements of the two models atpoint A are almost the same The dynamic response of thetwomodels at point C decreases a little with increasing tunnelspacing However as tunnel spacing increases the maximum
displacement of the two models at point A remains the sameand the results of two models become closer at point DThis is because as the tunnel spacing increases the effect ofthe other tunnel on point D decreases gradually And withthe increasing of the burying depth the dynamic responsesof points C and D have obvious downtrend Besides thedisplacement at the point C is larger than that at the point Din the model with two loads while the situation of the presentmodel is the opposite That is because the load in the othertunnel has a larger contribution to the dynamic response atpoint C than at point D
The comparison of the displacements and the vibrationvelocity between the present model and the model with bothtunnels loaded is shown in Figures 8 and 9 for differentload velocities For points C and D the dynamic responsesof the model with both tunnels loaded are obviously largerthan the model with only one tunnel loaded However theprediction results of the two models at point A are very closeAnd the maximum displacement at point C is smaller thanthat at point D in the model with only one tunnel loadedwhile the situation of the two-load model is the oppositewhich is explained in the previous section The differencebetween the maximum displacements of the two models isalmost unchanged at different load velocity However thedifference in the vibration velocity results of the two modelsat point D decreases as the velocity increases and reaches aminimum at 119888119888119904 = 09 From Figure 9 it can also be observedthat the maximum vibration velocities at points C and D are
Mathematical Problems in Engineering 7
Table 2 Parameters of the liner the saturated ground and the moving load
Saturated Soil Ground
Width a (m) Height b (m) Length l (m) Biot parameters 12057242 24 120 099
Spacing 119889 (m) Burying depth h (m) Porosity 119899 Poissonrsquos ratio ]9ndash24 6ndash18 0286 04
Youngrsquos modulus Darcy permeability coefficient Shear wave velocity 119888s Biot parameters119864 (Pa) 119896119863 (ms) (ms) 119872 (Nm)223 times 108 10 times 10minus5ndash10 times 10minus2 1911 574 times 109
Density of the bulk material 120588 (kgm3) Density of the pore fluid 120588f (kgm3)2178 times 103 10 times 103
Concrete LinerYoungrsquos modulus Poissonrsquos ratio Density of the bulk material outer radius Thickness119864 (Pa) ] 120588 (kgm3) r(m) t(m)
250 times 1010 02 2400 times 103 30 03
Moving Load Point load value (N) Load velocity (ms)20 times 104 1911ndash1911
0
4 45 5 55 6Tunnel spacing dr
point A(twin-tunnel)point C(twin-tunnel)point D(twin-tunnel)point A(single tunnel)point C(single tunnel)point D(single tunnel)
(b) Maximum displacement at the observation points for differentburying depth
Figure 5 Comparison of dynamic responses between a single-tunnel model and a twin-tunnel model
almost the same in the present model with only one tunnelloaded while the value at point C in the two-load model issignificantly larger than that at point D Because the load inthe other tunnel is closer to point C than point D it has agreater impact on the dynamic response of point C
To illustrate the necessity in introducing the saturatedground model the dynamic responses of the twin tunnelembedded in a single-phase ground are compared to thoseof the saturated ground model and presented in Figure 10The single-phase ground model is obtained by reducing thesaturated ground model after fixing the pore-fluid relativedisplacement and setting Biotrsquos parameters to zeros It isobserved that the dynamic responses of the elastic ground arelarger than those in the saturated ground The discrepancybetween the two ground models widens when the buryingdepth increases And as the load velocity increases the
difference of dynamic response between the two models atpoint D also becomes larger From the comparison it can beconcluded that the ground water has an obvious influence onthe vibration responses of the tunnel and the ground Theirvibration levels would be overestimated if the ground wateris not considered
53 e Influences of Load Velocity The influences of loadvelocity on the dynamic responses of the liner-tunnel-groundsystem are investigated in this section Six different velocities119888119888s = 01 03 05 07 09 and 10 are considered along withthe tunnel spacing 119889119903 = 4 and the burying depth ℎ119903 = 4
It is observed from Figure 11 that the displacements atpoints A and B change slightly with the increasing loadvelocity while the displacements at points C and D reacha peak at around 09 119888s Since C and D are on the ground
8 Mathematical Problems in Engineering
point C(twin-tunnel)point D(twin-tunnel)point C(single tunnel)point D(single tunnel)
Max
imum
disp
lace
men
t0 02 04 06 08 1
Load velocity s
minus365
minus360
minus355
minus350
minus345
minus340
minus335
minus330
UZlowast
(dB)
Figure 6 Comparison on maximum displacement at observation points between a single-tunnel model and a twin-tunnel model
(a) Maximum displacement at the observation points for differenttunnel spacing
3 35 4 45 5Burying depth hr
point C(one load)point D(one load)point C(two loads)point D(two loads)
Max
imum
disp
lace
men
t
minus45
minus40
minus35
minus30
minus25
minus20
minus15
minus10
UZlowast
(dB)
(b) Maximum displacement at the observation points for differentburying depth
Figure 7 Comparison of dynamic responses between the present model and a model with both tunnels loaded
surface their responses will be amplified when the loadvelocity approaches the Rayleigh wave velocity of the groundwhich is around 09 119888s The same results were reported inprevious studies on the critical velocity for a train runningon the ground surface [27 28] However points A and B areset on the concrete liner whose shear wave velocity is muchhigher than that of the soil Thus the load velocity is very lowcompared to the shear wave velocity of the concrete whichmakes the displacement of points A and B nonsensitive to thechange of load velocity
54 e Influence of Tunnel Spacing In this section theinfluence of the tunnel spacing on the dynamic responses isinvestigated by considering 6 different twin tunnel distancesie 119889119903 = 3 4 5 6 7 8 The load velocity is 03 119888119904 The
maximum vertical displacements at the observation pointswhen ℎ119903 = 4 are compared in Figure 12
The displacement at point A is significantly larger thanthose at other points since point A is traversed directlyby the point load When the tunnel spacing increases thedisplacements at points B and C decrease gradually Howeverthe displacements at points A and D remain almost the sameThis observation is within expectation because the positionsof points A and D remain the same while points B and C areshifted away from the load when the spacing increases Thedisplacements at points B and C vary greatly when the tunnelspacing 119889119903 increases from 4 to 6
55e Influence of Burying Depth In this section paramet-ric study is conducted for 5 different burying depths of the
Figure 9 Comparison of maximum vibration velocity at obser-vation points between a model containing one load and a modelcontaining two loads
twin tunnel ie ℎ119903 = 2 3 4 5 6 And the load velocity is setto 03 119888s
The variation of the maximum vertical displacements atthe observation points with respect to the burying depth ispresented in Figure 13 In this case the tunnel spacing is fixedas 119889119903 = 4 It is observed that the displacements at points AC and D decrease when ℎ119903 increases For the displacementat point B it increases slightly when ℎ119903 lt 4 however itdrops substantially when ℎ119903 increases further to 6 This
phenomenon is explained by the fact that when the tunnelis shallowly buried the waves generated by the source tunnel(ie the tunnel where the moving load is applied) impinge onthe ground surface at a large incident angle (resembling theglancing incident) correspondingly the reflected waves leavethe surface at a large angle and thus the surface reflectionregion at nonsourced tunnels is small resulting in a fractionof surface reflections contributing to the dynamic response atpoint B However the surface reflection region will expandwhen hr increases gradually but when the burying depthis large enough (ie ℎ119889 gt 5) the material and geometricaldamping are dominant which results in the decreasing of thedisplacement at point B
6 Conclusions
In this paper a 3D twin-tunnel model is presented to evaluatethe dynamic responses of saturated half-space induced bysubways The effects of spacing between two tunnels tunnelburied depth load moving speed and soil permeabilityon the vibration response are investigated Based on thederivation and numerical examples presented above thefollowing conclusions can be drawn
(1) The existence of the other tunnel has influences onthe vibration caused by the moving subway Andwhen the twin tunnels are both loaded the dynamicresponses cannot be calculated by simply summingup the results of a single tunnel model Therefore itis necessary to consider a twin-tunnel model whenpredicting the dynamic responses induced by sub-ways for all load velocities The dynamic responsesof the single-phase ground are larger than those inthe saturated ground which denotes that using asingle phase model to predict the vibrations is moreconservative
(2) The influence of the spacing between two tunnels onthe dynamic response of the tunnel without the loadapplied is greater than that on the ground surfaceWith the increase of the spacing 119889119903 the dynamicresponse of the adjacent tunnel decreases Tunnelburied depth has a great influence on the dynamicresponse of adjacent tunnel and the ground sur-face For points on the ground surface the dynamicresponse decreases with the increase of the tunnelburied depthDue to the presence of surface reflectionregion the dynamic response of adjacent tunnelincreases at first and then decreases as the burialdepth increases So more attention should be paid tothe spacing between tunnels and the tunnel burieddepth in the construction of subway
(3) The subways moving at a low speed will generate asmaller dynamic response in the saturated half-spacethan moving at a higher speed And when the loadvelocity approaches the Rayleigh wave velocity of theground which is around 09 119888119904 the responses of thesoil and liner will be amplified Therefore the subwaymoving speed is also an important parameter thataffects the environmental vibration
10 Mathematical Problems in Engineering
0
2 3 4 5 6Burying depth hr
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
Max
imum
vib
ratio
n ve
loci
ty
minus40
minus30
minus20
minus10
10
20
30V
Zlowast(d
B)
(a) Maximum vibration velocity at observation points for differentburying depth
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
Max
imum
disp
lace
men
t
01 03 05 07 09Load velocity s
minus40
minus35
minus30
minus25
minus20
minus15
minus10
minus5
0
UZlowast
(dB)
(b) Maximum displacement at observation points for different loadvelocity
minus30
minus20
minus10
0
10
20
30
Max
imum
vib
ratio
n ve
loci
ty
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
01 03 05 07 09Load velocity s
VZlowast
(dB)
(c) Maximum vibration velocity at observation points for differentload velocity
Figure 10 Comparisons of dynamic response at observation points between a saturated ground and an elastic ground
Appendix
The expressions of the elements in the mass matrix thedamping matrix the stiffness matrix and the force vector arepresented as follows
Figure 13 Vertical maximum displacements for different buryingdepth of tunnels
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the National Key RampD Programof China (Grant No 2016YFC0800200) the Projects ofInternational Cooperation and Exchanges NSFC (Grant No51620105008) National Science Foundation for Young Scien-tists of China (Grant No 51608482) and National NaturalScience Foundation of China (Grant No 51478424)
References
[1] A V Metrikine and A C W M Vrouwenvelder ldquoSurfaceground vibration due to a moving train in a tunnel two-dimensional modelrdquo Journal of Sound and Vibration vol 234no 1 pp 43ndash66 2000
[2] J A Forrest and H E M Hunt ldquoA three-dimensional tunnelmodel for calculation of train-induced ground vibrationrdquo Jour-nal of Sound and Vibration vol 294 no 4 pp 678ndash705 2006
[3] J A Forrest andH EMHunt ldquoGround vibration generated bytrains in underground tunnelsrdquo Journal of Sound and Vibrationvol 294 no 4 pp 706ndash736 2006
[4] K-H Xie G-B Liu and Z-Y Shi ldquoDynamic response ofpartially sealed circular tunnel in viscoelastic saturated soilrdquoSoil Dynamics and Earthquake Engineering vol 24 no 12 pp1003ndash1011 2004
[5] K-H Xie and G-B Liu ldquoTransient response of a sphericalcavity with a partially sealed shell embedded in viscoelasticsaturated soilrdquo Journal of Zhejiang University Science A vol 6no 3 pp 194ndash201 2005
[6] Z-H Yuan Y-Q Cai L Shi H-L Sun and Z-G CaoldquoResponse of rail structure and circular tunnel in saturatedsoil subjected to harmonic moving loadrdquo Chinese Journal ofGeotechnical Engineering vol 38 no 2 pp 311ndash322 2016
12 Mathematical Problems in Engineering
[7] H E M Hunt ldquoModelling of rail vehicles and track forcalculation of ground-vibration transmission into buildingsrdquoJournal of Sound and Vibration vol 193 no 1 pp 185ndash194 1996
[8] T Balendra K H Chua K W Lo and S L Lee ldquoSteady-statevibration of subway-soilbuilding systemrdquo Journal of EngineeringMechanics vol 115 no 1 pp 145ndash162 1989
[9] F-C Wang H Xia and H-R Zhang ldquoVibration effects ofsubway trains on surrounding buildingsrdquo Journal of NorthernJiaotong University vol 23 no 5 pp 45ndash48 1999
[10] Y Yang HHung and L Hsu ldquoGround vibrations due to under-ground trains considering soil-tunnel interactionrdquo Interactionand Multiscale Mechanics vol 1 no 1 pp 157ndash175 2007
[11] Y-E Zhang and Changshi-Pan ldquoTests and analyses of thedynamic response of tunnels subjected to passing train loadrdquoJournal of Shijiazhuang Railway Institute vol 6 no 2 pp 7ndash141993
[12] C J C Jones D J Thompson and M Petyt ldquoStudies usinga combined finite element and boundary element model forvibration propagation from railway tunnelsrdquo International Insti-tute of Acoustics and Vibration pp 2703ndash2710 2000
[13] W Gardien and H G Stuit ldquoModelling of soil vibrations fromrailway tunnelsrdquo Journal of Sound and Vibration vol 267 no 3pp 605ndash619 2003
[14] S Wolf ldquoPotential low frequency ground vibration (lt63 Hz)impacts from underground LRT operationsrdquo Journal of Soundand Vibration vol 267 no 3 pp 651ndash661 2003
[15] G Degrande D Clouteau R Othman et al ldquoA numericalmodel for ground-borne vibrations from underground railwaytraffic based on a periodic finite element-boundary elementformulationrdquo Journal of Sound and Vibration vol 293 no 3-5pp 645ndash666 2006
[16] W-F Liu W-N Liu S Gupta and G Degrande ldquoPrediction ofvibrations in the tunnel and free field due to passage of metrotrainsrdquo Engineering Mechanics vol 27 no 1 pp 250ndash256 2010
[17] K A Kuo H E M Hunt and M F M Hussein ldquoThe effectof a twin tunnel on the propagation of ground-borne vibrationfrom an underground railwayrdquo Journal of Sound and Vibrationvol 330 no 25 pp 6203ndash6222 2011
[18] Z-H Yuan Y-Q Cai W Yuan Y-L Xu and Z-G CaoldquoDynamic response of circular railway tunnel and track systemin saturated soil under moving train loadingrdquo Rock and SoilMechanics vol 38 no 4 pp 1003ndash1014 2017
[19] Z P Liao and H L Wong ldquoA transmitting boundary for thenumerical simulation of elasticwave propagationrdquo InternationalJournal of Soil Dynamics and Earthquake Engineering vol 3 pp133ndash144 1984
[20] L Shi P Wang Y Cai and Z Cao ldquoMulti-transmittingformula for finite element modeling of wave propagation in asaturated poroelastic mediumrdquo Soil Dynamics and EarthquakeEngineering vol 80 pp 11ndash24 2016
[21] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid I Low-frequency rangerdquoe Journal ofthe Acoustical Society of America vol 28 pp 168ndash178 1956
[22] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956
[23] O C Zienkiewicz C T Chang and P Bettess ldquoDrainedundrained consolidating and dynamic behaviour assumptionsin soilsrdquo Geotechnique vol 30 no 4 pp 385ndash395 1980
[24] L Shi H Sun Y Cai C Xu and P Wang ldquoValidity of fullydrained fully undrained and u-p formulations for modeling a
poroelastic half-space under a moving harmonic point loadrdquoSoil Dynamics and Earthquake Engineering vol 42 pp 292ndash3012012
[25] T Senjuntichai and R K N D Rajapakse ldquoTransient responseof a circular cavity in a poroelastic mediumrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 17 no 6 pp 357ndash383 1993
[26] Y-B Yang S-R Kuo and H-H Hung ldquoFrequency-independent infinite elements for analysing semi-infiniteproblemsrdquo International Journal for Numerical Methods inEngineering vol 39 no 20 pp 3553ndash3569 1996
[27] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic response of tracksystem and saturated soil in half space subjected to a movingtrain loadrdquo Chinese Journal of Rock Mechanics and Engineeringvol 26 no 8 pp 1705ndash1712 2007
[28] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic responses oftrack system and poroelastic soil under high-speed train loadrdquoJournal of Zhejiang University (Engineering Science) vol 42 no11 pp 2002ndash2008 2008
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
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Dierential EquationsInternational Journal of
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AnalysisInternational Journal of
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Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
6 Mathematical Problems in Engineering
D(C)
A(B)MTFMTF
Free and permeable
Fixed
z
y-Plane
l2
l
b
x
(a) Side view
D C
A B
z
MTFMTF
Fixed
Free and permeable
bＨ
d
tr
ℎ
a x-Plane y
(b) Cross section view
Figure 3 Schematic of the 3D computation model
D C
A
z
MTFMTF
Fixed
Free and permeable
b Ｈt r
ℎ
a x-Plane y
d2
(a) Single tunnel model
D C
A B
a
z
MTFMTF
Fixed
Free and permeable
b Ｈ Ｈ
d
tr
ℎ
x-Plane y
(b) Model with both tunnels loaded
Figure 4 Cross sections of single tunnel model and the model with both tunnels loaded
displacement at point C and D for both models decrease InFigure 5(b) it is observed that the maximum displacementsof single tunnelmodel are all smaller than those of the presentmodel when the burying depth changes from 3 to 5 Thedecrease rates for both models are almost the same
The comparison between the single tunnel model andthe present model is shown in Figure 6 under different loadvelocities It is observed from Figure 6 that the dynamicresponses of the twin-tunnel model are larger than those ofthe single tunnel model which is inconsistent with the resultsshown in Figure 5 And the difference of the two models atpoint C is greater than that at point D The differences reachpeak value when 119888119888s = 09 Since the dynamic response atpoint C is jointly affected by two tunnels while the dynamicresponse at point D is dominated by the tunnel loadedtherefore it is essential to consider the twin-tunnel model toobtain more accurate prediction results
Practically both metro tunnels are always loaded atthe same time due to the high operation frequency of thesubways The dynamic responses of the model with bothtunnel inverts loaded are studied in this section
In contrast to the result of the presentmodel the dynamicresponses of point C and point D due to two moving pointloads are larger in terms of the maximum displacement InFigure 7 the maximum displacements of the two models atpoint A are almost the same The dynamic response of thetwomodels at point C decreases a little with increasing tunnelspacing However as tunnel spacing increases the maximum
displacement of the two models at point A remains the sameand the results of two models become closer at point DThis is because as the tunnel spacing increases the effect ofthe other tunnel on point D decreases gradually And withthe increasing of the burying depth the dynamic responsesof points C and D have obvious downtrend Besides thedisplacement at the point C is larger than that at the point Din the model with two loads while the situation of the presentmodel is the opposite That is because the load in the othertunnel has a larger contribution to the dynamic response atpoint C than at point D
The comparison of the displacements and the vibrationvelocity between the present model and the model with bothtunnels loaded is shown in Figures 8 and 9 for differentload velocities For points C and D the dynamic responsesof the model with both tunnels loaded are obviously largerthan the model with only one tunnel loaded However theprediction results of the two models at point A are very closeAnd the maximum displacement at point C is smaller thanthat at point D in the model with only one tunnel loadedwhile the situation of the two-load model is the oppositewhich is explained in the previous section The differencebetween the maximum displacements of the two models isalmost unchanged at different load velocity However thedifference in the vibration velocity results of the two modelsat point D decreases as the velocity increases and reaches aminimum at 119888119888119904 = 09 From Figure 9 it can also be observedthat the maximum vibration velocities at points C and D are
Mathematical Problems in Engineering 7
Table 2 Parameters of the liner the saturated ground and the moving load
Saturated Soil Ground
Width a (m) Height b (m) Length l (m) Biot parameters 12057242 24 120 099
Spacing 119889 (m) Burying depth h (m) Porosity 119899 Poissonrsquos ratio ]9ndash24 6ndash18 0286 04
Youngrsquos modulus Darcy permeability coefficient Shear wave velocity 119888s Biot parameters119864 (Pa) 119896119863 (ms) (ms) 119872 (Nm)223 times 108 10 times 10minus5ndash10 times 10minus2 1911 574 times 109
Density of the bulk material 120588 (kgm3) Density of the pore fluid 120588f (kgm3)2178 times 103 10 times 103
Concrete LinerYoungrsquos modulus Poissonrsquos ratio Density of the bulk material outer radius Thickness119864 (Pa) ] 120588 (kgm3) r(m) t(m)
250 times 1010 02 2400 times 103 30 03
Moving Load Point load value (N) Load velocity (ms)20 times 104 1911ndash1911
0
4 45 5 55 6Tunnel spacing dr
point A(twin-tunnel)point C(twin-tunnel)point D(twin-tunnel)point A(single tunnel)point C(single tunnel)point D(single tunnel)
(b) Maximum displacement at the observation points for differentburying depth
Figure 5 Comparison of dynamic responses between a single-tunnel model and a twin-tunnel model
almost the same in the present model with only one tunnelloaded while the value at point C in the two-load model issignificantly larger than that at point D Because the load inthe other tunnel is closer to point C than point D it has agreater impact on the dynamic response of point C
To illustrate the necessity in introducing the saturatedground model the dynamic responses of the twin tunnelembedded in a single-phase ground are compared to thoseof the saturated ground model and presented in Figure 10The single-phase ground model is obtained by reducing thesaturated ground model after fixing the pore-fluid relativedisplacement and setting Biotrsquos parameters to zeros It isobserved that the dynamic responses of the elastic ground arelarger than those in the saturated ground The discrepancybetween the two ground models widens when the buryingdepth increases And as the load velocity increases the
difference of dynamic response between the two models atpoint D also becomes larger From the comparison it can beconcluded that the ground water has an obvious influence onthe vibration responses of the tunnel and the ground Theirvibration levels would be overestimated if the ground wateris not considered
53 e Influences of Load Velocity The influences of loadvelocity on the dynamic responses of the liner-tunnel-groundsystem are investigated in this section Six different velocities119888119888s = 01 03 05 07 09 and 10 are considered along withthe tunnel spacing 119889119903 = 4 and the burying depth ℎ119903 = 4
It is observed from Figure 11 that the displacements atpoints A and B change slightly with the increasing loadvelocity while the displacements at points C and D reacha peak at around 09 119888s Since C and D are on the ground
8 Mathematical Problems in Engineering
point C(twin-tunnel)point D(twin-tunnel)point C(single tunnel)point D(single tunnel)
Max
imum
disp
lace
men
t0 02 04 06 08 1
Load velocity s
minus365
minus360
minus355
minus350
minus345
minus340
minus335
minus330
UZlowast
(dB)
Figure 6 Comparison on maximum displacement at observation points between a single-tunnel model and a twin-tunnel model
(a) Maximum displacement at the observation points for differenttunnel spacing
3 35 4 45 5Burying depth hr
point C(one load)point D(one load)point C(two loads)point D(two loads)
Max
imum
disp
lace
men
t
minus45
minus40
minus35
minus30
minus25
minus20
minus15
minus10
UZlowast
(dB)
(b) Maximum displacement at the observation points for differentburying depth
Figure 7 Comparison of dynamic responses between the present model and a model with both tunnels loaded
surface their responses will be amplified when the loadvelocity approaches the Rayleigh wave velocity of the groundwhich is around 09 119888s The same results were reported inprevious studies on the critical velocity for a train runningon the ground surface [27 28] However points A and B areset on the concrete liner whose shear wave velocity is muchhigher than that of the soil Thus the load velocity is very lowcompared to the shear wave velocity of the concrete whichmakes the displacement of points A and B nonsensitive to thechange of load velocity
54 e Influence of Tunnel Spacing In this section theinfluence of the tunnel spacing on the dynamic responses isinvestigated by considering 6 different twin tunnel distancesie 119889119903 = 3 4 5 6 7 8 The load velocity is 03 119888119904 The
maximum vertical displacements at the observation pointswhen ℎ119903 = 4 are compared in Figure 12
The displacement at point A is significantly larger thanthose at other points since point A is traversed directlyby the point load When the tunnel spacing increases thedisplacements at points B and C decrease gradually Howeverthe displacements at points A and D remain almost the sameThis observation is within expectation because the positionsof points A and D remain the same while points B and C areshifted away from the load when the spacing increases Thedisplacements at points B and C vary greatly when the tunnelspacing 119889119903 increases from 4 to 6
55e Influence of Burying Depth In this section paramet-ric study is conducted for 5 different burying depths of the
Figure 9 Comparison of maximum vibration velocity at obser-vation points between a model containing one load and a modelcontaining two loads
twin tunnel ie ℎ119903 = 2 3 4 5 6 And the load velocity is setto 03 119888s
The variation of the maximum vertical displacements atthe observation points with respect to the burying depth ispresented in Figure 13 In this case the tunnel spacing is fixedas 119889119903 = 4 It is observed that the displacements at points AC and D decrease when ℎ119903 increases For the displacementat point B it increases slightly when ℎ119903 lt 4 however itdrops substantially when ℎ119903 increases further to 6 This
phenomenon is explained by the fact that when the tunnelis shallowly buried the waves generated by the source tunnel(ie the tunnel where the moving load is applied) impinge onthe ground surface at a large incident angle (resembling theglancing incident) correspondingly the reflected waves leavethe surface at a large angle and thus the surface reflectionregion at nonsourced tunnels is small resulting in a fractionof surface reflections contributing to the dynamic response atpoint B However the surface reflection region will expandwhen hr increases gradually but when the burying depthis large enough (ie ℎ119889 gt 5) the material and geometricaldamping are dominant which results in the decreasing of thedisplacement at point B
6 Conclusions
In this paper a 3D twin-tunnel model is presented to evaluatethe dynamic responses of saturated half-space induced bysubways The effects of spacing between two tunnels tunnelburied depth load moving speed and soil permeabilityon the vibration response are investigated Based on thederivation and numerical examples presented above thefollowing conclusions can be drawn
(1) The existence of the other tunnel has influences onthe vibration caused by the moving subway Andwhen the twin tunnels are both loaded the dynamicresponses cannot be calculated by simply summingup the results of a single tunnel model Therefore itis necessary to consider a twin-tunnel model whenpredicting the dynamic responses induced by sub-ways for all load velocities The dynamic responsesof the single-phase ground are larger than those inthe saturated ground which denotes that using asingle phase model to predict the vibrations is moreconservative
(2) The influence of the spacing between two tunnels onthe dynamic response of the tunnel without the loadapplied is greater than that on the ground surfaceWith the increase of the spacing 119889119903 the dynamicresponse of the adjacent tunnel decreases Tunnelburied depth has a great influence on the dynamicresponse of adjacent tunnel and the ground sur-face For points on the ground surface the dynamicresponse decreases with the increase of the tunnelburied depthDue to the presence of surface reflectionregion the dynamic response of adjacent tunnelincreases at first and then decreases as the burialdepth increases So more attention should be paid tothe spacing between tunnels and the tunnel burieddepth in the construction of subway
(3) The subways moving at a low speed will generate asmaller dynamic response in the saturated half-spacethan moving at a higher speed And when the loadvelocity approaches the Rayleigh wave velocity of theground which is around 09 119888119904 the responses of thesoil and liner will be amplified Therefore the subwaymoving speed is also an important parameter thataffects the environmental vibration
10 Mathematical Problems in Engineering
0
2 3 4 5 6Burying depth hr
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
Max
imum
vib
ratio
n ve
loci
ty
minus40
minus30
minus20
minus10
10
20
30V
Zlowast(d
B)
(a) Maximum vibration velocity at observation points for differentburying depth
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
Max
imum
disp
lace
men
t
01 03 05 07 09Load velocity s
minus40
minus35
minus30
minus25
minus20
minus15
minus10
minus5
0
UZlowast
(dB)
(b) Maximum displacement at observation points for different loadvelocity
minus30
minus20
minus10
0
10
20
30
Max
imum
vib
ratio
n ve
loci
ty
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
01 03 05 07 09Load velocity s
VZlowast
(dB)
(c) Maximum vibration velocity at observation points for differentload velocity
Figure 10 Comparisons of dynamic response at observation points between a saturated ground and an elastic ground
Appendix
The expressions of the elements in the mass matrix thedamping matrix the stiffness matrix and the force vector arepresented as follows
Figure 13 Vertical maximum displacements for different buryingdepth of tunnels
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the National Key RampD Programof China (Grant No 2016YFC0800200) the Projects ofInternational Cooperation and Exchanges NSFC (Grant No51620105008) National Science Foundation for Young Scien-tists of China (Grant No 51608482) and National NaturalScience Foundation of China (Grant No 51478424)
References
[1] A V Metrikine and A C W M Vrouwenvelder ldquoSurfaceground vibration due to a moving train in a tunnel two-dimensional modelrdquo Journal of Sound and Vibration vol 234no 1 pp 43ndash66 2000
[2] J A Forrest and H E M Hunt ldquoA three-dimensional tunnelmodel for calculation of train-induced ground vibrationrdquo Jour-nal of Sound and Vibration vol 294 no 4 pp 678ndash705 2006
[3] J A Forrest andH EMHunt ldquoGround vibration generated bytrains in underground tunnelsrdquo Journal of Sound and Vibrationvol 294 no 4 pp 706ndash736 2006
[4] K-H Xie G-B Liu and Z-Y Shi ldquoDynamic response ofpartially sealed circular tunnel in viscoelastic saturated soilrdquoSoil Dynamics and Earthquake Engineering vol 24 no 12 pp1003ndash1011 2004
[5] K-H Xie and G-B Liu ldquoTransient response of a sphericalcavity with a partially sealed shell embedded in viscoelasticsaturated soilrdquo Journal of Zhejiang University Science A vol 6no 3 pp 194ndash201 2005
[6] Z-H Yuan Y-Q Cai L Shi H-L Sun and Z-G CaoldquoResponse of rail structure and circular tunnel in saturatedsoil subjected to harmonic moving loadrdquo Chinese Journal ofGeotechnical Engineering vol 38 no 2 pp 311ndash322 2016
12 Mathematical Problems in Engineering
[7] H E M Hunt ldquoModelling of rail vehicles and track forcalculation of ground-vibration transmission into buildingsrdquoJournal of Sound and Vibration vol 193 no 1 pp 185ndash194 1996
[8] T Balendra K H Chua K W Lo and S L Lee ldquoSteady-statevibration of subway-soilbuilding systemrdquo Journal of EngineeringMechanics vol 115 no 1 pp 145ndash162 1989
[9] F-C Wang H Xia and H-R Zhang ldquoVibration effects ofsubway trains on surrounding buildingsrdquo Journal of NorthernJiaotong University vol 23 no 5 pp 45ndash48 1999
[10] Y Yang HHung and L Hsu ldquoGround vibrations due to under-ground trains considering soil-tunnel interactionrdquo Interactionand Multiscale Mechanics vol 1 no 1 pp 157ndash175 2007
[11] Y-E Zhang and Changshi-Pan ldquoTests and analyses of thedynamic response of tunnels subjected to passing train loadrdquoJournal of Shijiazhuang Railway Institute vol 6 no 2 pp 7ndash141993
[12] C J C Jones D J Thompson and M Petyt ldquoStudies usinga combined finite element and boundary element model forvibration propagation from railway tunnelsrdquo International Insti-tute of Acoustics and Vibration pp 2703ndash2710 2000
[13] W Gardien and H G Stuit ldquoModelling of soil vibrations fromrailway tunnelsrdquo Journal of Sound and Vibration vol 267 no 3pp 605ndash619 2003
[14] S Wolf ldquoPotential low frequency ground vibration (lt63 Hz)impacts from underground LRT operationsrdquo Journal of Soundand Vibration vol 267 no 3 pp 651ndash661 2003
[15] G Degrande D Clouteau R Othman et al ldquoA numericalmodel for ground-borne vibrations from underground railwaytraffic based on a periodic finite element-boundary elementformulationrdquo Journal of Sound and Vibration vol 293 no 3-5pp 645ndash666 2006
[16] W-F Liu W-N Liu S Gupta and G Degrande ldquoPrediction ofvibrations in the tunnel and free field due to passage of metrotrainsrdquo Engineering Mechanics vol 27 no 1 pp 250ndash256 2010
[17] K A Kuo H E M Hunt and M F M Hussein ldquoThe effectof a twin tunnel on the propagation of ground-borne vibrationfrom an underground railwayrdquo Journal of Sound and Vibrationvol 330 no 25 pp 6203ndash6222 2011
[18] Z-H Yuan Y-Q Cai W Yuan Y-L Xu and Z-G CaoldquoDynamic response of circular railway tunnel and track systemin saturated soil under moving train loadingrdquo Rock and SoilMechanics vol 38 no 4 pp 1003ndash1014 2017
[19] Z P Liao and H L Wong ldquoA transmitting boundary for thenumerical simulation of elasticwave propagationrdquo InternationalJournal of Soil Dynamics and Earthquake Engineering vol 3 pp133ndash144 1984
[20] L Shi P Wang Y Cai and Z Cao ldquoMulti-transmittingformula for finite element modeling of wave propagation in asaturated poroelastic mediumrdquo Soil Dynamics and EarthquakeEngineering vol 80 pp 11ndash24 2016
[21] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid I Low-frequency rangerdquoe Journal ofthe Acoustical Society of America vol 28 pp 168ndash178 1956
[22] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956
[23] O C Zienkiewicz C T Chang and P Bettess ldquoDrainedundrained consolidating and dynamic behaviour assumptionsin soilsrdquo Geotechnique vol 30 no 4 pp 385ndash395 1980
[24] L Shi H Sun Y Cai C Xu and P Wang ldquoValidity of fullydrained fully undrained and u-p formulations for modeling a
poroelastic half-space under a moving harmonic point loadrdquoSoil Dynamics and Earthquake Engineering vol 42 pp 292ndash3012012
[25] T Senjuntichai and R K N D Rajapakse ldquoTransient responseof a circular cavity in a poroelastic mediumrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 17 no 6 pp 357ndash383 1993
[26] Y-B Yang S-R Kuo and H-H Hung ldquoFrequency-independent infinite elements for analysing semi-infiniteproblemsrdquo International Journal for Numerical Methods inEngineering vol 39 no 20 pp 3553ndash3569 1996
[27] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic response of tracksystem and saturated soil in half space subjected to a movingtrain loadrdquo Chinese Journal of Rock Mechanics and Engineeringvol 26 no 8 pp 1705ndash1712 2007
[28] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic responses oftrack system and poroelastic soil under high-speed train loadrdquoJournal of Zhejiang University (Engineering Science) vol 42 no11 pp 2002ndash2008 2008
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
(b) Maximum displacement at the observation points for differentburying depth
Figure 5 Comparison of dynamic responses between a single-tunnel model and a twin-tunnel model
almost the same in the present model with only one tunnelloaded while the value at point C in the two-load model issignificantly larger than that at point D Because the load inthe other tunnel is closer to point C than point D it has agreater impact on the dynamic response of point C
To illustrate the necessity in introducing the saturatedground model the dynamic responses of the twin tunnelembedded in a single-phase ground are compared to thoseof the saturated ground model and presented in Figure 10The single-phase ground model is obtained by reducing thesaturated ground model after fixing the pore-fluid relativedisplacement and setting Biotrsquos parameters to zeros It isobserved that the dynamic responses of the elastic ground arelarger than those in the saturated ground The discrepancybetween the two ground models widens when the buryingdepth increases And as the load velocity increases the
difference of dynamic response between the two models atpoint D also becomes larger From the comparison it can beconcluded that the ground water has an obvious influence onthe vibration responses of the tunnel and the ground Theirvibration levels would be overestimated if the ground wateris not considered
53 e Influences of Load Velocity The influences of loadvelocity on the dynamic responses of the liner-tunnel-groundsystem are investigated in this section Six different velocities119888119888s = 01 03 05 07 09 and 10 are considered along withthe tunnel spacing 119889119903 = 4 and the burying depth ℎ119903 = 4
It is observed from Figure 11 that the displacements atpoints A and B change slightly with the increasing loadvelocity while the displacements at points C and D reacha peak at around 09 119888s Since C and D are on the ground
8 Mathematical Problems in Engineering
point C(twin-tunnel)point D(twin-tunnel)point C(single tunnel)point D(single tunnel)
Max
imum
disp
lace
men
t0 02 04 06 08 1
Load velocity s
minus365
minus360
minus355
minus350
minus345
minus340
minus335
minus330
UZlowast
(dB)
Figure 6 Comparison on maximum displacement at observation points between a single-tunnel model and a twin-tunnel model
(a) Maximum displacement at the observation points for differenttunnel spacing
3 35 4 45 5Burying depth hr
point C(one load)point D(one load)point C(two loads)point D(two loads)
Max
imum
disp
lace
men
t
minus45
minus40
minus35
minus30
minus25
minus20
minus15
minus10
UZlowast
(dB)
(b) Maximum displacement at the observation points for differentburying depth
Figure 7 Comparison of dynamic responses between the present model and a model with both tunnels loaded
surface their responses will be amplified when the loadvelocity approaches the Rayleigh wave velocity of the groundwhich is around 09 119888s The same results were reported inprevious studies on the critical velocity for a train runningon the ground surface [27 28] However points A and B areset on the concrete liner whose shear wave velocity is muchhigher than that of the soil Thus the load velocity is very lowcompared to the shear wave velocity of the concrete whichmakes the displacement of points A and B nonsensitive to thechange of load velocity
54 e Influence of Tunnel Spacing In this section theinfluence of the tunnel spacing on the dynamic responses isinvestigated by considering 6 different twin tunnel distancesie 119889119903 = 3 4 5 6 7 8 The load velocity is 03 119888119904 The
maximum vertical displacements at the observation pointswhen ℎ119903 = 4 are compared in Figure 12
The displacement at point A is significantly larger thanthose at other points since point A is traversed directlyby the point load When the tunnel spacing increases thedisplacements at points B and C decrease gradually Howeverthe displacements at points A and D remain almost the sameThis observation is within expectation because the positionsof points A and D remain the same while points B and C areshifted away from the load when the spacing increases Thedisplacements at points B and C vary greatly when the tunnelspacing 119889119903 increases from 4 to 6
55e Influence of Burying Depth In this section paramet-ric study is conducted for 5 different burying depths of the
Figure 9 Comparison of maximum vibration velocity at obser-vation points between a model containing one load and a modelcontaining two loads
twin tunnel ie ℎ119903 = 2 3 4 5 6 And the load velocity is setto 03 119888s
The variation of the maximum vertical displacements atthe observation points with respect to the burying depth ispresented in Figure 13 In this case the tunnel spacing is fixedas 119889119903 = 4 It is observed that the displacements at points AC and D decrease when ℎ119903 increases For the displacementat point B it increases slightly when ℎ119903 lt 4 however itdrops substantially when ℎ119903 increases further to 6 This
phenomenon is explained by the fact that when the tunnelis shallowly buried the waves generated by the source tunnel(ie the tunnel where the moving load is applied) impinge onthe ground surface at a large incident angle (resembling theglancing incident) correspondingly the reflected waves leavethe surface at a large angle and thus the surface reflectionregion at nonsourced tunnels is small resulting in a fractionof surface reflections contributing to the dynamic response atpoint B However the surface reflection region will expandwhen hr increases gradually but when the burying depthis large enough (ie ℎ119889 gt 5) the material and geometricaldamping are dominant which results in the decreasing of thedisplacement at point B
6 Conclusions
In this paper a 3D twin-tunnel model is presented to evaluatethe dynamic responses of saturated half-space induced bysubways The effects of spacing between two tunnels tunnelburied depth load moving speed and soil permeabilityon the vibration response are investigated Based on thederivation and numerical examples presented above thefollowing conclusions can be drawn
(1) The existence of the other tunnel has influences onthe vibration caused by the moving subway Andwhen the twin tunnels are both loaded the dynamicresponses cannot be calculated by simply summingup the results of a single tunnel model Therefore itis necessary to consider a twin-tunnel model whenpredicting the dynamic responses induced by sub-ways for all load velocities The dynamic responsesof the single-phase ground are larger than those inthe saturated ground which denotes that using asingle phase model to predict the vibrations is moreconservative
(2) The influence of the spacing between two tunnels onthe dynamic response of the tunnel without the loadapplied is greater than that on the ground surfaceWith the increase of the spacing 119889119903 the dynamicresponse of the adjacent tunnel decreases Tunnelburied depth has a great influence on the dynamicresponse of adjacent tunnel and the ground sur-face For points on the ground surface the dynamicresponse decreases with the increase of the tunnelburied depthDue to the presence of surface reflectionregion the dynamic response of adjacent tunnelincreases at first and then decreases as the burialdepth increases So more attention should be paid tothe spacing between tunnels and the tunnel burieddepth in the construction of subway
(3) The subways moving at a low speed will generate asmaller dynamic response in the saturated half-spacethan moving at a higher speed And when the loadvelocity approaches the Rayleigh wave velocity of theground which is around 09 119888119904 the responses of thesoil and liner will be amplified Therefore the subwaymoving speed is also an important parameter thataffects the environmental vibration
10 Mathematical Problems in Engineering
0
2 3 4 5 6Burying depth hr
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
Max
imum
vib
ratio
n ve
loci
ty
minus40
minus30
minus20
minus10
10
20
30V
Zlowast(d
B)
(a) Maximum vibration velocity at observation points for differentburying depth
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
Max
imum
disp
lace
men
t
01 03 05 07 09Load velocity s
minus40
minus35
minus30
minus25
minus20
minus15
minus10
minus5
0
UZlowast
(dB)
(b) Maximum displacement at observation points for different loadvelocity
minus30
minus20
minus10
0
10
20
30
Max
imum
vib
ratio
n ve
loci
ty
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
01 03 05 07 09Load velocity s
VZlowast
(dB)
(c) Maximum vibration velocity at observation points for differentload velocity
Figure 10 Comparisons of dynamic response at observation points between a saturated ground and an elastic ground
Appendix
The expressions of the elements in the mass matrix thedamping matrix the stiffness matrix and the force vector arepresented as follows
Figure 13 Vertical maximum displacements for different buryingdepth of tunnels
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the National Key RampD Programof China (Grant No 2016YFC0800200) the Projects ofInternational Cooperation and Exchanges NSFC (Grant No51620105008) National Science Foundation for Young Scien-tists of China (Grant No 51608482) and National NaturalScience Foundation of China (Grant No 51478424)
References
[1] A V Metrikine and A C W M Vrouwenvelder ldquoSurfaceground vibration due to a moving train in a tunnel two-dimensional modelrdquo Journal of Sound and Vibration vol 234no 1 pp 43ndash66 2000
[2] J A Forrest and H E M Hunt ldquoA three-dimensional tunnelmodel for calculation of train-induced ground vibrationrdquo Jour-nal of Sound and Vibration vol 294 no 4 pp 678ndash705 2006
[3] J A Forrest andH EMHunt ldquoGround vibration generated bytrains in underground tunnelsrdquo Journal of Sound and Vibrationvol 294 no 4 pp 706ndash736 2006
[4] K-H Xie G-B Liu and Z-Y Shi ldquoDynamic response ofpartially sealed circular tunnel in viscoelastic saturated soilrdquoSoil Dynamics and Earthquake Engineering vol 24 no 12 pp1003ndash1011 2004
[5] K-H Xie and G-B Liu ldquoTransient response of a sphericalcavity with a partially sealed shell embedded in viscoelasticsaturated soilrdquo Journal of Zhejiang University Science A vol 6no 3 pp 194ndash201 2005
[6] Z-H Yuan Y-Q Cai L Shi H-L Sun and Z-G CaoldquoResponse of rail structure and circular tunnel in saturatedsoil subjected to harmonic moving loadrdquo Chinese Journal ofGeotechnical Engineering vol 38 no 2 pp 311ndash322 2016
12 Mathematical Problems in Engineering
[7] H E M Hunt ldquoModelling of rail vehicles and track forcalculation of ground-vibration transmission into buildingsrdquoJournal of Sound and Vibration vol 193 no 1 pp 185ndash194 1996
[8] T Balendra K H Chua K W Lo and S L Lee ldquoSteady-statevibration of subway-soilbuilding systemrdquo Journal of EngineeringMechanics vol 115 no 1 pp 145ndash162 1989
[9] F-C Wang H Xia and H-R Zhang ldquoVibration effects ofsubway trains on surrounding buildingsrdquo Journal of NorthernJiaotong University vol 23 no 5 pp 45ndash48 1999
[10] Y Yang HHung and L Hsu ldquoGround vibrations due to under-ground trains considering soil-tunnel interactionrdquo Interactionand Multiscale Mechanics vol 1 no 1 pp 157ndash175 2007
[11] Y-E Zhang and Changshi-Pan ldquoTests and analyses of thedynamic response of tunnels subjected to passing train loadrdquoJournal of Shijiazhuang Railway Institute vol 6 no 2 pp 7ndash141993
[12] C J C Jones D J Thompson and M Petyt ldquoStudies usinga combined finite element and boundary element model forvibration propagation from railway tunnelsrdquo International Insti-tute of Acoustics and Vibration pp 2703ndash2710 2000
[13] W Gardien and H G Stuit ldquoModelling of soil vibrations fromrailway tunnelsrdquo Journal of Sound and Vibration vol 267 no 3pp 605ndash619 2003
[14] S Wolf ldquoPotential low frequency ground vibration (lt63 Hz)impacts from underground LRT operationsrdquo Journal of Soundand Vibration vol 267 no 3 pp 651ndash661 2003
[15] G Degrande D Clouteau R Othman et al ldquoA numericalmodel for ground-borne vibrations from underground railwaytraffic based on a periodic finite element-boundary elementformulationrdquo Journal of Sound and Vibration vol 293 no 3-5pp 645ndash666 2006
[16] W-F Liu W-N Liu S Gupta and G Degrande ldquoPrediction ofvibrations in the tunnel and free field due to passage of metrotrainsrdquo Engineering Mechanics vol 27 no 1 pp 250ndash256 2010
[17] K A Kuo H E M Hunt and M F M Hussein ldquoThe effectof a twin tunnel on the propagation of ground-borne vibrationfrom an underground railwayrdquo Journal of Sound and Vibrationvol 330 no 25 pp 6203ndash6222 2011
[18] Z-H Yuan Y-Q Cai W Yuan Y-L Xu and Z-G CaoldquoDynamic response of circular railway tunnel and track systemin saturated soil under moving train loadingrdquo Rock and SoilMechanics vol 38 no 4 pp 1003ndash1014 2017
[19] Z P Liao and H L Wong ldquoA transmitting boundary for thenumerical simulation of elasticwave propagationrdquo InternationalJournal of Soil Dynamics and Earthquake Engineering vol 3 pp133ndash144 1984
[20] L Shi P Wang Y Cai and Z Cao ldquoMulti-transmittingformula for finite element modeling of wave propagation in asaturated poroelastic mediumrdquo Soil Dynamics and EarthquakeEngineering vol 80 pp 11ndash24 2016
[21] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid I Low-frequency rangerdquoe Journal ofthe Acoustical Society of America vol 28 pp 168ndash178 1956
[22] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956
[23] O C Zienkiewicz C T Chang and P Bettess ldquoDrainedundrained consolidating and dynamic behaviour assumptionsin soilsrdquo Geotechnique vol 30 no 4 pp 385ndash395 1980
[24] L Shi H Sun Y Cai C Xu and P Wang ldquoValidity of fullydrained fully undrained and u-p formulations for modeling a
poroelastic half-space under a moving harmonic point loadrdquoSoil Dynamics and Earthquake Engineering vol 42 pp 292ndash3012012
[25] T Senjuntichai and R K N D Rajapakse ldquoTransient responseof a circular cavity in a poroelastic mediumrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 17 no 6 pp 357ndash383 1993
[26] Y-B Yang S-R Kuo and H-H Hung ldquoFrequency-independent infinite elements for analysing semi-infiniteproblemsrdquo International Journal for Numerical Methods inEngineering vol 39 no 20 pp 3553ndash3569 1996
[27] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic response of tracksystem and saturated soil in half space subjected to a movingtrain loadrdquo Chinese Journal of Rock Mechanics and Engineeringvol 26 no 8 pp 1705ndash1712 2007
[28] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic responses oftrack system and poroelastic soil under high-speed train loadrdquoJournal of Zhejiang University (Engineering Science) vol 42 no11 pp 2002ndash2008 2008
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
(a) Maximum displacement at the observation points for differenttunnel spacing
3 35 4 45 5Burying depth hr
point C(one load)point D(one load)point C(two loads)point D(two loads)
Max
imum
disp
lace
men
t
minus45
minus40
minus35
minus30
minus25
minus20
minus15
minus10
UZlowast
(dB)
(b) Maximum displacement at the observation points for differentburying depth
Figure 7 Comparison of dynamic responses between the present model and a model with both tunnels loaded
surface their responses will be amplified when the loadvelocity approaches the Rayleigh wave velocity of the groundwhich is around 09 119888s The same results were reported inprevious studies on the critical velocity for a train runningon the ground surface [27 28] However points A and B areset on the concrete liner whose shear wave velocity is muchhigher than that of the soil Thus the load velocity is very lowcompared to the shear wave velocity of the concrete whichmakes the displacement of points A and B nonsensitive to thechange of load velocity
54 e Influence of Tunnel Spacing In this section theinfluence of the tunnel spacing on the dynamic responses isinvestigated by considering 6 different twin tunnel distancesie 119889119903 = 3 4 5 6 7 8 The load velocity is 03 119888119904 The
maximum vertical displacements at the observation pointswhen ℎ119903 = 4 are compared in Figure 12
The displacement at point A is significantly larger thanthose at other points since point A is traversed directlyby the point load When the tunnel spacing increases thedisplacements at points B and C decrease gradually Howeverthe displacements at points A and D remain almost the sameThis observation is within expectation because the positionsof points A and D remain the same while points B and C areshifted away from the load when the spacing increases Thedisplacements at points B and C vary greatly when the tunnelspacing 119889119903 increases from 4 to 6
55e Influence of Burying Depth In this section paramet-ric study is conducted for 5 different burying depths of the
Figure 9 Comparison of maximum vibration velocity at obser-vation points between a model containing one load and a modelcontaining two loads
twin tunnel ie ℎ119903 = 2 3 4 5 6 And the load velocity is setto 03 119888s
The variation of the maximum vertical displacements atthe observation points with respect to the burying depth ispresented in Figure 13 In this case the tunnel spacing is fixedas 119889119903 = 4 It is observed that the displacements at points AC and D decrease when ℎ119903 increases For the displacementat point B it increases slightly when ℎ119903 lt 4 however itdrops substantially when ℎ119903 increases further to 6 This
phenomenon is explained by the fact that when the tunnelis shallowly buried the waves generated by the source tunnel(ie the tunnel where the moving load is applied) impinge onthe ground surface at a large incident angle (resembling theglancing incident) correspondingly the reflected waves leavethe surface at a large angle and thus the surface reflectionregion at nonsourced tunnels is small resulting in a fractionof surface reflections contributing to the dynamic response atpoint B However the surface reflection region will expandwhen hr increases gradually but when the burying depthis large enough (ie ℎ119889 gt 5) the material and geometricaldamping are dominant which results in the decreasing of thedisplacement at point B
6 Conclusions
In this paper a 3D twin-tunnel model is presented to evaluatethe dynamic responses of saturated half-space induced bysubways The effects of spacing between two tunnels tunnelburied depth load moving speed and soil permeabilityon the vibration response are investigated Based on thederivation and numerical examples presented above thefollowing conclusions can be drawn
(1) The existence of the other tunnel has influences onthe vibration caused by the moving subway Andwhen the twin tunnels are both loaded the dynamicresponses cannot be calculated by simply summingup the results of a single tunnel model Therefore itis necessary to consider a twin-tunnel model whenpredicting the dynamic responses induced by sub-ways for all load velocities The dynamic responsesof the single-phase ground are larger than those inthe saturated ground which denotes that using asingle phase model to predict the vibrations is moreconservative
(2) The influence of the spacing between two tunnels onthe dynamic response of the tunnel without the loadapplied is greater than that on the ground surfaceWith the increase of the spacing 119889119903 the dynamicresponse of the adjacent tunnel decreases Tunnelburied depth has a great influence on the dynamicresponse of adjacent tunnel and the ground sur-face For points on the ground surface the dynamicresponse decreases with the increase of the tunnelburied depthDue to the presence of surface reflectionregion the dynamic response of adjacent tunnelincreases at first and then decreases as the burialdepth increases So more attention should be paid tothe spacing between tunnels and the tunnel burieddepth in the construction of subway
(3) The subways moving at a low speed will generate asmaller dynamic response in the saturated half-spacethan moving at a higher speed And when the loadvelocity approaches the Rayleigh wave velocity of theground which is around 09 119888119904 the responses of thesoil and liner will be amplified Therefore the subwaymoving speed is also an important parameter thataffects the environmental vibration
10 Mathematical Problems in Engineering
0
2 3 4 5 6Burying depth hr
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
Max
imum
vib
ratio
n ve
loci
ty
minus40
minus30
minus20
minus10
10
20
30V
Zlowast(d
B)
(a) Maximum vibration velocity at observation points for differentburying depth
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
Max
imum
disp
lace
men
t
01 03 05 07 09Load velocity s
minus40
minus35
minus30
minus25
minus20
minus15
minus10
minus5
0
UZlowast
(dB)
(b) Maximum displacement at observation points for different loadvelocity
minus30
minus20
minus10
0
10
20
30
Max
imum
vib
ratio
n ve
loci
ty
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
01 03 05 07 09Load velocity s
VZlowast
(dB)
(c) Maximum vibration velocity at observation points for differentload velocity
Figure 10 Comparisons of dynamic response at observation points between a saturated ground and an elastic ground
Appendix
The expressions of the elements in the mass matrix thedamping matrix the stiffness matrix and the force vector arepresented as follows
Figure 13 Vertical maximum displacements for different buryingdepth of tunnels
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the National Key RampD Programof China (Grant No 2016YFC0800200) the Projects ofInternational Cooperation and Exchanges NSFC (Grant No51620105008) National Science Foundation for Young Scien-tists of China (Grant No 51608482) and National NaturalScience Foundation of China (Grant No 51478424)
References
[1] A V Metrikine and A C W M Vrouwenvelder ldquoSurfaceground vibration due to a moving train in a tunnel two-dimensional modelrdquo Journal of Sound and Vibration vol 234no 1 pp 43ndash66 2000
[2] J A Forrest and H E M Hunt ldquoA three-dimensional tunnelmodel for calculation of train-induced ground vibrationrdquo Jour-nal of Sound and Vibration vol 294 no 4 pp 678ndash705 2006
[3] J A Forrest andH EMHunt ldquoGround vibration generated bytrains in underground tunnelsrdquo Journal of Sound and Vibrationvol 294 no 4 pp 706ndash736 2006
[4] K-H Xie G-B Liu and Z-Y Shi ldquoDynamic response ofpartially sealed circular tunnel in viscoelastic saturated soilrdquoSoil Dynamics and Earthquake Engineering vol 24 no 12 pp1003ndash1011 2004
[5] K-H Xie and G-B Liu ldquoTransient response of a sphericalcavity with a partially sealed shell embedded in viscoelasticsaturated soilrdquo Journal of Zhejiang University Science A vol 6no 3 pp 194ndash201 2005
[6] Z-H Yuan Y-Q Cai L Shi H-L Sun and Z-G CaoldquoResponse of rail structure and circular tunnel in saturatedsoil subjected to harmonic moving loadrdquo Chinese Journal ofGeotechnical Engineering vol 38 no 2 pp 311ndash322 2016
12 Mathematical Problems in Engineering
[7] H E M Hunt ldquoModelling of rail vehicles and track forcalculation of ground-vibration transmission into buildingsrdquoJournal of Sound and Vibration vol 193 no 1 pp 185ndash194 1996
[8] T Balendra K H Chua K W Lo and S L Lee ldquoSteady-statevibration of subway-soilbuilding systemrdquo Journal of EngineeringMechanics vol 115 no 1 pp 145ndash162 1989
[9] F-C Wang H Xia and H-R Zhang ldquoVibration effects ofsubway trains on surrounding buildingsrdquo Journal of NorthernJiaotong University vol 23 no 5 pp 45ndash48 1999
[10] Y Yang HHung and L Hsu ldquoGround vibrations due to under-ground trains considering soil-tunnel interactionrdquo Interactionand Multiscale Mechanics vol 1 no 1 pp 157ndash175 2007
[11] Y-E Zhang and Changshi-Pan ldquoTests and analyses of thedynamic response of tunnels subjected to passing train loadrdquoJournal of Shijiazhuang Railway Institute vol 6 no 2 pp 7ndash141993
[12] C J C Jones D J Thompson and M Petyt ldquoStudies usinga combined finite element and boundary element model forvibration propagation from railway tunnelsrdquo International Insti-tute of Acoustics and Vibration pp 2703ndash2710 2000
[13] W Gardien and H G Stuit ldquoModelling of soil vibrations fromrailway tunnelsrdquo Journal of Sound and Vibration vol 267 no 3pp 605ndash619 2003
[14] S Wolf ldquoPotential low frequency ground vibration (lt63 Hz)impacts from underground LRT operationsrdquo Journal of Soundand Vibration vol 267 no 3 pp 651ndash661 2003
[15] G Degrande D Clouteau R Othman et al ldquoA numericalmodel for ground-borne vibrations from underground railwaytraffic based on a periodic finite element-boundary elementformulationrdquo Journal of Sound and Vibration vol 293 no 3-5pp 645ndash666 2006
[16] W-F Liu W-N Liu S Gupta and G Degrande ldquoPrediction ofvibrations in the tunnel and free field due to passage of metrotrainsrdquo Engineering Mechanics vol 27 no 1 pp 250ndash256 2010
[17] K A Kuo H E M Hunt and M F M Hussein ldquoThe effectof a twin tunnel on the propagation of ground-borne vibrationfrom an underground railwayrdquo Journal of Sound and Vibrationvol 330 no 25 pp 6203ndash6222 2011
[18] Z-H Yuan Y-Q Cai W Yuan Y-L Xu and Z-G CaoldquoDynamic response of circular railway tunnel and track systemin saturated soil under moving train loadingrdquo Rock and SoilMechanics vol 38 no 4 pp 1003ndash1014 2017
[19] Z P Liao and H L Wong ldquoA transmitting boundary for thenumerical simulation of elasticwave propagationrdquo InternationalJournal of Soil Dynamics and Earthquake Engineering vol 3 pp133ndash144 1984
[20] L Shi P Wang Y Cai and Z Cao ldquoMulti-transmittingformula for finite element modeling of wave propagation in asaturated poroelastic mediumrdquo Soil Dynamics and EarthquakeEngineering vol 80 pp 11ndash24 2016
[21] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid I Low-frequency rangerdquoe Journal ofthe Acoustical Society of America vol 28 pp 168ndash178 1956
[22] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956
[23] O C Zienkiewicz C T Chang and P Bettess ldquoDrainedundrained consolidating and dynamic behaviour assumptionsin soilsrdquo Geotechnique vol 30 no 4 pp 385ndash395 1980
[24] L Shi H Sun Y Cai C Xu and P Wang ldquoValidity of fullydrained fully undrained and u-p formulations for modeling a
poroelastic half-space under a moving harmonic point loadrdquoSoil Dynamics and Earthquake Engineering vol 42 pp 292ndash3012012
[25] T Senjuntichai and R K N D Rajapakse ldquoTransient responseof a circular cavity in a poroelastic mediumrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 17 no 6 pp 357ndash383 1993
[26] Y-B Yang S-R Kuo and H-H Hung ldquoFrequency-independent infinite elements for analysing semi-infiniteproblemsrdquo International Journal for Numerical Methods inEngineering vol 39 no 20 pp 3553ndash3569 1996
[27] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic response of tracksystem and saturated soil in half space subjected to a movingtrain loadrdquo Chinese Journal of Rock Mechanics and Engineeringvol 26 no 8 pp 1705ndash1712 2007
[28] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic responses oftrack system and poroelastic soil under high-speed train loadrdquoJournal of Zhejiang University (Engineering Science) vol 42 no11 pp 2002ndash2008 2008
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Figure 9 Comparison of maximum vibration velocity at obser-vation points between a model containing one load and a modelcontaining two loads
twin tunnel ie ℎ119903 = 2 3 4 5 6 And the load velocity is setto 03 119888s
The variation of the maximum vertical displacements atthe observation points with respect to the burying depth ispresented in Figure 13 In this case the tunnel spacing is fixedas 119889119903 = 4 It is observed that the displacements at points AC and D decrease when ℎ119903 increases For the displacementat point B it increases slightly when ℎ119903 lt 4 however itdrops substantially when ℎ119903 increases further to 6 This
phenomenon is explained by the fact that when the tunnelis shallowly buried the waves generated by the source tunnel(ie the tunnel where the moving load is applied) impinge onthe ground surface at a large incident angle (resembling theglancing incident) correspondingly the reflected waves leavethe surface at a large angle and thus the surface reflectionregion at nonsourced tunnels is small resulting in a fractionof surface reflections contributing to the dynamic response atpoint B However the surface reflection region will expandwhen hr increases gradually but when the burying depthis large enough (ie ℎ119889 gt 5) the material and geometricaldamping are dominant which results in the decreasing of thedisplacement at point B
6 Conclusions
In this paper a 3D twin-tunnel model is presented to evaluatethe dynamic responses of saturated half-space induced bysubways The effects of spacing between two tunnels tunnelburied depth load moving speed and soil permeabilityon the vibration response are investigated Based on thederivation and numerical examples presented above thefollowing conclusions can be drawn
(1) The existence of the other tunnel has influences onthe vibration caused by the moving subway Andwhen the twin tunnels are both loaded the dynamicresponses cannot be calculated by simply summingup the results of a single tunnel model Therefore itis necessary to consider a twin-tunnel model whenpredicting the dynamic responses induced by sub-ways for all load velocities The dynamic responsesof the single-phase ground are larger than those inthe saturated ground which denotes that using asingle phase model to predict the vibrations is moreconservative
(2) The influence of the spacing between two tunnels onthe dynamic response of the tunnel without the loadapplied is greater than that on the ground surfaceWith the increase of the spacing 119889119903 the dynamicresponse of the adjacent tunnel decreases Tunnelburied depth has a great influence on the dynamicresponse of adjacent tunnel and the ground sur-face For points on the ground surface the dynamicresponse decreases with the increase of the tunnelburied depthDue to the presence of surface reflectionregion the dynamic response of adjacent tunnelincreases at first and then decreases as the burialdepth increases So more attention should be paid tothe spacing between tunnels and the tunnel burieddepth in the construction of subway
(3) The subways moving at a low speed will generate asmaller dynamic response in the saturated half-spacethan moving at a higher speed And when the loadvelocity approaches the Rayleigh wave velocity of theground which is around 09 119888119904 the responses of thesoil and liner will be amplified Therefore the subwaymoving speed is also an important parameter thataffects the environmental vibration
10 Mathematical Problems in Engineering
0
2 3 4 5 6Burying depth hr
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
Max
imum
vib
ratio
n ve
loci
ty
minus40
minus30
minus20
minus10
10
20
30V
Zlowast(d
B)
(a) Maximum vibration velocity at observation points for differentburying depth
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
Max
imum
disp
lace
men
t
01 03 05 07 09Load velocity s
minus40
minus35
minus30
minus25
minus20
minus15
minus10
minus5
0
UZlowast
(dB)
(b) Maximum displacement at observation points for different loadvelocity
minus30
minus20
minus10
0
10
20
30
Max
imum
vib
ratio
n ve
loci
ty
point A (saturated ground)point D(saturated ground)point A (elastic ground)point D (elastic ground)
01 03 05 07 09Load velocity s
VZlowast
(dB)
(c) Maximum vibration velocity at observation points for differentload velocity
Figure 10 Comparisons of dynamic response at observation points between a saturated ground and an elastic ground
Appendix
The expressions of the elements in the mass matrix thedamping matrix the stiffness matrix and the force vector arepresented as follows
Figure 13 Vertical maximum displacements for different buryingdepth of tunnels
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the National Key RampD Programof China (Grant No 2016YFC0800200) the Projects ofInternational Cooperation and Exchanges NSFC (Grant No51620105008) National Science Foundation for Young Scien-tists of China (Grant No 51608482) and National NaturalScience Foundation of China (Grant No 51478424)
References
[1] A V Metrikine and A C W M Vrouwenvelder ldquoSurfaceground vibration due to a moving train in a tunnel two-dimensional modelrdquo Journal of Sound and Vibration vol 234no 1 pp 43ndash66 2000
[2] J A Forrest and H E M Hunt ldquoA three-dimensional tunnelmodel for calculation of train-induced ground vibrationrdquo Jour-nal of Sound and Vibration vol 294 no 4 pp 678ndash705 2006
[3] J A Forrest andH EMHunt ldquoGround vibration generated bytrains in underground tunnelsrdquo Journal of Sound and Vibrationvol 294 no 4 pp 706ndash736 2006
[4] K-H Xie G-B Liu and Z-Y Shi ldquoDynamic response ofpartially sealed circular tunnel in viscoelastic saturated soilrdquoSoil Dynamics and Earthquake Engineering vol 24 no 12 pp1003ndash1011 2004
[5] K-H Xie and G-B Liu ldquoTransient response of a sphericalcavity with a partially sealed shell embedded in viscoelasticsaturated soilrdquo Journal of Zhejiang University Science A vol 6no 3 pp 194ndash201 2005
[6] Z-H Yuan Y-Q Cai L Shi H-L Sun and Z-G CaoldquoResponse of rail structure and circular tunnel in saturatedsoil subjected to harmonic moving loadrdquo Chinese Journal ofGeotechnical Engineering vol 38 no 2 pp 311ndash322 2016
12 Mathematical Problems in Engineering
[7] H E M Hunt ldquoModelling of rail vehicles and track forcalculation of ground-vibration transmission into buildingsrdquoJournal of Sound and Vibration vol 193 no 1 pp 185ndash194 1996
[8] T Balendra K H Chua K W Lo and S L Lee ldquoSteady-statevibration of subway-soilbuilding systemrdquo Journal of EngineeringMechanics vol 115 no 1 pp 145ndash162 1989
[9] F-C Wang H Xia and H-R Zhang ldquoVibration effects ofsubway trains on surrounding buildingsrdquo Journal of NorthernJiaotong University vol 23 no 5 pp 45ndash48 1999
[10] Y Yang HHung and L Hsu ldquoGround vibrations due to under-ground trains considering soil-tunnel interactionrdquo Interactionand Multiscale Mechanics vol 1 no 1 pp 157ndash175 2007
[11] Y-E Zhang and Changshi-Pan ldquoTests and analyses of thedynamic response of tunnels subjected to passing train loadrdquoJournal of Shijiazhuang Railway Institute vol 6 no 2 pp 7ndash141993
[12] C J C Jones D J Thompson and M Petyt ldquoStudies usinga combined finite element and boundary element model forvibration propagation from railway tunnelsrdquo International Insti-tute of Acoustics and Vibration pp 2703ndash2710 2000
[13] W Gardien and H G Stuit ldquoModelling of soil vibrations fromrailway tunnelsrdquo Journal of Sound and Vibration vol 267 no 3pp 605ndash619 2003
[14] S Wolf ldquoPotential low frequency ground vibration (lt63 Hz)impacts from underground LRT operationsrdquo Journal of Soundand Vibration vol 267 no 3 pp 651ndash661 2003
[15] G Degrande D Clouteau R Othman et al ldquoA numericalmodel for ground-borne vibrations from underground railwaytraffic based on a periodic finite element-boundary elementformulationrdquo Journal of Sound and Vibration vol 293 no 3-5pp 645ndash666 2006
[16] W-F Liu W-N Liu S Gupta and G Degrande ldquoPrediction ofvibrations in the tunnel and free field due to passage of metrotrainsrdquo Engineering Mechanics vol 27 no 1 pp 250ndash256 2010
[17] K A Kuo H E M Hunt and M F M Hussein ldquoThe effectof a twin tunnel on the propagation of ground-borne vibrationfrom an underground railwayrdquo Journal of Sound and Vibrationvol 330 no 25 pp 6203ndash6222 2011
[18] Z-H Yuan Y-Q Cai W Yuan Y-L Xu and Z-G CaoldquoDynamic response of circular railway tunnel and track systemin saturated soil under moving train loadingrdquo Rock and SoilMechanics vol 38 no 4 pp 1003ndash1014 2017
[19] Z P Liao and H L Wong ldquoA transmitting boundary for thenumerical simulation of elasticwave propagationrdquo InternationalJournal of Soil Dynamics and Earthquake Engineering vol 3 pp133ndash144 1984
[20] L Shi P Wang Y Cai and Z Cao ldquoMulti-transmittingformula for finite element modeling of wave propagation in asaturated poroelastic mediumrdquo Soil Dynamics and EarthquakeEngineering vol 80 pp 11ndash24 2016
[21] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid I Low-frequency rangerdquoe Journal ofthe Acoustical Society of America vol 28 pp 168ndash178 1956
[22] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956
[23] O C Zienkiewicz C T Chang and P Bettess ldquoDrainedundrained consolidating and dynamic behaviour assumptionsin soilsrdquo Geotechnique vol 30 no 4 pp 385ndash395 1980
[24] L Shi H Sun Y Cai C Xu and P Wang ldquoValidity of fullydrained fully undrained and u-p formulations for modeling a
poroelastic half-space under a moving harmonic point loadrdquoSoil Dynamics and Earthquake Engineering vol 42 pp 292ndash3012012
[25] T Senjuntichai and R K N D Rajapakse ldquoTransient responseof a circular cavity in a poroelastic mediumrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 17 no 6 pp 357ndash383 1993
[26] Y-B Yang S-R Kuo and H-H Hung ldquoFrequency-independent infinite elements for analysing semi-infiniteproblemsrdquo International Journal for Numerical Methods inEngineering vol 39 no 20 pp 3553ndash3569 1996
[27] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic response of tracksystem and saturated soil in half space subjected to a movingtrain loadrdquo Chinese Journal of Rock Mechanics and Engineeringvol 26 no 8 pp 1705ndash1712 2007
[28] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic responses oftrack system and poroelastic soil under high-speed train loadrdquoJournal of Zhejiang University (Engineering Science) vol 42 no11 pp 2002ndash2008 2008
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Figure 13 Vertical maximum displacements for different buryingdepth of tunnels
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the National Key RampD Programof China (Grant No 2016YFC0800200) the Projects ofInternational Cooperation and Exchanges NSFC (Grant No51620105008) National Science Foundation for Young Scien-tists of China (Grant No 51608482) and National NaturalScience Foundation of China (Grant No 51478424)
References
[1] A V Metrikine and A C W M Vrouwenvelder ldquoSurfaceground vibration due to a moving train in a tunnel two-dimensional modelrdquo Journal of Sound and Vibration vol 234no 1 pp 43ndash66 2000
[2] J A Forrest and H E M Hunt ldquoA three-dimensional tunnelmodel for calculation of train-induced ground vibrationrdquo Jour-nal of Sound and Vibration vol 294 no 4 pp 678ndash705 2006
[3] J A Forrest andH EMHunt ldquoGround vibration generated bytrains in underground tunnelsrdquo Journal of Sound and Vibrationvol 294 no 4 pp 706ndash736 2006
[4] K-H Xie G-B Liu and Z-Y Shi ldquoDynamic response ofpartially sealed circular tunnel in viscoelastic saturated soilrdquoSoil Dynamics and Earthquake Engineering vol 24 no 12 pp1003ndash1011 2004
[5] K-H Xie and G-B Liu ldquoTransient response of a sphericalcavity with a partially sealed shell embedded in viscoelasticsaturated soilrdquo Journal of Zhejiang University Science A vol 6no 3 pp 194ndash201 2005
[6] Z-H Yuan Y-Q Cai L Shi H-L Sun and Z-G CaoldquoResponse of rail structure and circular tunnel in saturatedsoil subjected to harmonic moving loadrdquo Chinese Journal ofGeotechnical Engineering vol 38 no 2 pp 311ndash322 2016
12 Mathematical Problems in Engineering
[7] H E M Hunt ldquoModelling of rail vehicles and track forcalculation of ground-vibration transmission into buildingsrdquoJournal of Sound and Vibration vol 193 no 1 pp 185ndash194 1996
[8] T Balendra K H Chua K W Lo and S L Lee ldquoSteady-statevibration of subway-soilbuilding systemrdquo Journal of EngineeringMechanics vol 115 no 1 pp 145ndash162 1989
[9] F-C Wang H Xia and H-R Zhang ldquoVibration effects ofsubway trains on surrounding buildingsrdquo Journal of NorthernJiaotong University vol 23 no 5 pp 45ndash48 1999
[10] Y Yang HHung and L Hsu ldquoGround vibrations due to under-ground trains considering soil-tunnel interactionrdquo Interactionand Multiscale Mechanics vol 1 no 1 pp 157ndash175 2007
[11] Y-E Zhang and Changshi-Pan ldquoTests and analyses of thedynamic response of tunnels subjected to passing train loadrdquoJournal of Shijiazhuang Railway Institute vol 6 no 2 pp 7ndash141993
[12] C J C Jones D J Thompson and M Petyt ldquoStudies usinga combined finite element and boundary element model forvibration propagation from railway tunnelsrdquo International Insti-tute of Acoustics and Vibration pp 2703ndash2710 2000
[13] W Gardien and H G Stuit ldquoModelling of soil vibrations fromrailway tunnelsrdquo Journal of Sound and Vibration vol 267 no 3pp 605ndash619 2003
[14] S Wolf ldquoPotential low frequency ground vibration (lt63 Hz)impacts from underground LRT operationsrdquo Journal of Soundand Vibration vol 267 no 3 pp 651ndash661 2003
[15] G Degrande D Clouteau R Othman et al ldquoA numericalmodel for ground-borne vibrations from underground railwaytraffic based on a periodic finite element-boundary elementformulationrdquo Journal of Sound and Vibration vol 293 no 3-5pp 645ndash666 2006
[16] W-F Liu W-N Liu S Gupta and G Degrande ldquoPrediction ofvibrations in the tunnel and free field due to passage of metrotrainsrdquo Engineering Mechanics vol 27 no 1 pp 250ndash256 2010
[17] K A Kuo H E M Hunt and M F M Hussein ldquoThe effectof a twin tunnel on the propagation of ground-borne vibrationfrom an underground railwayrdquo Journal of Sound and Vibrationvol 330 no 25 pp 6203ndash6222 2011
[18] Z-H Yuan Y-Q Cai W Yuan Y-L Xu and Z-G CaoldquoDynamic response of circular railway tunnel and track systemin saturated soil under moving train loadingrdquo Rock and SoilMechanics vol 38 no 4 pp 1003ndash1014 2017
[19] Z P Liao and H L Wong ldquoA transmitting boundary for thenumerical simulation of elasticwave propagationrdquo InternationalJournal of Soil Dynamics and Earthquake Engineering vol 3 pp133ndash144 1984
[20] L Shi P Wang Y Cai and Z Cao ldquoMulti-transmittingformula for finite element modeling of wave propagation in asaturated poroelastic mediumrdquo Soil Dynamics and EarthquakeEngineering vol 80 pp 11ndash24 2016
[21] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid I Low-frequency rangerdquoe Journal ofthe Acoustical Society of America vol 28 pp 168ndash178 1956
[22] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956
[23] O C Zienkiewicz C T Chang and P Bettess ldquoDrainedundrained consolidating and dynamic behaviour assumptionsin soilsrdquo Geotechnique vol 30 no 4 pp 385ndash395 1980
[24] L Shi H Sun Y Cai C Xu and P Wang ldquoValidity of fullydrained fully undrained and u-p formulations for modeling a
poroelastic half-space under a moving harmonic point loadrdquoSoil Dynamics and Earthquake Engineering vol 42 pp 292ndash3012012
[25] T Senjuntichai and R K N D Rajapakse ldquoTransient responseof a circular cavity in a poroelastic mediumrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 17 no 6 pp 357ndash383 1993
[26] Y-B Yang S-R Kuo and H-H Hung ldquoFrequency-independent infinite elements for analysing semi-infiniteproblemsrdquo International Journal for Numerical Methods inEngineering vol 39 no 20 pp 3553ndash3569 1996
[27] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic response of tracksystem and saturated soil in half space subjected to a movingtrain loadrdquo Chinese Journal of Rock Mechanics and Engineeringvol 26 no 8 pp 1705ndash1712 2007
[28] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic responses oftrack system and poroelastic soil under high-speed train loadrdquoJournal of Zhejiang University (Engineering Science) vol 42 no11 pp 2002ndash2008 2008
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Figure 13 Vertical maximum displacements for different buryingdepth of tunnels
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the National Key RampD Programof China (Grant No 2016YFC0800200) the Projects ofInternational Cooperation and Exchanges NSFC (Grant No51620105008) National Science Foundation for Young Scien-tists of China (Grant No 51608482) and National NaturalScience Foundation of China (Grant No 51478424)
References
[1] A V Metrikine and A C W M Vrouwenvelder ldquoSurfaceground vibration due to a moving train in a tunnel two-dimensional modelrdquo Journal of Sound and Vibration vol 234no 1 pp 43ndash66 2000
[2] J A Forrest and H E M Hunt ldquoA three-dimensional tunnelmodel for calculation of train-induced ground vibrationrdquo Jour-nal of Sound and Vibration vol 294 no 4 pp 678ndash705 2006
[3] J A Forrest andH EMHunt ldquoGround vibration generated bytrains in underground tunnelsrdquo Journal of Sound and Vibrationvol 294 no 4 pp 706ndash736 2006
[4] K-H Xie G-B Liu and Z-Y Shi ldquoDynamic response ofpartially sealed circular tunnel in viscoelastic saturated soilrdquoSoil Dynamics and Earthquake Engineering vol 24 no 12 pp1003ndash1011 2004
[5] K-H Xie and G-B Liu ldquoTransient response of a sphericalcavity with a partially sealed shell embedded in viscoelasticsaturated soilrdquo Journal of Zhejiang University Science A vol 6no 3 pp 194ndash201 2005
[6] Z-H Yuan Y-Q Cai L Shi H-L Sun and Z-G CaoldquoResponse of rail structure and circular tunnel in saturatedsoil subjected to harmonic moving loadrdquo Chinese Journal ofGeotechnical Engineering vol 38 no 2 pp 311ndash322 2016
12 Mathematical Problems in Engineering
[7] H E M Hunt ldquoModelling of rail vehicles and track forcalculation of ground-vibration transmission into buildingsrdquoJournal of Sound and Vibration vol 193 no 1 pp 185ndash194 1996
[8] T Balendra K H Chua K W Lo and S L Lee ldquoSteady-statevibration of subway-soilbuilding systemrdquo Journal of EngineeringMechanics vol 115 no 1 pp 145ndash162 1989
[9] F-C Wang H Xia and H-R Zhang ldquoVibration effects ofsubway trains on surrounding buildingsrdquo Journal of NorthernJiaotong University vol 23 no 5 pp 45ndash48 1999
[10] Y Yang HHung and L Hsu ldquoGround vibrations due to under-ground trains considering soil-tunnel interactionrdquo Interactionand Multiscale Mechanics vol 1 no 1 pp 157ndash175 2007
[11] Y-E Zhang and Changshi-Pan ldquoTests and analyses of thedynamic response of tunnels subjected to passing train loadrdquoJournal of Shijiazhuang Railway Institute vol 6 no 2 pp 7ndash141993
[12] C J C Jones D J Thompson and M Petyt ldquoStudies usinga combined finite element and boundary element model forvibration propagation from railway tunnelsrdquo International Insti-tute of Acoustics and Vibration pp 2703ndash2710 2000
[13] W Gardien and H G Stuit ldquoModelling of soil vibrations fromrailway tunnelsrdquo Journal of Sound and Vibration vol 267 no 3pp 605ndash619 2003
[14] S Wolf ldquoPotential low frequency ground vibration (lt63 Hz)impacts from underground LRT operationsrdquo Journal of Soundand Vibration vol 267 no 3 pp 651ndash661 2003
[15] G Degrande D Clouteau R Othman et al ldquoA numericalmodel for ground-borne vibrations from underground railwaytraffic based on a periodic finite element-boundary elementformulationrdquo Journal of Sound and Vibration vol 293 no 3-5pp 645ndash666 2006
[16] W-F Liu W-N Liu S Gupta and G Degrande ldquoPrediction ofvibrations in the tunnel and free field due to passage of metrotrainsrdquo Engineering Mechanics vol 27 no 1 pp 250ndash256 2010
[17] K A Kuo H E M Hunt and M F M Hussein ldquoThe effectof a twin tunnel on the propagation of ground-borne vibrationfrom an underground railwayrdquo Journal of Sound and Vibrationvol 330 no 25 pp 6203ndash6222 2011
[18] Z-H Yuan Y-Q Cai W Yuan Y-L Xu and Z-G CaoldquoDynamic response of circular railway tunnel and track systemin saturated soil under moving train loadingrdquo Rock and SoilMechanics vol 38 no 4 pp 1003ndash1014 2017
[19] Z P Liao and H L Wong ldquoA transmitting boundary for thenumerical simulation of elasticwave propagationrdquo InternationalJournal of Soil Dynamics and Earthquake Engineering vol 3 pp133ndash144 1984
[20] L Shi P Wang Y Cai and Z Cao ldquoMulti-transmittingformula for finite element modeling of wave propagation in asaturated poroelastic mediumrdquo Soil Dynamics and EarthquakeEngineering vol 80 pp 11ndash24 2016
[21] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid I Low-frequency rangerdquoe Journal ofthe Acoustical Society of America vol 28 pp 168ndash178 1956
[22] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956
[23] O C Zienkiewicz C T Chang and P Bettess ldquoDrainedundrained consolidating and dynamic behaviour assumptionsin soilsrdquo Geotechnique vol 30 no 4 pp 385ndash395 1980
[24] L Shi H Sun Y Cai C Xu and P Wang ldquoValidity of fullydrained fully undrained and u-p formulations for modeling a
poroelastic half-space under a moving harmonic point loadrdquoSoil Dynamics and Earthquake Engineering vol 42 pp 292ndash3012012
[25] T Senjuntichai and R K N D Rajapakse ldquoTransient responseof a circular cavity in a poroelastic mediumrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 17 no 6 pp 357ndash383 1993
[26] Y-B Yang S-R Kuo and H-H Hung ldquoFrequency-independent infinite elements for analysing semi-infiniteproblemsrdquo International Journal for Numerical Methods inEngineering vol 39 no 20 pp 3553ndash3569 1996
[27] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic response of tracksystem and saturated soil in half space subjected to a movingtrain loadrdquo Chinese Journal of Rock Mechanics and Engineeringvol 26 no 8 pp 1705ndash1712 2007
[28] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic responses oftrack system and poroelastic soil under high-speed train loadrdquoJournal of Zhejiang University (Engineering Science) vol 42 no11 pp 2002ndash2008 2008
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
12 Mathematical Problems in Engineering
[7] H E M Hunt ldquoModelling of rail vehicles and track forcalculation of ground-vibration transmission into buildingsrdquoJournal of Sound and Vibration vol 193 no 1 pp 185ndash194 1996
[8] T Balendra K H Chua K W Lo and S L Lee ldquoSteady-statevibration of subway-soilbuilding systemrdquo Journal of EngineeringMechanics vol 115 no 1 pp 145ndash162 1989
[9] F-C Wang H Xia and H-R Zhang ldquoVibration effects ofsubway trains on surrounding buildingsrdquo Journal of NorthernJiaotong University vol 23 no 5 pp 45ndash48 1999
[10] Y Yang HHung and L Hsu ldquoGround vibrations due to under-ground trains considering soil-tunnel interactionrdquo Interactionand Multiscale Mechanics vol 1 no 1 pp 157ndash175 2007
[11] Y-E Zhang and Changshi-Pan ldquoTests and analyses of thedynamic response of tunnels subjected to passing train loadrdquoJournal of Shijiazhuang Railway Institute vol 6 no 2 pp 7ndash141993
[12] C J C Jones D J Thompson and M Petyt ldquoStudies usinga combined finite element and boundary element model forvibration propagation from railway tunnelsrdquo International Insti-tute of Acoustics and Vibration pp 2703ndash2710 2000
[13] W Gardien and H G Stuit ldquoModelling of soil vibrations fromrailway tunnelsrdquo Journal of Sound and Vibration vol 267 no 3pp 605ndash619 2003
[14] S Wolf ldquoPotential low frequency ground vibration (lt63 Hz)impacts from underground LRT operationsrdquo Journal of Soundand Vibration vol 267 no 3 pp 651ndash661 2003
[15] G Degrande D Clouteau R Othman et al ldquoA numericalmodel for ground-borne vibrations from underground railwaytraffic based on a periodic finite element-boundary elementformulationrdquo Journal of Sound and Vibration vol 293 no 3-5pp 645ndash666 2006
[16] W-F Liu W-N Liu S Gupta and G Degrande ldquoPrediction ofvibrations in the tunnel and free field due to passage of metrotrainsrdquo Engineering Mechanics vol 27 no 1 pp 250ndash256 2010
[17] K A Kuo H E M Hunt and M F M Hussein ldquoThe effectof a twin tunnel on the propagation of ground-borne vibrationfrom an underground railwayrdquo Journal of Sound and Vibrationvol 330 no 25 pp 6203ndash6222 2011
[18] Z-H Yuan Y-Q Cai W Yuan Y-L Xu and Z-G CaoldquoDynamic response of circular railway tunnel and track systemin saturated soil under moving train loadingrdquo Rock and SoilMechanics vol 38 no 4 pp 1003ndash1014 2017
[19] Z P Liao and H L Wong ldquoA transmitting boundary for thenumerical simulation of elasticwave propagationrdquo InternationalJournal of Soil Dynamics and Earthquake Engineering vol 3 pp133ndash144 1984
[20] L Shi P Wang Y Cai and Z Cao ldquoMulti-transmittingformula for finite element modeling of wave propagation in asaturated poroelastic mediumrdquo Soil Dynamics and EarthquakeEngineering vol 80 pp 11ndash24 2016
[21] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid I Low-frequency rangerdquoe Journal ofthe Acoustical Society of America vol 28 pp 168ndash178 1956
[22] M A Biot ldquoTheory of propagation of elastic waves in a fluid-saturated porous solid II Higher frequency rangerdquoe Journalof the Acoustical Society of America vol 28 pp 179ndash191 1956
[23] O C Zienkiewicz C T Chang and P Bettess ldquoDrainedundrained consolidating and dynamic behaviour assumptionsin soilsrdquo Geotechnique vol 30 no 4 pp 385ndash395 1980
[24] L Shi H Sun Y Cai C Xu and P Wang ldquoValidity of fullydrained fully undrained and u-p formulations for modeling a
poroelastic half-space under a moving harmonic point loadrdquoSoil Dynamics and Earthquake Engineering vol 42 pp 292ndash3012012
[25] T Senjuntichai and R K N D Rajapakse ldquoTransient responseof a circular cavity in a poroelastic mediumrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 17 no 6 pp 357ndash383 1993
[26] Y-B Yang S-R Kuo and H-H Hung ldquoFrequency-independent infinite elements for analysing semi-infiniteproblemsrdquo International Journal for Numerical Methods inEngineering vol 39 no 20 pp 3553ndash3569 1996
[27] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic response of tracksystem and saturated soil in half space subjected to a movingtrain loadrdquo Chinese Journal of Rock Mechanics and Engineeringvol 26 no 8 pp 1705ndash1712 2007
[28] H-L Sun Y-Q Cai and C-J Xu ldquoDynamic responses oftrack system and poroelastic soil under high-speed train loadrdquoJournal of Zhejiang University (Engineering Science) vol 42 no11 pp 2002ndash2008 2008
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
