effects of spatially varying seismic ground motions and ... · acceleration in the vertical...

Research ArticleEffects of Spatially Varying Seismic GroundMotions and IncidentAngles on Behavior of Long Tunnels

Yundong Zhou Yongxin Wu Ziheng Shangguan and Zhanbin Wang

Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering Hohai UniversityNanjing 210098 China

Correspondence should be addressed to Yongxin Wu yxwuhhu163com

Received 14 March 2018 Accepted 7 May 2018 Published 11 June 2018

Academic Editor Yongfeng Deng

Copyright copy 2018Yundong Zhou et al-is is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Seismic behavior of long circle tunnels is significantly influenced by the nature of input motion -is study based on the 3D finite-element method (FEM) evaluates the effects of spatially varying seismic ground motions and uniform input seismic ground motionsand their incident angles on the diameter strain rate and tensivecompressive principal stresses under different strata It is found that (1)the spatially varying seismic groundmotions induced larger diameter strain rate (radially deformation) than the uniform input seismicmotion (2) the spatially varying seismic ground motions had an asymmetric effect on the radial strain rate distributions and (3) therising incident angles changed the pure shear stress state into a complex stress state for tunnels under specified input motion

1 Introduction

Tunnels as an integral part of the infrastructure of modernsociety would suffer damages when subjected to dynamicloadings during earthquake activity for example in Kobeearthquake (1995) [1] Chi-Chi earthquake (1999) [2] Duzceearthquake (1999) [3] Mid-Niigata Prefecture earthquake(2004) [4] Wenchuan earthquake (2008) [5] and Tohokuearthquake (2011) [6] Nevertheless the number of largetunnels and underground structures had grown significantlyin recent decades-erefore seismic evaluation of tunnels inseismically active areas is critical during engineering design

Seismic behavior of tunnels has been widely studied bymany researchers [7ndash11] and these researches have con-centrated mostly on 2D analysis For the analysis of axial andbending deformations of tunnels it is most appropriate toutilize 3D models

However for the 3D analysis of seismic behavior oftunnel soil-tunnel analyses in the past were typically limitedto relatively small regions which made it difficult to fullyconsider the complex spatial features involved in such largestructures [12] Tunnels often had significant length andcould be built on different strata foundations which madethe seismic analysis of tunnels a complex problem and was

usually evaluated under idealized conditions by using nu-merical methods such as the finite-element method (FEM)

For the seismic analysis of complex stress distribution anddeformation of long-distance tunnels it is often more reliable toadopt three-dimensional (3D) methods With the rapid devel-opment of science and technology it is now possible to use high-performance computers to conduct large-scale 3D FEM seismicanalysis for tunnels-ere are three kinds of deformation such asaxial compressionextension longitudinal bending and ovallingracking occurring in tunnels during earthquake [10] Particu-larly the cross-sectional distortion of the tunnel can be related toseismic waves propagating along the tunnel

Yu et al [13 14] presented a multiscale 3D FEM analysisof long tunnels under seismic loads where the mechanicalcharacteristics of the tunnel segments and joints underartificial or recorded earthquake loads were evaluated indetail -e model in this paper would take into account notonly the motion distribution with time but also the spatialvariability (incoherency effect) the wave-passage effect andthe site-response effect

-is paper based on an earlier report by the authors [15]attempts to develop a new model for seismic analysis of longtunnels with multisupport excitations which properly ac-counts for the spatial variability-emain focus of this paper

HindawiAdvances in Civil EngineeringVolume 2018 Article ID 8195396 6 pageshttpsdoiorg10115520188195396

is to assess the influence ofmultisupport input earthquake wavesand uniform input earthquake waves as well as their incidentangles on the diameter strain rate and tensivecompressiveprincipal stresses under different strata A full-scale 3D finite-element model is built comprising geological data tunnel ge-ometry and so on to farthest simulate actual situation

2 Simulation of Seismic Ground Motions

21 Simulation of Uniform Seismic Ground Motion Considera zero mean Gaussian stationary seismic ground motionwith a target autospectral density S(ω) -e seismic groundmotions can be generated through the following expression

x(t) 21113944N

l1

S ωl( 1113857Δω1113969

cos ωlt + φl( 1113857 (1)

where N is the number of frequency intervals Δω ωuN isfrequency increment with ωu as the cutoff frequencyωl lΔω and the φlrsquos are statistically independent randomphase angles uniformly distributed between (0 2π] Equation (1)is valid if there is an upper cutoff frequency ωu above which thecontribution of the power spectral density (PSD) to the sim-ulations is negligible for practical purposes

22 Simulation of Spatially Varying Seismic GroundMotions-e variations in ground motion are caused by the followingfour sources (1) the ldquoincoherence effectrdquo (2) the ldquowave-passageeffectrdquo (3) the ldquosite-response effectrdquo and (4) the ldquoattenuationeffectrdquo -e spectral representation method is one of the mostwidely usedmethods in simulating the spatially varying seismicground motions

In practical application spatially correlated groundmotions can be considered as a one-dimensional n-variety(1D-nV) stochastic vector process X(t) with componentsxj(t) middot (j 1 2 n) Based on the spectral representationmethod the jth component of the ground motions can begenerated by [13]

xj(t) 2 1113944n

k11113944

N

l1Hjk ωl( 1113857

11138681113868111386811138681113868

11138681113868111386811138681113868Δω

radiccos ωltminusϕjk ωl( 1113857 + φkl1113960 1113961

ωl lΔω (l 0 1 K Nminus 1)

Δω ωu

N

(2)

whereωu is the upper cutoff frequency beyondwhich elementsof the power spectral can be assumed to be zero for eithermathematical or physical reasons and φjk are independentrandom phase angles uniformly distributed over (0 2π]|Hjk(ω)| and ϕjk(ω) are the modulus and phase parts ofHjk(ω) respectively which can be obtained by the root de-composition of power spectral density matrix as follows

S(ω) H(ω)HT(ω) (3)

where the superscript T denotes conjugate transpose

-e power spectral density matrix is given as

S(ω)

S11(ω) S12(ω) middot middot middot S1m(ω)


⋮ ⋮ ⋱ ⋮

Sm1(ω) Sm2(ω) middot middot middot Smm(ω)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4)

where Sjj(ω) is the autopower spectral density function andSjk(ω t) is the cross-power spectral density function whichcan be expressed as

Sjk(ω t) cjk(ω)11138681113868111386811138681113868

11138681113868111386811138681113868

Sjj(ω)Skk(ω)

1113969e

iθjk(ω) (5)

where |cjk(ω)| is the lagged coherence function representingldquoincoherencerdquo effect and θjk(ω) is composed of wave passage

However the ground motions simulated by the abovemethod are stationary while the actual seismic records arenonstationary -erefore to obtain nonstationary seismicground motion the way of multiplying an envelope functionis applied -e envelope function is as follows

f(t)

tt1( 11138572 tlt t1

1 t1 le tlt t2

eminusc tminust2( ) tge t2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(6)

where t1 t2 and c are three parameters describing the shapeof the envelope function In this study they are set to bet1 6 t2 10 and c 05 respectively

3 FE Model

-e tunnel model adopted in this paper was built withcircular appearance whose outside diameter is 10m insidediameter is 9m and length is 1000m It had a buried depthof 30m from the tunnel center to soil surface-e simulationsetup is shown in Figure 1 It consists of a 1000times 300times100mbox with the 1000m length circular tunnel -e space co-ordinates were built by taking the length direction as theZ-axis the width direction as the X-axis and the heightdirection as the Y-direction -e soil profile was modeled asfour layers of MohrndashCoulomb materials and tunnel as theelastic material All the property parameters used can befound in Tables 1 and 2 An infinite domain by using theartificial boundary was adopted [12] where both borders ofthe tunnel were fixed in the Z-direction

In this work the constitutive model of the tunnel is anelastic model which is given as follows

σ λI1δ + 2Gε (7)

in which σ is the stress tensor ε is the strain tensor I1 is thefirst strain invariant λ is the Lame constant and G is theshear modulus

-e MohrndashCoulomb yield criterion for soil is expressedas follows

m + 12

max1113874 σ1 minus σ21113868111386811138681113868

1113868111386811138681113868 + K σ1 + σ2( 1113857 σ1 minus σ31113868111386811138681113868

1113868111386811138681113868

+K σ1 + σ3( 1113857 σ2 minus σ31113868111386811138681113868

1113868111386811138681113868 + K σ2 + σ3( 11138571113875 Syc

(8)

2 Advances in Civil Engineering

where

m SycSyt

K mminus 1m + 1

(9)

and the parameters Syc and Syt are the yield stresses of thematerial in uniaxial compression and tension respectively

On the interface the deformation of the tunnel and soilis in concert and harmony

4 Computation and Analysis

In computation a regular 3D nite-element model was builtbased on geotechnical data tunnel geometry and so on etraveling waves velocity motivated from bedrock surface was500ms in the 3D spatially varying seismic ground motioneld e schematic diagram of dierent incident anglesused is shown in Figure 2 e simulated spatially varyingground motions are plotted in Figure 3

41 Eect on Radial Deformations e diameter strain rateΔd is dened as

Δd dprime minus dd

(10)

where d and dprime are the tunnel diameters before and afterdeformation

e positive rate value denotes that the tunnel crosssection undergoes tensile deformation in radial directionand the negative value corresponds to compressive de-formation In this section dmax is dened as themaximum ofΔd in the earthquake time process dmax in the verticaldirection is calculated from displacement time processes atthe vertex and nadir of tunnel in the Y-axis direction while theone in the lateral direction is obtained from processes at theright and left sides of the tunnel in the X-axis direction It wasfound from the simulation that lateralΔd and verticalΔdwereequal and opposite in the direction at any arbitrary timevidelicet the primary deformation of tunnel under seismicwaves was ovalling In addition the partial feature resultsunder uniform and spatially varying seismic ground motionsthat is the lateral dmax at Z 100m 300m 500m 700m and

X

Y

Z

Figure 1 Finite-element mesh model adopted in the 3D FE analyses for soil with long tunnels

Table 1 Properties of dierent soil layers

Layer number Youngrsquos modulus (MPa) Poissonrsquos ratio Density (kgm3) Cohesion (kPa) Internal friction angle (deg) ickness (m)1 3029 045 1760 108 142 2062 6135 036 1810 177 189 2393 6319 047 1870 22 229 2424 6251 036 1950 249 182 315

Table 2 Properties of the tunnel

Elastic modulus (kgm3) Poissonrsquos ratio Unit weight (kNm3)36e7 02 257 7 8

2 31 4 5 6

9 10 11 12

90deg 60deg45deg30deg

0degXYZ

Figure 2 e schematic diagram at dierent incident angles

Advances in Civil Engineering 3

900m with incident angles of 0deg 30deg 45deg 60deg and 90deg can befound in Tables 3 and 4

It could be observed that the lateral dmax increases withincreasing incident angles for both seismic input forms butthe maximum positions dier from each other For uniformseismic input simulations the lateral dmax position is at themiddle point (Z 500m) of the tunnel and the diameterstrain rates are symmetrically distributed on the left andright sides of the tunnel is can be attributed to thebalanced boundary conditions and uniform seismic wavesFor spatially varying seismic ground motions the lateraldmax is located between Z 700m and Z 900m and ex-hibits asymmetric features due to the nonuniform wavese table also implies that the lateral dmax for the wholetunnel under spatially varying seismic ground motions isgreater than that under uniform input ground motions

42 Eect on Stress Distributions e maximum principalshear (σtmax) and compressive (σcmax) stresses are commonlyused to analyze the eect of seismic input ground motionson tunnels e peak stresses under dierent incident anglesare illustrated in Table 5

It is indicated that the maximum tensile stress is identicalto the maximum compressive stress while the incident angleis 0deg for uniform seismic input tunnels is is owing to the

tunnel and soil prole placed in pure shear stress state Whenincident angles are equal to 30deg 45deg 60deg and 90deg the values ofmaximum tensile and compressive stresses are dierent be-cause of the complex shear and normal stress states induced byseismic load It can be thus concluded that the earthquakeacceleration in the vertical direction changes not only themaximum stress but also the stress characteristics In additionthe spatially varying seismic ground motion is another factorinuencing the stress state according to the dierent valuesbetween the maximum tensile stress and maximum com-pressive stress at an incident angle of 0deg under spatially varyingseismic ground motions Meanwhile the value of maximumtensilecompressive stresses decreases with the increasing in-cident angles e maximum tensile stress nephogram at anincident angle of 0deg for both uniformand spatially varying seismicground motions is respectively shown in Figures 4 and 5

0 4 8 12 16 20

04

02

00

ndash02

ndash04

t (s)

Acce

lera

tion

(g)

(a)

0 4 8 12 16 20

04

02

00

ndash02

ndash04

t (s)

Acc

eler

atio

n (m

s2 )

(b)

0 4 8 12 16 20

04

02

00

ndash02

ndash04

t (s)

Acc

eler

atio

n (g

)

(c)

Figure 3 e simulated spatially varying seismic ground motions (a) x 0m (b) x 500m and (c) x 1000m

Table 3 Lateral maximum diameter strain rate of tunnel underuniform seismic motion

Incidentangle Z 100m Z 300m Z 500m Z 700m Z 900m

0deg minus0036 minus0004 0 minus0004 minus003630deg 0087 0113 0118 0113 008745deg 0130 0158 0165 0158 013060deg 0171 0196 0204 0196 017190deg 0214 0226 0236 0226 0214

Table 4 Lateral maximum diameter strain rate of tunnel undermultisupport seismic motion


0deg 0022 minus0045 0033 0035 003930deg 0075 minus0125 minus0169 minus0169 minus009845deg 0111 minus0163 minus0225 minus0225 minus012360deg 0141 minus0193 minus0269 minus0269 minus014390deg 0170 minus0208 minus0293 minus0293 minus0155

Table 5 Maximum stresses of tunnel body at dierent incidentangles

Motion form Principalstress 0deg 30deg 45deg 60deg 90deg

Uniform σtmax 1160 1085 924 718 745σcmax minus1160 minus1058 minus1002 minus899 minus681

Multisupport σtmax 1774 1392 1129 894 666σcmax minus1884 minus1568 minus1337 minus1116 minus849


e red upper portion of the tunnel end (Z 1000m) inFigure 4 is the maximum tensile stress responding to the redlower portion near the tunnel end It is suggested that the spatiallyvarying seismic ground motion aects the stress distribution

5 Conclusions

Numerical simulation was an eective method for studyingthe tunnel responses under seismic event In this papera novel large-scale analytical method was established forestimating the seismic response of long tunnels within soilfoundations Based on the 3D FEM platform this methodcould oer a reliable way for investigating the nonlinear

seismic behavior of long tunnels Main ndings are sum-marized as follows

(1) e spatially varying seismic groundmotions inducedlarger diameter strain rates (radial deformation) thanthe uniform input seismic motion e maximumradial strain rate increased with the increasing in-cident angles for both seismic input forms

(2) For uniform seismic input simulations the maxi-mum radial strain rate occurred at the middle pointof the tunnel and the radial strain rates were sym-metrically distributed on the left and right sides ofthe tunnel Moreover the spatially varying seismic

X

Y

Z

ODB yizhibotanhuang31508328595673odb Abaqusstandard 611-1 Sun Sep 04 213851 GMT + 0800 2016 Step dynamicIncrement 125 step time = 1194Primary var S max principal

S max principal(Avg 75)

+1160e + 07+1064e + 07+9667e + 06+8699e + 06+7731e + 06+6763e + 06+5796e + 06+4828e + 06+3860e + 06+2892e + 06+1924e + 06+9560e + 05ndash1189e + 04

Figure 4 e maximum tensile stress nephogram at an incident angle of 0deg under uniform seismic input motion

X

Y

Z


+1774e + 07

+1626e + 07

+1478e + 07

+1330e + 07

+1182e + 07

+1034e + 07

+8863e + 06

+7384e + 06

+5905e + 06

+4426e + 06

+2947e + 06

+1467e + 06

ndash1168e + 04ODB xingbojiaxiangganbotanhuang31508330291047odb Abaqusstandard 611-1 Tue Sep 06 110407 GMT + 0800 2016Step dynamicIncrement 135 step time = 1294Primary var S max principal

Figure 5 e maximum tensile stress nephogram at an incident angle of 0deg under multisupport seismic input motion


ground motions had an asymmetric effect on theradial strain rate distributions

(3) -e rising incident angles changed the pure shearstress state into a complex stress state for tunnelsunder specified input motion -e spatially varyingseismic ground motions also influenced the stressstate relative to the uniform seismic inputMeanwhilethe values of maximum tensilecompressive stressesdecreased with the increasing incident angles

Data Availability

All the data supporting the conclusions of this study arepresented in the tables of the article -e code and details ofthe FEM for the analysis are available upon request from thecorresponding author

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-e supports by the National Natural Science Foundation ofChina (Grant no 41630638) the National Key Basic ResearchProgram of China (Grant no 2015CB057901) and the Fun-damental Research Funds for the Central Universities (Grantno 2018B13914) are greatly acknowledged

References

[1] H Iida T Hiroto N Yoshida et al ldquoDamage to Daikai subwaystationrdquo Soils and Foundations vol 36 pp 283ndash300 1996

[2] W L Wang T T Wang J J Su et al ldquoAssessment of damagein mountain tunnels due to the Taiwan Chi-Chi earthquakerdquoTunnelling and Underground Space Technology vol 16 no 3pp 133ndash150 2001

[3] C Menkiti R Mair and R Miles ldquoHighway tunnel perfor-mance during the 1999 Duzce earthquakerdquo in Proceedings ofthe Fifteenth International Conference on Soil Mechanics andGeotechnical Engineering Istanbul Turkey August 2001

[4] T Tobita S Iai M G Wang et al ldquoPreliminary report of theMid Niigata Prefecture Japan earthquake in 2004rdquo Journal ofJapan Society for Natural Disaster Science vol 23 no 4pp 595ndash602 2005 in Japanese

[5] Z Cao L-Q Hou X Yuan et al ldquoCharacteristics ofliquefaction-induced damages during Wenchuan Ms 80earthquakerdquo Rock and Soil Mechanics vol 31 no 11pp 3549ndash3555 2010 in Chinese

[6] S A Ashford R W Boulanger J L Donahue et al Geo-technical Quick Report on the Kanto Plain Region during theMarch 11 2011 Off Pacific Coast of Tohoku Earthquake JapanGEER Association Report No GEER Geotechnical ExtremeEvents Reconnaissance (GEER) Tokyo Japan 2011

[7] A Bobet ldquoEffect of pore water pressure on tunnel supportduring static and seismic loadingrdquo Tunnelling and Un-derground Space Technology vol 18 no 4 pp 377ndash393 2003

[8] S E Kattis D E Beskos and A H D Cheng ldquo2D dynamicresponse of unlined and lined tunnels in poroelastic soil toharmonic body wavesrdquo Earthquake Engineering amp StructuralDynamics vol 32 no 1 pp 97ndash110 2003

[9] S A S Youakim S E E El-Metewally and W F ChenldquoNonlinear analysis of tunnels in clayeysandy soil witha concrete liningrdquo Engineering Structures vol 22 no 2pp 707ndash722 2000

[10] A Amorosi and D Boldini ldquoNumerical modeling of thetransverse dynamic behavior of circular tunnels in clayeysoilsrdquo Soil Dynamics and Earthquake Engineering vol 29no 6 pp 1059ndash1072 2009

[11] S B Kojic and M Trifunac ldquoTransient pressures in hydro-technical tunnels during earthquakesrdquo Earthquake Engineeringand Structural Dynamics vol 16 no 4 pp 523ndash539 1988

[12] H Dobashi E Ochiai T Ichimura et al ldquo3D FE analysis ofseismic response of complicated large-scale ramp tunnel struc-turerdquo in Proceedings of the ECCOMAS Dematic Conference onComputational Methods in Structural Dynamics and EarthquakeEngineering pp 13ndash16 Rethymno Crete Greece 2007

[13] H Yu Y Yuan and A Bobet ldquoMultiscale method for longtunnels subjected to seismic loadingrdquo International Journalfor Numerical and Analytical Methods in Geomechanicsvol 37 no 4 pp 374ndash398 2013

[14] H Yu Y Yuan Z Qiao et al ldquoSeismic analysis of a longtunnel based on multi-scale methodrdquo Engineering Structuresvol 49 no 2 pp 572ndash587 2013

[15] W U Yongxin Y Gao and D Li ldquoSimulation of spatiallycorrelated earthquake ground motions for engineering pur-posesrdquo Earthquake Engineering and Engineering Vibrationvol 10 no 2 pp 163ndash173 2011


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Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation


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wwwhindawicom Volume 2018

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Submit your manuscripts atwwwhindawicom

is to assess the influence ofmultisupport input earthquake wavesand uniform input earthquake waves as well as their incidentangles on the diameter strain rate and tensivecompressiveprincipal stresses under different strata A full-scale 3D finite-element model is built comprising geological data tunnel ge-ometry and so on to farthest simulate actual situation

2 Simulation of Seismic Ground Motions

21 Simulation of Uniform Seismic Ground Motion Considera zero mean Gaussian stationary seismic ground motionwith a target autospectral density S(ω) -e seismic groundmotions can be generated through the following expression

x(t) 21113944N

l1

S ωl( 1113857Δω1113969

cos ωlt + φl( 1113857 (1)

where N is the number of frequency intervals Δω ωuN isfrequency increment with ωu as the cutoff frequencyωl lΔω and the φlrsquos are statistically independent randomphase angles uniformly distributed between (0 2π] Equation (1)is valid if there is an upper cutoff frequency ωu above which thecontribution of the power spectral density (PSD) to the sim-ulations is negligible for practical purposes

22 Simulation of Spatially Varying Seismic GroundMotions-e variations in ground motion are caused by the followingfour sources (1) the ldquoincoherence effectrdquo (2) the ldquowave-passageeffectrdquo (3) the ldquosite-response effectrdquo and (4) the ldquoattenuationeffectrdquo -e spectral representation method is one of the mostwidely usedmethods in simulating the spatially varying seismicground motions

In practical application spatially correlated groundmotions can be considered as a one-dimensional n-variety(1D-nV) stochastic vector process X(t) with componentsxj(t) middot (j 1 2 n) Based on the spectral representationmethod the jth component of the ground motions can begenerated by [13]

xj(t) 2 1113944n

k11113944

N

l1Hjk ωl( 1113857

11138681113868111386811138681113868

11138681113868111386811138681113868Δω

radiccos ωltminusϕjk ωl( 1113857 + φkl1113960 1113961

ωl lΔω (l 0 1 K Nminus 1)

Δω ωu

N

(2)

whereωu is the upper cutoff frequency beyondwhich elementsof the power spectral can be assumed to be zero for eithermathematical or physical reasons and φjk are independentrandom phase angles uniformly distributed over (0 2π]|Hjk(ω)| and ϕjk(ω) are the modulus and phase parts ofHjk(ω) respectively which can be obtained by the root de-composition of power spectral density matrix as follows

S(ω) H(ω)HT(ω) (3)

where the superscript T denotes conjugate transpose

-e power spectral density matrix is given as

S(ω)



⋮ ⋮ ⋱ ⋮

Sm1(ω) Sm2(ω) middot middot middot Smm(ω)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4)

where Sjj(ω) is the autopower spectral density function andSjk(ω t) is the cross-power spectral density function whichcan be expressed as

Sjk(ω t) cjk(ω)11138681113868111386811138681113868

11138681113868111386811138681113868

Sjj(ω)Skk(ω)

1113969e

iθjk(ω) (5)

where |cjk(ω)| is the lagged coherence function representingldquoincoherencerdquo effect and θjk(ω) is composed of wave passage

However the ground motions simulated by the abovemethod are stationary while the actual seismic records arenonstationary -erefore to obtain nonstationary seismicground motion the way of multiplying an envelope functionis applied -e envelope function is as follows

f(t)

tt1( 11138572 tlt t1

1 t1 le tlt t2

eminusc tminust2( ) tge t2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(6)

where t1 t2 and c are three parameters describing the shapeof the envelope function In this study they are set to bet1 6 t2 10 and c 05 respectively

3 FE Model

-e tunnel model adopted in this paper was built withcircular appearance whose outside diameter is 10m insidediameter is 9m and length is 1000m It had a buried depthof 30m from the tunnel center to soil surface-e simulationsetup is shown in Figure 1 It consists of a 1000times 300times100mbox with the 1000m length circular tunnel -e space co-ordinates were built by taking the length direction as theZ-axis the width direction as the X-axis and the heightdirection as the Y-direction -e soil profile was modeled asfour layers of MohrndashCoulomb materials and tunnel as theelastic material All the property parameters used can befound in Tables 1 and 2 An infinite domain by using theartificial boundary was adopted [12] where both borders ofthe tunnel were fixed in the Z-direction

In this work the constitutive model of the tunnel is anelastic model which is given as follows

σ λI1δ + 2Gε (7)

in which σ is the stress tensor ε is the strain tensor I1 is thefirst strain invariant λ is the Lame constant and G is theshear modulus

-e MohrndashCoulomb yield criterion for soil is expressedas follows

m + 12

max1113874 σ1 minus σ21113868111386811138681113868

1113868111386811138681113868 + K σ1 + σ2( 1113857 σ1 minus σ31113868111386811138681113868

1113868111386811138681113868

+K σ1 + σ3( 1113857 σ2 minus σ31113868111386811138681113868

1113868111386811138681113868 + K σ2 + σ3( 11138571113875 Syc

(8)


where

m SycSyt

K mminus 1m + 1

(9)






Δd dprime minus dd

(10)



X

Y

Z






2 31 4 5 6

9 10 11 12


0degXYZ








0 4 8 12 16 20

04

02

00

ndash02

ndash04

t (s)

Acce

lera

tion

(g)

(a)

0 4 8 12 16 20

04

02

00

ndash02

ndash04

t (s)

Acc

eler

atio

n (m

s2 )

(b)

0 4 8 12 16 20

04

02

00

ndash02

ndash04

t (s)

Acc

eler

atio

n (g

)

(c)














5 Conclusions





X

Y

Z





X

Y

Z


+1774e + 07

+1626e + 07

+1478e + 07

+1330e + 07

+1182e + 07

+1034e + 07

+8863e + 06

+7384e + 06

+5905e + 06

+4426e + 06

+2947e + 06

+1467e + 06






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


where

m SycSyt

K mminus 1m + 1

(9)






Δd dprime minus dd

(10)



X

Y

Z






2 31 4 5 6

9 10 11 12


0degXYZ








0 4 8 12 16 20

04

02

00

ndash02

ndash04

t (s)

Acce

lera

tion

(g)

(a)

0 4 8 12 16 20

04

02

00

ndash02

ndash04

t (s)

Acc

eler

atio

n (m

s2 )

(b)

0 4 8 12 16 20

04

02

00

ndash02

ndash04

t (s)

Acc

eler

atio

n (g

)

(c)














5 Conclusions





X

Y

Z





X

Y

Z


+1774e + 07

+1626e + 07

+1478e + 07

+1330e + 07

+1182e + 07

+1034e + 07

+8863e + 06

+7384e + 06

+5905e + 06

+4426e + 06

+2947e + 06

+1467e + 06






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







0 4 8 12 16 20

04

02

00

ndash02

ndash04

t (s)

Acce

lera

tion

(g)

(a)

0 4 8 12 16 20

04

02

00

ndash02

ndash04

t (s)

Acc

eler

atio

n (m

s2 )

(b)

0 4 8 12 16 20

04

02

00

ndash02

ndash04

t (s)

Acc

eler

atio

n (g

)

(c)














5 Conclusions





X

Y

Z





X

Y

Z


+1774e + 07

+1626e + 07

+1478e + 07

+1330e + 07

+1182e + 07

+1034e + 07

+8863e + 06

+7384e + 06

+5905e + 06

+4426e + 06

+2947e + 06

+1467e + 06






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 Conclusions





X

Y

Z





X

Y

Z


+1774e + 07

+1626e + 07

+1478e + 07

+1330e + 07

+1182e + 07

+1034e + 07

+8863e + 06

+7384e + 06

+5905e + 06

+4426e + 06

+2947e + 06

+1467e + 06






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




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


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