StudyonStructuralServicePerformanceofHeavy-Haul ...downloads.hindawi.com/journals/ace/2018/3510979.pdfunsprung mass, track irregularity, driving speed, load pe-riodicity, movement

Research ArticleStudy on Structural Service Performance of Heavy-HaulRailway Tunnel with Voided Base

Zhiqiang Zhang Bowen Zeng Chaolong Dai and Wanping He

Key Laboratory of Transportation Tunnel Engineering Ministry of Education Southwest Jiaotong UniversityChengdu 610031 Sichuan China

Correspondence should be addressed to Zhiqiang Zhang clarkswjtueducn

Received 11 May 2018 Accepted 15 August 2018 Published 11 October 2018

Academic Editor Qianbing Zhang

Copyright copy 2018 Zhiqiang Zhang et al )is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

)e structural design of heavy-haul railway tunnels still follows the design method of ordinary railway tunnels Most of them donot take into account the influence of large axle load of 30 t or more let alone such problems as void of surrounding rock underlong-term dynamic loads In order to analyze the dynamic response of heavy-haul railway tunnels under long-term reciprocatingcyclic dynamic loads considering the factors such as axle load of vehicle body unsprungmass and track irregularity the vibrationload time-history curve of heavy-haul railway trains is determined the three-dimensional dynamics coupling model of dynamicload-tunnel-surrounding rock is established and the fatigue life of the structure under different void conditions is analyzed basedon the S-N curve of concrete According to the study the loading unloading and vibration caused by train passing will lead tofluctuations in the vertical displacement response of the monitoring point )e peaks and valleys of the response time-historycurve correspond to the effect of the train wheels rolling through When the void is 6m wide and 10 cm thick the verticaldisplacement of the inverted arch is increased by about 9 times the peak velocity of the inverted arch is increased by about 38times and the maximum principal stress is increased by about 473 compared with the condition without void With the samevoid thickness the vertical displacement and velocity curves of the inverted arch vary significantly with the increase of the voidwidth)e width of the base void has a significant effect on the fatigue life of the structure of heavy-haul railway tunnels Based onthe operation requirement of 100-year service life the ultimate void width is 2m

1 Introduction

Heavy-haul railways feature great transport capacity andsignificant economic and social benefits )ey represent thelevel of a countryrsquos freight capacity Since the 1960s heavy-haul railways have gradually dominated the railway trans-portation in various countries Unlike traditional railwaysthe under-track structure and surrounding rock of heavy-haul railways will bear greater loads due to the significantincrease in the axle load of trains Combined with a series offactors such as stability of soft surrounding rocks and trackirregularity heavy-haul railway tunnels are more easilydamaged under long-term vibration of trains and newrequirements for the structure of heavy-load railway tunnelsare therefore put forward )e impact of heavy-haul trainson the base structure is mainly manifested in the increase of

the load on the track and the under-track base foundationSuch load is different not only from the static load but alsofrom the seismic load It is a reciprocating cyclic load that isapplied for a long time )e study on the mathematicalexpression and time-history curve of the train excitationload is mainly based on field measurements and theoreticalanalysis Jenkins et al [1] established the dynamic analysismodel of depressed joint track and defined the two objectivespecial forces in wheel-rail impact vibration high-frequencyimpact force P2 and low-frequency response force P1Lamaran et al [2] analyzed forced vibration of rail andsimulated the train load as the sum of the static load andadditional dynamic load associated with wheel weight railsupport condition sleeper spacing and vehicle speed Oneet al [3] carried out actual measurements and proved thatthe wheel-rail acting force is severer in a medium frequency

HindawiAdvances in Civil EngineeringVolume 2018 Article ID 3510979 12 pageshttpsdoiorg10115520183510979

range and mainly affects the dynamic response of the vehiclebody in a high-frequency range

)e dynamic response of tunnel base structure undersuch cyclic dynamic load is also a key to the scholarsrsquo study[4 5] In the early days when the computational capacity waslimited a reasonable analytical model was the main researchapproach Metrikine et al [6] established a typical planeanalytical model and analyzed the response of the structureunder constant point load covariant point load and randompoint load by simulating the tunnel in viscoelastic soil withthe Euler beam Balendra et al [7] established a semi-analytical semiplane strain model and adopted the sub-structure method to reduce the complexity of the problemForrest et al [8 9] established a pipe-in-pine model andanalyzed the vibration effect during train passing by takingthe outer elastomeric pipe with infinite radius as the stratumthe thin inner pipe as the lining and the floating plate insidethe thin shell as the filling (the plate and the shell areconnected by a spring) )e model was later improved byHussein et al [10] With the improvement of computerperformance and computational capacity more and morescholars have proposed numerical models suitable for trainvibration )e finite element method was widely used tostudy the tunnel vibration response [11ndash13] Because bothfinite element and boundary element have their own ad-vantages in dealing with lining structure and infinite soilPeng et al [14 15] carried out numerical simulation for theunderlying concrete structure of tunnel under the impactload of train vibration with the nonlinear dynamic finiteelement theory and predicted the service life of the un-derlying structure by using the Tepfers concrete single-logarithmic fatigue equation A finite element-boundaryelement coupling model was established to analyze theimpact of train vibration on adjacent buildings [16ndash18]

Under long-term vibration loads of heavy-haul trainsthe tunnel base will inevitably be subjected to cumulativefatigue damage or even failure and void of surrounding rockDue to the large discreteness of concrete and many otherinfluence factors there is no effective method to predict thefatigue life of concrete and existing studies usually predictthe fatigue life of concrete with the cumulative damagetheory Bazant et al [19] and Mazars et al [20] establishedthe early scalar fatigue damage model for concrete withtheories related to fracture mechanics Fardis et al [21]further optimized the model and adapted it to concretestrain softening by modifying the flexibility matrix equationof bounding surface Abu-Lebdeh et al [22] proposeda constitutive model considering the plasticity of concreteand the damage boundary )e model simulates the state ofstiffness degradation and softening of concrete under thecyclic action multiaxial stress Wu et al [23] introducedtensile and shear damage variables to describe concretestiffness degradation at a macroscale and derived themathematical expressions of plastic strain evolution law andHelmholtz free energy Based on the theoretical system ofHelmholtz free energy Peng et al [24] conducted an ex-perimental study for concrete of inverted arch under dif-ferent vibration amplitudes in a coupling environment ofstatic and cyclic loads and figured out the static load level

that can cause concrete damage and failure as well as thequantitative expression of damage evolution trends at dif-ferent stages Wang et al [25] conducted the impact com-pression test for many times on 5 types of support concreteof different ages with SHPB test equipment )e peakdamage at stable age indicates that the statistical damagemodel of the Weibull distribution can well reflect thedamage characteristics of the material

During the service of heavy-haul railway tunnels thefatigue damage and void of surrounding rock of the basestructure are not independent of each other but are con-comitant and there are incentive and superposition effectsbetween the two Structural damage [26] will lead to decreaseof stiffness and greater deformation under stress which willincur greater load and compression on the surrounding rockof the base as well as continuous accumulation of plasticdeformation [27 28] and aggravated void of the surroundingrock thus changing the support conditions of the structureand causing greater damage to the structure subjected toupper dynamic loads )is paper studies the characteristicsof the full-cycle temporal-spatial response of the basestructure of heavy-haul railway tunnels under repeatedimpact by establishing a three-dimensional numerical dy-namics coupling model of dynamic load-tunnel-surrounding rock and predicts the fatigue life of the tun-nel structure under different void conditions by carrying outfailure evaluation for the base structure under damage-voidexcitation based on the S-N curve of concrete It is found thatduring the service of heavy-haul railway tunnels the pres-ence of voided base has a significant influence on the dis-placement and velocity response of the lining structure aswell as the distribution and magnitude of the principal stresson the lining In addition the void width of the base hasa great influence on various indicators of the lining struc-ture while the influence of void thickness is minor

2 Numerical Model of Heavy-HaulRailway Tunnels

21Determination of Train Excitation Load )e vibration ofa train is under the joint influence of the train and the trackIt is mainly caused by the abrasion of wheels the state of railjoints and geometric irregularity On the vertical longitu-dinal plane of the wheel-rail system there are a wide varietyof vibration sources local unevenness of the existing railsurface such as rail head crushing or rail surface peeling andperiodic system irregularities such as waveform trackcorrugation and eccentric wheels Particularly abrupt ir-regularities are more common at such places as depressedjoints joints with height difference large rail joints andwelds In addition defects such as loose sleepers or hardenedballast bed may occur to the under-track foundationresulting in dynamic irregularity of uneven elasticity andthereby creating an excitation load

Considering the influence factors of train vibration andcombined with the management value of track geometricirregularity in the UK the vertical excitation force of thetrain can be simulated by an excitation function [1 29] )isfunction takes into account the axle load of vehicle body

2 Advances in Civil Engineering

unsprung mass track irregularity driving speed load pe-riodicity movement and combination of loads and theeect of rails and sleepers on the dispersion and trans-mission of loads as shown in the following formula [30]

F(t) p0 + p1 sin ω1t( ) + p2 sin ω2t( ) + p3 sin ω3t( ) (1)

where p0 is the static load of wheels and p1 p2 and p3 arethe vibration loads corresponding to the typical values of①ndash③ in Table 1 Taking M0 as the unsprung mass of thetrain then the corresponding vibration load should be

pi M0aiω2i (i 1 2 3) (2)

where ai is the typical rise of arch and ωi is the circularfrequency of irregularity vibration wavelength at corre-sponding speed corresponding to①② and③ in Table 1respectively e formula is

ωi 2π]Li (i 1 2 3) (3)

where ] is the running speed of train and Li is the typicalwavelength corresponding to ① ② and ③ in Table 1

e management value of track geometric irregularity inthe UK is shown in Table 1 [31]

When calculating the train load the unilateral staticwheel weight should be taken the vehicle load is 150kN andthe unsprung mass is taken as M01200kg Generally thespeed of heavy-haul railways does not exceed 100kmherefore the vibration load of the train at a speed]100kmh is taken into account for the simulation of thetime-history curve of train excitation load as shown inFigure 1

22 Material Mechanics Parameters and Boundary Condition

221 Material Mechanics Parameters During dynamicanalysis the rock mass medium is taken as an ideal elastic-plastic material which follows the MohrndashCoulomb strengthcriterion and the large-strain deformation mode is adoptede yield function is

fs σ1 minus σ3Nϕ + 2cNϕ

radic

ft σ3 minus σt

(4)

where σ1 σ2 and σ3 are the principal stresses of the material(σ1 le σ2 le σ3 and the compressive stress is negative) c is thecohesive force σt is the tensile strength of the material andNϕ is related to the internal friction angle and its expressionis

Nϕ 1 + sinϕ1minus sinϕ

(5)

Shear damage occurs when the stress on a certain pointwithin the rock mass is positive (fs gt 0) tensile damageoccurs when ft gt 0

e potential function in the MohrndashCoulomb modelis described by gs and gt dening the shear plastic owand tensile plastic ow respectively gs Follows the non-associated ow rule while gt follows the associated ow rule

gs σ1 minus σ3Nψ

gt minusσ3 (6)

where ψ is the dilatancy angle and Nψ 1 + sinϕ1minus sinϕe function h is dened in the σ1 minus σ3 stress plane

h σ3 minus σt + ap σ1 minus σ

p( ) (7)

where ap 1 +N2

ϕ

radic+Nϕ and σp σtNϕ minus 2c

Nϕ

radic

e positive and negative zones indicated by the func-tion h are taken as the criteria for determining the tensile andshear plastic ow zones as shown in Figure 2

If the stress point falls into zone 1 then the shear yieldoccurs and the stress should be adjusted to the curvecorresponding to the yield function fs 0 according to theow rule determined by the shear potential function gs if thestress point falls into zone 2 then the tensile yield occursand the stress should be adjusted to the curve correspondingto the yield function ft 0 according to the ow rule de-termined by the tensile potential function gt

Composite lining is used for the tunnel support struc-ture C25 shotcrete is used for the primary support C30form concrete is used for the secondary lining and thesurrounding rocks are of red-bed soft rock See Table 2 forthe specic material parameters of the model

222 Dynamic Boundary Condition Viscous boundarywhich was rst proposed by Lysmer and Kuhlemeyer [32] is

Table 1 Management value of track geometric irregularity in theUK

Control conditions Wavelength (m) Versine (mm)① Ride performance 5000 1600

2000 9001000 500

② Additional dynamicload on the track 500 250

200 060100 030

③ Corrugated wear 050 010005 0005

240

210

180

150

120

90

60

30

0

ndash300 1 2

Time (s)3 4

Load

(kN

)

Figure 1 Time-history curve of train excitation load

Advances in Civil Engineering 3

adopted as the boundary condition of the model )iscondition is equivalent to the situation where independentdampers are separately set at the normal and tangentialpositions of the artificial boundary to absorb the scatteredwave by dissipating the energy of the vibration wave with thegenerated viscous damping force proportional to the movingspeed )is method is easy to be implemented in dynamictime-domain analysis and its validity has been widelyproved in finite element and finite difference methods )enormal and tangential forces provided by the dampers are

tn minusρCpvn

ts minusρCsvs1113896 (8)

where ρ is the density of rock mass media Cp and Cs are thewave velocities of longitudinal and transverse waves in rockmass media respectively and vn and vs are the normal andtangential components of the particle vibration velocity dueto fluctuations on the artificial boundary

In the calculation model a viscous boundary is appliedto the bottom and the left and right boundaries and normalconstraint is imposed on the front and rear boundaries )esoil mass above the model is the free surface

)e depth of the tunnel is taken as 5597m and theheight and width of the model are 100m With reference tothe wheel-rail excitation force function provided in Formula(1) the duration of train excitation load applied on themodel is 35 s In the calculation of the train excitation loadapplied on the model the concentrated force is convertedinto surface force loaded on the subgrade to simulate thetrain passing through the tunnel See Figure 3 for the cal-culation model

23Analysis ofHeavy-HaulRailwayTunnelwithVoidedBase)e structure of the tunnel base is in an initial state at thebeginning With the increased times of train load thesurrounding rock of the base is gradually voided )is mayoccur symmetrically from the position near the track (voidcondition 1) or from the center of the inverted arch (voidcondition 2) With the occurrence of the void the basestructure of the tunnel gradually cracks from elastic de-formation With the occurrence of cracks combined withthe weakened constraint on the structure caused by voidthe base structure will be subjected to greater deformation

Table 2 Physical mechanics properties of model

Material type Density (kgm3) Elastic modulus (GPa) Poissonrsquos ratio Friction Angle (deg) Cohesive force (MPa)Subgrade 2500 6 03 39 1Base slab 2500 36 03 36 085Primary support 2500 255 02 587 13Secondary lining 2500 28 02 60 16Red-bed soft rock 2150 36 03 33 045

Zone 1 Zone 2

ndash

ndash

ndash

+

+

+

h=

0

fs = 0ft = 0

σ1

σ3

Figure 2 Schematic diagram for tensile and shear plastic flow zones in the Mohrndashcoulomb model

1

Block group

2

3

4

5

6

7

8

9

Figure 3 Numerical calculation model


under train load and such deformation will accelerate theformation of void in turn (void condition 3) Due to con-tinuous development of void the support condition of thebase structure keeps changing and the number of crackskeeps increasing Such structural damage and void of basedevelop with and promote each other finally resulting infailure of base structure See Figure 4 for the evolutionconcept

In order to analyze the influence of the base void(Figure 5) on the dynamic response of the base structure thecalculation model compares and analyzes 8 void cases asshown in Table 3 )e dynamic response of the structurecaused by the train load is mainly at the inverted arch)erefore monitoring points are arranged near the invertedarch the arch springing and the void area of the tunnel asshown in Figure 6

24 Prediction Methods for Fatigue Life of ConcreteFatigue refers to the structural change process in which thelocal permanent and progressive development of the ma-terial occurs under the condition that alternating stress andstrain are applied on a certain point or at some points of thematerial Fatigue failure is the result that the fatigue damageis accumulated to a critical value It may cause the accu-mulation of cracks or the bending fracture of the materialafter certain times of alternating loads are applied [33] )efatigue life refers to the cycle times that the material isapplied with stress or strain before the fatigue failure ap-pears or the time between the beginning of the loading andthe occurrence of the fracture

Under the action of the cyclic load the material damageof the base structure of heavy-haul railway tunnels is moreand more serious the material properties deteriorategradually with the increase of cycle times and thus thebearing capacity of the components decreases gradually andfinally damage appears )e cumulative fatigue damagemodel of concrete can be established to predict the servicelife)e prediction can provide a reference for the evaluationof the residual service life of the base structure of the existingheavy-haul railway tunnel and the maintenance and re-inforcement of the damaged structure )e existing methodsfor fatigue life prediction of concrete mainly includenominal stress method energy method local stress andstrain method and stress field intensity method )e mostwidely used method is the nominal stress method which isalso known as S-N curve method )is method is mainlyapplicable to the calculation of high-cycle fatigue life )especific analytical steps of the nominal stress method are asfollows

(1) Determine the load spectrum and apply it to thestructure to obtain the nominal stress spectrum ofthe structure

(2) Determine the fatigue dangerous part of the struc-ture and its nominal stress spectrum

(3) Establish the S-N curve equation of the structure orcomponent

(4) Determine the fatigue life of the structure

3 Results and Discussion

31 Effects on Displacement and Velocity of Base under VoidCondition )e displacement and velocity response oftunnel structure under train load is mainly vertical and theresponse of the inverted arch is the most intense It is foundthat the displacement and velocity of the monitoring point Ain the center of the inverted arch are the largest throughmonitoring the vertical displacement and velocity of eachmonitoring point)e point A is taken as the research objectthe vertical displacement and velocity time-history curvesunder different void dimensions are extracted as shown inFigures 7 and 8

)rough the analysis on vertical displacement and ve-locity time-history curves for the inverted arch of the tunnelthe findings are as follows

)e effects of loading unloading and vibration gener-ated by train passing cause the fluctuation of vertical dis-placement response at lining monitoring points )e peaksand valleys of the time-history curve for each responsecorrespond to the rolling through effect of train wheels

)e vertical displacement and velocity curves of theinverted arch vary obviously with the increase of void widthwhich shows that the void width of the base under the railhas a great effect on the displacement and velocity of theinverted arch However such variation is minor with theincrease of the void thickness which shows that the voidthickness of the base under the rail has little effect on thedisplacement of the inverted arch)e peak values of verticaldisplacement and velocity of the inverted arch in differentcases are shown in Table 4

)e curves for relationship between the peak values ofdisplacement and velocity of the inverted arch and the voidarea of the base are shown in Figures 9 and 10

)e peak value of vertical displacement of the invertedarch reaches 26mm and the vertical velocity reaches82 cms when the void width of base reaches 6m )evertical displacement is increased by about 9 times and thepeak value of velocity is increased by about 38 timescompared with the condition without void )us it can beseen that the vertical displacement and velocity of theinverted arch are obviously affected by the void of base

)e curve variation trends in Figures 9 and 10 also provethat the void depth has a greater effect on the displacementand velocity responses of the base structure than the voidthickness )e peak value variations for vertical displace-ment and velocity of inverted arch are around 1mm and04 cms respectively with the increase of void thickness Inorder to verify the rationality of this result the peak value ofvertical displacement and velocity of the inverted arch iscalculated in the case of void width of 12m since the voidwidth is almost impossible to be greater than 12m and thecalculation results are shown in Table 5

It can be seen from Table 5 that in the case of a voidwidth of 12m displacement and velocity remain almostunchanged )erefore the void thickness has little effect onthe peak value of displacement and velocity of the invertedarch when the void width is in the range between 5m and12m


32 Eects on Maximum Principal Stress of Base under VoidCondition e curves for maximum principal stress at theinverted arch in dierent cases are shown in Figure 11

As shown in Figure 11 the response curve of maximumprincipal stress of the inverted arch varies greatly with the

Cracks in the structure Cracks in the structure

Crack

Crack

CrackVoid

Void Void

Crack

Structure

Initial state

Void state 1

Void state 3

Void state 2

Basement rock

Interaction ofcumulative damage

and void


and void

Figure 4 Concept diagram for formation and evolution of base void

Subgrade

Invert filling

Secondary lining

Void areaBasement rock

Void width Void thickness

Basement rock

Figure 5 Schematic diagram for under-track base void of the tunnel

Table 3 Description of calculation cases of void

Calculationcases

Voidconditions

Void dimensionsVoid width

(m)Void thickness

(m)Case 1 Base without void mdash mdashCase 2 Base with void 2 10Case 3 Base with void 3 10Case 4 Base with void 4 10Case 5 Base with void 5 10Case 6 Base with void 6 10Case 7 Base with void 4 20Case 8 Base with void 4 30

E2

D2 C2 B2 A B1 C1 D1

E1

A monitoring point near the invertE1 E2 monitoring points near the arch foot

B1 B2 C1 C2 D1 D2 monitoring points near void zone

Figure 6 Schematic diagram for arrangement of monitoringpoints of structure


increase of void width which shows that the void width ofthe base under the rail has a great effect on the principalstress of the inverted arch However the curve varies slightlywith the increase of the void thickness which shows that thevoid thickness of the base under the rail has little effect onthe principal stress of the inverted arch

)e peak values of the maximum principal stress of theinverted arch in different cases are shown in Table 6 )evariation of the peak value corresponding to the expansionof the void area of base is shown in Figure 12

It can be seen from Table 6 and Figure 12 that themaximum principal stress of the inverted arch is affected

5

0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash3000 05 10 15 20 25 30 35

Time (s)

Vert

ical

disp

lace

men

t (m

m)

Without void2m3m

4m5m6m

(a)

00 05 10 15 20 25 30 35

0ndash2ndash4ndash6ndash8

ndash10ndash12ndash14Ve

rtic

al d

ispla

cem

ent (

mm

)

Time (s)

10cm20cm30cm

(b)

Figure 7 Time-history curves for vertical displacement of the inverted arch (a) Vertical displacement of the inverted arch under differentvoid widths (with a void thickness of 10 cm) (b) Vertical displacement of the inverted arch under different void thicknesses (with a voidwidth of 4m)

00 05 10 15 20 25 30 35

86420

ndash2ndash4ndash6ndash8

ndash10

Time (s)

Vert

ical

spee

d (c

ms

)

Without void2m3m

4m5m6m

(a)

00 05 10 15 20 25 30 35

43210


Time (s)

Vert

ical

spee

d (c

ms

)

10cm20cm30cm

(b)

Figure 8 Time-history curves for vertical velocity of the inverted arch under different void cases (a) Vertical velocity of the inverted archunder different void widths (with a void thickness of 10 cm) (b) Vertical velocity of the inverted arch under different void thicknesses (witha void width of 4m)

Table 4 Peak values of vertical displacement and velocity of the inverted arch

Case Void conditionsVoid dimensions

Peak value of vibration displacement (cm) Peak value of velocity (cms)Void width (m) Void thickness (cm)

1 Without void mdash mdash minus026 minus1672 Void 2 10 minus061 minus213 Void 3 10 minus084 minus254 Void 4 10 minus104 minus315 Void 5 10 minus127 minus446 Void 6 10 minus26 minus827 Void 4 20 minus114 minus358 Void 4 30 minus12 minus39


ndash30

ndash25

ndash20

ndash15

ndash10

ndash05

001 2 3 4 5 6 7

Void width (m)

Max

imum

ver

tical

disp

lace

men

t (cm

)

(a)

10Void thickness (cm)

Max

imum

ver

tical

disp

lace

men

t (cm

)

15 20 25 30

ndash25

ndash20

ndash15

ndash10

ndash05

00

(b)

Figure 9 Peak value variation curve for vertical displacement of the inverted arch (a) void thickness is 10 cm (b) void width is 4m

10

ndash10

ndash8

ndash6

ndash4

ndash2

02 3 4 5 6 7

Void width (m)

Max

imum

ver

tical

spee

d (c

ms

)

(a)

ndash40

ndash38

ndash36

ndash34

ndash32

ndash3010 15 20 25 30

Void thickness (cm)

Max

imum

ver

tical

spee

d (c

ms

)

(b)

Figure 10 Peak value variation curve for vertical velocity of the inverted arch (a) void thickness is 10 cm (b) void width is 4m

Table 5 Peak value of vertical displacement and velocity (in the case of void width of 12m)

Void conditionsVoid dimensions

Peak value of displacement (cm) Peak value of velocity (cms)Void width (m) Void thickness (cm)

Void 12 15 minus42 minus126Void 12 20 minus43 minus128Void 12 25 minus43 minus129

20

15

10

05

00

Time (s)

Max

imum

prin

cipa

l stre

ss (M

Pa)

00 05 10 15 20 25 30 35

Without void2m3m

4m5m6m

(a)

14

12

10

08

06

0400 05 10 15M

axim

um p

rinci

pal s

tress

(MPa

)

Time (s)20 25 30 35

10cm20cm30cm

(b)

Figure 11 Time-history curves for maximum principal stress at the inverted arch (a) Maximum principal stress of the inverted arch underdifferent void widths (with a void thickness of 10 cm) (b) Maximum principal stress of the inverted arch under different void thicknesses(with a void width of 4m)


obviously by the void of base under the rail and that themaximum principal stress of the inverted arch varies ob-viously with the increase of the void width)e peak value ofmaximum principal stress of the inverted arch reaches165MPa when the void width of the base reaches 6mincreased by 053MPa compared with the condition withoutvoid )us it can be seen that the maximum principal stressof the inverted arch is affected obviously by the void width ofthe base )e peak value variation for maximum principalstress of the inverted arch is around 002MPa with thegradual increase of void thickness Like the displacementand velocity the maximum principal stress of the invertedarch in the case of void width of 12m is also obtained inTable 7

Table 7 indicates that when the void width is 12m thepeak value of maximum principal stress of the inverted archis approximately 24MPa regardless of the void thickness)us it can be seen that void thickness has little effect on thepeak value of principal stress of the inverted arch when thevoid width is in the range between 5m and 12m

33 Analysis on Fatigue Life of Heavy-Haul Railway TunnelStructure under Train Vibration It can be concludedthrough the analysis in the previous sections that the heavy-haul railway tunnel structure has the most obvious dynamicresponse at the inverted arch Meanwhile based on theanalysis of the maximum principal stress the structure issubjected to tensile stress at the inverted arch only and it issubjected to compressive stress at other parts )e level ofcompressive stress is small for concrete and it is generallyconsidered that it will not cause fatigue failure )ereforethe part that ultimately controls the fatigue life of heavy-haulrailway tunnel structure under train load is the part wherethe inverted arch is subjected to tensile stress and themaximum stress of this part has been determined

)e most important step in the nominal stress method isto determine the fatigue life S-N curve of the structure or

component At present the fatigue performance data of allkinds of components are obtained by a large number of testsso that the S-N curve is determined Many scholars havestudied the fatigue performance of concrete material intension Tepfers et al [34] put forward an equation similar tothat of concrete compressive fatigue to describe concretetensile fatigue on the basis of fatigue tests such as axialtension and splitting as shown in the following formula

Stmax at minus bt 1minusRt( 1113857lgN (9)

where Stmax σtmaxft Rt σtminσtmax σtmax upper limitvalue of fatigue tensile stress σtmin lower limit value offatigue tensile stress ft static axial tensile strength ofconcrete at bt coefficients determined by the fatigue testwhich are generally taken as at 10 and bt 00685 and Nlimit loading times

)e above equation is more accurate in describing thefatigue life of concrete under axial tension But the base ofheavy-haul railway tunnels is a structure subjected tobending tension and its stress condition is more compli-cated than a structure subjected to axial tension Sidoroffet al [35] have proved that the above formula is also ap-plicable to the bending tensile fatigue S-N curve of concretebut the taken value of the coefficient in the fatigue test isdifferent )e bending tensile fatigue equation of concrete isobtained as follows [36 37]

Table 6 Peak values of maximum principal stress of the inverted arch (MPa)

Case Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8Peak values of maximum principal stress 112 120 133 141 158 165 143 144

1 2 3 4 5 6 7Void width (m)

Max

imum

prin

cipa

l stre

ss (M

Pa) 20

19181716151413121110

(a)

10 15 20 25 30Void ickness (cm)

Max

imum

prin

cipa

l stre

ss (M

Pa) 150

148

146

144

142

140

(b)

Figure 12 Peak value variation curve for maximum principal stress of the inverted arch (a) void thickness is 10 cm (b) void width is 4m

Table 7 Peak value of maximum principal stress (in the case ofvoid width of 12m)

Voidconditions

Void dimensionsVoidwidth(m)

Void thickness(cm)

Peak values ofmaximum

principal stress (MPa)Void 12 15 244Void 12 20 247Void 12 25 249


Srmax ar minus brlgN (10)

where Srmax σrmaxfr σrmax maximum tensile stress ofconcrete at lower edge under constant-amplitude repeatedload σtmin lower limit value of fatigue tensile stress frstatic bending tensile strength of concrete ar br coefficientsdetermined by the fatigue test which are generally taken asar 094ndash135 and br 0045ndash0118

)e time-history curves for the maximum principalstress of the inverted arch under different void conditionshave been obtained through the calculation and analysis forthe dynamic response of the structure under different voidconditions Table 8 shows the statistical results of themaximum and minimum values

)e maximum and minimum values of the maximumprincipal stress are respectively regarded as the maximumand the minimum values of the tensile stress of the structure)e bending tensile fatigue life of the concrete is calculatedwith the formula (10) where ar 094 and br 0049 Ra-phael et al [38] has proved that the bending tensile strengthof concrete is generally 35 higher than the straight tensilestrength based on many test results )erefore the fatiguelife of concrete is calculated by taking fr 22MPa andsetting Case 1 as the example

Srmax σrmaxft 11222

0509

lgN 094minus 0509

0049 880

(11)

)e load action times corresponding to the fatiguefailure

N 631 times 108 (12)

)us it can be seen from the above calculation that thefatigue life of the structure under the condition without voidis longer

Similarly the calculation results of the rest of the casesare statistically shown in Table 9 )e variation of fatigue lifein different cases is shown in Figure 13

It can be seen from Table 9 and Figure 13 that the voidwidth of the base has an obvious effect on the fatigue life ofheavy-haul railway tunnel structure and the fatigue life ofthe structure decreases sharply especially in the early stage ofvoid development )e variation of fatigue life tends to begentle when the void width reaches a certain value It can be

seen from Case 3 that the action times of fatigue load for thestructure is only 708times106 when the void reaches 3m If it isstill estimated as per that the wheel set load of theDatongndashQinhuangdao Railway is 1000 times per day theservice life of the tunnel is only 1940 years )e allowableultimate void width of base is 2m based on the operationrequirement of 100-year service life

)e void thickness of the base will not have an obviouseffect on the service life of the heavy-haul railway tunnelstructure with certain void width and the fatigue life of thestructure is maintained at a longer level

Table 8 Maximum principal stress of the inverted arch in different cases (MPa)

Case Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8

Maximum principal stress Max 112 12 133 141 158 165 143 144Min 0295 0314 0376 0415 0481 0508 0422 0430

Table 9 Statistical list of fatigue life in various cases

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8lgN 880 805 685 610 453 388 592 583N 631E8 112E8 708E6 125E6 339E4 759E3 832E5 676E5

1 2 3 4 5 6Case

Ntimes

108

8

7

6

5

4

3

2

1

0

(a)

Ntimes

108

4 6 8Case

00200018001600140012001000080006000400020000

(b)

Figure 13 Law curve for structure fatigue life varying with basevoid range


4 Conclusions

In this paper the dynamic response characteristics of theheavy-haul railway tunnel structure under repeated impactsare studied based on the three-dimensional numerical dy-namics coupling model of dynamic load-tunnel-surroundingrock and the fatigue life of the tunnel structure under dif-ferent void conditions is predicted based on the S-N curve ofconcrete )e main conclusions are as follows

(1) )e effects of loading unloading and vibrationgenerated by train passing cause the fluctuation ofvertical displacement response at lining monitoringpoints )e peaks and valleys of the time-historycurve for each response correspond to the rollingthrough effect of train wheels

(2) )e dynamic response of base with void is muchlarger than that without voidWhen the void width is6m and the void thickness is 10 cm the verticaldisplacement of the inverted arch is about 10 timesthat under the condition without void the peakvelocity is about 49 times that under the conditionwithout void and the maximum principal stress isincreased by about 473

(3) )e dynamic response of base is greatly affected bythe void width under the condition of base with voidWith the same void thickness when the void width is6m the peak values of displacement and velocity ofthe inverted arch are respectively about 426 and390 times the values when the void width is 2m

(4) )e void width of base has an obvious effect onthe fatigue life of the heavy-haul railway tunnelstructure )e ultimate void width is 2m accordingto the S-N curve of concrete under bending tensilecondition and based on the operation requirement of100-year service life

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there are no conflicts of interestsregarding the publication of this paper

Acknowledgments

)e authors are very grateful for the funding from theNational Key Research and Development Program of China(2016YFC0802202) and the General Program of NationalNatural Science Foundation of China (51678503)

References

[1] H H Jenkins J E Stephenson G A Clayton GWMorlandand D Lyon ldquo)e effect of track and vehicle parameters onwheelrail vertical dynamic forcesrdquo Railway EngineeringJournal vol 3 no 1 pp 2ndash16 1974

[2] G Lamaran and M Derdas ldquoEvaluation of dynamics load onrail track sleepers based on vehicle-track modeling andanalysisrdquo International Journal of Structural Stability vol 2no 3 2002

[3] K One and M Yamada ldquoAnalysis of railway track vibrationrdquoJournal of Sound and Vibration vol 130 no 2 pp 269ndash2971989

[4] J Lai S He J Qiu et al ldquoCharacteristics of seismic disastersand aseismic measures of tunnels in Wenchuan earthquakerdquoEnvironmental Earth Sciences vol 76 no 94 2017

[5] J Qiu X Wang and J Lai ldquoResponse characteristics andpreventions for seismic subsidence of loess in NorthwestChinardquo Natural Hazards vol 92 no 3 pp 1909ndash1935 2018

[6] A V Metrikine and A C W M Vrouwenvelder ldquoSurfaceground vibration due to moving train in a tunnel two-dimensional modelrdquo Journal of Sound and Vibrationvol 234 no 1 pp 43ndash66 2000

[7] T Balendra C G Koh and Y C Ho ldquoDynamic response ofbuildings due to trains in underground tunnelsrdquo EarthquakeEngineering and Structural Dynamics vol 20 no 3pp 275ndash291 1991

[8] J A Forrest and H E M Hunt ldquoA three-dimensional modelfor calculation of train-induced ground vibrationrdquo Journal ofSound and Vibration vol 294 no 1 pp 678ndash705 2006

[9] J A Forrest and H E M Hunt ldquoGround vibration generatedby trains in underground tunnelsrdquo Journal of Sound andVibration vol 294 no 4 pp 706ndash736 2006

[10] M F M Hussein and H E M Hunt ldquoA numerical model forcalculating vibration from a railway tunnel embedded ina full-spacerdquo Journal of Sound and Vibration vol 305 no 3pp 401ndash431 2007

[11] R )iede and H G Natke ldquo)e influence of thicknessvariation of subway walls on the vibration emission generatedby subway trafficrdquo in Proceedings of Soil Dynamics andEarthquake Engineering V International Conference SoilDynamics and Earthquake Engineering Karlsruhe GermanySeptember 1991

[12] F Guan and I D Moore ldquo)ree-dimensional dynamic re-sponse of twin cavities due to traveling loadsrdquo Journal ofEngineering Mechanics vol 120 no 3 pp 637ndash651 1994

[13] K H Chua K W Lo and T Balendra ldquoBuilding responsedue to subway train trafficrdquo Journal of Geotechnical Engi-neering vol 121 no 11 pp 747ndash754 1995

[14] L M Peng C H Shi J Huang and S L Liu ldquoStudy on thefatigue life of the tunnel bed structure under train loadsrdquoJournal of the China Railway Society no 1 pp 82ndash85 2007

[15] L M Peng L C Huang and S L Liu ldquoStudy on dynamicresponse of railway tunnel bedding structurerdquo Journal ofVibration and Shock vol 25 no 5 pp 175ndash179 2006

[16] D Clouteau M Arnst T M G Al-Hussaini et al ldquoFreefieldvibrations due to dynamic loading on a tunnel embedded ina stratifiedmediumrdquo Journal of Sound and Vibration vol 283no 1 pp 173ndash199 2005

[17] Z Zhang X Shi X Wang and H Li ldquoStability of NATMtunnel faces in soft surrounding rocksrdquo Computers andGeotechnics vol 96 pp 90ndash102 2018

[18] Z Zhang H Li H Yang and BWang ldquoFailuremodes and faceinstability of shallow tunnels under soft groundsrdquo InternationalJournal of Damage Mechanics article 105678951877313 2018

[19] Z P Bazant and H O Byung ldquoStrain-rate effect in rapidtriaxial loading of concreterdquo ACI Materials Journal vol 180pp 764ndash782 1982


[20] J Mazars ldquoDescription of micro- and macroscale damage ofconcrete structuresrdquo Engineering Fracture Mechanics vol 25no 5-6 pp 729ndash737 1985

[21] M N Fardis and E S Chen ldquoA cyclic multiaxial model forconcreterdquo Computational Mechanics vol 1 no 4 pp 301ndash315 1986

[22] T M Abu-Lebdeh ldquoPlasticity-damage model for concreteunder cyclic multiaxial loadingrdquo Journal of EngineeringMechanics ASCE vol 119 no 7 pp 1465ndash1485 1993

[23] J Y Wu J L Li and F Rui ldquoAn energy release rate-basedplastic-damage model for concreterdquo International Journal ofSolids and Structures vol 43 no 3 pp 583ndash612 2006

[24] L M Peng N Liu and C H Shi ldquoExperimental study ondynamic amplitudes for cumulate damage characteristics oftunnel invert concreterdquo Journal of Railway Science and En-gineering vol 13 no 6 pp 1091ndash1099 2016

[25] S M Wang X B Li F Q Gong and J J Zhu ldquoExperimentalstudy on mechanical properties of different ages concreteunder static and dynamic loadrdquo Engineering Mechanicsvol 30 no 2 pp 143ndash149 2013

[26] Q Yan H Chen W Chen et al ldquoDynamic characteristic andfatigue accumulative damage of a cross shield tunnel structureunder vibration loadrdquo Shock and Vibration vol 2018pp 1ndash14 2004

[27] J Qiu X Wang S He et al ldquo)e catastrophic landside inMaoxian county Sichuan SW China on June 24 2017rdquoNatural Hazards vol 89 no 3 pp 1485ndash1493 2017

[28] J Qiu Y Xie H Fan Z Wang and Y Zhang ldquoCentrifugemodelling of twin-tunnelling induced ground movements inloess stratardquo Arabian Journal of Geosciences vol 10 no 493pp 1ndash14 2017

[29] T M Dawn and C G Stanworth ldquoGround vibrations frompassing trainsrdquo Journal of Sound and Vibration vol 66 no 3pp 355ndash362 1979

[30] C Pan and G N Pande ldquoPreliminary deterministic finiteelement study on a tunnel driven in loess subjected to trainloadingrdquo China Civil Engineering Journal vol 17 no 4pp 19ndash28 1984

[31] B Liang e Dynamic Characteristics of High Speed RailwaySubgrade and Geosyntheticsrsquo Applications PhD DissertationSouthwest Jiaotong University Chengdu China 1998

[32] J Lysmer and R L Kuhlemeyer ldquoFinite dynamic model forinfinite mediardquo Journal of the Engineering Mechanics ASCEvol 95 no 4 pp 859ndash877 1969

[33] J M Hanson C A Ballinger and D Linger ldquoConsiderationsfor design of concrete structures subjected to fatigue loadingrdquoACI Structural Journal vol 71 1974

[34] R Tepfers ldquoTensile fatigue strength of plain concreterdquo Journalof the American Concrete Institute vol 76 no 8 pp 919ndash9331979

[35] F Sidoroff Description of Anisotropic Damage Application toElasticity pp 237ndash244 Springer Heidelberg Germany 1981

[36] J W Murdock and C E Kesler ldquoEffect of range of stress onfatigue strength of plain concrete beamsrdquo Journal of AmericanConcrete Institute vol 55 no 2 pp 221ndash232 1959

[37] B H Oh ldquoFatigue analysis of plain concrete in flexurerdquoJournal of Structural Engineering vol 112 no 2 pp 273ndash2881986

[38] J M Raphael ldquoTensile strength of concreterdquo Journal of theAmerican Concrete Institute vol 81 no 2 pp 158ndash165 1984


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Shock and Vibration


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Submit your manuscripts atwwwhindawicom

range and mainly affects the dynamic response of the vehiclebody in a high-frequency range

)e dynamic response of tunnel base structure undersuch cyclic dynamic load is also a key to the scholarsrsquo study[4 5] In the early days when the computational capacity waslimited a reasonable analytical model was the main researchapproach Metrikine et al [6] established a typical planeanalytical model and analyzed the response of the structureunder constant point load covariant point load and randompoint load by simulating the tunnel in viscoelastic soil withthe Euler beam Balendra et al [7] established a semi-analytical semiplane strain model and adopted the sub-structure method to reduce the complexity of the problemForrest et al [8 9] established a pipe-in-pine model andanalyzed the vibration effect during train passing by takingthe outer elastomeric pipe with infinite radius as the stratumthe thin inner pipe as the lining and the floating plate insidethe thin shell as the filling (the plate and the shell areconnected by a spring) )e model was later improved byHussein et al [10] With the improvement of computerperformance and computational capacity more and morescholars have proposed numerical models suitable for trainvibration )e finite element method was widely used tostudy the tunnel vibration response [11ndash13] Because bothfinite element and boundary element have their own ad-vantages in dealing with lining structure and infinite soilPeng et al [14 15] carried out numerical simulation for theunderlying concrete structure of tunnel under the impactload of train vibration with the nonlinear dynamic finiteelement theory and predicted the service life of the un-derlying structure by using the Tepfers concrete single-logarithmic fatigue equation A finite element-boundaryelement coupling model was established to analyze theimpact of train vibration on adjacent buildings [16ndash18]

Under long-term vibration loads of heavy-haul trainsthe tunnel base will inevitably be subjected to cumulativefatigue damage or even failure and void of surrounding rockDue to the large discreteness of concrete and many otherinfluence factors there is no effective method to predict thefatigue life of concrete and existing studies usually predictthe fatigue life of concrete with the cumulative damagetheory Bazant et al [19] and Mazars et al [20] establishedthe early scalar fatigue damage model for concrete withtheories related to fracture mechanics Fardis et al [21]further optimized the model and adapted it to concretestrain softening by modifying the flexibility matrix equationof bounding surface Abu-Lebdeh et al [22] proposeda constitutive model considering the plasticity of concreteand the damage boundary )e model simulates the state ofstiffness degradation and softening of concrete under thecyclic action multiaxial stress Wu et al [23] introducedtensile and shear damage variables to describe concretestiffness degradation at a macroscale and derived themathematical expressions of plastic strain evolution law andHelmholtz free energy Based on the theoretical system ofHelmholtz free energy Peng et al [24] conducted an ex-perimental study for concrete of inverted arch under dif-ferent vibration amplitudes in a coupling environment ofstatic and cyclic loads and figured out the static load level

that can cause concrete damage and failure as well as thequantitative expression of damage evolution trends at dif-ferent stages Wang et al [25] conducted the impact com-pression test for many times on 5 types of support concreteof different ages with SHPB test equipment )e peakdamage at stable age indicates that the statistical damagemodel of the Weibull distribution can well reflect thedamage characteristics of the material

During the service of heavy-haul railway tunnels thefatigue damage and void of surrounding rock of the basestructure are not independent of each other but are con-comitant and there are incentive and superposition effectsbetween the two Structural damage [26] will lead to decreaseof stiffness and greater deformation under stress which willincur greater load and compression on the surrounding rockof the base as well as continuous accumulation of plasticdeformation [27 28] and aggravated void of the surroundingrock thus changing the support conditions of the structureand causing greater damage to the structure subjected toupper dynamic loads )is paper studies the characteristicsof the full-cycle temporal-spatial response of the basestructure of heavy-haul railway tunnels under repeatedimpact by establishing a three-dimensional numerical dy-namics coupling model of dynamic load-tunnel-surrounding rock and predicts the fatigue life of the tun-nel structure under different void conditions by carrying outfailure evaluation for the base structure under damage-voidexcitation based on the S-N curve of concrete It is found thatduring the service of heavy-haul railway tunnels the pres-ence of voided base has a significant influence on the dis-placement and velocity response of the lining structure aswell as the distribution and magnitude of the principal stresson the lining In addition the void width of the base hasa great influence on various indicators of the lining struc-ture while the influence of void thickness is minor

2 Numerical Model of Heavy-HaulRailway Tunnels

21Determination of Train Excitation Load )e vibration ofa train is under the joint influence of the train and the trackIt is mainly caused by the abrasion of wheels the state of railjoints and geometric irregularity On the vertical longitu-dinal plane of the wheel-rail system there are a wide varietyof vibration sources local unevenness of the existing railsurface such as rail head crushing or rail surface peeling andperiodic system irregularities such as waveform trackcorrugation and eccentric wheels Particularly abrupt ir-regularities are more common at such places as depressedjoints joints with height difference large rail joints andwelds In addition defects such as loose sleepers or hardenedballast bed may occur to the under-track foundationresulting in dynamic irregularity of uneven elasticity andthereby creating an excitation load

Considering the influence factors of train vibration andcombined with the management value of track geometricirregularity in the UK the vertical excitation force of thetrain can be simulated by an excitation function [1 29] )isfunction takes into account the axle load of vehicle body





pi M0aiω2i (i 1 2 3) (2)


ωi 2π]Li (i 1 2 3) (3)







radic

ft σ3 minus σt

(4)



(5)



gs σ1 minus σ3Nψ

gt minusσ3 (6)



p( ) (7)

where ap 1 +N2

ϕ


Nϕ

radic







2000 9001000 500


200 060100 030


240

210

180

150

120

90

60

30

0

ndash300 1 2

Time (s)3 4

Load

(kN

)




tn minusρCpvn








Zone 1 Zone 2

ndash

ndash

ndash

+

+

+

h=

0

fs = 0ft = 0

σ1

σ3


1

Block group

2

3

4

5

6

7

8

9
























Crack

Crack

CrackVoid

Void Void

Crack

Structure

Initial state

Void state 1

Void state 3

Void state 2

Basement rock


and void


and void


Subgrade

Invert filling

Secondary lining



Basement rock



Calculationcases

Voidconditions


(m)Void thickness


E2

D2 C2 B2 A B1 C1 D1

E1








5

0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash3000 05 10 15 20 25 30 35

Time (s)

Vert

ical

disp

lace

men

t (m

m)

Without void2m3m

4m5m6m

(a)

00 05 10 15 20 25 30 35



rtic

al d

ispla

cem

ent (

mm

)

Time (s)

10cm20cm30cm

(b)


00 05 10 15 20 25 30 35

86420


ndash10

Time (s)

Vert

ical

spee

d (c

ms

)

Without void2m3m

4m5m6m

(a)

00 05 10 15 20 25 30 35

43210


Time (s)

Vert

ical

spee

d (c

ms

)

10cm20cm30cm

(b)







ndash30

ndash25

ndash20

ndash15

ndash10

ndash05

001 2 3 4 5 6 7

Void width (m)

Max

imum

ver

tical

disp

lace

men

t (cm

)

(a)


Max

imum

ver

tical

disp

lace

men

t (cm

)

15 20 25 30

ndash25

ndash20

ndash15

ndash10

ndash05

00

(b)


10

ndash10

ndash8

ndash6

ndash4

ndash2

02 3 4 5 6 7

Void width (m)

Max

imum

ver

tical

spee

d (c

ms

)

(a)

ndash40

ndash38

ndash36

ndash34

ndash32

ndash3010 15 20 25 30

Void thickness (cm)

Max

imum

ver

tical

spee

d (c

ms

)

(b)






20

15

10

05

00

Time (s)

Max

imum

prin

cipa

l stre

ss (M

Pa)

00 05 10 15 20 25 30 35

Without void2m3m

4m5m6m

(a)

14

12

10

08

06

0400 05 10 15M

axim

um p

rinci

pal s

tress

(MPa

)

Time (s)20 25 30 35

10cm20cm30cm

(b)













1 2 3 4 5 6 7Void width (m)

Max

imum

prin

cipa

l stre

ss (M

Pa) 20

19181716151413121110

(a)


Max

imum

prin

cipa

l stre

ss (M

Pa) 150

148

146

144

142

140

(b)



Voidconditions


Void thickness(cm)









0509

lgN 094minus 0509

0049 880

(11)


N 631 times 108 (12)











1 2 3 4 5 6Case

Ntimes

108

8

7

6

5

4

3

2

1

0

(a)

Ntimes

108

4 6 8Case

00200018001600140012001000080006000400020000

(b)



4 Conclusions






Data Availability




Acknowledgments


References











































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





pi M0aiω2i (i 1 2 3) (2)


ωi 2π]Li (i 1 2 3) (3)







radic

ft σ3 minus σt

(4)



(5)



gs σ1 minus σ3Nψ

gt minusσ3 (6)



p( ) (7)

where ap 1 +N2

ϕ


Nϕ

radic







2000 9001000 500


200 060100 030


240

210

180

150

120

90

60

30

0

ndash300 1 2

Time (s)3 4

Load

(kN

)




tn minusρCpvn








Zone 1 Zone 2

ndash

ndash

ndash

+

+

+

h=

0

fs = 0ft = 0

σ1

σ3


1

Block group

2

3

4

5

6

7

8

9
























Crack

Crack

CrackVoid

Void Void

Crack

Structure

Initial state

Void state 1

Void state 3

Void state 2

Basement rock


and void


and void


Subgrade

Invert filling

Secondary lining



Basement rock



Calculationcases

Voidconditions


(m)Void thickness


E2

D2 C2 B2 A B1 C1 D1

E1








5

0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash3000 05 10 15 20 25 30 35

Time (s)

Vert

ical

disp

lace

men

t (m

m)

Without void2m3m

4m5m6m

(a)

00 05 10 15 20 25 30 35



rtic

al d

ispla

cem

ent (

mm

)

Time (s)

10cm20cm30cm

(b)


00 05 10 15 20 25 30 35

86420


ndash10

Time (s)

Vert

ical

spee

d (c

ms

)

Without void2m3m

4m5m6m

(a)

00 05 10 15 20 25 30 35

43210


Time (s)

Vert

ical

spee

d (c

ms

)

10cm20cm30cm

(b)







ndash30

ndash25

ndash20

ndash15

ndash10

ndash05

001 2 3 4 5 6 7

Void width (m)

Max

imum

ver

tical

disp

lace

men

t (cm

)

(a)


Max

imum

ver

tical

disp

lace

men

t (cm

)

15 20 25 30

ndash25

ndash20

ndash15

ndash10

ndash05

00

(b)


10

ndash10

ndash8

ndash6

ndash4

ndash2

02 3 4 5 6 7

Void width (m)

Max

imum

ver

tical

spee

d (c

ms

)

(a)

ndash40

ndash38

ndash36

ndash34

ndash32

ndash3010 15 20 25 30

Void thickness (cm)

Max

imum

ver

tical

spee

d (c

ms

)

(b)






20

15

10

05

00

Time (s)

Max

imum

prin

cipa

l stre

ss (M

Pa)

00 05 10 15 20 25 30 35

Without void2m3m

4m5m6m

(a)

14

12

10

08

06

0400 05 10 15M

axim

um p

rinci

pal s

tress

(MPa

)

Time (s)20 25 30 35

10cm20cm30cm

(b)













1 2 3 4 5 6 7Void width (m)

Max

imum

prin

cipa

l stre

ss (M

Pa) 20

19181716151413121110

(a)


Max

imum

prin

cipa

l stre

ss (M

Pa) 150

148

146

144

142

140

(b)



Voidconditions


Void thickness(cm)









0509

lgN 094minus 0509

0049 880

(11)


N 631 times 108 (12)











1 2 3 4 5 6Case

Ntimes

108

8

7

6

5

4

3

2

1

0

(a)

Ntimes

108

4 6 8Case

00200018001600140012001000080006000400020000

(b)



4 Conclusions






Data Availability




Acknowledgments


References











































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



tn minusρCpvn








Zone 1 Zone 2

ndash

ndash

ndash

+

+

+

h=

0

fs = 0ft = 0

σ1

σ3


1

Block group

2

3

4

5

6

7

8

9
























Crack

Crack

CrackVoid

Void Void

Crack

Structure

Initial state

Void state 1

Void state 3

Void state 2

Basement rock


and void


and void


Subgrade

Invert filling

Secondary lining



Basement rock



Calculationcases

Voidconditions


(m)Void thickness


E2

D2 C2 B2 A B1 C1 D1

E1








5

0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash3000 05 10 15 20 25 30 35

Time (s)

Vert

ical

disp

lace

men

t (m

m)

Without void2m3m

4m5m6m

(a)

00 05 10 15 20 25 30 35



rtic

al d

ispla

cem

ent (

mm

)

Time (s)

10cm20cm30cm

(b)


00 05 10 15 20 25 30 35

86420


ndash10

Time (s)

Vert

ical

spee

d (c

ms

)

Without void2m3m

4m5m6m

(a)

00 05 10 15 20 25 30 35

43210


Time (s)

Vert

ical

spee

d (c

ms

)

10cm20cm30cm

(b)







ndash30

ndash25

ndash20

ndash15

ndash10

ndash05

001 2 3 4 5 6 7

Void width (m)

Max

imum

ver

tical

disp

lace

men

t (cm

)

(a)


Max

imum

ver

tical

disp

lace

men

t (cm

)

15 20 25 30

ndash25

ndash20

ndash15

ndash10

ndash05

00

(b)


10

ndash10

ndash8

ndash6

ndash4

ndash2

02 3 4 5 6 7

Void width (m)

Max

imum

ver

tical

spee

d (c

ms

)

(a)

ndash40

ndash38

ndash36

ndash34

ndash32

ndash3010 15 20 25 30

Void thickness (cm)

Max

imum

ver

tical

spee

d (c

ms

)

(b)






20

15

10

05

00

Time (s)

Max

imum

prin

cipa

l stre

ss (M

Pa)

00 05 10 15 20 25 30 35

Without void2m3m

4m5m6m

(a)

14

12

10

08

06

0400 05 10 15M

axim

um p

rinci

pal s

tress

(MPa

)

Time (s)20 25 30 35

10cm20cm30cm

(b)













1 2 3 4 5 6 7Void width (m)

Max

imum

prin

cipa

l stre

ss (M

Pa) 20

19181716151413121110

(a)


Max

imum

prin

cipa

l stre

ss (M

Pa) 150

148

146

144

142

140

(b)



Voidconditions


Void thickness(cm)









0509

lgN 094minus 0509

0049 880

(11)


N 631 times 108 (12)











1 2 3 4 5 6Case

Ntimes

108

8

7

6

5

4

3

2

1

0

(a)

Ntimes

108

4 6 8Case

00200018001600140012001000080006000400020000

(b)



4 Conclusions






Data Availability




Acknowledgments


References











































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia























Crack

Crack

CrackVoid

Void Void

Crack

Structure

Initial state

Void state 1

Void state 3

Void state 2

Basement rock


and void


and void


Subgrade

Invert filling

Secondary lining



Basement rock



Calculationcases

Voidconditions


(m)Void thickness


E2

D2 C2 B2 A B1 C1 D1

E1








5

0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash3000 05 10 15 20 25 30 35

Time (s)

Vert

ical

disp

lace

men

t (m

m)

Without void2m3m

4m5m6m

(a)

00 05 10 15 20 25 30 35



rtic

al d

ispla

cem

ent (

mm

)

Time (s)

10cm20cm30cm

(b)


00 05 10 15 20 25 30 35

86420


ndash10

Time (s)

Vert

ical

spee

d (c

ms

)

Without void2m3m

4m5m6m

(a)

00 05 10 15 20 25 30 35

43210


Time (s)

Vert

ical

spee

d (c

ms

)

10cm20cm30cm

(b)







ndash30

ndash25

ndash20

ndash15

ndash10

ndash05

001 2 3 4 5 6 7

Void width (m)

Max

imum

ver

tical

disp

lace

men

t (cm

)

(a)


Max

imum

ver

tical

disp

lace

men

t (cm

)

15 20 25 30

ndash25

ndash20

ndash15

ndash10

ndash05

00

(b)


10

ndash10

ndash8

ndash6

ndash4

ndash2

02 3 4 5 6 7

Void width (m)

Max

imum

ver

tical

spee

d (c

ms

)

(a)

ndash40

ndash38

ndash36

ndash34

ndash32

ndash3010 15 20 25 30

Void thickness (cm)

Max

imum

ver

tical

spee

d (c

ms

)

(b)






20

15

10

05

00

Time (s)

Max

imum

prin

cipa

l stre

ss (M

Pa)

00 05 10 15 20 25 30 35

Without void2m3m

4m5m6m

(a)

14

12

10

08

06

0400 05 10 15M

axim

um p

rinci

pal s

tress

(MPa

)

Time (s)20 25 30 35

10cm20cm30cm

(b)













1 2 3 4 5 6 7Void width (m)

Max

imum

prin

cipa

l stre

ss (M

Pa) 20

19181716151413121110

(a)


Max

imum

prin

cipa

l stre

ss (M

Pa) 150

148

146

144

142

140

(b)



Voidconditions


Void thickness(cm)









0509

lgN 094minus 0509

0049 880

(11)


N 631 times 108 (12)











1 2 3 4 5 6Case

Ntimes

108

8

7

6

5

4

3

2

1

0

(a)

Ntimes

108

4 6 8Case

00200018001600140012001000080006000400020000

(b)



4 Conclusions






Data Availability




Acknowledgments


References











































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





Crack

Crack

CrackVoid

Void Void

Crack

Structure

Initial state

Void state 1

Void state 3

Void state 2

Basement rock


and void


and void


Subgrade

Invert filling

Secondary lining



Basement rock



Calculationcases

Voidconditions


(m)Void thickness


E2

D2 C2 B2 A B1 C1 D1

E1








5

0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash3000 05 10 15 20 25 30 35

Time (s)

Vert

ical

disp

lace

men

t (m

m)

Without void2m3m

4m5m6m

(a)

00 05 10 15 20 25 30 35



rtic

al d

ispla

cem

ent (

mm

)

Time (s)

10cm20cm30cm

(b)


00 05 10 15 20 25 30 35

86420


ndash10

Time (s)

Vert

ical

spee

d (c

ms

)

Without void2m3m

4m5m6m

(a)

00 05 10 15 20 25 30 35

43210


Time (s)

Vert

ical

spee

d (c

ms

)

10cm20cm30cm

(b)







ndash30

ndash25

ndash20

ndash15

ndash10

ndash05

001 2 3 4 5 6 7

Void width (m)

Max

imum

ver

tical

disp

lace

men

t (cm

)

(a)


Max

imum

ver

tical

disp

lace

men

t (cm

)

15 20 25 30

ndash25

ndash20

ndash15

ndash10

ndash05

00

(b)


10

ndash10

ndash8

ndash6

ndash4

ndash2

02 3 4 5 6 7

Void width (m)

Max

imum

ver

tical

spee

d (c

ms

)

(a)

ndash40

ndash38

ndash36

ndash34

ndash32

ndash3010 15 20 25 30

Void thickness (cm)

Max

imum

ver

tical

spee

d (c

ms

)

(b)






20

15

10

05

00

Time (s)

Max

imum

prin

cipa

l stre

ss (M

Pa)

00 05 10 15 20 25 30 35

Without void2m3m

4m5m6m

(a)

14

12

10

08

06

0400 05 10 15M

axim

um p

rinci

pal s

tress

(MPa

)

Time (s)20 25 30 35

10cm20cm30cm

(b)













1 2 3 4 5 6 7Void width (m)

Max

imum

prin

cipa

l stre

ss (M

Pa) 20

19181716151413121110

(a)


Max

imum

prin

cipa

l stre

ss (M

Pa) 150

148

146

144

142

140

(b)



Voidconditions


Void thickness(cm)









0509

lgN 094minus 0509

0049 880

(11)


N 631 times 108 (12)











1 2 3 4 5 6Case

Ntimes

108

8

7

6

5

4

3

2

1

0

(a)

Ntimes

108

4 6 8Case

00200018001600140012001000080006000400020000

(b)



4 Conclusions






Data Availability




Acknowledgments


References











































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





5

0

ndash5

ndash10

ndash15

ndash20

ndash25

ndash3000 05 10 15 20 25 30 35

Time (s)

Vert

ical

disp

lace

men

t (m

m)

Without void2m3m

4m5m6m

(a)

00 05 10 15 20 25 30 35



rtic

al d

ispla

cem

ent (

mm

)

Time (s)

10cm20cm30cm

(b)


00 05 10 15 20 25 30 35

86420


ndash10

Time (s)

Vert

ical

spee

d (c

ms

)

Without void2m3m

4m5m6m

(a)

00 05 10 15 20 25 30 35

43210


Time (s)

Vert

ical

spee

d (c

ms

)

10cm20cm30cm

(b)







ndash30

ndash25

ndash20

ndash15

ndash10

ndash05

001 2 3 4 5 6 7

Void width (m)

Max

imum

ver

tical

disp

lace

men

t (cm

)

(a)


Max

imum

ver

tical

disp

lace

men

t (cm

)

15 20 25 30

ndash25

ndash20

ndash15

ndash10

ndash05

00

(b)


10

ndash10

ndash8

ndash6

ndash4

ndash2

02 3 4 5 6 7

Void width (m)

Max

imum

ver

tical

spee

d (c

ms

)

(a)

ndash40

ndash38

ndash36

ndash34

ndash32

ndash3010 15 20 25 30

Void thickness (cm)

Max

imum

ver

tical

spee

d (c

ms

)

(b)






20

15

10

05

00

Time (s)

Max

imum

prin

cipa

l stre

ss (M

Pa)

00 05 10 15 20 25 30 35

Without void2m3m

4m5m6m

(a)

14

12

10

08

06

0400 05 10 15M

axim

um p

rinci

pal s

tress

(MPa

)

Time (s)20 25 30 35

10cm20cm30cm

(b)













1 2 3 4 5 6 7Void width (m)

Max

imum

prin

cipa

l stre

ss (M

Pa) 20

19181716151413121110

(a)


Max

imum

prin

cipa

l stre

ss (M

Pa) 150

148

146

144

142

140

(b)



Voidconditions


Void thickness(cm)









0509

lgN 094minus 0509

0049 880

(11)


N 631 times 108 (12)











1 2 3 4 5 6Case

Ntimes

108

8

7

6

5

4

3

2

1

0

(a)

Ntimes

108

4 6 8Case

00200018001600140012001000080006000400020000

(b)



4 Conclusions






Data Availability




Acknowledgments


References











































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


ndash30

ndash25

ndash20

ndash15

ndash10

ndash05

001 2 3 4 5 6 7

Void width (m)

Max

imum

ver

tical

disp

lace

men

t (cm

)

(a)


Max

imum

ver

tical

disp

lace

men

t (cm

)

15 20 25 30

ndash25

ndash20

ndash15

ndash10

ndash05

00

(b)


10

ndash10

ndash8

ndash6

ndash4

ndash2

02 3 4 5 6 7

Void width (m)

Max

imum

ver

tical

spee

d (c

ms

)

(a)

ndash40

ndash38

ndash36

ndash34

ndash32

ndash3010 15 20 25 30

Void thickness (cm)

Max

imum

ver

tical

spee

d (c

ms

)

(b)






20

15

10

05

00

Time (s)

Max

imum

prin

cipa

l stre

ss (M

Pa)

00 05 10 15 20 25 30 35

Without void2m3m

4m5m6m

(a)

14

12

10

08

06

0400 05 10 15M

axim

um p

rinci

pal s

tress

(MPa

)

Time (s)20 25 30 35

10cm20cm30cm

(b)













1 2 3 4 5 6 7Void width (m)

Max

imum

prin

cipa

l stre

ss (M

Pa) 20

19181716151413121110

(a)


Max

imum

prin

cipa

l stre

ss (M

Pa) 150

148

146

144

142

140

(b)



Voidconditions


Void thickness(cm)









0509

lgN 094minus 0509

0049 880

(11)


N 631 times 108 (12)











1 2 3 4 5 6Case

Ntimes

108

8

7

6

5

4

3

2

1

0

(a)

Ntimes

108

4 6 8Case

00200018001600140012001000080006000400020000

(b)



4 Conclusions






Data Availability




Acknowledgments


References











































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia












1 2 3 4 5 6 7Void width (m)

Max

imum

prin

cipa

l stre

ss (M

Pa) 20

19181716151413121110

(a)


Max

imum

prin

cipa

l stre

ss (M

Pa) 150

148

146

144

142

140

(b)



Voidconditions


Void thickness(cm)









0509

lgN 094minus 0509

0049 880

(11)


N 631 times 108 (12)











1 2 3 4 5 6Case

Ntimes

108

8

7

6

5

4

3

2

1

0

(a)

Ntimes

108

4 6 8Case

00200018001600140012001000080006000400020000

(b)



4 Conclusions






Data Availability




Acknowledgments


References











































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







0509

lgN 094minus 0509

0049 880

(11)


N 631 times 108 (12)











1 2 3 4 5 6Case

Ntimes

108

8

7

6

5

4

3

2

1

0

(a)

Ntimes

108

4 6 8Case

00200018001600140012001000080006000400020000

(b)



4 Conclusions






Data Availability




Acknowledgments


References











































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


4 Conclusions






Data Availability




Acknowledgments


References











































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia
























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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