Research ArticleDynamic Response of Underground Circular Lining TunnelsSubjected to Incident P Waves

Hua Xu12 Tianbin Li2 Jingsong Xu1 and Yingjun Wang3

1 Key Laboratory of Transportation Tunnel Engineering Ministry of Education School of Civil EngineeringSouthwest Jiaotong University Chengdu Sichuan 610031 China

2 State Key Laboratory of Geohazard Prevention and Geoenvironment Protection Chengdu University of TechnologyChengdu Sichuan 610059 China

3Department of Structural Engineering University of California-San Diego La Jolla CA 92093 USA

Correspondence should be addressed to Hua Xu xuhua8318gmailcom

Received 6 August 2014 Revised 29 October 2014 Accepted 29 October 2014 Published 20 November 2014

Academic Editor Yan-Jun Liu

Copyright copy 2014 Hua Xu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Dynamic stress concentration in tunnels and underground structures during earthquakes often leads to serious structural damageA series solution of wave equation for dynamic response of underground circular lining tunnels subjected to incident plane P wavesis presented by Fourier-Bessel series expansion method in this paper The deformation and stress fields of the whole medium ofsurrounding rock and tunnel were obtained by solving the equations of seismic wave propagation in an elastic half space Basedon the assumption of a large circular arc a series of solutions for dynamic stress were deduced by using a wave function expansionapproach for a circular lining tunnel in an elastic half space rock medium subjected to incident plane P waves Then the dynamicresponse of the circular lining tunnel was obtained by solving a series of algebraic equations after imposing its boundary conditionsfor displacement and stress of the circular lining tunnel The effects of different factors on circular lining rock tunnels includingincident frequency incident angle buried depth rock conditions and lining stiffness were derived and several application examplesare presented The results may provide a good reference for studies on the dynamic response and aseismic design of tunnels andunderground structures

1 Introduction

Large earthquakes (ie the Kobe earthquake in Japan 1995the Chi-Chi earthquake in Taiwan 1999 the Wenchuanearthquake of magnitude Ms = 80 in China 2008) hadcaused a large number of seismic damages of tunnels andunderground structures [1] This is inconsistent with thetraditional concept [2] that underground structures were lesssusceptible to seismic damage during earthquakes becausethe seismic acceleration of underground structures wasless than that of structures on the ground surface Thedynamic response of tunnels and underground structuresduring earthquakes is a process of stress wave propagationreflection and interaction For rock lining tunnels due to thesimilar stiffness between the tunnel lining and the surround-ing rock medium it is generally thought that the motion oftunnel lining will follow that of the surrounding rock which

means that the interaction between the surrounding rock andtunnel lining can be ignored Built on this assumption thefluctuation field and stress field of the surrounding rock andtunnel lining are solved by the wave equation of seismic wavepropagation in an elastic half space rock medium [3]

Theoretical studies on dynamic response and aseismicdesign of tunnels and underground structures may be cat-egorized into numerical methods and analytical methods[4] The numerical methods include finite difference finiteelement and boundary element methods and the analyticalmethods are mainly referred to as wave function expansionmethod which are based on the theoretical analysis of elasticwave scattering and dynamic stress concentration Althoughnumerical methods can be used for arbitrary-shaped tunnelsanalytical solutions are still valuable for providing insight intothe formation mechanism of the problems and for checkingaccuracies of numerical methods

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 297424 11 pageshttpdxdoiorg1011552014297424

2 Mathematical Problems in Engineering

In the 1970s using the wave function expansion methodPao and Mow [5] initiatively studied dynamic stress con-centration of single cavity in the whole space for incidentelastic plane P waves Later Lee et al [6ndash9] extended thesolution to half space based on the assumption of largecircular arc and provided the analytical solutions consideringsingle cavity in half space for incident plane P and SVwaves which is much more complicated due to the wavemode conversionManoogian and Lee [10] used the weightedresidual method to study the problem of scattering anddiffraction of plane SH-waves by including a different elasticmedium of arbitrary shape in a half plane Davis et al [11]studied the transversal response of underground cylindricalcavities to incident SV waves and derived analytical solutionsto evaluate the dynamic response of a flexible buried pipeduring the Northridge earthquake Just recently Liang et al[12ndash14] Ji et al [15] and You and Liang [16] investigatedthe dynamic stress concentration of a cylindrical lined cavityin an elastic half space for incident plane P and SV wavesand derived the series solution to study the amplification ofground surface motion due to underground group cavitiesfor incident plane P waves Kouretzis et al [17] employedthe 3-D shell theory in order to derive analytical expressionsfor the distribution along the cross section of axial hoopand shear strains for long cylindrical underground structures(buried pipelines and tunnels) subjected to seismic shearwave excitation Esmaeili et al [18] used hybrid boundaryand finite element method (FEM) to study the dynamicresponse of lined circular tunnel subjected to plane P andSV harmonic seismic waves Xu et al [19] used Fourier-Bessel series expansion method to deduce a series solution ofwave equation for dynamic response of underground circularlining tunnels and approximately studied dynamic stressconcentration of lining with some influence factors

Based on the assumption of a large circular arc a seriessolution of dynamic stress is deduced by using a wavefunction expansion approach for a circular lining tunnelin an elastic half space rock medium subjected to incidentplane P waves Compared to previous studies the boundaryconditions of displacement and stress of circular lining tunnelare updated to adapt different surrounding rock conditionscomputational parameters are reconfigured and some casesare analyzed again by using MATLAB program to obtainmore precise results This paper focuses on the effect of thedynamic stress response on circular lining rock tunnels withdifferent factors including incident frequency incident angleburied depth rock conditions and lining stiffness The aimand novelty in this paper mainly include that (1) the effect oflow-frequency and high-frequency contents of seismic waveson the lining is estimated (2) the most unfavorable seismicwave incident angle to tunnels is obtained by comparing thedynamic stress response of the lining subjected to seismicwaves with different incident angle (3) the effect of burieddepth on the dynamic stress response of the lining in differentsurrounding rock conditions is investigated and the depthboundary where the dynamic stress response of the liningis significantly reduced is put forward (4) the effect of theelastic modulus ratio between the lining and the surroundingrock is estimated to obtain the optimal elastic modulus ratio

The rock medium

Half space surface o

o1

o2

y

y1

y2

R

R1

R2

r1

r2

H

120595s2

120595s1

1205951112059512

P waves

x

120579120572

1205792

D12

x2

x1

The lining

120601(r)

120601s2

120601s1

1206011112060112

120595(r)

Figure 1 The model of half space rock medium and tunnel lining

These results may provide a good reference for studies onthe dynamic response and aseismic design of tunnels andunderground structures

2 Model for Series Solution of Wave Equation

The schematic model is shown in Figure 1 consisting ofhalf space rock medium and a tunnel lining with burieddepth internal diameter and external diameter of the tunnellining being denoted by119867 119877

1 and 119877

2 respectively The half

plane rock medium and tunnel lining are assumed to beelastic isotropic and homogeneous The material propertiesare characterized by the Lame constants 120582

119904 120582119897 120583119904 and 120583

119897

and the longitudinal and transverse wave velocities for elastichalf space medium and tunnel lining are denoted by 119888

120572119904 119888120572119897

119888120573119904 and 119888

120573119897 where subscript 119904 indicates half plane medium

119897 indicates tunnel lining 120572 indicates longitudinal wave (iecompressive wave) and 120573 indicates transverse wave (ieshear wave)The large circular model is used here to simulatethe surface of the half space (Figure 1) It was shown thatwhenthe circular becomes larger and larger the solution convergesto the exact solution [8 9]

21 Analyzing Wave Function of Free-Field in Half SpaceWith the circular frequency of 120596 and the incident angle of 120579

120572

thewave potential functions of the incident plane Pwaves andreflected plane P and SV waves can be expressed in Fourier-Bessel series in the coordinates of 119900

1as [12]

120601(119894+119903)

(1199031 1205791) =

infin

sum

119899=0

119869119899(1198961199041205721199031) (1198600119899

cos 1198991205791+ 1198610119899

sin 1198991205791)

120595(119903)(1199031 1205791) =

infin

sum

119899=0

119869119899(1198961199041205731199031) (1198620119899

sin 1198991205791+ 1198630119899

cos 1198991205791)

(1)

Mathematical Problems in Engineering 3

where 119896119904120572

= 120596119888120572119904

is the longitudinal wave number 119896119904120573

=

120596119888120573119904is the transverse wave number and 119869

119899(119909) is the first type

of Bessel function where the time factor exp(minus119894120596119905) has beenomittedThe coefficients in formula (1) are defined as follows

1198600119899

1198610119899

= 120576119899119894119899cos 119899120579

120572

sin 119899120579120572

times [plusmn (minus1)119899 exp (minus119894119896

119904120572ℎ cos 120579

120572) + 1198961exp (119894119896

119904120572ℎ cos 120579

120572)]

1198620119899

1198630119899

= 1205761198991198941198991198962sin 119899120579

120573

cos 119899120579120573

exp (119894119896119904120573ℎ cos 120579

120573)

(2)

where 119894 is the imaginary unit 120576119899= 1 for 119899 = 0 and 120576

119899= 2

for 119899 ge 1 Symbol 120579120573is the reflection angle of reflected SV

waves which satisfies the equation sin 120579120572119888120572119904

= sin 120579120573119888120573119904

The reflection coefficients 1198961and 1198962are given by

1198961=

sin 2120579120572sin 2120579

120573minus (119888120572119904119888120573119904)2

cos22120579120573

sin 2120579120572sin 2120579

120573+ (119888120572119904119888120573119904)2

cos22120579120573

1198962=

minus2 sin 2120579120572cos 2120579

120573

sin 2120579120572sin 2120579

120573+ (119888120572119904119888120573119904)2

cos22120579120573

(3)

In O2coordinate system the wave potential functions of

the incident plane Pwaves and reflected plane P and SVwavesare similar to O

1coordinate system but are not presented

here

22 Analyzing Scattering Field at the Interface between TunnelLining and the Half Space The presence of the interfacebetween the circular tunnel lining and the half space leads toscattering P waves 120593

11(1199031 1205791) and SVwaves120595

11(1199031 1205791) which

can be expressed as series forms as below [15]

12060111(1199031 1205791) =

infin

sum

119899=0

119869119899(11989611205721199031) (119860(1)

11119899cos 119899120579

1+ 119861(1)

11119899sin 119899120579

1)

12059511(1199031 1205791) =

infin

sum

119899=0

119869119899(11989611205731199031) (119862(1)

11119899sin 119899120579

1+ 119863(1)

11119899cos 119899120579

1)

12060112(1199031 1205791) =

infin

sum

119899=0

1198671

119899(11989611205721199031) (119860(1)

12119899cos 119899120579

1+ 119861(1)

12119899sin 119899120579

1)

12059512(1199031 1205791) =

infin

sum

119899=0

1198671

119899(11989611205731199031) (119862(1)

12119899sin 119899120579

1+ 119863(1)

12119899cos 119899120579

1)

(4)

where 119869119899(119909) and 119867

(1)

119899(119909) are the first class Bessel function

and the Hankel function respectively The wave numbers ofP waves and SV waves of tunnel lining are 119896

1120572= 120596119888

1205721

and 1198961120573

= 1205961198881205731 respectively In the lining medium the

expressions of P waves and SV waves are complete Through

the coordinate transformation 1206011199042(1199032 1205792) and 120595

1199042(1199032 1205792) can

be converted into the Bessel series form in the coordinatesystem of 119903

1minus 1205791and 119869

119899(1198961199041205721199031) and 119867

(1)

119899(1198961199041205721199031) in the

coordinate system 1199031minus 1205791are a set of basis functions of the

wave equationSimilarly for the presence of circular tunnel lining there

are scattering P waves 1206011199041(1199031 1205791) and SV waves 120595

1199041(1199031 1205791)

which are caused at the interface between the tunnel liningand the half space Built on the assumption of large circulararc the scattering P waves and SV waves are denoted as12060112(1199031 1205791) and 120595

1199042(1199032 1205792) They can be expressed as series

forms

1206011199041(1199031 1205791)

=

infin

sum

119899=0

119867(1)

119899(1198961199041205721199031) (119860(1)

1199041119899cos 119899120579

1+ 119861(1)

1199041119899sin 119899120579

1)

1205951199041(1199031 1205791)

=

infin

sum

119899=0

119867(1)

119899(1198961199041205731199031) (119862(1)

1199041119899sin 119899120579

1+ 119863(1)

1199041119899cos 119899120579

1)

1206011199042(1199032 1205792)

=

infin

sum

119898=0

119869119898(1198961199041205721199032) (119860(2)

1199042119898cos119898120579

2+ 119861(2)

1199042119898sin119898120579

2)

1205951199042(1199032 1205792)

=

infin

sum

119898=0

119869119898(1198961199041205731199032) (119862(2)

1199042119898sin119898120579

2+ 119863(2)

1199042119898cos119898120579

2)

(5)

where 119896119904120572

= 120596119888120572119904

and 119896119904120573

= 120596119888120573119904

are the wave numbersof P waves and SV waves in the half space From a physicalpoint of view 120601

1199041(1199031 1205791) and120595

1199041(1199031 1205791) are represented by the

outward propagation waves from 1199001point while 120601

1199042(1199032 1205792)

and 1205951199042(1199032 1205792) are generated by the surface of the half space

in the large circular arcTherefore the expressions of P wavesand SV waves are complete in the half space medium

The wave potential function in the lining medium can beexpressed as

1206011= 12060111+ 12060112 120595

1= 12059511+ 12059512 (6)

Thewave potential function in the half spacemedium canbe written as

120601119904= 120601(119894+119903)

+ 1206011199041+ 1206011199042 120595

119904= 120595(119903)+ 1205951199041+ 1205951199042 (7)

23 Solving Series Solution after Imposing Boundary Condi-tions Under a plane strain assumption there are two familiesof boundary condition (1) the stress at the surface of the halfspace and the tunnel lining inner surface is zero (2) the stressand displacement are continuous at the interface of the tunnellining and the half space So the boundary conditions of the

4 Mathematical Problems in Engineering

stress and displacement of the tunnel lining are summarizedas

120591119904

119903119903= 120591119904

119903120579= 0 119903

2= 119887

1205911

119903119903= 1205911

119903120579= 0 119903

1= 1198861

1205911

119903119903= 120591119904

119903119903 120591

1

119903120579= 120591119904

119903120579

1199061

119903= 119906119904

119903 119906

1

120579= 119906119904

120579 119903

2= 1198862

(8)

Based on the boundary conditions in (8) the coefficientsin (4) and (5) can be obtained The wave potential functions1206011and120595

1 in the tunnel lining and120601

119904and120595

119904 in the half space

can be solved by (6) and (7) The stress and displacementexpressions under a plane strain assumption for incidentplane P waves can be described by

119906119903=120597120601

120597119903+1

119903

120597120593

120597120579 119906

120579=1

119903

120597120601

120597120579minus120597120593

120597119903

120591119903119903= 120582nabla2120601 + 2120583 [

1205972120601

1205971199032+120597

120597119903(1

119903

120597120593

120597120579)]

120591119903120579= 2120583(

1

119903

1205972120601

120597119903120597120579minus1

1199032

120597120601

120597120579) + 120583[

1

1199032

1205972120593

1205971205792minus 119903

120597

120597119903(1

119903

120597120593

120597119903)]

120591120579120579= 120582nabla2120601 + 2120583 [

1

119903(120597120601

120597119903+1

119903

1205972120601

1205971205792) +

1

119903(1

119903

120597120593

120597120579minus1205972120593

120597119903120597120579)]

(9)

After imposing the boundary conditions of 120591119904119903119903= 120591119904

119903120579= 0

and 1205911119903119903= 1205911

119903120579= 0 (10) and (11) can be obtained by combining

(1) (2) and (9) [5]

infin

sum

119899=0

[

[

1198641(1)

11(119899 1198861) 1198641(1)

12(119899 1198861)

1198641(1)

21(119899 1198861) 1198641(1)

22(119899 1198861)

]

]

119860(1)

11119899

119862(1)

11119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

1198641(2)

11(119899 1198861) 1198641(2)

12(119899 1198861)

1198641(2)

21(119899 1198861) 1198641(2)

22(119899 1198861)

]

]

119860(1)

12119899

119862(1)

12119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

1198641(1)

11(119899 1198861) 1198641(1)

12(119899 1198861)

1198641(1)

21(119899 1198861) 1198641(1)

22(119899 1198861)

]

]

119861(1)

11119899

119863(1)

11119899

(sin 119899120579

1

cos 1198991205791

)

+

infin

sum

119899=0

[

[

1198641(2)

11(119899 1198861) 1198641(2)

12(119899 1198861)

1198641(2)

21(119899 1198861) 1198641(2)

22(119899 1198861)

]

]

119861(1)

12119899

119863(1)

12119899

(sin 119899120579

1

cos 1198991205791

)

= 0

0

(10)

infin

sum

119898=0

[

[

119864119904(2)

11(119898 119887) 119864

119904(2)

12(119898 119887)

119864119904(2)minus

21(119898 119887) 119864

119904(2)

22(119898 119887)

]

]

119860(2)

1199041119898

119862(2)

1199041119898

(cos119898120579

2

sin1198981205792

)

+

infin

sum

119898=0

[

[

119864119904(1)

11(119898 119887) 119864

119904(1)

12(119898 119887)

119864119904(1)minus

21(119898 119887) 119864

119904(1)

22(119898 119887)

]

]

119860(2)

1199042119898

119862(2)

1199042119898

(cos119898120579

2

sin1198981205792

)

+

infin

sum

119898=0

[119864119904(2)

11(119898 119887) 119864

119904(2)minus

12(119898 119887)

119864119904(2)

21(119898 119887) 119864

119904(2)

22(119898 119887)

]

119861(2)

1199041119898

119863(2)

1199041119898

(sin119898120579

2

cos1198981205792

)

+

infin

sum

119898=0

[

[

119864119904(1)

11(119898 119887) 119864

119904(1)minus

12(119898 119887)

119864119904(1)

21(119898 119887) 119864

119904(1)

22(119898 119887)

]

]

119861(2)

1199041119898

119863(2)

1199041119898

(sin119898120579

2

cos1198981205792

)

= 0

0

(11)

where

119864(119894)

11(119899 119903) = (119899

2+ 119899 minus

1

21198962

1205731199032)119862119899(119896120572119903) minus 119896

120572119903119862119899minus1

(119896120572119903)

119864(119894)∓

12(119899 119903) = ∓119899 [minus (119899 + 1) 119862

119899(119896120572119903) + 119896

120573119903119862119899minus1

(119896120573119903)]

119864(119894)∓

21(119899 119903) = ∓119899 [minus (119899 + 1) 119862

119899(119896120572119903) + 119896

120572119903119862119899minus1

(119896120572119903)]

119864(119894)

22(119899 119903) = minus (119899

2+ 119899 minus

1

21198962

1205731199032)119862119899(119896120573119903) minus 119896

120573119903119862119899minus1

(119896120573119903)

(12)

When 119894 = 1 119862119899(119909) is a function of 119869

119899(119909) when 119894 = 2 119862

119899(119909) is

the function of119867(1)119899(119909) For the half plane referred to as ldquo119904rdquo

119896120572and 119896120573can be replaced by 119896

119904120572and 119896119904120573 respectively For the

tunnel lining referred to as ldquo1rdquo 119896120572and 119896120573can be replaced by

1198961120572

and 1198961120573 respectively Potential functions 120601(119894+119903) 120595(119903) 120601(119903)

and 120595(119894+119903)satisfy the stress boundary in the half spaceApplying the boundary conditions of 1199061

119903= 119906119904

119903and 1199061

120579=

119906119904

120579 the following equation can be obtained by combining (1)

(2) and (9)

infin

sum

119899=0

[

[

1198681(1)

11(119899 1198862) 1198681(1)

12(119899 1198862)

1198681(1)minus

21(119899 1198862) 1198681(1)

22(119899 1198862)

]

]

119860(1)

11119899

119862(1)

11119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

1198681(2)

11(119899 1198862) 1198681(2)

12(119899 1198862)

1198681(2)minus

21(119899 1198862) 1198681(2)

22(119899 1198862)

]

]

119860(1)

12119899

119862(1)

12119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

1198681(1)

11(119899 1198862) 1198681(1)minus

12(119899 1198862)

1198681(1)

21(119899 1198862) 1198681(1)

22(119899 1198862)

]

]

119861(1)

11119899

119863(1)

11119899

(sin 119899120579

1

cos 1198991205791

)

+

infin

sum

119899=0

[

[

1198681(2)

11(119899 1198862) 1198681(2)minus

12(119899 1198862)

1198681(2)

21(119899 1198862) 1198681(2)

22(119899 1198862)

]

]

119861(1)

12119899

119863(1)

12119899

(sin 119899120579

1

cos 1198991205791

)

Mathematical Problems in Engineering 5

=

infin

sum

119899=0

[

[

119868119904(1)

11(119899 1198862) 119868119904(1)

12(119899 1198862)

119868119904(1)minus

21(119899 1198862) 119868119904(1)

22(119899 1198862)

]

]

1198600119899+ 119860(1)

1199042119899

1198620119899+ 119862(1)

1199042119899

times (cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

119868119904(2)

11(119899 1198862) 119868119904(2)

12(119899 1198862)

119868119904(2)minus

21(119899 1198862) 119868119904(2)

22(119899 1198862)

]

]

119860(1)

1199041119899

119862(1)

1199041119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

119868119904(1)

11(119899 1198862) 119868119904(1)minus

12(119899 1198862)

119868119904(1)minus

21(119899 1198862) 119868119904(1)

22(119899 1198862)

]

]

1198610119899+ 119861(1)

1199042119899

1198630119899+ 119863(1)

1199042119899

times (sin 119899120579

1

cos 1198991205791

)

+

infin

sum

119899=0

[

[

119868119904(2)

11(119899 1198862) 119868119904(2)minus

12(119899 1198862)

119868119904(2)minus

21(119899 1198862) 119868119904(2)

22(119899 1198862)

]

]

119861(1)

1199041119899

119863(1)

1199041119899

(sin 119899120579

1

cos 1198991205791

)

(13)

where

119868(119894)

11(119899 119903) = minus119899119862

119899(119896120572119903) + 119896

120572119903119862119899minus1

(119896120572119903)

119868(119894)∓

12(119899 119903) = ∓119899119862

119899(119896120573119903)

119868(119894)∓

21(119899 119903) = ∓119899119862

119899(119896120572119903)

119868(119894)

22(119899 119903) = 119899119862

119899(119896120573119903) minus 119896

120573119903119862119899minus1

(119896120573119903)

(14)

Applying the boundary conditions of 1205911119903119903= 120591119904

119903119903and 1205911119903120579= 120591119904

119903120579

the following equation can be obtained by combining (1) (2)and (9)

infin

sum

119899=0

1205831[

[

1198641(1)

11(119899 1198862) 1198641(1)

12(119899 1198862)

1198641(1)minus

21(119899 1198862) 1198641(1)

22(119899 1198862)

]

]

119860(1)

11119899

119862(1)

11119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

1205831[

[

1198641(2)

11(119899 1198862) 1198641(2)

12(119899 1198862)

1198641(2)minus

21(119899 1198862) 1198641(2)

22(119899 1198862)

]

]

119860(1)

12119899

119862(1)

12119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

1205831[

[

1198641(1)

11(119899 1198862) 1198641(1)minus

12(119899 1198862)

1198641(1)

21(119899 1198862) 1198641(1)

22(119899 1198862)

]

]

119861(1)

11119899

119863(1)

11119899

(sin 119899120579

1

cos 1198991205791

)

+ 1205831

infin

sum

119899=0

[

[

1198641(2)

11(119899 1198861) 1198641(2)minus

12(119899 1198861)

1198641(2)

21(119899 1198861) 1198641(2)

22(119899 1198861)

]

]

119861(1)

12119899

119863(1)

12119899

(sin 119899120579

1

cos 1198991205791

)

=

infin

sum

119899=0

120583119904[

[

119864119904(1)

11(119899 1198862) 119864119904(1)

12(119899 1198862)

119864119904(1)minus

21(119899 1198862) 119864119904(1)

22(119899 1198862)

]

]

1198600119899+ 119860(1)

1199042119899

1198620119899+ 119862(1)

1199042119899

times (cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

120583119904[

[

119864119904(2)

11(119899 1198862) 119864119904(2)

12(119899 1198862)

119864119904(2)minus

21(119899 1198862) 119864119904(2)

22(119899 1198862)

]

]

119860(1)

1199041119899

119862(1)

1199041119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

120583119904[

[

119864119904(1)

11(119899 1198862) 119864119904(1)

12(119899 1198862)

119864119904(1)minus

21(119899 1198862) 119864119904(1)

22(119899 1198862)

]

]

1198610119899+ 119861(1)

1199042119899

1198630119899+ 119863(1)

1199042119899

times (sin 119899120579

1

cos 1198991205791

)

+

infin

sum

119899=0

120583119904[

[

119864119904(2)

11(119899 1198862) 119864119904(2)minus

12(119899 1198862)

119864119904(2)minus

21(119899 1198862) 119864119904(2)

22(119899 1198862)

]

]

119861(1)

1199041119899

119863(1)

1199041119899

(sin 119899120579

1

cos 1198991205791

)

(15)

Solving linear systemof (10) (11) (13) and (15) all coefficientsin the wave functions can be obtained in the coordinatesystem of 119903 minus 120579

1 The wave potential functions 120593

1and 120595

1in

the tunnel lining and120593119904and120595

119904in the half space can be solved

The stress and displacement functions of the tunnel lining forincident plane P waves can be solved from (9)

Generally under the action of steady state P wavesthe dynamic stress distribution of tunnel lining can bestudied through solving the coefficient of dynamic stressconcentration of the inner surface of tunnel lining on thetoroidal direction [13 14]

120591lowast

120579120579can be obtained from

120591lowast

120579120579=

10038161003816100381610038161003816100381610038161003816

120591120579120579

1205910

10038161003816100381610038161003816100381610038161003816

(16)

where 1205910is the maximum value of dynamic stress of tunnel

liningThrough solving the dimensionless coefficient of 120591lowast

120579120579

which is the coefficient of dynamic stress concentration of lin-ing on the toroidal direction the dynamic stress distributionand variation in the tunnel lining for incidence P waves canbe obtained

3 Calculating Examples and Analysis Results

Based on the analytical solutions of the seismic wave prop-agation in an elastic medium and the study of propagationbehavior of seismic waves in the rock and soil mediumseismic motion parameters geotechnical properties and lin-ing materials significantly affect the dynamic stress responseof tunnels and underground structures It is necessary toinvestigate the effect of various parameters including inci-dent frequency content incident angle buried depth rockmaterial properties and the lining stiffness by using theproposed series solutions

Basic assumptions of the application examples employedin this paper are as follows [13] (1) the outside diameter ofcircular tunnel is 119877

2= 12119877

1with a lining thickness of 02119877

1

(2) the buried depth is represented by the ratio (1198671198771) of the

distance from the tunnel center to the ground surface and theinternal radius of tunnel (3) the elastic modulus of tunnellining and surrounding rock is 119890

1and 119890119904 respectively (4) the

6 Mathematical Problems in Engineering

0

30

6090

120

150

210

270300

330

0

20

180

240

40

60

20

40

60

|120591lowast 120579120579|

120578 = 05

120578 = 10120578 = 20

(a) ℎ1 = 20m

120

150

300

330

0

30

6090

180

210

240270

0

30

15

15

30

45

45

120578 = 05

120578 = 10120578 = 20

|120591lowast 120579120579|

(b) ℎ1 = 100m

Figure 2 Dynamic stress concentration coefficient of tunnel lining for different incident frequency and buried depth

lining stiffness 120575 is represented by the shear wave velocityratio of the lining and the rock half space (5) the Poissonrsquosratio of rockmedium and lining is assumed to be 025 (6) thelongitudinal wave velocity of these twomedium is 1732 timesof the shear wave velocity (7) the radius of the large circulararc is 119877 = 119863

12+ 119867 and the distance from the centers O

2to

O1is taken as 119863

12= 1041198771in the example which is large

enough to simulate the half space to ensure that the solutionconverges to the exact solution

31 Effect of Incident Frequency on the Dynamic Stress Concen-tration Coefficient for the Tunnel Lining The dimensionlessfrequency 120578 is defined as the ratio of inner diameter of thetunnel and the incident wave length [15]

120578 =21198771

120582119904120573

=

1198961199041205731198771

120587 (17)

where 120582119904120573is the shear wave length of the half space rock

Considering the effect of different incident frequencycontents of P waves on dynamic response of tunnel liningthree typical cases (120578 = 05 10 and 20) are carried out forincidence P waves from vertical upward direction

Figure 2 shows how the toroidal dynamic stress concen-tration coefficient for the tunnel lining (120591lowast

120579120579) changes with

different incident frequency of P waves for two buried depthsℎ1= 20m and ℎ

1= 100m It is shown that with increasing

incident frequency 120591lowast120579120579

decreases gradually and 120591lowast120579120579

at low-frequency P waves is about 10 times larger than that athigh-frequency Therefore low-frequency seismic waves aremore detrimental to the tunnel lining than high-frequencyAlso the increase of incident frequency leads to increasinglycomplex distribution of the toroidal dynamic stress in thelining and the less principal stress concentration direction at

low-frequency conditions is gradually transformed intomorestress concentration directions at high-frequency conditions

32 Effect of Incident Angle on the Dynamic Stress Con-centration Coefficient of Lining The stress concentrationcoefficients of tunnels and underground structures are verysensitive to incident angle of seismic waves Small changesof the incident angle may lead to a significant change ofthe stress and deformation at various locations of the tunnelstructures Herein studies of three representative cases (ievertical incidencewith incident angle 120579120572=0∘ a small incidentangle 120579120572= 30∘ and large incident angle 120579120572= 60∘) were carriedout for frequency coefficients of 120578 = 1 and 120578 = 2 Figure 3displays the distribution of toroidal dynamic stress of in thelining for three different incident angles

It is observed that the coefficient of toroidal dynamicstress concentration decreases gradually with increasing inci-dent angle but the change in the amplitude is small inboth dimensionless frequency values 120578 = 1 and 120578 = 2 It isworth noting that for vertically incident P waves or a smallincident angle (120579120572 = 0∘ sim30∘) the coefficients of dynamicstress concentration of lining are very complex and unevenThe conclusion here agrees well with the results of shakingtable tests [20]

33 Effect of Buried Depth on the Dynamic Stress Concentra-tion Coefficient of Lining To demonstrate the effect of theburied depth and the material properties two different typesof rockmass (soft and hard rockmass) are chosen to study theseismic response of the tunnel lining within a buried depthrange from 10m to 150m

Case 1 (soft rock as surrounding rock medium of tunnel withelastic modulus 119890

119904= 2GPa) Figure 4 shows the distribution

Mathematical Problems in Engineering 7

0

3

3

6

9

6

9

0

30

6090

120

150

210

270300

330

180

240

120579120572 = 0∘

120579120572 = 30∘

120579120572 = 60∘

|120591lowast 120579120579|

(a) 120578 = 1

0

3

3

6

9

6

9

0

30

6090

120

150

210

270300

330

180

240

120579120572 = 0∘

120579120572 = 30∘

120579120572 = 60∘

|120591lowast 120579120579|

(b) 120578 = 2

Figure 3 Dynamic stress concentration coefficient of tunnel lining with different incident angles

of toroidal dynamic stress in the lining with different burieddepth in a range of 10mndash150m With the increase of theburied depths the dynamic stress concentration coefficientof the lining does not converge to the situation of singleround hole inwhole spaceThedistributions of lining toroidaldynamic stress with different values of buried depth are verycomplex and uneven and no directionality can be obviouslyobserved

As Figure 5 shows when the buried depth in the softsurrounding rock is within 100m the maximum dynamicstress response of the lining varies with buried depth andtwo peak values can be observed at the buried depth ofℎ1= 30m and 100m When the buried depth is greater

than 100m (ℎ1gt 100m) the coefficient of dynamic stress

concentration of lining does not vary significantly with a highlevel of about 65 It appears that with an increasing burieddepth the decreasing trend of dynamic stress response is notpronounced in the soft surrounding rock and the dynamicstress of tunnel lining may still be high in the case of alarge buried depth There are three main reasons (1) thedifference in stiffness is large between tunnel lining and softsurrounding rock (2) the degree of stress concentration oftunnel lining is high and dynamic stress of the lining doesnot decrease obviously with increase of the buried depth and(3) the pressure of surrounding rock increases gradually withincrease of the buried depth In this case the soft surroundingrock has a low strength large porosity many joints andfractures and low capability of self-arching and these rockproperties lead to large tunnel dynamic response

Case 2 (hard surrounding rock medium with elastic modulus119890119904= 20GPa) Regarding the case with hard surrounding

rock medium no obvious directionality can be observed in

terms of the distributions of toroidal dynamic stress causedby P waves but the dynamic stress response of lining ismore pronounced than the case of soft rock as illustrated inFigure 6

Figure 7 shows that when the tunnel buried depth isless than 100m the coefficient of maximum dynamic stressconcentration in the tunnel lining is generally high rangingfrom 60 to 85 which indicates that the response of dynamicstress of the lining is large at the portal segment of hard rocktunnel and the tunnel lining may be damaged easily duringa large earthquake When the tunnel buried depth is greaterthan 75m the coefficient of dynamic stress concentrationis reduced significantly The coefficient of dynamic stressconcentration starts to reach a constant level after the burieddepth is greater than 100m It appears that the lining dynamicstress is small and tends to stabilize at the large buried depthsegment of hard rock tunnel and the tunnel liningmay be lesssusceptible to damage during a large earthquakeTherefore itis concluded that under the hard rock conditions the tunnelburied depth (ie thickness of the overlying rock) is nota major factor that affects the dynamic stress and seismicdamages of tunnel lining when the tunnel buried depth isgreater than 100m This conclusion is consistent with thefindings of Gao et al [21] which states that the dynamicstress concentration coefficient tends to be constant when theburied depth is greater than 25119877

1and the buried depth of

251198771approaches 100m

34 Effect of Elastic Modulus on the Dynamic Stress Con-centration Coefficient of Lining The elastic modulus (119890

1)

of the lining can be considered an index for the elasticdeformation of tunnel lining and the stresses caused by

8 Mathematical Problems in Engineering

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 10mh1 = 20mh1 = 30m

|120591lowast 120579120579|

(a) ℎ1 = 10sim30m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 40mh1 = 50mh1 = 75m

|120591lowast 120579120579|

(b) ℎ1 = 40sim75m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 100mh1 = 125mh1 = 150m

|120591lowast 120579120579|

(c) ℎ1 = 100sim150m

Figure 4 Dynamic stress concentration coefficient of tunnel lining with different values of buried depth in the soft surrounding rock

0123456789

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150Buried depth h1 (m)

|120591lowast 120579120579|

Figure 5 The maximum dynamic stress concentration coefficientof tunnel lining with different buried depths in the soft surroundingrock

elastic deformation increase with increasing elastic modu-lus This also means that at a given level of stresses thelining deformation decreases with increasing lining stiffnessWe focus on the effect of lining elastic modulus on thedynamic stress concentration coefficient of the lining ina soft surrounding rock Figure 8 shows the coefficientsof dynamic stress concentration (120591lowast

120579120579) with different lining

elastic modulus in a soft surrounding rock (119890119904= 2GPa)

As shown in Figure 8 the coefficient of dynamicstress concentration of the lining increases gradually withincreasing lining elastic modulus and the greater the elasticmodulus is the more uneven the distribution of the dynamicstress coefficient is The different adaptabilities of lining

Mathematical Problems in Engineering 9

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 10mh1 = 20mh1 = 30m

|120591lowast 120579120579|

(a) ℎ1 = 10sim30m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 40mh1 = 50mh1 = 75m

|120591lowast 120579120579|

(b) ℎ1 = 40sim75m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 100mh1 = 125mh1 = 150m

|120591lowast 120579120579|

(c) ℎ1 = 100sim150m

Figure 6 Dynamic stress concentration coefficient of tunnel lining with different buried depths in hard surrounding rock

0123456789

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150Buried depth h1 (m)

|120591lowast 120579120579|

Figure 7Themaximumdynamic stress concentration coefficient oftunnel lining with different buried depths in hard surrounding rock

with different stiffness to dynamic stresses may lead to thevariation of the dynamic stress of lining Herein the followingexample is adopted to illustrate this aspect When the elasticmodulus of the lining is 119890

1= 10Gpa and soft surrounding

rock is 119890119904= 2Gpa (ie 119890

1= 5119890119904) the maximum value of

dynamic stress concentration coefficient of the lining is morethan 100 and the variation of dynamic stress of lining acrossall directions is very complexTherefore under the premise ofmeeting the bearing capacity and deformation of the liningappropriately soft lining is recommended to be adopted inthe tunnel structure to mitigate the seismic damage of tunnellining and the ratio between the elastic modulus of the liningand soft surrounding rock should be less than 5

10 Mathematical Problems in Engineering

0

10

20

30

10

20

30

0

30

6090

120

150

210

270300

330

180

240

e1 = 2Gpae1 = 4Gpa

|120591lowast 120579120579|

(a) 1198901 = 2sim4GPa

0

100

100

200

300

400

200

300

400

0

30

6090

120

150

180

210

240270

300

330

e1 = 6Gpae1 = 8Gpa

e1 = 10Gpae1 = 20Gpa

|120591lowast 120579120579|

(b) 1198901 = 6sim20GPa

Figure 8 Dynamic stress concentration coefficient of tunnel lining with different lining elastic modulus in the soft surrounding rock

4 Conclusions

A series solution of wave equation for dynamic response ofunderground circular lining tunnels subjected to incidentplane P waves is presented by Fourier-Bessel series expansionmethod in this paper The effects on circular lining rocktunnels of different factors including incident frequencyincident angle buried depth rock conditions and liningstiffness are studied through the proposed series solution Inthis study the following conclusions are drawn

(1) In circular lining rock tunnels the coefficient oftoroidal dynamic stress concentration and dynamicstress decrease gradually with the increase of incidentfrequency of P waves 120591lowast

120579120579for low-frequency P waves

is about 10 times larger than that for high-frequencyindicating that low-frequency contents of seismicwaves are more detrimental to the tunnel linings thanhigh-frequency contents

(2) The incident angle of P waves has a significantinfluence on dynamic response of the tunnel liningWhen seismic waves propagate vertically or with asmall incident angle (120579120572 = 0∘ sim30∘) the distributionof the coefficients of dynamic stress concentration inthe tunnel lining is very complex and uneven whichis detrimental to the safety of the tunnel lining

(3) When the buried depth of a tunnel in the soft sur-rounding rock is in a range of 0sim100m the decreasingtendency of dynamic stress response of the liningwith increasing tunnel buried depth is not obviouseven if the buried depth is greater than 100m thecoefficient of dynamic stress concentration of thelining still remains at a high level of about 65 In thissituation tunnels crossing the fault zone or the high

stress segment may get damaged seriously duringlarge earthquake

(4) When the tunnel buried depth in hard surroundingrock is less than 100m the coefficient of dynamicstress concentration in the tunnel lining is highranging from 60 to 85 While the buried depth isgreater than 100m the dynamic stress of the lining issmall and tends to stabilize When the tunnel burieddepth is more than 100m the tunnel buried depth isnot a major factor that affects the dynamic stress

(5) The coefficient of dynamic stress concentration inthe lining increases gradually with increasing elasticmodulus of the lining and the greater the elasticmodulus is the more uneven the dynamic stresscoefficient is Therefore under the premise of meet-ing the bearing capacity and deformation of liningappropriate soft lining is recommended to be adoptedin the tunnel structures to mitigate seismic damageof tunnel lining and the ratio between the elasticmodulus of the lining and soft surrounding rockshould be less than 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of China(Grant no 20110184120033) the Open Foundation of StateKey Laboratory of Geohazard Prevention and Geoenviron-ment Protection (Chengdu University of Technology Grant

Mathematical Problems in Engineering 11

no SKLGP2011K014) and the Science Foundation for Excel-lent Youth Scholars of Southwest JiaotongUniversity of China(Grant no SWJTU11BR003)

References

[1] T B Li ldquoDamage to mountain tunnels related to theWenchuanearthquake and some suggestions for aseismic tunnel construc-tionrdquo Bulletin of Engineering Geology and the Environment vol71 no 2 pp 297ndash308 2012

[2] I Towhata Geotechnical Earthquake Engineering SpringerBerlin Germany 2008

[3] V W Lee and M D Trifunac ldquoResponse of tunnels to incidentSH-wavesrdquo Journal of the Engineering Mechanics Division vol105 no 4 pp 643ndash659 1979

[4] Y L Zheng L D Yang and W Y Li Seismic Design of Under-ground Structure Tongji University Press Shanghai China2005

[5] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak and CompanyNew York NY USA 1973

[6] V W Lee and H Cao ldquoDiffraction of SV waves by circularcanyons of various depthsrdquo Journal of Engineering Mechanicsvol 115 no 9 pp 2035ndash2056 1989

[7] H Cao and V W Lee ldquoScattering and diffraction of planeP waves by circular cylindrical canyons with variable depth-to-width ratiordquo International Journal of Soil Dynamics andEarthquake Engineering vol 9 no 3 pp 141ndash150 1990

[8] VW Lee and J Karl ldquoDiffraction of SV waves by undergroundcircular cylindrical cavitiesrdquo Soil Dynamics and EarthquakeEngineering vol 11 no 8 pp 445ndash456 1992

[9] V W Lee and J Karl ldquoOn deformation of near a circularunderground cavity subjected to incident plane P wavesrdquoEuropean Journal of Earthquake Engineering vol 7 no 1 pp 29ndash35 1993

[10] M Manoogian and V W Lee ldquoDiffraction of SH-waves bysubsurface inclusions of arbitrary shaperdquo Journal of EngineeringMechanics vol 122 no 2 pp 123ndash129 1996

[11] C A Davis V W Lee and J P Bardet ldquoTransverse responceof underground cavities and pipes to incident SV wavesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 383ndash410 2001

[12] J W Liang H Zhang and V W Lee ldquoAnalytical solution fordynamic stress concentration of underground cavities underincident plane P wavesrdquo Chinese Journal of Geotechnical Engi-neering vol 26 no 6 pp 815ndash819 2004

[13] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (I) analytical solutionrdquoEarthquake Engineering and Engineering Vibration vol 27 no1 pp 1ndash6 2007

[14] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (II) numerical resultsrdquoEarthquake Engineering and Engineering Vibration vol 27 no2 pp 1ndash11 2007

[15] X D Ji J W Liang and J J Yang ldquoDynamic stress concen-tration of an underground cylindrical lined cavity subjected toincident plane P and SV wavesrdquo Journal of Tianjin UniversityScience and Technology vol 39 no 5 pp 511ndash517 2006

[16] H B You and J W Liang ldquoScattering of plane SV waves by acavity in a layered half-spacerdquo Rock and Soil Mechanics vol 27no 3 pp 383ndash388 2006

[17] G P Kouretzis G D Bouckovalas and C J Gantes ldquo3-D shell analysis of cylindrical underground structures underseismic shear (S) wave actionrdquo Soil Dynamics and EarthquakeEngineering vol 26 no 10 pp 909ndash921 2006

[18] M Esmaeili S Vahdani andANoorzad ldquoDynamic response oflined circular tunnel to plane harmonic wavesrdquo Tunnelling andUnderground Space Technology vol 11 no 17 pp 1ndash9 2005

[19] H Xu T B Li and L Q Li ldquoResearch on dynamic responseof underground circular lining tunnel under the action of Pwavesrdquo Applied Mechanics and Materials vol 99-100 pp 181ndash189 2011

[20] H Xu T B Li DWang Y S Li and Z H Lin ldquoStudy of seismicresponses of mountain tunnels with 3D shaking table modeltestrdquo Chinese Journal of Rock Mechanics and Engineering vol32 no 9 pp 1762ndash1771 2013

[21] G Y Gao Y S Li and T B Li ldquoDynamic analysis of shallowburied circular lining tunnel with vertical incident P waverdquoCentral South Highway Engineering vol 32 no 2 pp 56ndash602007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

In the 1970s using the wave function expansion methodPao and Mow [5] initiatively studied dynamic stress con-centration of single cavity in the whole space for incidentelastic plane P waves Later Lee et al [6ndash9] extended thesolution to half space based on the assumption of largecircular arc and provided the analytical solutions consideringsingle cavity in half space for incident plane P and SVwaves which is much more complicated due to the wavemode conversionManoogian and Lee [10] used the weightedresidual method to study the problem of scattering anddiffraction of plane SH-waves by including a different elasticmedium of arbitrary shape in a half plane Davis et al [11]studied the transversal response of underground cylindricalcavities to incident SV waves and derived analytical solutionsto evaluate the dynamic response of a flexible buried pipeduring the Northridge earthquake Just recently Liang et al[12ndash14] Ji et al [15] and You and Liang [16] investigatedthe dynamic stress concentration of a cylindrical lined cavityin an elastic half space for incident plane P and SV wavesand derived the series solution to study the amplification ofground surface motion due to underground group cavitiesfor incident plane P waves Kouretzis et al [17] employedthe 3-D shell theory in order to derive analytical expressionsfor the distribution along the cross section of axial hoopand shear strains for long cylindrical underground structures(buried pipelines and tunnels) subjected to seismic shearwave excitation Esmaeili et al [18] used hybrid boundaryand finite element method (FEM) to study the dynamicresponse of lined circular tunnel subjected to plane P andSV harmonic seismic waves Xu et al [19] used Fourier-Bessel series expansion method to deduce a series solution ofwave equation for dynamic response of underground circularlining tunnels and approximately studied dynamic stressconcentration of lining with some influence factors

Based on the assumption of a large circular arc a seriessolution of dynamic stress is deduced by using a wavefunction expansion approach for a circular lining tunnelin an elastic half space rock medium subjected to incidentplane P waves Compared to previous studies the boundaryconditions of displacement and stress of circular lining tunnelare updated to adapt different surrounding rock conditionscomputational parameters are reconfigured and some casesare analyzed again by using MATLAB program to obtainmore precise results This paper focuses on the effect of thedynamic stress response on circular lining rock tunnels withdifferent factors including incident frequency incident angleburied depth rock conditions and lining stiffness The aimand novelty in this paper mainly include that (1) the effect oflow-frequency and high-frequency contents of seismic waveson the lining is estimated (2) the most unfavorable seismicwave incident angle to tunnels is obtained by comparing thedynamic stress response of the lining subjected to seismicwaves with different incident angle (3) the effect of burieddepth on the dynamic stress response of the lining in differentsurrounding rock conditions is investigated and the depthboundary where the dynamic stress response of the liningis significantly reduced is put forward (4) the effect of theelastic modulus ratio between the lining and the surroundingrock is estimated to obtain the optimal elastic modulus ratio

The rock medium

Half space surface o

o1

o2

y

y1

y2

R

R1

R2

r1

r2

H

120595s2

120595s1

1205951112059512

P waves

x

120579120572

1205792

D12

x2

x1

The lining

120601(r)

120601s2

120601s1

1206011112060112

120595(r)

Figure 1 The model of half space rock medium and tunnel lining

These results may provide a good reference for studies onthe dynamic response and aseismic design of tunnels andunderground structures

2 Model for Series Solution of Wave Equation

The schematic model is shown in Figure 1 consisting ofhalf space rock medium and a tunnel lining with burieddepth internal diameter and external diameter of the tunnellining being denoted by119867 119877

1 and 119877

2 respectively The half

plane rock medium and tunnel lining are assumed to beelastic isotropic and homogeneous The material propertiesare characterized by the Lame constants 120582

119904 120582119897 120583119904 and 120583

119897

and the longitudinal and transverse wave velocities for elastichalf space medium and tunnel lining are denoted by 119888

120572119904 119888120572119897

119888120573119904 and 119888

120573119897 where subscript 119904 indicates half plane medium

119897 indicates tunnel lining 120572 indicates longitudinal wave (iecompressive wave) and 120573 indicates transverse wave (ieshear wave)The large circular model is used here to simulatethe surface of the half space (Figure 1) It was shown thatwhenthe circular becomes larger and larger the solution convergesto the exact solution [8 9]

21 Analyzing Wave Function of Free-Field in Half SpaceWith the circular frequency of 120596 and the incident angle of 120579

120572

thewave potential functions of the incident plane Pwaves andreflected plane P and SV waves can be expressed in Fourier-Bessel series in the coordinates of 119900

1as [12]

120601(119894+119903)

(1199031 1205791) =

infin

sum

119899=0

119869119899(1198961199041205721199031) (1198600119899

cos 1198991205791+ 1198610119899

sin 1198991205791)

120595(119903)(1199031 1205791) =

infin

sum

119899=0

119869119899(1198961199041205731199031) (1198620119899

sin 1198991205791+ 1198630119899

cos 1198991205791)

(1)

Mathematical Problems in Engineering 3

where 119896119904120572

= 120596119888120572119904

is the longitudinal wave number 119896119904120573

=

120596119888120573119904is the transverse wave number and 119869

119899(119909) is the first type

of Bessel function where the time factor exp(minus119894120596119905) has beenomittedThe coefficients in formula (1) are defined as follows

1198600119899

1198610119899

= 120576119899119894119899cos 119899120579

120572

sin 119899120579120572

times [plusmn (minus1)119899 exp (minus119894119896

119904120572ℎ cos 120579

120572) + 1198961exp (119894119896

119904120572ℎ cos 120579

120572)]

1198620119899

1198630119899

= 1205761198991198941198991198962sin 119899120579

120573

cos 119899120579120573

exp (119894119896119904120573ℎ cos 120579

120573)

(2)

where 119894 is the imaginary unit 120576119899= 1 for 119899 = 0 and 120576

119899= 2

for 119899 ge 1 Symbol 120579120573is the reflection angle of reflected SV

waves which satisfies the equation sin 120579120572119888120572119904

= sin 120579120573119888120573119904

The reflection coefficients 1198961and 1198962are given by

1198961=

sin 2120579120572sin 2120579

120573minus (119888120572119904119888120573119904)2

cos22120579120573

sin 2120579120572sin 2120579

120573+ (119888120572119904119888120573119904)2

cos22120579120573

1198962=

minus2 sin 2120579120572cos 2120579

120573

sin 2120579120572sin 2120579

120573+ (119888120572119904119888120573119904)2

cos22120579120573

(3)

In O2coordinate system the wave potential functions of

the incident plane Pwaves and reflected plane P and SVwavesare similar to O

1coordinate system but are not presented

here

22 Analyzing Scattering Field at the Interface between TunnelLining and the Half Space The presence of the interfacebetween the circular tunnel lining and the half space leads toscattering P waves 120593

11(1199031 1205791) and SVwaves120595

11(1199031 1205791) which

can be expressed as series forms as below [15]

12060111(1199031 1205791) =

infin

sum

119899=0

119869119899(11989611205721199031) (119860(1)

11119899cos 119899120579

1+ 119861(1)

11119899sin 119899120579

1)

12059511(1199031 1205791) =

infin

sum

119899=0

119869119899(11989611205731199031) (119862(1)

11119899sin 119899120579

1+ 119863(1)

11119899cos 119899120579

1)

12060112(1199031 1205791) =

infin

sum

119899=0

1198671

119899(11989611205721199031) (119860(1)

12119899cos 119899120579

1+ 119861(1)

12119899sin 119899120579

1)

12059512(1199031 1205791) =

infin

sum

119899=0

1198671

119899(11989611205731199031) (119862(1)

12119899sin 119899120579

1+ 119863(1)

12119899cos 119899120579

1)

(4)

where 119869119899(119909) and 119867

(1)

119899(119909) are the first class Bessel function

and the Hankel function respectively The wave numbers ofP waves and SV waves of tunnel lining are 119896

1120572= 120596119888

1205721

and 1198961120573

= 1205961198881205731 respectively In the lining medium the

expressions of P waves and SV waves are complete Through

the coordinate transformation 1206011199042(1199032 1205792) and 120595

1199042(1199032 1205792) can

be converted into the Bessel series form in the coordinatesystem of 119903

1minus 1205791and 119869

119899(1198961199041205721199031) and 119867

(1)

119899(1198961199041205721199031) in the

coordinate system 1199031minus 1205791are a set of basis functions of the

wave equationSimilarly for the presence of circular tunnel lining there

are scattering P waves 1206011199041(1199031 1205791) and SV waves 120595

1199041(1199031 1205791)

which are caused at the interface between the tunnel liningand the half space Built on the assumption of large circulararc the scattering P waves and SV waves are denoted as12060112(1199031 1205791) and 120595

1199042(1199032 1205792) They can be expressed as series

forms

1206011199041(1199031 1205791)

=

infin

sum

119899=0

119867(1)

119899(1198961199041205721199031) (119860(1)

1199041119899cos 119899120579

1+ 119861(1)

1199041119899sin 119899120579

1)

1205951199041(1199031 1205791)

=

infin

sum

119899=0

119867(1)

119899(1198961199041205731199031) (119862(1)

1199041119899sin 119899120579

1+ 119863(1)

1199041119899cos 119899120579

1)

1206011199042(1199032 1205792)

=

infin

sum

119898=0

119869119898(1198961199041205721199032) (119860(2)

1199042119898cos119898120579

2+ 119861(2)

1199042119898sin119898120579

2)

1205951199042(1199032 1205792)

=

infin

sum

119898=0

119869119898(1198961199041205731199032) (119862(2)

1199042119898sin119898120579

2+ 119863(2)

1199042119898cos119898120579

2)

(5)

where 119896119904120572

= 120596119888120572119904

and 119896119904120573

= 120596119888120573119904

are the wave numbersof P waves and SV waves in the half space From a physicalpoint of view 120601

1199041(1199031 1205791) and120595

1199041(1199031 1205791) are represented by the

outward propagation waves from 1199001point while 120601

1199042(1199032 1205792)

and 1205951199042(1199032 1205792) are generated by the surface of the half space

in the large circular arcTherefore the expressions of P wavesand SV waves are complete in the half space medium

The wave potential function in the lining medium can beexpressed as

1206011= 12060111+ 12060112 120595

1= 12059511+ 12059512 (6)

Thewave potential function in the half spacemedium canbe written as

120601119904= 120601(119894+119903)

+ 1206011199041+ 1206011199042 120595

119904= 120595(119903)+ 1205951199041+ 1205951199042 (7)

23 Solving Series Solution after Imposing Boundary Condi-tions Under a plane strain assumption there are two familiesof boundary condition (1) the stress at the surface of the halfspace and the tunnel lining inner surface is zero (2) the stressand displacement are continuous at the interface of the tunnellining and the half space So the boundary conditions of the

4 Mathematical Problems in Engineering

stress and displacement of the tunnel lining are summarizedas

120591119904

119903119903= 120591119904

119903120579= 0 119903

2= 119887

1205911

119903119903= 1205911

119903120579= 0 119903

1= 1198861

1205911

119903119903= 120591119904

119903119903 120591

1

119903120579= 120591119904

119903120579

1199061

119903= 119906119904

119903 119906

1

120579= 119906119904

120579 119903

2= 1198862

(8)

Based on the boundary conditions in (8) the coefficientsin (4) and (5) can be obtained The wave potential functions1206011and120595

1 in the tunnel lining and120601

119904and120595

119904 in the half space

can be solved by (6) and (7) The stress and displacementexpressions under a plane strain assumption for incidentplane P waves can be described by

119906119903=120597120601

120597119903+1

119903

120597120593

120597120579 119906

120579=1

119903

120597120601

120597120579minus120597120593

120597119903

120591119903119903= 120582nabla2120601 + 2120583 [

1205972120601

1205971199032+120597

120597119903(1

119903

120597120593

120597120579)]

120591119903120579= 2120583(

1

119903

1205972120601

120597119903120597120579minus1

1199032

120597120601

120597120579) + 120583[

1

1199032

1205972120593

1205971205792minus 119903

120597

120597119903(1

119903

120597120593

120597119903)]

120591120579120579= 120582nabla2120601 + 2120583 [

1

119903(120597120601

120597119903+1

119903

1205972120601

1205971205792) +

1

119903(1

119903

120597120593

120597120579minus1205972120593

120597119903120597120579)]

(9)

After imposing the boundary conditions of 120591119904119903119903= 120591119904

119903120579= 0

and 1205911119903119903= 1205911

119903120579= 0 (10) and (11) can be obtained by combining

(1) (2) and (9) [5]

infin

sum

119899=0

[

[

1198641(1)

11(119899 1198861) 1198641(1)

12(119899 1198861)

1198641(1)

21(119899 1198861) 1198641(1)

22(119899 1198861)

]

]

119860(1)

11119899

119862(1)

11119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

1198641(2)

11(119899 1198861) 1198641(2)

12(119899 1198861)

1198641(2)

21(119899 1198861) 1198641(2)

22(119899 1198861)

]

]

119860(1)

12119899

119862(1)

12119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

1198641(1)

11(119899 1198861) 1198641(1)

12(119899 1198861)

1198641(1)

21(119899 1198861) 1198641(1)

22(119899 1198861)

]

]

119861(1)

11119899

119863(1)

11119899

(sin 119899120579

1

cos 1198991205791

)

+

infin

sum

119899=0

[

[

1198641(2)

11(119899 1198861) 1198641(2)

12(119899 1198861)

1198641(2)

21(119899 1198861) 1198641(2)

22(119899 1198861)

]

]

119861(1)

12119899

119863(1)

12119899

(sin 119899120579

1

cos 1198991205791

)

= 0

0

(10)

infin

sum

119898=0

[

[

119864119904(2)

11(119898 119887) 119864

119904(2)

12(119898 119887)

119864119904(2)minus

21(119898 119887) 119864

119904(2)

22(119898 119887)

]

]

119860(2)

1199041119898

119862(2)

1199041119898

(cos119898120579

2

sin1198981205792

)

+

infin

sum

119898=0

[

[

119864119904(1)

11(119898 119887) 119864

119904(1)

12(119898 119887)

119864119904(1)minus

21(119898 119887) 119864

119904(1)

22(119898 119887)

]

]

119860(2)

1199042119898

119862(2)

1199042119898

(cos119898120579

2

sin1198981205792

)

+

infin

sum

119898=0

[119864119904(2)

11(119898 119887) 119864

119904(2)minus

12(119898 119887)

119864119904(2)

21(119898 119887) 119864

119904(2)

22(119898 119887)

]

119861(2)

1199041119898

119863(2)

1199041119898

(sin119898120579

2

cos1198981205792

)

+

infin

sum

119898=0

[

[

119864119904(1)

11(119898 119887) 119864

119904(1)minus

12(119898 119887)

119864119904(1)

21(119898 119887) 119864

119904(1)

22(119898 119887)

]

]

119861(2)

1199041119898

119863(2)

1199041119898

(sin119898120579

2

cos1198981205792

)

= 0

0

(11)

where

119864(119894)

11(119899 119903) = (119899

2+ 119899 minus

1

21198962

1205731199032)119862119899(119896120572119903) minus 119896

120572119903119862119899minus1

(119896120572119903)

119864(119894)∓

12(119899 119903) = ∓119899 [minus (119899 + 1) 119862

119899(119896120572119903) + 119896

120573119903119862119899minus1

(119896120573119903)]

119864(119894)∓

21(119899 119903) = ∓119899 [minus (119899 + 1) 119862

119899(119896120572119903) + 119896

120572119903119862119899minus1

(119896120572119903)]

119864(119894)

22(119899 119903) = minus (119899

2+ 119899 minus

1

21198962

1205731199032)119862119899(119896120573119903) minus 119896

120573119903119862119899minus1

(119896120573119903)

(12)

When 119894 = 1 119862119899(119909) is a function of 119869

119899(119909) when 119894 = 2 119862

119899(119909) is

the function of119867(1)119899(119909) For the half plane referred to as ldquo119904rdquo

119896120572and 119896120573can be replaced by 119896

119904120572and 119896119904120573 respectively For the

tunnel lining referred to as ldquo1rdquo 119896120572and 119896120573can be replaced by

1198961120572

and 1198961120573 respectively Potential functions 120601(119894+119903) 120595(119903) 120601(119903)

and 120595(119894+119903)satisfy the stress boundary in the half spaceApplying the boundary conditions of 1199061

119903= 119906119904

119903and 1199061

120579=

119906119904

120579 the following equation can be obtained by combining (1)

(2) and (9)

infin

sum

119899=0

[

[

1198681(1)

11(119899 1198862) 1198681(1)

12(119899 1198862)

1198681(1)minus

21(119899 1198862) 1198681(1)

22(119899 1198862)

]

]

119860(1)

11119899

119862(1)

11119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

1198681(2)

11(119899 1198862) 1198681(2)

12(119899 1198862)

1198681(2)minus

21(119899 1198862) 1198681(2)

22(119899 1198862)

]

]

119860(1)

12119899

119862(1)

12119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

1198681(1)

11(119899 1198862) 1198681(1)minus

12(119899 1198862)

1198681(1)

21(119899 1198862) 1198681(1)

22(119899 1198862)

]

]

119861(1)

11119899

119863(1)

11119899

(sin 119899120579

1

cos 1198991205791

)

+

infin

sum

119899=0

[

[

1198681(2)

11(119899 1198862) 1198681(2)minus

12(119899 1198862)

1198681(2)

21(119899 1198862) 1198681(2)

22(119899 1198862)

]

]

119861(1)

12119899

119863(1)

12119899

(sin 119899120579

1

cos 1198991205791

)

Mathematical Problems in Engineering 5

=

infin

sum

119899=0

[

[

119868119904(1)

11(119899 1198862) 119868119904(1)

12(119899 1198862)

119868119904(1)minus

21(119899 1198862) 119868119904(1)

22(119899 1198862)

]

]

1198600119899+ 119860(1)

1199042119899

1198620119899+ 119862(1)

1199042119899

times (cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

119868119904(2)

11(119899 1198862) 119868119904(2)

12(119899 1198862)

119868119904(2)minus

21(119899 1198862) 119868119904(2)

22(119899 1198862)

]

]

119860(1)

1199041119899

119862(1)

1199041119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

119868119904(1)

11(119899 1198862) 119868119904(1)minus

12(119899 1198862)

119868119904(1)minus

21(119899 1198862) 119868119904(1)

22(119899 1198862)

]

]

1198610119899+ 119861(1)

1199042119899

1198630119899+ 119863(1)

1199042119899

times (sin 119899120579

1

cos 1198991205791

)

+

infin

sum

119899=0

[

[

119868119904(2)

11(119899 1198862) 119868119904(2)minus

12(119899 1198862)

119868119904(2)minus

21(119899 1198862) 119868119904(2)

22(119899 1198862)

]

]

119861(1)

1199041119899

119863(1)

1199041119899

(sin 119899120579

1

cos 1198991205791

)

(13)

where

119868(119894)

11(119899 119903) = minus119899119862

119899(119896120572119903) + 119896

120572119903119862119899minus1

(119896120572119903)

119868(119894)∓

12(119899 119903) = ∓119899119862

119899(119896120573119903)

119868(119894)∓

21(119899 119903) = ∓119899119862

119899(119896120572119903)

119868(119894)

22(119899 119903) = 119899119862

119899(119896120573119903) minus 119896

120573119903119862119899minus1

(119896120573119903)

(14)

Applying the boundary conditions of 1205911119903119903= 120591119904

119903119903and 1205911119903120579= 120591119904

119903120579

the following equation can be obtained by combining (1) (2)and (9)

infin

sum

119899=0

1205831[

[

1198641(1)

11(119899 1198862) 1198641(1)

12(119899 1198862)

1198641(1)minus

21(119899 1198862) 1198641(1)

22(119899 1198862)

]

]

119860(1)

11119899

119862(1)

11119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

1205831[

[

1198641(2)

11(119899 1198862) 1198641(2)

12(119899 1198862)

1198641(2)minus

21(119899 1198862) 1198641(2)

22(119899 1198862)

]

]

119860(1)

12119899

119862(1)

12119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

1205831[

[

1198641(1)

11(119899 1198862) 1198641(1)minus

12(119899 1198862)

1198641(1)

21(119899 1198862) 1198641(1)

22(119899 1198862)

]

]

119861(1)

11119899

119863(1)

11119899

(sin 119899120579

1

cos 1198991205791

)

+ 1205831

infin

sum

119899=0

[

[

1198641(2)

11(119899 1198861) 1198641(2)minus

12(119899 1198861)

1198641(2)

21(119899 1198861) 1198641(2)

22(119899 1198861)

]

]

119861(1)

12119899

119863(1)

12119899

(sin 119899120579

1

cos 1198991205791

)

=

infin

sum

119899=0

120583119904[

[

119864119904(1)

11(119899 1198862) 119864119904(1)

12(119899 1198862)

119864119904(1)minus

21(119899 1198862) 119864119904(1)

22(119899 1198862)

]

]

1198600119899+ 119860(1)

1199042119899

1198620119899+ 119862(1)

1199042119899

times (cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

120583119904[

[

119864119904(2)

11(119899 1198862) 119864119904(2)

12(119899 1198862)

119864119904(2)minus

21(119899 1198862) 119864119904(2)

22(119899 1198862)

]

]

119860(1)

1199041119899

119862(1)

1199041119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

120583119904[

[

119864119904(1)

11(119899 1198862) 119864119904(1)

12(119899 1198862)

119864119904(1)minus

21(119899 1198862) 119864119904(1)

22(119899 1198862)

]

]

1198610119899+ 119861(1)

1199042119899

1198630119899+ 119863(1)

1199042119899

times (sin 119899120579

1

cos 1198991205791

)

+

infin

sum

119899=0

120583119904[

[

119864119904(2)

11(119899 1198862) 119864119904(2)minus

12(119899 1198862)

119864119904(2)minus

21(119899 1198862) 119864119904(2)

22(119899 1198862)

]

]

119861(1)

1199041119899

119863(1)

1199041119899

(sin 119899120579

1

cos 1198991205791

)

(15)

Solving linear systemof (10) (11) (13) and (15) all coefficientsin the wave functions can be obtained in the coordinatesystem of 119903 minus 120579

1 The wave potential functions 120593

1and 120595

1in

the tunnel lining and120593119904and120595

119904in the half space can be solved

The stress and displacement functions of the tunnel lining forincident plane P waves can be solved from (9)

Generally under the action of steady state P wavesthe dynamic stress distribution of tunnel lining can bestudied through solving the coefficient of dynamic stressconcentration of the inner surface of tunnel lining on thetoroidal direction [13 14]

120591lowast

120579120579can be obtained from

120591lowast

120579120579=

10038161003816100381610038161003816100381610038161003816

120591120579120579

1205910

10038161003816100381610038161003816100381610038161003816

(16)

where 1205910is the maximum value of dynamic stress of tunnel

liningThrough solving the dimensionless coefficient of 120591lowast

120579120579

which is the coefficient of dynamic stress concentration of lin-ing on the toroidal direction the dynamic stress distributionand variation in the tunnel lining for incidence P waves canbe obtained

3 Calculating Examples and Analysis Results

Based on the analytical solutions of the seismic wave prop-agation in an elastic medium and the study of propagationbehavior of seismic waves in the rock and soil mediumseismic motion parameters geotechnical properties and lin-ing materials significantly affect the dynamic stress responseof tunnels and underground structures It is necessary toinvestigate the effect of various parameters including inci-dent frequency content incident angle buried depth rockmaterial properties and the lining stiffness by using theproposed series solutions

Basic assumptions of the application examples employedin this paper are as follows [13] (1) the outside diameter ofcircular tunnel is 119877

2= 12119877

1with a lining thickness of 02119877

1

(2) the buried depth is represented by the ratio (1198671198771) of the

distance from the tunnel center to the ground surface and theinternal radius of tunnel (3) the elastic modulus of tunnellining and surrounding rock is 119890

1and 119890119904 respectively (4) the

6 Mathematical Problems in Engineering

0

30

6090

120

150

210

270300

330

0

20

180

240

40

60

20

40

60

|120591lowast 120579120579|

120578 = 05

120578 = 10120578 = 20

(a) ℎ1 = 20m

120

150

300

330

0

30

6090

180

210

240270

0

30

15

15

30

45

45

120578 = 05

120578 = 10120578 = 20

|120591lowast 120579120579|

(b) ℎ1 = 100m

Figure 2 Dynamic stress concentration coefficient of tunnel lining for different incident frequency and buried depth

lining stiffness 120575 is represented by the shear wave velocityratio of the lining and the rock half space (5) the Poissonrsquosratio of rockmedium and lining is assumed to be 025 (6) thelongitudinal wave velocity of these twomedium is 1732 timesof the shear wave velocity (7) the radius of the large circulararc is 119877 = 119863

12+ 119867 and the distance from the centers O

2to

O1is taken as 119863

12= 1041198771in the example which is large

enough to simulate the half space to ensure that the solutionconverges to the exact solution

31 Effect of Incident Frequency on the Dynamic Stress Concen-tration Coefficient for the Tunnel Lining The dimensionlessfrequency 120578 is defined as the ratio of inner diameter of thetunnel and the incident wave length [15]

120578 =21198771

120582119904120573

=

1198961199041205731198771

120587 (17)

where 120582119904120573is the shear wave length of the half space rock

Considering the effect of different incident frequencycontents of P waves on dynamic response of tunnel liningthree typical cases (120578 = 05 10 and 20) are carried out forincidence P waves from vertical upward direction

Figure 2 shows how the toroidal dynamic stress concen-tration coefficient for the tunnel lining (120591lowast

120579120579) changes with

different incident frequency of P waves for two buried depthsℎ1= 20m and ℎ

1= 100m It is shown that with increasing

incident frequency 120591lowast120579120579

decreases gradually and 120591lowast120579120579

at low-frequency P waves is about 10 times larger than that athigh-frequency Therefore low-frequency seismic waves aremore detrimental to the tunnel lining than high-frequencyAlso the increase of incident frequency leads to increasinglycomplex distribution of the toroidal dynamic stress in thelining and the less principal stress concentration direction at

low-frequency conditions is gradually transformed intomorestress concentration directions at high-frequency conditions

32 Effect of Incident Angle on the Dynamic Stress Con-centration Coefficient of Lining The stress concentrationcoefficients of tunnels and underground structures are verysensitive to incident angle of seismic waves Small changesof the incident angle may lead to a significant change ofthe stress and deformation at various locations of the tunnelstructures Herein studies of three representative cases (ievertical incidencewith incident angle 120579120572=0∘ a small incidentangle 120579120572= 30∘ and large incident angle 120579120572= 60∘) were carriedout for frequency coefficients of 120578 = 1 and 120578 = 2 Figure 3displays the distribution of toroidal dynamic stress of in thelining for three different incident angles

It is observed that the coefficient of toroidal dynamicstress concentration decreases gradually with increasing inci-dent angle but the change in the amplitude is small inboth dimensionless frequency values 120578 = 1 and 120578 = 2 It isworth noting that for vertically incident P waves or a smallincident angle (120579120572 = 0∘ sim30∘) the coefficients of dynamicstress concentration of lining are very complex and unevenThe conclusion here agrees well with the results of shakingtable tests [20]

33 Effect of Buried Depth on the Dynamic Stress Concentra-tion Coefficient of Lining To demonstrate the effect of theburied depth and the material properties two different typesof rockmass (soft and hard rockmass) are chosen to study theseismic response of the tunnel lining within a buried depthrange from 10m to 150m

Case 1 (soft rock as surrounding rock medium of tunnel withelastic modulus 119890

119904= 2GPa) Figure 4 shows the distribution

Mathematical Problems in Engineering 7

0

3

3

6

9

6

9

0

30

6090

120

150

210

270300

330

180

240

120579120572 = 0∘

120579120572 = 30∘

120579120572 = 60∘

|120591lowast 120579120579|

(a) 120578 = 1

0

3

3

6

9

6

9

0

30

6090

120

150

210

270300

330

180

240

120579120572 = 0∘

120579120572 = 30∘

120579120572 = 60∘

|120591lowast 120579120579|

(b) 120578 = 2

Figure 3 Dynamic stress concentration coefficient of tunnel lining with different incident angles

of toroidal dynamic stress in the lining with different burieddepth in a range of 10mndash150m With the increase of theburied depths the dynamic stress concentration coefficientof the lining does not converge to the situation of singleround hole inwhole spaceThedistributions of lining toroidaldynamic stress with different values of buried depth are verycomplex and uneven and no directionality can be obviouslyobserved

As Figure 5 shows when the buried depth in the softsurrounding rock is within 100m the maximum dynamicstress response of the lining varies with buried depth andtwo peak values can be observed at the buried depth ofℎ1= 30m and 100m When the buried depth is greater

than 100m (ℎ1gt 100m) the coefficient of dynamic stress

concentration of lining does not vary significantly with a highlevel of about 65 It appears that with an increasing burieddepth the decreasing trend of dynamic stress response is notpronounced in the soft surrounding rock and the dynamicstress of tunnel lining may still be high in the case of alarge buried depth There are three main reasons (1) thedifference in stiffness is large between tunnel lining and softsurrounding rock (2) the degree of stress concentration oftunnel lining is high and dynamic stress of the lining doesnot decrease obviously with increase of the buried depth and(3) the pressure of surrounding rock increases gradually withincrease of the buried depth In this case the soft surroundingrock has a low strength large porosity many joints andfractures and low capability of self-arching and these rockproperties lead to large tunnel dynamic response

Case 2 (hard surrounding rock medium with elastic modulus119890119904= 20GPa) Regarding the case with hard surrounding

rock medium no obvious directionality can be observed in

terms of the distributions of toroidal dynamic stress causedby P waves but the dynamic stress response of lining ismore pronounced than the case of soft rock as illustrated inFigure 6

Figure 7 shows that when the tunnel buried depth isless than 100m the coefficient of maximum dynamic stressconcentration in the tunnel lining is generally high rangingfrom 60 to 85 which indicates that the response of dynamicstress of the lining is large at the portal segment of hard rocktunnel and the tunnel lining may be damaged easily duringa large earthquake When the tunnel buried depth is greaterthan 75m the coefficient of dynamic stress concentrationis reduced significantly The coefficient of dynamic stressconcentration starts to reach a constant level after the burieddepth is greater than 100m It appears that the lining dynamicstress is small and tends to stabilize at the large buried depthsegment of hard rock tunnel and the tunnel liningmay be lesssusceptible to damage during a large earthquakeTherefore itis concluded that under the hard rock conditions the tunnelburied depth (ie thickness of the overlying rock) is nota major factor that affects the dynamic stress and seismicdamages of tunnel lining when the tunnel buried depth isgreater than 100m This conclusion is consistent with thefindings of Gao et al [21] which states that the dynamicstress concentration coefficient tends to be constant when theburied depth is greater than 25119877

1and the buried depth of

251198771approaches 100m

34 Effect of Elastic Modulus on the Dynamic Stress Con-centration Coefficient of Lining The elastic modulus (119890

1)

of the lining can be considered an index for the elasticdeformation of tunnel lining and the stresses caused by

8 Mathematical Problems in Engineering

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 10mh1 = 20mh1 = 30m

|120591lowast 120579120579|

(a) ℎ1 = 10sim30m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 40mh1 = 50mh1 = 75m

|120591lowast 120579120579|

(b) ℎ1 = 40sim75m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 100mh1 = 125mh1 = 150m

|120591lowast 120579120579|

(c) ℎ1 = 100sim150m

Figure 4 Dynamic stress concentration coefficient of tunnel lining with different values of buried depth in the soft surrounding rock

0123456789

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150Buried depth h1 (m)

|120591lowast 120579120579|

Figure 5 The maximum dynamic stress concentration coefficientof tunnel lining with different buried depths in the soft surroundingrock

elastic deformation increase with increasing elastic modu-lus This also means that at a given level of stresses thelining deformation decreases with increasing lining stiffnessWe focus on the effect of lining elastic modulus on thedynamic stress concentration coefficient of the lining ina soft surrounding rock Figure 8 shows the coefficientsof dynamic stress concentration (120591lowast

120579120579) with different lining

elastic modulus in a soft surrounding rock (119890119904= 2GPa)

As shown in Figure 8 the coefficient of dynamicstress concentration of the lining increases gradually withincreasing lining elastic modulus and the greater the elasticmodulus is the more uneven the distribution of the dynamicstress coefficient is The different adaptabilities of lining

Mathematical Problems in Engineering 9

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 10mh1 = 20mh1 = 30m

|120591lowast 120579120579|

(a) ℎ1 = 10sim30m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 40mh1 = 50mh1 = 75m

|120591lowast 120579120579|

(b) ℎ1 = 40sim75m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 100mh1 = 125mh1 = 150m

|120591lowast 120579120579|

(c) ℎ1 = 100sim150m

Figure 6 Dynamic stress concentration coefficient of tunnel lining with different buried depths in hard surrounding rock

0123456789

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150Buried depth h1 (m)

|120591lowast 120579120579|

Figure 7Themaximumdynamic stress concentration coefficient oftunnel lining with different buried depths in hard surrounding rock

with different stiffness to dynamic stresses may lead to thevariation of the dynamic stress of lining Herein the followingexample is adopted to illustrate this aspect When the elasticmodulus of the lining is 119890

1= 10Gpa and soft surrounding

rock is 119890119904= 2Gpa (ie 119890

1= 5119890119904) the maximum value of

dynamic stress concentration coefficient of the lining is morethan 100 and the variation of dynamic stress of lining acrossall directions is very complexTherefore under the premise ofmeeting the bearing capacity and deformation of the liningappropriately soft lining is recommended to be adopted inthe tunnel structure to mitigate the seismic damage of tunnellining and the ratio between the elastic modulus of the liningand soft surrounding rock should be less than 5

10 Mathematical Problems in Engineering

0

10

20

30

10

20

30

0

30

6090

120

150

210

270300

330

180

240

e1 = 2Gpae1 = 4Gpa

|120591lowast 120579120579|

(a) 1198901 = 2sim4GPa

0

100

100

200

300

400

200

300

400

0

30

6090

120

150

180

210

240270

300

330

e1 = 6Gpae1 = 8Gpa

e1 = 10Gpae1 = 20Gpa

|120591lowast 120579120579|

(b) 1198901 = 6sim20GPa

Figure 8 Dynamic stress concentration coefficient of tunnel lining with different lining elastic modulus in the soft surrounding rock

4 Conclusions

A series solution of wave equation for dynamic response ofunderground circular lining tunnels subjected to incidentplane P waves is presented by Fourier-Bessel series expansionmethod in this paper The effects on circular lining rocktunnels of different factors including incident frequencyincident angle buried depth rock conditions and liningstiffness are studied through the proposed series solution Inthis study the following conclusions are drawn

(1) In circular lining rock tunnels the coefficient oftoroidal dynamic stress concentration and dynamicstress decrease gradually with the increase of incidentfrequency of P waves 120591lowast

120579120579for low-frequency P waves

is about 10 times larger than that for high-frequencyindicating that low-frequency contents of seismicwaves are more detrimental to the tunnel linings thanhigh-frequency contents

(2) The incident angle of P waves has a significantinfluence on dynamic response of the tunnel liningWhen seismic waves propagate vertically or with asmall incident angle (120579120572 = 0∘ sim30∘) the distributionof the coefficients of dynamic stress concentration inthe tunnel lining is very complex and uneven whichis detrimental to the safety of the tunnel lining

(3) When the buried depth of a tunnel in the soft sur-rounding rock is in a range of 0sim100m the decreasingtendency of dynamic stress response of the liningwith increasing tunnel buried depth is not obviouseven if the buried depth is greater than 100m thecoefficient of dynamic stress concentration of thelining still remains at a high level of about 65 In thissituation tunnels crossing the fault zone or the high

stress segment may get damaged seriously duringlarge earthquake

(4) When the tunnel buried depth in hard surroundingrock is less than 100m the coefficient of dynamicstress concentration in the tunnel lining is highranging from 60 to 85 While the buried depth isgreater than 100m the dynamic stress of the lining issmall and tends to stabilize When the tunnel burieddepth is more than 100m the tunnel buried depth isnot a major factor that affects the dynamic stress

(5) The coefficient of dynamic stress concentration inthe lining increases gradually with increasing elasticmodulus of the lining and the greater the elasticmodulus is the more uneven the dynamic stresscoefficient is Therefore under the premise of meet-ing the bearing capacity and deformation of liningappropriate soft lining is recommended to be adoptedin the tunnel structures to mitigate seismic damageof tunnel lining and the ratio between the elasticmodulus of the lining and soft surrounding rockshould be less than 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of China(Grant no 20110184120033) the Open Foundation of StateKey Laboratory of Geohazard Prevention and Geoenviron-ment Protection (Chengdu University of Technology Grant

Mathematical Problems in Engineering 11

no SKLGP2011K014) and the Science Foundation for Excel-lent Youth Scholars of Southwest JiaotongUniversity of China(Grant no SWJTU11BR003)

References

[1] T B Li ldquoDamage to mountain tunnels related to theWenchuanearthquake and some suggestions for aseismic tunnel construc-tionrdquo Bulletin of Engineering Geology and the Environment vol71 no 2 pp 297ndash308 2012

[2] I Towhata Geotechnical Earthquake Engineering SpringerBerlin Germany 2008

[3] V W Lee and M D Trifunac ldquoResponse of tunnels to incidentSH-wavesrdquo Journal of the Engineering Mechanics Division vol105 no 4 pp 643ndash659 1979

[4] Y L Zheng L D Yang and W Y Li Seismic Design of Under-ground Structure Tongji University Press Shanghai China2005

[5] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak and CompanyNew York NY USA 1973

[6] V W Lee and H Cao ldquoDiffraction of SV waves by circularcanyons of various depthsrdquo Journal of Engineering Mechanicsvol 115 no 9 pp 2035ndash2056 1989

[7] H Cao and V W Lee ldquoScattering and diffraction of planeP waves by circular cylindrical canyons with variable depth-to-width ratiordquo International Journal of Soil Dynamics andEarthquake Engineering vol 9 no 3 pp 141ndash150 1990

[8] VW Lee and J Karl ldquoDiffraction of SV waves by undergroundcircular cylindrical cavitiesrdquo Soil Dynamics and EarthquakeEngineering vol 11 no 8 pp 445ndash456 1992

[9] V W Lee and J Karl ldquoOn deformation of near a circularunderground cavity subjected to incident plane P wavesrdquoEuropean Journal of Earthquake Engineering vol 7 no 1 pp 29ndash35 1993

[10] M Manoogian and V W Lee ldquoDiffraction of SH-waves bysubsurface inclusions of arbitrary shaperdquo Journal of EngineeringMechanics vol 122 no 2 pp 123ndash129 1996

[11] C A Davis V W Lee and J P Bardet ldquoTransverse responceof underground cavities and pipes to incident SV wavesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 383ndash410 2001

[12] J W Liang H Zhang and V W Lee ldquoAnalytical solution fordynamic stress concentration of underground cavities underincident plane P wavesrdquo Chinese Journal of Geotechnical Engi-neering vol 26 no 6 pp 815ndash819 2004

[13] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (I) analytical solutionrdquoEarthquake Engineering and Engineering Vibration vol 27 no1 pp 1ndash6 2007

[14] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (II) numerical resultsrdquoEarthquake Engineering and Engineering Vibration vol 27 no2 pp 1ndash11 2007

[15] X D Ji J W Liang and J J Yang ldquoDynamic stress concen-tration of an underground cylindrical lined cavity subjected toincident plane P and SV wavesrdquo Journal of Tianjin UniversityScience and Technology vol 39 no 5 pp 511ndash517 2006

[16] H B You and J W Liang ldquoScattering of plane SV waves by acavity in a layered half-spacerdquo Rock and Soil Mechanics vol 27no 3 pp 383ndash388 2006

[17] G P Kouretzis G D Bouckovalas and C J Gantes ldquo3-D shell analysis of cylindrical underground structures underseismic shear (S) wave actionrdquo Soil Dynamics and EarthquakeEngineering vol 26 no 10 pp 909ndash921 2006

[18] M Esmaeili S Vahdani andANoorzad ldquoDynamic response oflined circular tunnel to plane harmonic wavesrdquo Tunnelling andUnderground Space Technology vol 11 no 17 pp 1ndash9 2005

[19] H Xu T B Li and L Q Li ldquoResearch on dynamic responseof underground circular lining tunnel under the action of Pwavesrdquo Applied Mechanics and Materials vol 99-100 pp 181ndash189 2011

[20] H Xu T B Li DWang Y S Li and Z H Lin ldquoStudy of seismicresponses of mountain tunnels with 3D shaking table modeltestrdquo Chinese Journal of Rock Mechanics and Engineering vol32 no 9 pp 1762ndash1771 2013

[21] G Y Gao Y S Li and T B Li ldquoDynamic analysis of shallowburied circular lining tunnel with vertical incident P waverdquoCentral South Highway Engineering vol 32 no 2 pp 56ndash602007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

where 119896119904120572

= 120596119888120572119904

is the longitudinal wave number 119896119904120573

=

120596119888120573119904is the transverse wave number and 119869

119899(119909) is the first type

of Bessel function where the time factor exp(minus119894120596119905) has beenomittedThe coefficients in formula (1) are defined as follows

1198600119899

1198610119899

= 120576119899119894119899cos 119899120579

120572

sin 119899120579120572

times [plusmn (minus1)119899 exp (minus119894119896

119904120572ℎ cos 120579

120572) + 1198961exp (119894119896

119904120572ℎ cos 120579

120572)]

1198620119899

1198630119899

= 1205761198991198941198991198962sin 119899120579

120573

cos 119899120579120573

exp (119894119896119904120573ℎ cos 120579

120573)

(2)

where 119894 is the imaginary unit 120576119899= 1 for 119899 = 0 and 120576

119899= 2

for 119899 ge 1 Symbol 120579120573is the reflection angle of reflected SV

waves which satisfies the equation sin 120579120572119888120572119904

= sin 120579120573119888120573119904

The reflection coefficients 1198961and 1198962are given by

1198961=

sin 2120579120572sin 2120579

120573minus (119888120572119904119888120573119904)2

cos22120579120573

sin 2120579120572sin 2120579

120573+ (119888120572119904119888120573119904)2

cos22120579120573

1198962=

minus2 sin 2120579120572cos 2120579

120573

sin 2120579120572sin 2120579

120573+ (119888120572119904119888120573119904)2

cos22120579120573

(3)

In O2coordinate system the wave potential functions of

the incident plane Pwaves and reflected plane P and SVwavesare similar to O

1coordinate system but are not presented

here

22 Analyzing Scattering Field at the Interface between TunnelLining and the Half Space The presence of the interfacebetween the circular tunnel lining and the half space leads toscattering P waves 120593

11(1199031 1205791) and SVwaves120595

11(1199031 1205791) which

can be expressed as series forms as below [15]

12060111(1199031 1205791) =

infin

sum

119899=0

119869119899(11989611205721199031) (119860(1)

11119899cos 119899120579

1+ 119861(1)

11119899sin 119899120579

1)

12059511(1199031 1205791) =

infin

sum

119899=0

119869119899(11989611205731199031) (119862(1)

11119899sin 119899120579

1+ 119863(1)

11119899cos 119899120579

1)

12060112(1199031 1205791) =

infin

sum

119899=0

1198671

119899(11989611205721199031) (119860(1)

12119899cos 119899120579

1+ 119861(1)

12119899sin 119899120579

1)

12059512(1199031 1205791) =

infin

sum

119899=0

1198671

119899(11989611205731199031) (119862(1)

12119899sin 119899120579

1+ 119863(1)

12119899cos 119899120579

1)

(4)

where 119869119899(119909) and 119867

(1)

119899(119909) are the first class Bessel function

and the Hankel function respectively The wave numbers ofP waves and SV waves of tunnel lining are 119896

1120572= 120596119888

1205721

and 1198961120573

= 1205961198881205731 respectively In the lining medium the

expressions of P waves and SV waves are complete Through

the coordinate transformation 1206011199042(1199032 1205792) and 120595

1199042(1199032 1205792) can

be converted into the Bessel series form in the coordinatesystem of 119903

1minus 1205791and 119869

119899(1198961199041205721199031) and 119867

(1)

119899(1198961199041205721199031) in the

coordinate system 1199031minus 1205791are a set of basis functions of the

wave equationSimilarly for the presence of circular tunnel lining there

are scattering P waves 1206011199041(1199031 1205791) and SV waves 120595

1199041(1199031 1205791)

which are caused at the interface between the tunnel liningand the half space Built on the assumption of large circulararc the scattering P waves and SV waves are denoted as12060112(1199031 1205791) and 120595

1199042(1199032 1205792) They can be expressed as series

forms

1206011199041(1199031 1205791)

=

infin

sum

119899=0

119867(1)

119899(1198961199041205721199031) (119860(1)

1199041119899cos 119899120579

1+ 119861(1)

1199041119899sin 119899120579

1)

1205951199041(1199031 1205791)

=

infin

sum

119899=0

119867(1)

119899(1198961199041205731199031) (119862(1)

1199041119899sin 119899120579

1+ 119863(1)

1199041119899cos 119899120579

1)

1206011199042(1199032 1205792)

=

infin

sum

119898=0

119869119898(1198961199041205721199032) (119860(2)

1199042119898cos119898120579

2+ 119861(2)

1199042119898sin119898120579

2)

1205951199042(1199032 1205792)

=

infin

sum

119898=0

119869119898(1198961199041205731199032) (119862(2)

1199042119898sin119898120579

2+ 119863(2)

1199042119898cos119898120579

2)

(5)

where 119896119904120572

= 120596119888120572119904

and 119896119904120573

= 120596119888120573119904

are the wave numbersof P waves and SV waves in the half space From a physicalpoint of view 120601

1199041(1199031 1205791) and120595

1199041(1199031 1205791) are represented by the

outward propagation waves from 1199001point while 120601

1199042(1199032 1205792)

and 1205951199042(1199032 1205792) are generated by the surface of the half space

in the large circular arcTherefore the expressions of P wavesand SV waves are complete in the half space medium

The wave potential function in the lining medium can beexpressed as

1206011= 12060111+ 12060112 120595

1= 12059511+ 12059512 (6)

Thewave potential function in the half spacemedium canbe written as

120601119904= 120601(119894+119903)

+ 1206011199041+ 1206011199042 120595

119904= 120595(119903)+ 1205951199041+ 1205951199042 (7)

23 Solving Series Solution after Imposing Boundary Condi-tions Under a plane strain assumption there are two familiesof boundary condition (1) the stress at the surface of the halfspace and the tunnel lining inner surface is zero (2) the stressand displacement are continuous at the interface of the tunnellining and the half space So the boundary conditions of the

4 Mathematical Problems in Engineering

stress and displacement of the tunnel lining are summarizedas

120591119904

119903119903= 120591119904

119903120579= 0 119903

2= 119887

1205911

119903119903= 1205911

119903120579= 0 119903

1= 1198861

1205911

119903119903= 120591119904

119903119903 120591

1

119903120579= 120591119904

119903120579

1199061

119903= 119906119904

119903 119906

1

120579= 119906119904

120579 119903

2= 1198862

(8)

Based on the boundary conditions in (8) the coefficientsin (4) and (5) can be obtained The wave potential functions1206011and120595

1 in the tunnel lining and120601

119904and120595

119904 in the half space

can be solved by (6) and (7) The stress and displacementexpressions under a plane strain assumption for incidentplane P waves can be described by

119906119903=120597120601

120597119903+1

119903

120597120593

120597120579 119906

120579=1

119903

120597120601

120597120579minus120597120593

120597119903

120591119903119903= 120582nabla2120601 + 2120583 [

1205972120601

1205971199032+120597

120597119903(1

119903

120597120593

120597120579)]

120591119903120579= 2120583(

1

119903

1205972120601

120597119903120597120579minus1

1199032

120597120601

120597120579) + 120583[

1

1199032

1205972120593

1205971205792minus 119903

120597

120597119903(1

119903

120597120593

120597119903)]

120591120579120579= 120582nabla2120601 + 2120583 [

1

119903(120597120601

120597119903+1

119903

1205972120601

1205971205792) +

1

119903(1

119903

120597120593

120597120579minus1205972120593

120597119903120597120579)]

(9)

After imposing the boundary conditions of 120591119904119903119903= 120591119904

119903120579= 0

and 1205911119903119903= 1205911

119903120579= 0 (10) and (11) can be obtained by combining

(1) (2) and (9) [5]

infin

sum

119899=0

[

[

1198641(1)

11(119899 1198861) 1198641(1)

12(119899 1198861)

1198641(1)

21(119899 1198861) 1198641(1)

22(119899 1198861)

]

]

119860(1)

11119899

119862(1)

11119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

1198641(2)

11(119899 1198861) 1198641(2)

12(119899 1198861)

1198641(2)

21(119899 1198861) 1198641(2)

22(119899 1198861)

]

]

119860(1)

12119899

119862(1)

12119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

1198641(1)

11(119899 1198861) 1198641(1)

12(119899 1198861)

1198641(1)

21(119899 1198861) 1198641(1)

22(119899 1198861)

]

]

119861(1)

11119899

119863(1)

11119899

(sin 119899120579

1

cos 1198991205791

)

+

infin

sum

119899=0

[

[

1198641(2)

11(119899 1198861) 1198641(2)

12(119899 1198861)

1198641(2)

21(119899 1198861) 1198641(2)

22(119899 1198861)

]

]

119861(1)

12119899

119863(1)

12119899

(sin 119899120579

1

cos 1198991205791

)

= 0

0

(10)

infin

sum

119898=0

[

[

119864119904(2)

11(119898 119887) 119864

119904(2)

12(119898 119887)

119864119904(2)minus

21(119898 119887) 119864

119904(2)

22(119898 119887)

]

]

119860(2)

1199041119898

119862(2)

1199041119898

(cos119898120579

2

sin1198981205792

)

+

infin

sum

119898=0

[

[

119864119904(1)

11(119898 119887) 119864

119904(1)

12(119898 119887)

119864119904(1)minus

21(119898 119887) 119864

119904(1)

22(119898 119887)

]

]

119860(2)

1199042119898

119862(2)

1199042119898

(cos119898120579

2

sin1198981205792

)

+

infin

sum

119898=0

[119864119904(2)

11(119898 119887) 119864

119904(2)minus

12(119898 119887)

119864119904(2)

21(119898 119887) 119864

119904(2)

22(119898 119887)

]

119861(2)

1199041119898

119863(2)

1199041119898

(sin119898120579

2

cos1198981205792

)

+

infin

sum

119898=0

[

[

119864119904(1)

11(119898 119887) 119864

119904(1)minus

12(119898 119887)

119864119904(1)

21(119898 119887) 119864

119904(1)

22(119898 119887)

]

]

119861(2)

1199041119898

119863(2)

1199041119898

(sin119898120579

2

cos1198981205792

)

= 0

0

(11)

where

119864(119894)

11(119899 119903) = (119899

2+ 119899 minus

1

21198962

1205731199032)119862119899(119896120572119903) minus 119896

120572119903119862119899minus1

(119896120572119903)

119864(119894)∓

12(119899 119903) = ∓119899 [minus (119899 + 1) 119862

119899(119896120572119903) + 119896

120573119903119862119899minus1

(119896120573119903)]

119864(119894)∓

21(119899 119903) = ∓119899 [minus (119899 + 1) 119862

119899(119896120572119903) + 119896

120572119903119862119899minus1

(119896120572119903)]

119864(119894)

22(119899 119903) = minus (119899

2+ 119899 minus

1

21198962

1205731199032)119862119899(119896120573119903) minus 119896

120573119903119862119899minus1

(119896120573119903)

(12)

When 119894 = 1 119862119899(119909) is a function of 119869

119899(119909) when 119894 = 2 119862

119899(119909) is

the function of119867(1)119899(119909) For the half plane referred to as ldquo119904rdquo

119896120572and 119896120573can be replaced by 119896

119904120572and 119896119904120573 respectively For the

tunnel lining referred to as ldquo1rdquo 119896120572and 119896120573can be replaced by

1198961120572

and 1198961120573 respectively Potential functions 120601(119894+119903) 120595(119903) 120601(119903)

and 120595(119894+119903)satisfy the stress boundary in the half spaceApplying the boundary conditions of 1199061

119903= 119906119904

119903and 1199061

120579=

119906119904

120579 the following equation can be obtained by combining (1)

(2) and (9)

infin

sum

119899=0

[

[

1198681(1)

11(119899 1198862) 1198681(1)

12(119899 1198862)

1198681(1)minus

21(119899 1198862) 1198681(1)

22(119899 1198862)

]

]

119860(1)

11119899

119862(1)

11119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

1198681(2)

11(119899 1198862) 1198681(2)

12(119899 1198862)

1198681(2)minus

21(119899 1198862) 1198681(2)

22(119899 1198862)

]

]

119860(1)

12119899

119862(1)

12119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

1198681(1)

11(119899 1198862) 1198681(1)minus

12(119899 1198862)

1198681(1)

21(119899 1198862) 1198681(1)

22(119899 1198862)

]

]

119861(1)

11119899

119863(1)

11119899

(sin 119899120579

1

cos 1198991205791

)

+

infin

sum

119899=0

[

[

1198681(2)

11(119899 1198862) 1198681(2)minus

12(119899 1198862)

1198681(2)

21(119899 1198862) 1198681(2)

22(119899 1198862)

]

]

119861(1)

12119899

119863(1)

12119899

(sin 119899120579

1

cos 1198991205791

)

Mathematical Problems in Engineering 5

=

infin

sum

119899=0

[

[

119868119904(1)

11(119899 1198862) 119868119904(1)

12(119899 1198862)

119868119904(1)minus

21(119899 1198862) 119868119904(1)

22(119899 1198862)

]

]

1198600119899+ 119860(1)

1199042119899

1198620119899+ 119862(1)

1199042119899

times (cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

119868119904(2)

11(119899 1198862) 119868119904(2)

12(119899 1198862)

119868119904(2)minus

21(119899 1198862) 119868119904(2)

22(119899 1198862)

]

]

119860(1)

1199041119899

119862(1)

1199041119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

119868119904(1)

11(119899 1198862) 119868119904(1)minus

12(119899 1198862)

119868119904(1)minus

21(119899 1198862) 119868119904(1)

22(119899 1198862)

]

]

1198610119899+ 119861(1)

1199042119899

1198630119899+ 119863(1)

1199042119899

times (sin 119899120579

1

cos 1198991205791

)

+

infin

sum

119899=0

[

[

119868119904(2)

11(119899 1198862) 119868119904(2)minus

12(119899 1198862)

119868119904(2)minus

21(119899 1198862) 119868119904(2)

22(119899 1198862)

]

]

119861(1)

1199041119899

119863(1)

1199041119899

(sin 119899120579

1

cos 1198991205791

)

(13)

where

119868(119894)

11(119899 119903) = minus119899119862

119899(119896120572119903) + 119896

120572119903119862119899minus1

(119896120572119903)

119868(119894)∓

12(119899 119903) = ∓119899119862

119899(119896120573119903)

119868(119894)∓

21(119899 119903) = ∓119899119862

119899(119896120572119903)

119868(119894)

22(119899 119903) = 119899119862

119899(119896120573119903) minus 119896

120573119903119862119899minus1

(119896120573119903)

(14)

Applying the boundary conditions of 1205911119903119903= 120591119904

119903119903and 1205911119903120579= 120591119904

119903120579

the following equation can be obtained by combining (1) (2)and (9)

infin

sum

119899=0

1205831[

[

1198641(1)

11(119899 1198862) 1198641(1)

12(119899 1198862)

1198641(1)minus

21(119899 1198862) 1198641(1)

22(119899 1198862)

]

]

119860(1)

11119899

119862(1)

11119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

1205831[

[

1198641(2)

11(119899 1198862) 1198641(2)

12(119899 1198862)

1198641(2)minus

21(119899 1198862) 1198641(2)

22(119899 1198862)

]

]

119860(1)

12119899

119862(1)

12119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

1205831[

[

1198641(1)

11(119899 1198862) 1198641(1)minus

12(119899 1198862)

1198641(1)

21(119899 1198862) 1198641(1)

22(119899 1198862)

]

]

119861(1)

11119899

119863(1)

11119899

(sin 119899120579

1

cos 1198991205791

)

+ 1205831

infin

sum

119899=0

[

[

1198641(2)

11(119899 1198861) 1198641(2)minus

12(119899 1198861)

1198641(2)

21(119899 1198861) 1198641(2)

22(119899 1198861)

]

]

119861(1)

12119899

119863(1)

12119899

(sin 119899120579

1

cos 1198991205791

)

=

infin

sum

119899=0

120583119904[

[

119864119904(1)

11(119899 1198862) 119864119904(1)

12(119899 1198862)

119864119904(1)minus

21(119899 1198862) 119864119904(1)

22(119899 1198862)

]

]

1198600119899+ 119860(1)

1199042119899

1198620119899+ 119862(1)

1199042119899

times (cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

120583119904[

[

119864119904(2)

11(119899 1198862) 119864119904(2)

12(119899 1198862)

119864119904(2)minus

21(119899 1198862) 119864119904(2)

22(119899 1198862)

]

]

119860(1)

1199041119899

119862(1)

1199041119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

120583119904[

[

119864119904(1)

11(119899 1198862) 119864119904(1)

12(119899 1198862)

119864119904(1)minus

21(119899 1198862) 119864119904(1)

22(119899 1198862)

]

]

1198610119899+ 119861(1)

1199042119899

1198630119899+ 119863(1)

1199042119899

times (sin 119899120579

1

cos 1198991205791

)

+

infin

sum

119899=0

120583119904[

[

119864119904(2)

11(119899 1198862) 119864119904(2)minus

12(119899 1198862)

119864119904(2)minus

21(119899 1198862) 119864119904(2)

22(119899 1198862)

]

]

119861(1)

1199041119899

119863(1)

1199041119899

(sin 119899120579

1

cos 1198991205791

)

(15)

Solving linear systemof (10) (11) (13) and (15) all coefficientsin the wave functions can be obtained in the coordinatesystem of 119903 minus 120579

1 The wave potential functions 120593

1and 120595

1in

the tunnel lining and120593119904and120595

119904in the half space can be solved

The stress and displacement functions of the tunnel lining forincident plane P waves can be solved from (9)

Generally under the action of steady state P wavesthe dynamic stress distribution of tunnel lining can bestudied through solving the coefficient of dynamic stressconcentration of the inner surface of tunnel lining on thetoroidal direction [13 14]

120591lowast

120579120579can be obtained from

120591lowast

120579120579=

10038161003816100381610038161003816100381610038161003816

120591120579120579

1205910

10038161003816100381610038161003816100381610038161003816

(16)

where 1205910is the maximum value of dynamic stress of tunnel

liningThrough solving the dimensionless coefficient of 120591lowast

120579120579

which is the coefficient of dynamic stress concentration of lin-ing on the toroidal direction the dynamic stress distributionand variation in the tunnel lining for incidence P waves canbe obtained

3 Calculating Examples and Analysis Results

Based on the analytical solutions of the seismic wave prop-agation in an elastic medium and the study of propagationbehavior of seismic waves in the rock and soil mediumseismic motion parameters geotechnical properties and lin-ing materials significantly affect the dynamic stress responseof tunnels and underground structures It is necessary toinvestigate the effect of various parameters including inci-dent frequency content incident angle buried depth rockmaterial properties and the lining stiffness by using theproposed series solutions

Basic assumptions of the application examples employedin this paper are as follows [13] (1) the outside diameter ofcircular tunnel is 119877

2= 12119877

1with a lining thickness of 02119877

1

(2) the buried depth is represented by the ratio (1198671198771) of the

distance from the tunnel center to the ground surface and theinternal radius of tunnel (3) the elastic modulus of tunnellining and surrounding rock is 119890

1and 119890119904 respectively (4) the

6 Mathematical Problems in Engineering

0

30

6090

120

150

210

270300

330

0

20

180

240

40

60

20

40

60

|120591lowast 120579120579|

120578 = 05

120578 = 10120578 = 20

(a) ℎ1 = 20m

120

150

300

330

0

30

6090

180

210

240270

0

30

15

15

30

45

45

120578 = 05

120578 = 10120578 = 20

|120591lowast 120579120579|

(b) ℎ1 = 100m

Figure 2 Dynamic stress concentration coefficient of tunnel lining for different incident frequency and buried depth

lining stiffness 120575 is represented by the shear wave velocityratio of the lining and the rock half space (5) the Poissonrsquosratio of rockmedium and lining is assumed to be 025 (6) thelongitudinal wave velocity of these twomedium is 1732 timesof the shear wave velocity (7) the radius of the large circulararc is 119877 = 119863

12+ 119867 and the distance from the centers O

2to

O1is taken as 119863

12= 1041198771in the example which is large

enough to simulate the half space to ensure that the solutionconverges to the exact solution

31 Effect of Incident Frequency on the Dynamic Stress Concen-tration Coefficient for the Tunnel Lining The dimensionlessfrequency 120578 is defined as the ratio of inner diameter of thetunnel and the incident wave length [15]

120578 =21198771

120582119904120573

=

1198961199041205731198771

120587 (17)

where 120582119904120573is the shear wave length of the half space rock

Considering the effect of different incident frequencycontents of P waves on dynamic response of tunnel liningthree typical cases (120578 = 05 10 and 20) are carried out forincidence P waves from vertical upward direction

Figure 2 shows how the toroidal dynamic stress concen-tration coefficient for the tunnel lining (120591lowast

120579120579) changes with

different incident frequency of P waves for two buried depthsℎ1= 20m and ℎ

1= 100m It is shown that with increasing

incident frequency 120591lowast120579120579

decreases gradually and 120591lowast120579120579

at low-frequency P waves is about 10 times larger than that athigh-frequency Therefore low-frequency seismic waves aremore detrimental to the tunnel lining than high-frequencyAlso the increase of incident frequency leads to increasinglycomplex distribution of the toroidal dynamic stress in thelining and the less principal stress concentration direction at

low-frequency conditions is gradually transformed intomorestress concentration directions at high-frequency conditions

32 Effect of Incident Angle on the Dynamic Stress Con-centration Coefficient of Lining The stress concentrationcoefficients of tunnels and underground structures are verysensitive to incident angle of seismic waves Small changesof the incident angle may lead to a significant change ofthe stress and deformation at various locations of the tunnelstructures Herein studies of three representative cases (ievertical incidencewith incident angle 120579120572=0∘ a small incidentangle 120579120572= 30∘ and large incident angle 120579120572= 60∘) were carriedout for frequency coefficients of 120578 = 1 and 120578 = 2 Figure 3displays the distribution of toroidal dynamic stress of in thelining for three different incident angles

It is observed that the coefficient of toroidal dynamicstress concentration decreases gradually with increasing inci-dent angle but the change in the amplitude is small inboth dimensionless frequency values 120578 = 1 and 120578 = 2 It isworth noting that for vertically incident P waves or a smallincident angle (120579120572 = 0∘ sim30∘) the coefficients of dynamicstress concentration of lining are very complex and unevenThe conclusion here agrees well with the results of shakingtable tests [20]

33 Effect of Buried Depth on the Dynamic Stress Concentra-tion Coefficient of Lining To demonstrate the effect of theburied depth and the material properties two different typesof rockmass (soft and hard rockmass) are chosen to study theseismic response of the tunnel lining within a buried depthrange from 10m to 150m

Case 1 (soft rock as surrounding rock medium of tunnel withelastic modulus 119890

119904= 2GPa) Figure 4 shows the distribution

Mathematical Problems in Engineering 7

0

3

3

6

9

6

9

0

30

6090

120

150

210

270300

330

180

240

120579120572 = 0∘

120579120572 = 30∘

120579120572 = 60∘

|120591lowast 120579120579|

(a) 120578 = 1

0

3

3

6

9

6

9

0

30

6090

120

150

210

270300

330

180

240

120579120572 = 0∘

120579120572 = 30∘

120579120572 = 60∘

|120591lowast 120579120579|

(b) 120578 = 2

Figure 3 Dynamic stress concentration coefficient of tunnel lining with different incident angles

of toroidal dynamic stress in the lining with different burieddepth in a range of 10mndash150m With the increase of theburied depths the dynamic stress concentration coefficientof the lining does not converge to the situation of singleround hole inwhole spaceThedistributions of lining toroidaldynamic stress with different values of buried depth are verycomplex and uneven and no directionality can be obviouslyobserved

As Figure 5 shows when the buried depth in the softsurrounding rock is within 100m the maximum dynamicstress response of the lining varies with buried depth andtwo peak values can be observed at the buried depth ofℎ1= 30m and 100m When the buried depth is greater

than 100m (ℎ1gt 100m) the coefficient of dynamic stress

concentration of lining does not vary significantly with a highlevel of about 65 It appears that with an increasing burieddepth the decreasing trend of dynamic stress response is notpronounced in the soft surrounding rock and the dynamicstress of tunnel lining may still be high in the case of alarge buried depth There are three main reasons (1) thedifference in stiffness is large between tunnel lining and softsurrounding rock (2) the degree of stress concentration oftunnel lining is high and dynamic stress of the lining doesnot decrease obviously with increase of the buried depth and(3) the pressure of surrounding rock increases gradually withincrease of the buried depth In this case the soft surroundingrock has a low strength large porosity many joints andfractures and low capability of self-arching and these rockproperties lead to large tunnel dynamic response

Case 2 (hard surrounding rock medium with elastic modulus119890119904= 20GPa) Regarding the case with hard surrounding

rock medium no obvious directionality can be observed in

terms of the distributions of toroidal dynamic stress causedby P waves but the dynamic stress response of lining ismore pronounced than the case of soft rock as illustrated inFigure 6

Figure 7 shows that when the tunnel buried depth isless than 100m the coefficient of maximum dynamic stressconcentration in the tunnel lining is generally high rangingfrom 60 to 85 which indicates that the response of dynamicstress of the lining is large at the portal segment of hard rocktunnel and the tunnel lining may be damaged easily duringa large earthquake When the tunnel buried depth is greaterthan 75m the coefficient of dynamic stress concentrationis reduced significantly The coefficient of dynamic stressconcentration starts to reach a constant level after the burieddepth is greater than 100m It appears that the lining dynamicstress is small and tends to stabilize at the large buried depthsegment of hard rock tunnel and the tunnel liningmay be lesssusceptible to damage during a large earthquakeTherefore itis concluded that under the hard rock conditions the tunnelburied depth (ie thickness of the overlying rock) is nota major factor that affects the dynamic stress and seismicdamages of tunnel lining when the tunnel buried depth isgreater than 100m This conclusion is consistent with thefindings of Gao et al [21] which states that the dynamicstress concentration coefficient tends to be constant when theburied depth is greater than 25119877

1and the buried depth of

251198771approaches 100m

34 Effect of Elastic Modulus on the Dynamic Stress Con-centration Coefficient of Lining The elastic modulus (119890

1)

of the lining can be considered an index for the elasticdeformation of tunnel lining and the stresses caused by

8 Mathematical Problems in Engineering

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 10mh1 = 20mh1 = 30m

|120591lowast 120579120579|

(a) ℎ1 = 10sim30m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 40mh1 = 50mh1 = 75m

|120591lowast 120579120579|

(b) ℎ1 = 40sim75m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 100mh1 = 125mh1 = 150m

|120591lowast 120579120579|

(c) ℎ1 = 100sim150m

Figure 4 Dynamic stress concentration coefficient of tunnel lining with different values of buried depth in the soft surrounding rock

0123456789

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150Buried depth h1 (m)

|120591lowast 120579120579|

Figure 5 The maximum dynamic stress concentration coefficientof tunnel lining with different buried depths in the soft surroundingrock

elastic deformation increase with increasing elastic modu-lus This also means that at a given level of stresses thelining deformation decreases with increasing lining stiffnessWe focus on the effect of lining elastic modulus on thedynamic stress concentration coefficient of the lining ina soft surrounding rock Figure 8 shows the coefficientsof dynamic stress concentration (120591lowast

120579120579) with different lining

elastic modulus in a soft surrounding rock (119890119904= 2GPa)

As shown in Figure 8 the coefficient of dynamicstress concentration of the lining increases gradually withincreasing lining elastic modulus and the greater the elasticmodulus is the more uneven the distribution of the dynamicstress coefficient is The different adaptabilities of lining

Mathematical Problems in Engineering 9

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 10mh1 = 20mh1 = 30m

|120591lowast 120579120579|

(a) ℎ1 = 10sim30m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 40mh1 = 50mh1 = 75m

|120591lowast 120579120579|

(b) ℎ1 = 40sim75m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 100mh1 = 125mh1 = 150m

|120591lowast 120579120579|

(c) ℎ1 = 100sim150m

Figure 6 Dynamic stress concentration coefficient of tunnel lining with different buried depths in hard surrounding rock

0123456789

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150Buried depth h1 (m)

|120591lowast 120579120579|

Figure 7Themaximumdynamic stress concentration coefficient oftunnel lining with different buried depths in hard surrounding rock

with different stiffness to dynamic stresses may lead to thevariation of the dynamic stress of lining Herein the followingexample is adopted to illustrate this aspect When the elasticmodulus of the lining is 119890

1= 10Gpa and soft surrounding

rock is 119890119904= 2Gpa (ie 119890

1= 5119890119904) the maximum value of

dynamic stress concentration coefficient of the lining is morethan 100 and the variation of dynamic stress of lining acrossall directions is very complexTherefore under the premise ofmeeting the bearing capacity and deformation of the liningappropriately soft lining is recommended to be adopted inthe tunnel structure to mitigate the seismic damage of tunnellining and the ratio between the elastic modulus of the liningand soft surrounding rock should be less than 5

10 Mathematical Problems in Engineering

0

10

20

30

10

20

30

0

30

6090

120

150

210

270300

330

180

240

e1 = 2Gpae1 = 4Gpa

|120591lowast 120579120579|

(a) 1198901 = 2sim4GPa

0

100

100

200

300

400

200

300

400

0

30

6090

120

150

180

210

240270

300

330

e1 = 6Gpae1 = 8Gpa

e1 = 10Gpae1 = 20Gpa

|120591lowast 120579120579|

(b) 1198901 = 6sim20GPa

Figure 8 Dynamic stress concentration coefficient of tunnel lining with different lining elastic modulus in the soft surrounding rock

4 Conclusions

A series solution of wave equation for dynamic response ofunderground circular lining tunnels subjected to incidentplane P waves is presented by Fourier-Bessel series expansionmethod in this paper The effects on circular lining rocktunnels of different factors including incident frequencyincident angle buried depth rock conditions and liningstiffness are studied through the proposed series solution Inthis study the following conclusions are drawn

(1) In circular lining rock tunnels the coefficient oftoroidal dynamic stress concentration and dynamicstress decrease gradually with the increase of incidentfrequency of P waves 120591lowast

120579120579for low-frequency P waves

is about 10 times larger than that for high-frequencyindicating that low-frequency contents of seismicwaves are more detrimental to the tunnel linings thanhigh-frequency contents

(2) The incident angle of P waves has a significantinfluence on dynamic response of the tunnel liningWhen seismic waves propagate vertically or with asmall incident angle (120579120572 = 0∘ sim30∘) the distributionof the coefficients of dynamic stress concentration inthe tunnel lining is very complex and uneven whichis detrimental to the safety of the tunnel lining

(3) When the buried depth of a tunnel in the soft sur-rounding rock is in a range of 0sim100m the decreasingtendency of dynamic stress response of the liningwith increasing tunnel buried depth is not obviouseven if the buried depth is greater than 100m thecoefficient of dynamic stress concentration of thelining still remains at a high level of about 65 In thissituation tunnels crossing the fault zone or the high

stress segment may get damaged seriously duringlarge earthquake

(4) When the tunnel buried depth in hard surroundingrock is less than 100m the coefficient of dynamicstress concentration in the tunnel lining is highranging from 60 to 85 While the buried depth isgreater than 100m the dynamic stress of the lining issmall and tends to stabilize When the tunnel burieddepth is more than 100m the tunnel buried depth isnot a major factor that affects the dynamic stress

(5) The coefficient of dynamic stress concentration inthe lining increases gradually with increasing elasticmodulus of the lining and the greater the elasticmodulus is the more uneven the dynamic stresscoefficient is Therefore under the premise of meet-ing the bearing capacity and deformation of liningappropriate soft lining is recommended to be adoptedin the tunnel structures to mitigate seismic damageof tunnel lining and the ratio between the elasticmodulus of the lining and soft surrounding rockshould be less than 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of China(Grant no 20110184120033) the Open Foundation of StateKey Laboratory of Geohazard Prevention and Geoenviron-ment Protection (Chengdu University of Technology Grant

Mathematical Problems in Engineering 11

no SKLGP2011K014) and the Science Foundation for Excel-lent Youth Scholars of Southwest JiaotongUniversity of China(Grant no SWJTU11BR003)

References

[1] T B Li ldquoDamage to mountain tunnels related to theWenchuanearthquake and some suggestions for aseismic tunnel construc-tionrdquo Bulletin of Engineering Geology and the Environment vol71 no 2 pp 297ndash308 2012

[2] I Towhata Geotechnical Earthquake Engineering SpringerBerlin Germany 2008

[3] V W Lee and M D Trifunac ldquoResponse of tunnels to incidentSH-wavesrdquo Journal of the Engineering Mechanics Division vol105 no 4 pp 643ndash659 1979

[4] Y L Zheng L D Yang and W Y Li Seismic Design of Under-ground Structure Tongji University Press Shanghai China2005

[5] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak and CompanyNew York NY USA 1973

[6] V W Lee and H Cao ldquoDiffraction of SV waves by circularcanyons of various depthsrdquo Journal of Engineering Mechanicsvol 115 no 9 pp 2035ndash2056 1989

[7] H Cao and V W Lee ldquoScattering and diffraction of planeP waves by circular cylindrical canyons with variable depth-to-width ratiordquo International Journal of Soil Dynamics andEarthquake Engineering vol 9 no 3 pp 141ndash150 1990

[8] VW Lee and J Karl ldquoDiffraction of SV waves by undergroundcircular cylindrical cavitiesrdquo Soil Dynamics and EarthquakeEngineering vol 11 no 8 pp 445ndash456 1992

[9] V W Lee and J Karl ldquoOn deformation of near a circularunderground cavity subjected to incident plane P wavesrdquoEuropean Journal of Earthquake Engineering vol 7 no 1 pp 29ndash35 1993

[10] M Manoogian and V W Lee ldquoDiffraction of SH-waves bysubsurface inclusions of arbitrary shaperdquo Journal of EngineeringMechanics vol 122 no 2 pp 123ndash129 1996

[11] C A Davis V W Lee and J P Bardet ldquoTransverse responceof underground cavities and pipes to incident SV wavesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 383ndash410 2001

[12] J W Liang H Zhang and V W Lee ldquoAnalytical solution fordynamic stress concentration of underground cavities underincident plane P wavesrdquo Chinese Journal of Geotechnical Engi-neering vol 26 no 6 pp 815ndash819 2004

[13] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (I) analytical solutionrdquoEarthquake Engineering and Engineering Vibration vol 27 no1 pp 1ndash6 2007

[14] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (II) numerical resultsrdquoEarthquake Engineering and Engineering Vibration vol 27 no2 pp 1ndash11 2007

[15] X D Ji J W Liang and J J Yang ldquoDynamic stress concen-tration of an underground cylindrical lined cavity subjected toincident plane P and SV wavesrdquo Journal of Tianjin UniversityScience and Technology vol 39 no 5 pp 511ndash517 2006

[16] H B You and J W Liang ldquoScattering of plane SV waves by acavity in a layered half-spacerdquo Rock and Soil Mechanics vol 27no 3 pp 383ndash388 2006

[17] G P Kouretzis G D Bouckovalas and C J Gantes ldquo3-D shell analysis of cylindrical underground structures underseismic shear (S) wave actionrdquo Soil Dynamics and EarthquakeEngineering vol 26 no 10 pp 909ndash921 2006

[18] M Esmaeili S Vahdani andANoorzad ldquoDynamic response oflined circular tunnel to plane harmonic wavesrdquo Tunnelling andUnderground Space Technology vol 11 no 17 pp 1ndash9 2005

[19] H Xu T B Li and L Q Li ldquoResearch on dynamic responseof underground circular lining tunnel under the action of Pwavesrdquo Applied Mechanics and Materials vol 99-100 pp 181ndash189 2011

[20] H Xu T B Li DWang Y S Li and Z H Lin ldquoStudy of seismicresponses of mountain tunnels with 3D shaking table modeltestrdquo Chinese Journal of Rock Mechanics and Engineering vol32 no 9 pp 1762ndash1771 2013

[21] G Y Gao Y S Li and T B Li ldquoDynamic analysis of shallowburied circular lining tunnel with vertical incident P waverdquoCentral South Highway Engineering vol 32 no 2 pp 56ndash602007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

stress and displacement of the tunnel lining are summarizedas

120591119904

119903119903= 120591119904

119903120579= 0 119903

2= 119887

1205911

119903119903= 1205911

119903120579= 0 119903

1= 1198861

1205911

119903119903= 120591119904

119903119903 120591

1

119903120579= 120591119904

119903120579

1199061

119903= 119906119904

119903 119906

1

120579= 119906119904

120579 119903

2= 1198862

(8)

Based on the boundary conditions in (8) the coefficientsin (4) and (5) can be obtained The wave potential functions1206011and120595

1 in the tunnel lining and120601

119904and120595

119904 in the half space

can be solved by (6) and (7) The stress and displacementexpressions under a plane strain assumption for incidentplane P waves can be described by

119906119903=120597120601

120597119903+1

119903

120597120593

120597120579 119906

120579=1

119903

120597120601

120597120579minus120597120593

120597119903

120591119903119903= 120582nabla2120601 + 2120583 [

1205972120601

1205971199032+120597

120597119903(1

119903

120597120593

120597120579)]

120591119903120579= 2120583(

1

119903

1205972120601

120597119903120597120579minus1

1199032

120597120601

120597120579) + 120583[

1

1199032

1205972120593

1205971205792minus 119903

120597

120597119903(1

119903

120597120593

120597119903)]

120591120579120579= 120582nabla2120601 + 2120583 [

1

119903(120597120601

120597119903+1

119903

1205972120601

1205971205792) +

1

119903(1

119903

120597120593

120597120579minus1205972120593

120597119903120597120579)]

(9)

After imposing the boundary conditions of 120591119904119903119903= 120591119904

119903120579= 0

and 1205911119903119903= 1205911

119903120579= 0 (10) and (11) can be obtained by combining

(1) (2) and (9) [5]

infin

sum

119899=0

[

[

1198641(1)

11(119899 1198861) 1198641(1)

12(119899 1198861)

1198641(1)

21(119899 1198861) 1198641(1)

22(119899 1198861)

]

]

119860(1)

11119899

119862(1)

11119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

1198641(2)

11(119899 1198861) 1198641(2)

12(119899 1198861)

1198641(2)

21(119899 1198861) 1198641(2)

22(119899 1198861)

]

]

119860(1)

12119899

119862(1)

12119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

1198641(1)

11(119899 1198861) 1198641(1)

12(119899 1198861)

1198641(1)

21(119899 1198861) 1198641(1)

22(119899 1198861)

]

]

119861(1)

11119899

119863(1)

11119899

(sin 119899120579

1

cos 1198991205791

)

+

infin

sum

119899=0

[

[

1198641(2)

11(119899 1198861) 1198641(2)

12(119899 1198861)

1198641(2)

21(119899 1198861) 1198641(2)

22(119899 1198861)

]

]

119861(1)

12119899

119863(1)

12119899

(sin 119899120579

1

cos 1198991205791

)

= 0

0

(10)

infin

sum

119898=0

[

[

119864119904(2)

11(119898 119887) 119864

119904(2)

12(119898 119887)

119864119904(2)minus

21(119898 119887) 119864

119904(2)

22(119898 119887)

]

]

119860(2)

1199041119898

119862(2)

1199041119898

(cos119898120579

2

sin1198981205792

)

+

infin

sum

119898=0

[

[

119864119904(1)

11(119898 119887) 119864

119904(1)

12(119898 119887)

119864119904(1)minus

21(119898 119887) 119864

119904(1)

22(119898 119887)

]

]

119860(2)

1199042119898

119862(2)

1199042119898

(cos119898120579

2

sin1198981205792

)

+

infin

sum

119898=0

[119864119904(2)

11(119898 119887) 119864

119904(2)minus

12(119898 119887)

119864119904(2)

21(119898 119887) 119864

119904(2)

22(119898 119887)

]

119861(2)

1199041119898

119863(2)

1199041119898

(sin119898120579

2

cos1198981205792

)

+

infin

sum

119898=0

[

[

119864119904(1)

11(119898 119887) 119864

119904(1)minus

12(119898 119887)

119864119904(1)

21(119898 119887) 119864

119904(1)

22(119898 119887)

]

]

119861(2)

1199041119898

119863(2)

1199041119898

(sin119898120579

2

cos1198981205792

)

= 0

0

(11)

where

119864(119894)

11(119899 119903) = (119899

2+ 119899 minus

1

21198962

1205731199032)119862119899(119896120572119903) minus 119896

120572119903119862119899minus1

(119896120572119903)

119864(119894)∓

12(119899 119903) = ∓119899 [minus (119899 + 1) 119862

119899(119896120572119903) + 119896

120573119903119862119899minus1

(119896120573119903)]

119864(119894)∓

21(119899 119903) = ∓119899 [minus (119899 + 1) 119862

119899(119896120572119903) + 119896

120572119903119862119899minus1

(119896120572119903)]

119864(119894)

22(119899 119903) = minus (119899

2+ 119899 minus

1

21198962

1205731199032)119862119899(119896120573119903) minus 119896

120573119903119862119899minus1

(119896120573119903)

(12)

When 119894 = 1 119862119899(119909) is a function of 119869

119899(119909) when 119894 = 2 119862

119899(119909) is

the function of119867(1)119899(119909) For the half plane referred to as ldquo119904rdquo

119896120572and 119896120573can be replaced by 119896

119904120572and 119896119904120573 respectively For the

tunnel lining referred to as ldquo1rdquo 119896120572and 119896120573can be replaced by

1198961120572

and 1198961120573 respectively Potential functions 120601(119894+119903) 120595(119903) 120601(119903)

and 120595(119894+119903)satisfy the stress boundary in the half spaceApplying the boundary conditions of 1199061

119903= 119906119904

119903and 1199061

120579=

119906119904

120579 the following equation can be obtained by combining (1)

(2) and (9)

infin

sum

119899=0

[

[

1198681(1)

11(119899 1198862) 1198681(1)

12(119899 1198862)

1198681(1)minus

21(119899 1198862) 1198681(1)

22(119899 1198862)

]

]

119860(1)

11119899

119862(1)

11119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

1198681(2)

11(119899 1198862) 1198681(2)

12(119899 1198862)

1198681(2)minus

21(119899 1198862) 1198681(2)

22(119899 1198862)

]

]

119860(1)

12119899

119862(1)

12119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

1198681(1)

11(119899 1198862) 1198681(1)minus

12(119899 1198862)

1198681(1)

21(119899 1198862) 1198681(1)

22(119899 1198862)

]

]

119861(1)

11119899

119863(1)

11119899

(sin 119899120579

1

cos 1198991205791

)

+

infin

sum

119899=0

[

[

1198681(2)

11(119899 1198862) 1198681(2)minus

12(119899 1198862)

1198681(2)

21(119899 1198862) 1198681(2)

22(119899 1198862)

]

]

119861(1)

12119899

119863(1)

12119899

(sin 119899120579

1

cos 1198991205791

)

Mathematical Problems in Engineering 5

=

infin

sum

119899=0

[

[

119868119904(1)

11(119899 1198862) 119868119904(1)

12(119899 1198862)

119868119904(1)minus

21(119899 1198862) 119868119904(1)

22(119899 1198862)

]

]

1198600119899+ 119860(1)

1199042119899

1198620119899+ 119862(1)

1199042119899

times (cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

119868119904(2)

11(119899 1198862) 119868119904(2)

12(119899 1198862)

119868119904(2)minus

21(119899 1198862) 119868119904(2)

22(119899 1198862)

]

]

119860(1)

1199041119899

119862(1)

1199041119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

119868119904(1)

11(119899 1198862) 119868119904(1)minus

12(119899 1198862)

119868119904(1)minus

21(119899 1198862) 119868119904(1)

22(119899 1198862)

]

]

1198610119899+ 119861(1)

1199042119899

1198630119899+ 119863(1)

1199042119899

times (sin 119899120579

1

cos 1198991205791

)

+

infin

sum

119899=0

[

[

119868119904(2)

11(119899 1198862) 119868119904(2)minus

12(119899 1198862)

119868119904(2)minus

21(119899 1198862) 119868119904(2)

22(119899 1198862)

]

]

119861(1)

1199041119899

119863(1)

1199041119899

(sin 119899120579

1

cos 1198991205791

)

(13)

where

119868(119894)

11(119899 119903) = minus119899119862

119899(119896120572119903) + 119896

120572119903119862119899minus1

(119896120572119903)

119868(119894)∓

12(119899 119903) = ∓119899119862

119899(119896120573119903)

119868(119894)∓

21(119899 119903) = ∓119899119862

119899(119896120572119903)

119868(119894)

22(119899 119903) = 119899119862

119899(119896120573119903) minus 119896

120573119903119862119899minus1

(119896120573119903)

(14)

Applying the boundary conditions of 1205911119903119903= 120591119904

119903119903and 1205911119903120579= 120591119904

119903120579

the following equation can be obtained by combining (1) (2)and (9)

infin

sum

119899=0

1205831[

[

1198641(1)

11(119899 1198862) 1198641(1)

12(119899 1198862)

1198641(1)minus

21(119899 1198862) 1198641(1)

22(119899 1198862)

]

]

119860(1)

11119899

119862(1)

11119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

1205831[

[

1198641(2)

11(119899 1198862) 1198641(2)

12(119899 1198862)

1198641(2)minus

21(119899 1198862) 1198641(2)

22(119899 1198862)

]

]

119860(1)

12119899

119862(1)

12119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

1205831[

[

1198641(1)

11(119899 1198862) 1198641(1)minus

12(119899 1198862)

1198641(1)

21(119899 1198862) 1198641(1)

22(119899 1198862)

]

]

119861(1)

11119899

119863(1)

11119899

(sin 119899120579

1

cos 1198991205791

)

+ 1205831

infin

sum

119899=0

[

[

1198641(2)

11(119899 1198861) 1198641(2)minus

12(119899 1198861)

1198641(2)

21(119899 1198861) 1198641(2)

22(119899 1198861)

]

]

119861(1)

12119899

119863(1)

12119899

(sin 119899120579

1

cos 1198991205791

)

=

infin

sum

119899=0

120583119904[

[

119864119904(1)

11(119899 1198862) 119864119904(1)

12(119899 1198862)

119864119904(1)minus

21(119899 1198862) 119864119904(1)

22(119899 1198862)

]

]

1198600119899+ 119860(1)

1199042119899

1198620119899+ 119862(1)

1199042119899

times (cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

120583119904[

[

119864119904(2)

11(119899 1198862) 119864119904(2)

12(119899 1198862)

119864119904(2)minus

21(119899 1198862) 119864119904(2)

22(119899 1198862)

]

]

119860(1)

1199041119899

119862(1)

1199041119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

120583119904[

[

119864119904(1)

11(119899 1198862) 119864119904(1)

12(119899 1198862)

119864119904(1)minus

21(119899 1198862) 119864119904(1)

22(119899 1198862)

]

]

1198610119899+ 119861(1)

1199042119899

1198630119899+ 119863(1)

1199042119899

times (sin 119899120579

1

cos 1198991205791

)

+

infin

sum

119899=0

120583119904[

[

119864119904(2)

11(119899 1198862) 119864119904(2)minus

12(119899 1198862)

119864119904(2)minus

21(119899 1198862) 119864119904(2)

22(119899 1198862)

]

]

119861(1)

1199041119899

119863(1)

1199041119899

(sin 119899120579

1

cos 1198991205791

)

(15)

Solving linear systemof (10) (11) (13) and (15) all coefficientsin the wave functions can be obtained in the coordinatesystem of 119903 minus 120579

1 The wave potential functions 120593

1and 120595

1in

the tunnel lining and120593119904and120595

119904in the half space can be solved

The stress and displacement functions of the tunnel lining forincident plane P waves can be solved from (9)

Generally under the action of steady state P wavesthe dynamic stress distribution of tunnel lining can bestudied through solving the coefficient of dynamic stressconcentration of the inner surface of tunnel lining on thetoroidal direction [13 14]

120591lowast

120579120579can be obtained from

120591lowast

120579120579=

10038161003816100381610038161003816100381610038161003816

120591120579120579

1205910

10038161003816100381610038161003816100381610038161003816

(16)

where 1205910is the maximum value of dynamic stress of tunnel

liningThrough solving the dimensionless coefficient of 120591lowast

120579120579

which is the coefficient of dynamic stress concentration of lin-ing on the toroidal direction the dynamic stress distributionand variation in the tunnel lining for incidence P waves canbe obtained

3 Calculating Examples and Analysis Results

Based on the analytical solutions of the seismic wave prop-agation in an elastic medium and the study of propagationbehavior of seismic waves in the rock and soil mediumseismic motion parameters geotechnical properties and lin-ing materials significantly affect the dynamic stress responseof tunnels and underground structures It is necessary toinvestigate the effect of various parameters including inci-dent frequency content incident angle buried depth rockmaterial properties and the lining stiffness by using theproposed series solutions

Basic assumptions of the application examples employedin this paper are as follows [13] (1) the outside diameter ofcircular tunnel is 119877

2= 12119877

1with a lining thickness of 02119877

1

(2) the buried depth is represented by the ratio (1198671198771) of the

distance from the tunnel center to the ground surface and theinternal radius of tunnel (3) the elastic modulus of tunnellining and surrounding rock is 119890

1and 119890119904 respectively (4) the

6 Mathematical Problems in Engineering

0

30

6090

120

150

210

270300

330

0

20

180

240

40

60

20

40

60

|120591lowast 120579120579|

120578 = 05

120578 = 10120578 = 20

(a) ℎ1 = 20m

120

150

300

330

0

30

6090

180

210

240270

0

30

15

15

30

45

45

120578 = 05

120578 = 10120578 = 20

|120591lowast 120579120579|

(b) ℎ1 = 100m

Figure 2 Dynamic stress concentration coefficient of tunnel lining for different incident frequency and buried depth

lining stiffness 120575 is represented by the shear wave velocityratio of the lining and the rock half space (5) the Poissonrsquosratio of rockmedium and lining is assumed to be 025 (6) thelongitudinal wave velocity of these twomedium is 1732 timesof the shear wave velocity (7) the radius of the large circulararc is 119877 = 119863

12+ 119867 and the distance from the centers O

2to

O1is taken as 119863

12= 1041198771in the example which is large

enough to simulate the half space to ensure that the solutionconverges to the exact solution

31 Effect of Incident Frequency on the Dynamic Stress Concen-tration Coefficient for the Tunnel Lining The dimensionlessfrequency 120578 is defined as the ratio of inner diameter of thetunnel and the incident wave length [15]

120578 =21198771

120582119904120573

=

1198961199041205731198771

120587 (17)

where 120582119904120573is the shear wave length of the half space rock

Considering the effect of different incident frequencycontents of P waves on dynamic response of tunnel liningthree typical cases (120578 = 05 10 and 20) are carried out forincidence P waves from vertical upward direction

Figure 2 shows how the toroidal dynamic stress concen-tration coefficient for the tunnel lining (120591lowast

120579120579) changes with

different incident frequency of P waves for two buried depthsℎ1= 20m and ℎ

1= 100m It is shown that with increasing

incident frequency 120591lowast120579120579

decreases gradually and 120591lowast120579120579

at low-frequency P waves is about 10 times larger than that athigh-frequency Therefore low-frequency seismic waves aremore detrimental to the tunnel lining than high-frequencyAlso the increase of incident frequency leads to increasinglycomplex distribution of the toroidal dynamic stress in thelining and the less principal stress concentration direction at

low-frequency conditions is gradually transformed intomorestress concentration directions at high-frequency conditions

32 Effect of Incident Angle on the Dynamic Stress Con-centration Coefficient of Lining The stress concentrationcoefficients of tunnels and underground structures are verysensitive to incident angle of seismic waves Small changesof the incident angle may lead to a significant change ofthe stress and deformation at various locations of the tunnelstructures Herein studies of three representative cases (ievertical incidencewith incident angle 120579120572=0∘ a small incidentangle 120579120572= 30∘ and large incident angle 120579120572= 60∘) were carriedout for frequency coefficients of 120578 = 1 and 120578 = 2 Figure 3displays the distribution of toroidal dynamic stress of in thelining for three different incident angles

It is observed that the coefficient of toroidal dynamicstress concentration decreases gradually with increasing inci-dent angle but the change in the amplitude is small inboth dimensionless frequency values 120578 = 1 and 120578 = 2 It isworth noting that for vertically incident P waves or a smallincident angle (120579120572 = 0∘ sim30∘) the coefficients of dynamicstress concentration of lining are very complex and unevenThe conclusion here agrees well with the results of shakingtable tests [20]

33 Effect of Buried Depth on the Dynamic Stress Concentra-tion Coefficient of Lining To demonstrate the effect of theburied depth and the material properties two different typesof rockmass (soft and hard rockmass) are chosen to study theseismic response of the tunnel lining within a buried depthrange from 10m to 150m

Case 1 (soft rock as surrounding rock medium of tunnel withelastic modulus 119890

119904= 2GPa) Figure 4 shows the distribution

Mathematical Problems in Engineering 7

0

3

3

6

9

6

9

0

30

6090

120

150

210

270300

330

180

240

120579120572 = 0∘

120579120572 = 30∘

120579120572 = 60∘

|120591lowast 120579120579|

(a) 120578 = 1

0

3

3

6

9

6

9

0

30

6090

120

150

210

270300

330

180

240

120579120572 = 0∘

120579120572 = 30∘

120579120572 = 60∘

|120591lowast 120579120579|

(b) 120578 = 2

Figure 3 Dynamic stress concentration coefficient of tunnel lining with different incident angles

of toroidal dynamic stress in the lining with different burieddepth in a range of 10mndash150m With the increase of theburied depths the dynamic stress concentration coefficientof the lining does not converge to the situation of singleround hole inwhole spaceThedistributions of lining toroidaldynamic stress with different values of buried depth are verycomplex and uneven and no directionality can be obviouslyobserved

As Figure 5 shows when the buried depth in the softsurrounding rock is within 100m the maximum dynamicstress response of the lining varies with buried depth andtwo peak values can be observed at the buried depth ofℎ1= 30m and 100m When the buried depth is greater

than 100m (ℎ1gt 100m) the coefficient of dynamic stress

concentration of lining does not vary significantly with a highlevel of about 65 It appears that with an increasing burieddepth the decreasing trend of dynamic stress response is notpronounced in the soft surrounding rock and the dynamicstress of tunnel lining may still be high in the case of alarge buried depth There are three main reasons (1) thedifference in stiffness is large between tunnel lining and softsurrounding rock (2) the degree of stress concentration oftunnel lining is high and dynamic stress of the lining doesnot decrease obviously with increase of the buried depth and(3) the pressure of surrounding rock increases gradually withincrease of the buried depth In this case the soft surroundingrock has a low strength large porosity many joints andfractures and low capability of self-arching and these rockproperties lead to large tunnel dynamic response

Case 2 (hard surrounding rock medium with elastic modulus119890119904= 20GPa) Regarding the case with hard surrounding

rock medium no obvious directionality can be observed in

terms of the distributions of toroidal dynamic stress causedby P waves but the dynamic stress response of lining ismore pronounced than the case of soft rock as illustrated inFigure 6

Figure 7 shows that when the tunnel buried depth isless than 100m the coefficient of maximum dynamic stressconcentration in the tunnel lining is generally high rangingfrom 60 to 85 which indicates that the response of dynamicstress of the lining is large at the portal segment of hard rocktunnel and the tunnel lining may be damaged easily duringa large earthquake When the tunnel buried depth is greaterthan 75m the coefficient of dynamic stress concentrationis reduced significantly The coefficient of dynamic stressconcentration starts to reach a constant level after the burieddepth is greater than 100m It appears that the lining dynamicstress is small and tends to stabilize at the large buried depthsegment of hard rock tunnel and the tunnel liningmay be lesssusceptible to damage during a large earthquakeTherefore itis concluded that under the hard rock conditions the tunnelburied depth (ie thickness of the overlying rock) is nota major factor that affects the dynamic stress and seismicdamages of tunnel lining when the tunnel buried depth isgreater than 100m This conclusion is consistent with thefindings of Gao et al [21] which states that the dynamicstress concentration coefficient tends to be constant when theburied depth is greater than 25119877

1and the buried depth of

251198771approaches 100m

34 Effect of Elastic Modulus on the Dynamic Stress Con-centration Coefficient of Lining The elastic modulus (119890

1)

of the lining can be considered an index for the elasticdeformation of tunnel lining and the stresses caused by

8 Mathematical Problems in Engineering

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 10mh1 = 20mh1 = 30m

|120591lowast 120579120579|

(a) ℎ1 = 10sim30m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 40mh1 = 50mh1 = 75m

|120591lowast 120579120579|

(b) ℎ1 = 40sim75m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 100mh1 = 125mh1 = 150m

|120591lowast 120579120579|

(c) ℎ1 = 100sim150m

Figure 4 Dynamic stress concentration coefficient of tunnel lining with different values of buried depth in the soft surrounding rock

0123456789

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150Buried depth h1 (m)

|120591lowast 120579120579|

Figure 5 The maximum dynamic stress concentration coefficientof tunnel lining with different buried depths in the soft surroundingrock

elastic deformation increase with increasing elastic modu-lus This also means that at a given level of stresses thelining deformation decreases with increasing lining stiffnessWe focus on the effect of lining elastic modulus on thedynamic stress concentration coefficient of the lining ina soft surrounding rock Figure 8 shows the coefficientsof dynamic stress concentration (120591lowast

120579120579) with different lining

elastic modulus in a soft surrounding rock (119890119904= 2GPa)

As shown in Figure 8 the coefficient of dynamicstress concentration of the lining increases gradually withincreasing lining elastic modulus and the greater the elasticmodulus is the more uneven the distribution of the dynamicstress coefficient is The different adaptabilities of lining

Mathematical Problems in Engineering 9

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 10mh1 = 20mh1 = 30m

|120591lowast 120579120579|

(a) ℎ1 = 10sim30m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 40mh1 = 50mh1 = 75m

|120591lowast 120579120579|

(b) ℎ1 = 40sim75m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 100mh1 = 125mh1 = 150m

|120591lowast 120579120579|

(c) ℎ1 = 100sim150m

Figure 6 Dynamic stress concentration coefficient of tunnel lining with different buried depths in hard surrounding rock

0123456789

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150Buried depth h1 (m)

|120591lowast 120579120579|

Figure 7Themaximumdynamic stress concentration coefficient oftunnel lining with different buried depths in hard surrounding rock

with different stiffness to dynamic stresses may lead to thevariation of the dynamic stress of lining Herein the followingexample is adopted to illustrate this aspect When the elasticmodulus of the lining is 119890

1= 10Gpa and soft surrounding

rock is 119890119904= 2Gpa (ie 119890

1= 5119890119904) the maximum value of

dynamic stress concentration coefficient of the lining is morethan 100 and the variation of dynamic stress of lining acrossall directions is very complexTherefore under the premise ofmeeting the bearing capacity and deformation of the liningappropriately soft lining is recommended to be adopted inthe tunnel structure to mitigate the seismic damage of tunnellining and the ratio between the elastic modulus of the liningand soft surrounding rock should be less than 5

10 Mathematical Problems in Engineering

0

10

20

30

10

20

30

0

30

6090

120

150

210

270300

330

180

240

e1 = 2Gpae1 = 4Gpa

|120591lowast 120579120579|

(a) 1198901 = 2sim4GPa

0

100

100

200

300

400

200

300

400

0

30

6090

120

150

180

210

240270

300

330

e1 = 6Gpae1 = 8Gpa

e1 = 10Gpae1 = 20Gpa

|120591lowast 120579120579|

(b) 1198901 = 6sim20GPa

Figure 8 Dynamic stress concentration coefficient of tunnel lining with different lining elastic modulus in the soft surrounding rock

4 Conclusions

A series solution of wave equation for dynamic response ofunderground circular lining tunnels subjected to incidentplane P waves is presented by Fourier-Bessel series expansionmethod in this paper The effects on circular lining rocktunnels of different factors including incident frequencyincident angle buried depth rock conditions and liningstiffness are studied through the proposed series solution Inthis study the following conclusions are drawn

(1) In circular lining rock tunnels the coefficient oftoroidal dynamic stress concentration and dynamicstress decrease gradually with the increase of incidentfrequency of P waves 120591lowast

120579120579for low-frequency P waves

is about 10 times larger than that for high-frequencyindicating that low-frequency contents of seismicwaves are more detrimental to the tunnel linings thanhigh-frequency contents

(2) The incident angle of P waves has a significantinfluence on dynamic response of the tunnel liningWhen seismic waves propagate vertically or with asmall incident angle (120579120572 = 0∘ sim30∘) the distributionof the coefficients of dynamic stress concentration inthe tunnel lining is very complex and uneven whichis detrimental to the safety of the tunnel lining

(3) When the buried depth of a tunnel in the soft sur-rounding rock is in a range of 0sim100m the decreasingtendency of dynamic stress response of the liningwith increasing tunnel buried depth is not obviouseven if the buried depth is greater than 100m thecoefficient of dynamic stress concentration of thelining still remains at a high level of about 65 In thissituation tunnels crossing the fault zone or the high

stress segment may get damaged seriously duringlarge earthquake

(4) When the tunnel buried depth in hard surroundingrock is less than 100m the coefficient of dynamicstress concentration in the tunnel lining is highranging from 60 to 85 While the buried depth isgreater than 100m the dynamic stress of the lining issmall and tends to stabilize When the tunnel burieddepth is more than 100m the tunnel buried depth isnot a major factor that affects the dynamic stress

(5) The coefficient of dynamic stress concentration inthe lining increases gradually with increasing elasticmodulus of the lining and the greater the elasticmodulus is the more uneven the dynamic stresscoefficient is Therefore under the premise of meet-ing the bearing capacity and deformation of liningappropriate soft lining is recommended to be adoptedin the tunnel structures to mitigate seismic damageof tunnel lining and the ratio between the elasticmodulus of the lining and soft surrounding rockshould be less than 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of China(Grant no 20110184120033) the Open Foundation of StateKey Laboratory of Geohazard Prevention and Geoenviron-ment Protection (Chengdu University of Technology Grant

Mathematical Problems in Engineering 11

no SKLGP2011K014) and the Science Foundation for Excel-lent Youth Scholars of Southwest JiaotongUniversity of China(Grant no SWJTU11BR003)

References

[1] T B Li ldquoDamage to mountain tunnels related to theWenchuanearthquake and some suggestions for aseismic tunnel construc-tionrdquo Bulletin of Engineering Geology and the Environment vol71 no 2 pp 297ndash308 2012

[2] I Towhata Geotechnical Earthquake Engineering SpringerBerlin Germany 2008

[3] V W Lee and M D Trifunac ldquoResponse of tunnels to incidentSH-wavesrdquo Journal of the Engineering Mechanics Division vol105 no 4 pp 643ndash659 1979

[4] Y L Zheng L D Yang and W Y Li Seismic Design of Under-ground Structure Tongji University Press Shanghai China2005

[5] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak and CompanyNew York NY USA 1973

[6] V W Lee and H Cao ldquoDiffraction of SV waves by circularcanyons of various depthsrdquo Journal of Engineering Mechanicsvol 115 no 9 pp 2035ndash2056 1989

[7] H Cao and V W Lee ldquoScattering and diffraction of planeP waves by circular cylindrical canyons with variable depth-to-width ratiordquo International Journal of Soil Dynamics andEarthquake Engineering vol 9 no 3 pp 141ndash150 1990

[8] VW Lee and J Karl ldquoDiffraction of SV waves by undergroundcircular cylindrical cavitiesrdquo Soil Dynamics and EarthquakeEngineering vol 11 no 8 pp 445ndash456 1992

[9] V W Lee and J Karl ldquoOn deformation of near a circularunderground cavity subjected to incident plane P wavesrdquoEuropean Journal of Earthquake Engineering vol 7 no 1 pp 29ndash35 1993

[10] M Manoogian and V W Lee ldquoDiffraction of SH-waves bysubsurface inclusions of arbitrary shaperdquo Journal of EngineeringMechanics vol 122 no 2 pp 123ndash129 1996

[11] C A Davis V W Lee and J P Bardet ldquoTransverse responceof underground cavities and pipes to incident SV wavesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 383ndash410 2001

[12] J W Liang H Zhang and V W Lee ldquoAnalytical solution fordynamic stress concentration of underground cavities underincident plane P wavesrdquo Chinese Journal of Geotechnical Engi-neering vol 26 no 6 pp 815ndash819 2004

[13] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (I) analytical solutionrdquoEarthquake Engineering and Engineering Vibration vol 27 no1 pp 1ndash6 2007

[14] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (II) numerical resultsrdquoEarthquake Engineering and Engineering Vibration vol 27 no2 pp 1ndash11 2007

[15] X D Ji J W Liang and J J Yang ldquoDynamic stress concen-tration of an underground cylindrical lined cavity subjected toincident plane P and SV wavesrdquo Journal of Tianjin UniversityScience and Technology vol 39 no 5 pp 511ndash517 2006

[16] H B You and J W Liang ldquoScattering of plane SV waves by acavity in a layered half-spacerdquo Rock and Soil Mechanics vol 27no 3 pp 383ndash388 2006

[17] G P Kouretzis G D Bouckovalas and C J Gantes ldquo3-D shell analysis of cylindrical underground structures underseismic shear (S) wave actionrdquo Soil Dynamics and EarthquakeEngineering vol 26 no 10 pp 909ndash921 2006

[18] M Esmaeili S Vahdani andANoorzad ldquoDynamic response oflined circular tunnel to plane harmonic wavesrdquo Tunnelling andUnderground Space Technology vol 11 no 17 pp 1ndash9 2005

[19] H Xu T B Li and L Q Li ldquoResearch on dynamic responseof underground circular lining tunnel under the action of Pwavesrdquo Applied Mechanics and Materials vol 99-100 pp 181ndash189 2011

[20] H Xu T B Li DWang Y S Li and Z H Lin ldquoStudy of seismicresponses of mountain tunnels with 3D shaking table modeltestrdquo Chinese Journal of Rock Mechanics and Engineering vol32 no 9 pp 1762ndash1771 2013

[21] G Y Gao Y S Li and T B Li ldquoDynamic analysis of shallowburied circular lining tunnel with vertical incident P waverdquoCentral South Highway Engineering vol 32 no 2 pp 56ndash602007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

=

infin

sum

119899=0

[

[

119868119904(1)

11(119899 1198862) 119868119904(1)

12(119899 1198862)

119868119904(1)minus

21(119899 1198862) 119868119904(1)

22(119899 1198862)

]

]

1198600119899+ 119860(1)

1199042119899

1198620119899+ 119862(1)

1199042119899

times (cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

119868119904(2)

11(119899 1198862) 119868119904(2)

12(119899 1198862)

119868119904(2)minus

21(119899 1198862) 119868119904(2)

22(119899 1198862)

]

]

119860(1)

1199041119899

119862(1)

1199041119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

[

[

119868119904(1)

11(119899 1198862) 119868119904(1)minus

12(119899 1198862)

119868119904(1)minus

21(119899 1198862) 119868119904(1)

22(119899 1198862)

]

]

1198610119899+ 119861(1)

1199042119899

1198630119899+ 119863(1)

1199042119899

times (sin 119899120579

1

cos 1198991205791

)

+

infin

sum

119899=0

[

[

119868119904(2)

11(119899 1198862) 119868119904(2)minus

12(119899 1198862)

119868119904(2)minus

21(119899 1198862) 119868119904(2)

22(119899 1198862)

]

]

119861(1)

1199041119899

119863(1)

1199041119899

(sin 119899120579

1

cos 1198991205791

)

(13)

where

119868(119894)

11(119899 119903) = minus119899119862

119899(119896120572119903) + 119896

120572119903119862119899minus1

(119896120572119903)

119868(119894)∓

12(119899 119903) = ∓119899119862

119899(119896120573119903)

119868(119894)∓

21(119899 119903) = ∓119899119862

119899(119896120572119903)

119868(119894)

22(119899 119903) = 119899119862

119899(119896120573119903) minus 119896

120573119903119862119899minus1

(119896120573119903)

(14)

Applying the boundary conditions of 1205911119903119903= 120591119904

119903119903and 1205911119903120579= 120591119904

119903120579

the following equation can be obtained by combining (1) (2)and (9)

infin

sum

119899=0

1205831[

[

1198641(1)

11(119899 1198862) 1198641(1)

12(119899 1198862)

1198641(1)minus

21(119899 1198862) 1198641(1)

22(119899 1198862)

]

]

119860(1)

11119899

119862(1)

11119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

1205831[

[

1198641(2)

11(119899 1198862) 1198641(2)

12(119899 1198862)

1198641(2)minus

21(119899 1198862) 1198641(2)

22(119899 1198862)

]

]

119860(1)

12119899

119862(1)

12119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

1205831[

[

1198641(1)

11(119899 1198862) 1198641(1)minus

12(119899 1198862)

1198641(1)

21(119899 1198862) 1198641(1)

22(119899 1198862)

]

]

119861(1)

11119899

119863(1)

11119899

(sin 119899120579

1

cos 1198991205791

)

+ 1205831

infin

sum

119899=0

[

[

1198641(2)

11(119899 1198861) 1198641(2)minus

12(119899 1198861)

1198641(2)

21(119899 1198861) 1198641(2)

22(119899 1198861)

]

]

119861(1)

12119899

119863(1)

12119899

(sin 119899120579

1

cos 1198991205791

)

=

infin

sum

119899=0

120583119904[

[

119864119904(1)

11(119899 1198862) 119864119904(1)

12(119899 1198862)

119864119904(1)minus

21(119899 1198862) 119864119904(1)

22(119899 1198862)

]

]

1198600119899+ 119860(1)

1199042119899

1198620119899+ 119862(1)

1199042119899

times (cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

120583119904[

[

119864119904(2)

11(119899 1198862) 119864119904(2)

12(119899 1198862)

119864119904(2)minus

21(119899 1198862) 119864119904(2)

22(119899 1198862)

]

]

119860(1)

1199041119899

119862(1)

1199041119899

(cos 119899120579

1

sin 1198991205791

)

+

infin

sum

119899=0

120583119904[

[

119864119904(1)

11(119899 1198862) 119864119904(1)

12(119899 1198862)

119864119904(1)minus

21(119899 1198862) 119864119904(1)

22(119899 1198862)

]

]

1198610119899+ 119861(1)

1199042119899

1198630119899+ 119863(1)

1199042119899

times (sin 119899120579

1

cos 1198991205791

)

+

infin

sum

119899=0

120583119904[

[

119864119904(2)

11(119899 1198862) 119864119904(2)minus

12(119899 1198862)

119864119904(2)minus

21(119899 1198862) 119864119904(2)

22(119899 1198862)

]

]

119861(1)

1199041119899

119863(1)

1199041119899

(sin 119899120579

1

cos 1198991205791

)

(15)

Solving linear systemof (10) (11) (13) and (15) all coefficientsin the wave functions can be obtained in the coordinatesystem of 119903 minus 120579

1 The wave potential functions 120593

1and 120595

1in

the tunnel lining and120593119904and120595

119904in the half space can be solved

The stress and displacement functions of the tunnel lining forincident plane P waves can be solved from (9)

Generally under the action of steady state P wavesthe dynamic stress distribution of tunnel lining can bestudied through solving the coefficient of dynamic stressconcentration of the inner surface of tunnel lining on thetoroidal direction [13 14]

120591lowast

120579120579can be obtained from

120591lowast

120579120579=

10038161003816100381610038161003816100381610038161003816

120591120579120579

1205910

10038161003816100381610038161003816100381610038161003816

(16)

where 1205910is the maximum value of dynamic stress of tunnel

liningThrough solving the dimensionless coefficient of 120591lowast

120579120579

which is the coefficient of dynamic stress concentration of lin-ing on the toroidal direction the dynamic stress distributionand variation in the tunnel lining for incidence P waves canbe obtained

3 Calculating Examples and Analysis Results

Based on the analytical solutions of the seismic wave prop-agation in an elastic medium and the study of propagationbehavior of seismic waves in the rock and soil mediumseismic motion parameters geotechnical properties and lin-ing materials significantly affect the dynamic stress responseof tunnels and underground structures It is necessary toinvestigate the effect of various parameters including inci-dent frequency content incident angle buried depth rockmaterial properties and the lining stiffness by using theproposed series solutions

Basic assumptions of the application examples employedin this paper are as follows [13] (1) the outside diameter ofcircular tunnel is 119877

2= 12119877

1with a lining thickness of 02119877

1

(2) the buried depth is represented by the ratio (1198671198771) of the

distance from the tunnel center to the ground surface and theinternal radius of tunnel (3) the elastic modulus of tunnellining and surrounding rock is 119890

1and 119890119904 respectively (4) the

6 Mathematical Problems in Engineering

0

30

6090

120

150

210

270300

330

0

20

180

240

40

60

20

40

60

|120591lowast 120579120579|

120578 = 05

120578 = 10120578 = 20

(a) ℎ1 = 20m

120

150

300

330

0

30

6090

180

210

240270

0

30

15

15

30

45

45

120578 = 05

120578 = 10120578 = 20

|120591lowast 120579120579|

(b) ℎ1 = 100m

Figure 2 Dynamic stress concentration coefficient of tunnel lining for different incident frequency and buried depth

lining stiffness 120575 is represented by the shear wave velocityratio of the lining and the rock half space (5) the Poissonrsquosratio of rockmedium and lining is assumed to be 025 (6) thelongitudinal wave velocity of these twomedium is 1732 timesof the shear wave velocity (7) the radius of the large circulararc is 119877 = 119863

12+ 119867 and the distance from the centers O

2to

O1is taken as 119863

12= 1041198771in the example which is large

enough to simulate the half space to ensure that the solutionconverges to the exact solution

31 Effect of Incident Frequency on the Dynamic Stress Concen-tration Coefficient for the Tunnel Lining The dimensionlessfrequency 120578 is defined as the ratio of inner diameter of thetunnel and the incident wave length [15]

120578 =21198771

120582119904120573

=

1198961199041205731198771

120587 (17)

where 120582119904120573is the shear wave length of the half space rock

Considering the effect of different incident frequencycontents of P waves on dynamic response of tunnel liningthree typical cases (120578 = 05 10 and 20) are carried out forincidence P waves from vertical upward direction

Figure 2 shows how the toroidal dynamic stress concen-tration coefficient for the tunnel lining (120591lowast

120579120579) changes with

different incident frequency of P waves for two buried depthsℎ1= 20m and ℎ

1= 100m It is shown that with increasing

incident frequency 120591lowast120579120579

decreases gradually and 120591lowast120579120579

at low-frequency P waves is about 10 times larger than that athigh-frequency Therefore low-frequency seismic waves aremore detrimental to the tunnel lining than high-frequencyAlso the increase of incident frequency leads to increasinglycomplex distribution of the toroidal dynamic stress in thelining and the less principal stress concentration direction at

low-frequency conditions is gradually transformed intomorestress concentration directions at high-frequency conditions

32 Effect of Incident Angle on the Dynamic Stress Con-centration Coefficient of Lining The stress concentrationcoefficients of tunnels and underground structures are verysensitive to incident angle of seismic waves Small changesof the incident angle may lead to a significant change ofthe stress and deformation at various locations of the tunnelstructures Herein studies of three representative cases (ievertical incidencewith incident angle 120579120572=0∘ a small incidentangle 120579120572= 30∘ and large incident angle 120579120572= 60∘) were carriedout for frequency coefficients of 120578 = 1 and 120578 = 2 Figure 3displays the distribution of toroidal dynamic stress of in thelining for three different incident angles

It is observed that the coefficient of toroidal dynamicstress concentration decreases gradually with increasing inci-dent angle but the change in the amplitude is small inboth dimensionless frequency values 120578 = 1 and 120578 = 2 It isworth noting that for vertically incident P waves or a smallincident angle (120579120572 = 0∘ sim30∘) the coefficients of dynamicstress concentration of lining are very complex and unevenThe conclusion here agrees well with the results of shakingtable tests [20]

33 Effect of Buried Depth on the Dynamic Stress Concentra-tion Coefficient of Lining To demonstrate the effect of theburied depth and the material properties two different typesof rockmass (soft and hard rockmass) are chosen to study theseismic response of the tunnel lining within a buried depthrange from 10m to 150m

Case 1 (soft rock as surrounding rock medium of tunnel withelastic modulus 119890

119904= 2GPa) Figure 4 shows the distribution

Mathematical Problems in Engineering 7

0

3

3

6

9

6

9

0

30

6090

120

150

210

270300

330

180

240

120579120572 = 0∘

120579120572 = 30∘

120579120572 = 60∘

|120591lowast 120579120579|

(a) 120578 = 1

0

3

3

6

9

6

9

0

30

6090

120

150

210

270300

330

180

240

120579120572 = 0∘

120579120572 = 30∘

120579120572 = 60∘

|120591lowast 120579120579|

(b) 120578 = 2

Figure 3 Dynamic stress concentration coefficient of tunnel lining with different incident angles

of toroidal dynamic stress in the lining with different burieddepth in a range of 10mndash150m With the increase of theburied depths the dynamic stress concentration coefficientof the lining does not converge to the situation of singleround hole inwhole spaceThedistributions of lining toroidaldynamic stress with different values of buried depth are verycomplex and uneven and no directionality can be obviouslyobserved

As Figure 5 shows when the buried depth in the softsurrounding rock is within 100m the maximum dynamicstress response of the lining varies with buried depth andtwo peak values can be observed at the buried depth ofℎ1= 30m and 100m When the buried depth is greater

than 100m (ℎ1gt 100m) the coefficient of dynamic stress

concentration of lining does not vary significantly with a highlevel of about 65 It appears that with an increasing burieddepth the decreasing trend of dynamic stress response is notpronounced in the soft surrounding rock and the dynamicstress of tunnel lining may still be high in the case of alarge buried depth There are three main reasons (1) thedifference in stiffness is large between tunnel lining and softsurrounding rock (2) the degree of stress concentration oftunnel lining is high and dynamic stress of the lining doesnot decrease obviously with increase of the buried depth and(3) the pressure of surrounding rock increases gradually withincrease of the buried depth In this case the soft surroundingrock has a low strength large porosity many joints andfractures and low capability of self-arching and these rockproperties lead to large tunnel dynamic response

Case 2 (hard surrounding rock medium with elastic modulus119890119904= 20GPa) Regarding the case with hard surrounding

rock medium no obvious directionality can be observed in

terms of the distributions of toroidal dynamic stress causedby P waves but the dynamic stress response of lining ismore pronounced than the case of soft rock as illustrated inFigure 6

Figure 7 shows that when the tunnel buried depth isless than 100m the coefficient of maximum dynamic stressconcentration in the tunnel lining is generally high rangingfrom 60 to 85 which indicates that the response of dynamicstress of the lining is large at the portal segment of hard rocktunnel and the tunnel lining may be damaged easily duringa large earthquake When the tunnel buried depth is greaterthan 75m the coefficient of dynamic stress concentrationis reduced significantly The coefficient of dynamic stressconcentration starts to reach a constant level after the burieddepth is greater than 100m It appears that the lining dynamicstress is small and tends to stabilize at the large buried depthsegment of hard rock tunnel and the tunnel liningmay be lesssusceptible to damage during a large earthquakeTherefore itis concluded that under the hard rock conditions the tunnelburied depth (ie thickness of the overlying rock) is nota major factor that affects the dynamic stress and seismicdamages of tunnel lining when the tunnel buried depth isgreater than 100m This conclusion is consistent with thefindings of Gao et al [21] which states that the dynamicstress concentration coefficient tends to be constant when theburied depth is greater than 25119877

1and the buried depth of

251198771approaches 100m

34 Effect of Elastic Modulus on the Dynamic Stress Con-centration Coefficient of Lining The elastic modulus (119890

1)

of the lining can be considered an index for the elasticdeformation of tunnel lining and the stresses caused by

8 Mathematical Problems in Engineering

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 10mh1 = 20mh1 = 30m

|120591lowast 120579120579|

(a) ℎ1 = 10sim30m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 40mh1 = 50mh1 = 75m

|120591lowast 120579120579|

(b) ℎ1 = 40sim75m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 100mh1 = 125mh1 = 150m

|120591lowast 120579120579|

(c) ℎ1 = 100sim150m

Figure 4 Dynamic stress concentration coefficient of tunnel lining with different values of buried depth in the soft surrounding rock

0123456789

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150Buried depth h1 (m)

|120591lowast 120579120579|

Figure 5 The maximum dynamic stress concentration coefficientof tunnel lining with different buried depths in the soft surroundingrock

elastic deformation increase with increasing elastic modu-lus This also means that at a given level of stresses thelining deformation decreases with increasing lining stiffnessWe focus on the effect of lining elastic modulus on thedynamic stress concentration coefficient of the lining ina soft surrounding rock Figure 8 shows the coefficientsof dynamic stress concentration (120591lowast

120579120579) with different lining

elastic modulus in a soft surrounding rock (119890119904= 2GPa)

As shown in Figure 8 the coefficient of dynamicstress concentration of the lining increases gradually withincreasing lining elastic modulus and the greater the elasticmodulus is the more uneven the distribution of the dynamicstress coefficient is The different adaptabilities of lining

Mathematical Problems in Engineering 9

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 10mh1 = 20mh1 = 30m

|120591lowast 120579120579|

(a) ℎ1 = 10sim30m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 40mh1 = 50mh1 = 75m

|120591lowast 120579120579|

(b) ℎ1 = 40sim75m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 100mh1 = 125mh1 = 150m

|120591lowast 120579120579|

(c) ℎ1 = 100sim150m

Figure 6 Dynamic stress concentration coefficient of tunnel lining with different buried depths in hard surrounding rock

0123456789

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150Buried depth h1 (m)

|120591lowast 120579120579|

Figure 7Themaximumdynamic stress concentration coefficient oftunnel lining with different buried depths in hard surrounding rock

with different stiffness to dynamic stresses may lead to thevariation of the dynamic stress of lining Herein the followingexample is adopted to illustrate this aspect When the elasticmodulus of the lining is 119890

1= 10Gpa and soft surrounding

rock is 119890119904= 2Gpa (ie 119890

1= 5119890119904) the maximum value of

dynamic stress concentration coefficient of the lining is morethan 100 and the variation of dynamic stress of lining acrossall directions is very complexTherefore under the premise ofmeeting the bearing capacity and deformation of the liningappropriately soft lining is recommended to be adopted inthe tunnel structure to mitigate the seismic damage of tunnellining and the ratio between the elastic modulus of the liningand soft surrounding rock should be less than 5

10 Mathematical Problems in Engineering

0

10

20

30

10

20

30

0

30

6090

120

150

210

270300

330

180

240

e1 = 2Gpae1 = 4Gpa

|120591lowast 120579120579|

(a) 1198901 = 2sim4GPa

0

100

100

200

300

400

200

300

400

0

30

6090

120

150

180

210

240270

300

330

e1 = 6Gpae1 = 8Gpa

e1 = 10Gpae1 = 20Gpa

|120591lowast 120579120579|

(b) 1198901 = 6sim20GPa

Figure 8 Dynamic stress concentration coefficient of tunnel lining with different lining elastic modulus in the soft surrounding rock

4 Conclusions

A series solution of wave equation for dynamic response ofunderground circular lining tunnels subjected to incidentplane P waves is presented by Fourier-Bessel series expansionmethod in this paper The effects on circular lining rocktunnels of different factors including incident frequencyincident angle buried depth rock conditions and liningstiffness are studied through the proposed series solution Inthis study the following conclusions are drawn

(1) In circular lining rock tunnels the coefficient oftoroidal dynamic stress concentration and dynamicstress decrease gradually with the increase of incidentfrequency of P waves 120591lowast

120579120579for low-frequency P waves

is about 10 times larger than that for high-frequencyindicating that low-frequency contents of seismicwaves are more detrimental to the tunnel linings thanhigh-frequency contents

(2) The incident angle of P waves has a significantinfluence on dynamic response of the tunnel liningWhen seismic waves propagate vertically or with asmall incident angle (120579120572 = 0∘ sim30∘) the distributionof the coefficients of dynamic stress concentration inthe tunnel lining is very complex and uneven whichis detrimental to the safety of the tunnel lining

(3) When the buried depth of a tunnel in the soft sur-rounding rock is in a range of 0sim100m the decreasingtendency of dynamic stress response of the liningwith increasing tunnel buried depth is not obviouseven if the buried depth is greater than 100m thecoefficient of dynamic stress concentration of thelining still remains at a high level of about 65 In thissituation tunnels crossing the fault zone or the high

stress segment may get damaged seriously duringlarge earthquake

(4) When the tunnel buried depth in hard surroundingrock is less than 100m the coefficient of dynamicstress concentration in the tunnel lining is highranging from 60 to 85 While the buried depth isgreater than 100m the dynamic stress of the lining issmall and tends to stabilize When the tunnel burieddepth is more than 100m the tunnel buried depth isnot a major factor that affects the dynamic stress

(5) The coefficient of dynamic stress concentration inthe lining increases gradually with increasing elasticmodulus of the lining and the greater the elasticmodulus is the more uneven the dynamic stresscoefficient is Therefore under the premise of meet-ing the bearing capacity and deformation of liningappropriate soft lining is recommended to be adoptedin the tunnel structures to mitigate seismic damageof tunnel lining and the ratio between the elasticmodulus of the lining and soft surrounding rockshould be less than 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of China(Grant no 20110184120033) the Open Foundation of StateKey Laboratory of Geohazard Prevention and Geoenviron-ment Protection (Chengdu University of Technology Grant

Mathematical Problems in Engineering 11

no SKLGP2011K014) and the Science Foundation for Excel-lent Youth Scholars of Southwest JiaotongUniversity of China(Grant no SWJTU11BR003)

References

[1] T B Li ldquoDamage to mountain tunnels related to theWenchuanearthquake and some suggestions for aseismic tunnel construc-tionrdquo Bulletin of Engineering Geology and the Environment vol71 no 2 pp 297ndash308 2012

[2] I Towhata Geotechnical Earthquake Engineering SpringerBerlin Germany 2008

[3] V W Lee and M D Trifunac ldquoResponse of tunnels to incidentSH-wavesrdquo Journal of the Engineering Mechanics Division vol105 no 4 pp 643ndash659 1979

[4] Y L Zheng L D Yang and W Y Li Seismic Design of Under-ground Structure Tongji University Press Shanghai China2005

[5] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak and CompanyNew York NY USA 1973

[6] V W Lee and H Cao ldquoDiffraction of SV waves by circularcanyons of various depthsrdquo Journal of Engineering Mechanicsvol 115 no 9 pp 2035ndash2056 1989

[7] H Cao and V W Lee ldquoScattering and diffraction of planeP waves by circular cylindrical canyons with variable depth-to-width ratiordquo International Journal of Soil Dynamics andEarthquake Engineering vol 9 no 3 pp 141ndash150 1990

[8] VW Lee and J Karl ldquoDiffraction of SV waves by undergroundcircular cylindrical cavitiesrdquo Soil Dynamics and EarthquakeEngineering vol 11 no 8 pp 445ndash456 1992

[9] V W Lee and J Karl ldquoOn deformation of near a circularunderground cavity subjected to incident plane P wavesrdquoEuropean Journal of Earthquake Engineering vol 7 no 1 pp 29ndash35 1993

[10] M Manoogian and V W Lee ldquoDiffraction of SH-waves bysubsurface inclusions of arbitrary shaperdquo Journal of EngineeringMechanics vol 122 no 2 pp 123ndash129 1996

[11] C A Davis V W Lee and J P Bardet ldquoTransverse responceof underground cavities and pipes to incident SV wavesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 383ndash410 2001

[12] J W Liang H Zhang and V W Lee ldquoAnalytical solution fordynamic stress concentration of underground cavities underincident plane P wavesrdquo Chinese Journal of Geotechnical Engi-neering vol 26 no 6 pp 815ndash819 2004

[13] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (I) analytical solutionrdquoEarthquake Engineering and Engineering Vibration vol 27 no1 pp 1ndash6 2007

[14] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (II) numerical resultsrdquoEarthquake Engineering and Engineering Vibration vol 27 no2 pp 1ndash11 2007

[15] X D Ji J W Liang and J J Yang ldquoDynamic stress concen-tration of an underground cylindrical lined cavity subjected toincident plane P and SV wavesrdquo Journal of Tianjin UniversityScience and Technology vol 39 no 5 pp 511ndash517 2006

[16] H B You and J W Liang ldquoScattering of plane SV waves by acavity in a layered half-spacerdquo Rock and Soil Mechanics vol 27no 3 pp 383ndash388 2006

[17] G P Kouretzis G D Bouckovalas and C J Gantes ldquo3-D shell analysis of cylindrical underground structures underseismic shear (S) wave actionrdquo Soil Dynamics and EarthquakeEngineering vol 26 no 10 pp 909ndash921 2006

[18] M Esmaeili S Vahdani andANoorzad ldquoDynamic response oflined circular tunnel to plane harmonic wavesrdquo Tunnelling andUnderground Space Technology vol 11 no 17 pp 1ndash9 2005

[19] H Xu T B Li and L Q Li ldquoResearch on dynamic responseof underground circular lining tunnel under the action of Pwavesrdquo Applied Mechanics and Materials vol 99-100 pp 181ndash189 2011

[20] H Xu T B Li DWang Y S Li and Z H Lin ldquoStudy of seismicresponses of mountain tunnels with 3D shaking table modeltestrdquo Chinese Journal of Rock Mechanics and Engineering vol32 no 9 pp 1762ndash1771 2013

[21] G Y Gao Y S Li and T B Li ldquoDynamic analysis of shallowburied circular lining tunnel with vertical incident P waverdquoCentral South Highway Engineering vol 32 no 2 pp 56ndash602007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0

30

6090

120

150

210

270300

330

0

20

180

240

40

60

20

40

60

|120591lowast 120579120579|

120578 = 05

120578 = 10120578 = 20

(a) ℎ1 = 20m

120

150

300

330

0

30

6090

180

210

240270

0

30

15

15

30

45

45

120578 = 05

120578 = 10120578 = 20

|120591lowast 120579120579|

(b) ℎ1 = 100m

Figure 2 Dynamic stress concentration coefficient of tunnel lining for different incident frequency and buried depth

lining stiffness 120575 is represented by the shear wave velocityratio of the lining and the rock half space (5) the Poissonrsquosratio of rockmedium and lining is assumed to be 025 (6) thelongitudinal wave velocity of these twomedium is 1732 timesof the shear wave velocity (7) the radius of the large circulararc is 119877 = 119863

12+ 119867 and the distance from the centers O

2to

O1is taken as 119863

12= 1041198771in the example which is large

enough to simulate the half space to ensure that the solutionconverges to the exact solution

31 Effect of Incident Frequency on the Dynamic Stress Concen-tration Coefficient for the Tunnel Lining The dimensionlessfrequency 120578 is defined as the ratio of inner diameter of thetunnel and the incident wave length [15]

120578 =21198771

120582119904120573

=

1198961199041205731198771

120587 (17)

where 120582119904120573is the shear wave length of the half space rock

Considering the effect of different incident frequencycontents of P waves on dynamic response of tunnel liningthree typical cases (120578 = 05 10 and 20) are carried out forincidence P waves from vertical upward direction

Figure 2 shows how the toroidal dynamic stress concen-tration coefficient for the tunnel lining (120591lowast

120579120579) changes with

different incident frequency of P waves for two buried depthsℎ1= 20m and ℎ

1= 100m It is shown that with increasing

incident frequency 120591lowast120579120579

decreases gradually and 120591lowast120579120579

at low-frequency P waves is about 10 times larger than that athigh-frequency Therefore low-frequency seismic waves aremore detrimental to the tunnel lining than high-frequencyAlso the increase of incident frequency leads to increasinglycomplex distribution of the toroidal dynamic stress in thelining and the less principal stress concentration direction at

low-frequency conditions is gradually transformed intomorestress concentration directions at high-frequency conditions

32 Effect of Incident Angle on the Dynamic Stress Con-centration Coefficient of Lining The stress concentrationcoefficients of tunnels and underground structures are verysensitive to incident angle of seismic waves Small changesof the incident angle may lead to a significant change ofthe stress and deformation at various locations of the tunnelstructures Herein studies of three representative cases (ievertical incidencewith incident angle 120579120572=0∘ a small incidentangle 120579120572= 30∘ and large incident angle 120579120572= 60∘) were carriedout for frequency coefficients of 120578 = 1 and 120578 = 2 Figure 3displays the distribution of toroidal dynamic stress of in thelining for three different incident angles

It is observed that the coefficient of toroidal dynamicstress concentration decreases gradually with increasing inci-dent angle but the change in the amplitude is small inboth dimensionless frequency values 120578 = 1 and 120578 = 2 It isworth noting that for vertically incident P waves or a smallincident angle (120579120572 = 0∘ sim30∘) the coefficients of dynamicstress concentration of lining are very complex and unevenThe conclusion here agrees well with the results of shakingtable tests [20]

33 Effect of Buried Depth on the Dynamic Stress Concentra-tion Coefficient of Lining To demonstrate the effect of theburied depth and the material properties two different typesof rockmass (soft and hard rockmass) are chosen to study theseismic response of the tunnel lining within a buried depthrange from 10m to 150m

Case 1 (soft rock as surrounding rock medium of tunnel withelastic modulus 119890

119904= 2GPa) Figure 4 shows the distribution

Mathematical Problems in Engineering 7

0

3

3

6

9

6

9

0

30

6090

120

150

210

270300

330

180

240

120579120572 = 0∘

120579120572 = 30∘

120579120572 = 60∘

|120591lowast 120579120579|

(a) 120578 = 1

0

3

3

6

9

6

9

0

30

6090

120

150

210

270300

330

180

240

120579120572 = 0∘

120579120572 = 30∘

120579120572 = 60∘

|120591lowast 120579120579|

(b) 120578 = 2

Figure 3 Dynamic stress concentration coefficient of tunnel lining with different incident angles

of toroidal dynamic stress in the lining with different burieddepth in a range of 10mndash150m With the increase of theburied depths the dynamic stress concentration coefficientof the lining does not converge to the situation of singleround hole inwhole spaceThedistributions of lining toroidaldynamic stress with different values of buried depth are verycomplex and uneven and no directionality can be obviouslyobserved

As Figure 5 shows when the buried depth in the softsurrounding rock is within 100m the maximum dynamicstress response of the lining varies with buried depth andtwo peak values can be observed at the buried depth ofℎ1= 30m and 100m When the buried depth is greater

than 100m (ℎ1gt 100m) the coefficient of dynamic stress

concentration of lining does not vary significantly with a highlevel of about 65 It appears that with an increasing burieddepth the decreasing trend of dynamic stress response is notpronounced in the soft surrounding rock and the dynamicstress of tunnel lining may still be high in the case of alarge buried depth There are three main reasons (1) thedifference in stiffness is large between tunnel lining and softsurrounding rock (2) the degree of stress concentration oftunnel lining is high and dynamic stress of the lining doesnot decrease obviously with increase of the buried depth and(3) the pressure of surrounding rock increases gradually withincrease of the buried depth In this case the soft surroundingrock has a low strength large porosity many joints andfractures and low capability of self-arching and these rockproperties lead to large tunnel dynamic response

Case 2 (hard surrounding rock medium with elastic modulus119890119904= 20GPa) Regarding the case with hard surrounding

rock medium no obvious directionality can be observed in

terms of the distributions of toroidal dynamic stress causedby P waves but the dynamic stress response of lining ismore pronounced than the case of soft rock as illustrated inFigure 6

Figure 7 shows that when the tunnel buried depth isless than 100m the coefficient of maximum dynamic stressconcentration in the tunnel lining is generally high rangingfrom 60 to 85 which indicates that the response of dynamicstress of the lining is large at the portal segment of hard rocktunnel and the tunnel lining may be damaged easily duringa large earthquake When the tunnel buried depth is greaterthan 75m the coefficient of dynamic stress concentrationis reduced significantly The coefficient of dynamic stressconcentration starts to reach a constant level after the burieddepth is greater than 100m It appears that the lining dynamicstress is small and tends to stabilize at the large buried depthsegment of hard rock tunnel and the tunnel liningmay be lesssusceptible to damage during a large earthquakeTherefore itis concluded that under the hard rock conditions the tunnelburied depth (ie thickness of the overlying rock) is nota major factor that affects the dynamic stress and seismicdamages of tunnel lining when the tunnel buried depth isgreater than 100m This conclusion is consistent with thefindings of Gao et al [21] which states that the dynamicstress concentration coefficient tends to be constant when theburied depth is greater than 25119877

1and the buried depth of

251198771approaches 100m

34 Effect of Elastic Modulus on the Dynamic Stress Con-centration Coefficient of Lining The elastic modulus (119890

1)

of the lining can be considered an index for the elasticdeformation of tunnel lining and the stresses caused by

8 Mathematical Problems in Engineering

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 10mh1 = 20mh1 = 30m

|120591lowast 120579120579|

(a) ℎ1 = 10sim30m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 40mh1 = 50mh1 = 75m

|120591lowast 120579120579|

(b) ℎ1 = 40sim75m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 100mh1 = 125mh1 = 150m

|120591lowast 120579120579|

(c) ℎ1 = 100sim150m

Figure 4 Dynamic stress concentration coefficient of tunnel lining with different values of buried depth in the soft surrounding rock

0123456789

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150Buried depth h1 (m)

|120591lowast 120579120579|

Figure 5 The maximum dynamic stress concentration coefficientof tunnel lining with different buried depths in the soft surroundingrock

elastic deformation increase with increasing elastic modu-lus This also means that at a given level of stresses thelining deformation decreases with increasing lining stiffnessWe focus on the effect of lining elastic modulus on thedynamic stress concentration coefficient of the lining ina soft surrounding rock Figure 8 shows the coefficientsof dynamic stress concentration (120591lowast

120579120579) with different lining

elastic modulus in a soft surrounding rock (119890119904= 2GPa)

As shown in Figure 8 the coefficient of dynamicstress concentration of the lining increases gradually withincreasing lining elastic modulus and the greater the elasticmodulus is the more uneven the distribution of the dynamicstress coefficient is The different adaptabilities of lining

Mathematical Problems in Engineering 9

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 10mh1 = 20mh1 = 30m

|120591lowast 120579120579|

(a) ℎ1 = 10sim30m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 40mh1 = 50mh1 = 75m

|120591lowast 120579120579|

(b) ℎ1 = 40sim75m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 100mh1 = 125mh1 = 150m

|120591lowast 120579120579|

(c) ℎ1 = 100sim150m

Figure 6 Dynamic stress concentration coefficient of tunnel lining with different buried depths in hard surrounding rock

0123456789

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150Buried depth h1 (m)

|120591lowast 120579120579|

Figure 7Themaximumdynamic stress concentration coefficient oftunnel lining with different buried depths in hard surrounding rock

with different stiffness to dynamic stresses may lead to thevariation of the dynamic stress of lining Herein the followingexample is adopted to illustrate this aspect When the elasticmodulus of the lining is 119890

1= 10Gpa and soft surrounding

rock is 119890119904= 2Gpa (ie 119890

1= 5119890119904) the maximum value of

dynamic stress concentration coefficient of the lining is morethan 100 and the variation of dynamic stress of lining acrossall directions is very complexTherefore under the premise ofmeeting the bearing capacity and deformation of the liningappropriately soft lining is recommended to be adopted inthe tunnel structure to mitigate the seismic damage of tunnellining and the ratio between the elastic modulus of the liningand soft surrounding rock should be less than 5

10 Mathematical Problems in Engineering

0

10

20

30

10

20

30

0

30

6090

120

150

210

270300

330

180

240

e1 = 2Gpae1 = 4Gpa

|120591lowast 120579120579|

(a) 1198901 = 2sim4GPa

0

100

100

200

300

400

200

300

400

0

30

6090

120

150

180

210

240270

300

330

e1 = 6Gpae1 = 8Gpa

e1 = 10Gpae1 = 20Gpa

|120591lowast 120579120579|

(b) 1198901 = 6sim20GPa

Figure 8 Dynamic stress concentration coefficient of tunnel lining with different lining elastic modulus in the soft surrounding rock

4 Conclusions

A series solution of wave equation for dynamic response ofunderground circular lining tunnels subjected to incidentplane P waves is presented by Fourier-Bessel series expansionmethod in this paper The effects on circular lining rocktunnels of different factors including incident frequencyincident angle buried depth rock conditions and liningstiffness are studied through the proposed series solution Inthis study the following conclusions are drawn

(1) In circular lining rock tunnels the coefficient oftoroidal dynamic stress concentration and dynamicstress decrease gradually with the increase of incidentfrequency of P waves 120591lowast

120579120579for low-frequency P waves

is about 10 times larger than that for high-frequencyindicating that low-frequency contents of seismicwaves are more detrimental to the tunnel linings thanhigh-frequency contents

(2) The incident angle of P waves has a significantinfluence on dynamic response of the tunnel liningWhen seismic waves propagate vertically or with asmall incident angle (120579120572 = 0∘ sim30∘) the distributionof the coefficients of dynamic stress concentration inthe tunnel lining is very complex and uneven whichis detrimental to the safety of the tunnel lining

(3) When the buried depth of a tunnel in the soft sur-rounding rock is in a range of 0sim100m the decreasingtendency of dynamic stress response of the liningwith increasing tunnel buried depth is not obviouseven if the buried depth is greater than 100m thecoefficient of dynamic stress concentration of thelining still remains at a high level of about 65 In thissituation tunnels crossing the fault zone or the high

stress segment may get damaged seriously duringlarge earthquake

(4) When the tunnel buried depth in hard surroundingrock is less than 100m the coefficient of dynamicstress concentration in the tunnel lining is highranging from 60 to 85 While the buried depth isgreater than 100m the dynamic stress of the lining issmall and tends to stabilize When the tunnel burieddepth is more than 100m the tunnel buried depth isnot a major factor that affects the dynamic stress

(5) The coefficient of dynamic stress concentration inthe lining increases gradually with increasing elasticmodulus of the lining and the greater the elasticmodulus is the more uneven the dynamic stresscoefficient is Therefore under the premise of meet-ing the bearing capacity and deformation of liningappropriate soft lining is recommended to be adoptedin the tunnel structures to mitigate seismic damageof tunnel lining and the ratio between the elasticmodulus of the lining and soft surrounding rockshould be less than 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of China(Grant no 20110184120033) the Open Foundation of StateKey Laboratory of Geohazard Prevention and Geoenviron-ment Protection (Chengdu University of Technology Grant

Mathematical Problems in Engineering 11

no SKLGP2011K014) and the Science Foundation for Excel-lent Youth Scholars of Southwest JiaotongUniversity of China(Grant no SWJTU11BR003)

References

[1] T B Li ldquoDamage to mountain tunnels related to theWenchuanearthquake and some suggestions for aseismic tunnel construc-tionrdquo Bulletin of Engineering Geology and the Environment vol71 no 2 pp 297ndash308 2012

[2] I Towhata Geotechnical Earthquake Engineering SpringerBerlin Germany 2008

[3] V W Lee and M D Trifunac ldquoResponse of tunnels to incidentSH-wavesrdquo Journal of the Engineering Mechanics Division vol105 no 4 pp 643ndash659 1979

[4] Y L Zheng L D Yang and W Y Li Seismic Design of Under-ground Structure Tongji University Press Shanghai China2005

[5] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak and CompanyNew York NY USA 1973

[6] V W Lee and H Cao ldquoDiffraction of SV waves by circularcanyons of various depthsrdquo Journal of Engineering Mechanicsvol 115 no 9 pp 2035ndash2056 1989

[7] H Cao and V W Lee ldquoScattering and diffraction of planeP waves by circular cylindrical canyons with variable depth-to-width ratiordquo International Journal of Soil Dynamics andEarthquake Engineering vol 9 no 3 pp 141ndash150 1990

[8] VW Lee and J Karl ldquoDiffraction of SV waves by undergroundcircular cylindrical cavitiesrdquo Soil Dynamics and EarthquakeEngineering vol 11 no 8 pp 445ndash456 1992

[9] V W Lee and J Karl ldquoOn deformation of near a circularunderground cavity subjected to incident plane P wavesrdquoEuropean Journal of Earthquake Engineering vol 7 no 1 pp 29ndash35 1993

[10] M Manoogian and V W Lee ldquoDiffraction of SH-waves bysubsurface inclusions of arbitrary shaperdquo Journal of EngineeringMechanics vol 122 no 2 pp 123ndash129 1996

[11] C A Davis V W Lee and J P Bardet ldquoTransverse responceof underground cavities and pipes to incident SV wavesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 383ndash410 2001

[12] J W Liang H Zhang and V W Lee ldquoAnalytical solution fordynamic stress concentration of underground cavities underincident plane P wavesrdquo Chinese Journal of Geotechnical Engi-neering vol 26 no 6 pp 815ndash819 2004

[13] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (I) analytical solutionrdquoEarthquake Engineering and Engineering Vibration vol 27 no1 pp 1ndash6 2007

[14] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (II) numerical resultsrdquoEarthquake Engineering and Engineering Vibration vol 27 no2 pp 1ndash11 2007

[15] X D Ji J W Liang and J J Yang ldquoDynamic stress concen-tration of an underground cylindrical lined cavity subjected toincident plane P and SV wavesrdquo Journal of Tianjin UniversityScience and Technology vol 39 no 5 pp 511ndash517 2006

[16] H B You and J W Liang ldquoScattering of plane SV waves by acavity in a layered half-spacerdquo Rock and Soil Mechanics vol 27no 3 pp 383ndash388 2006

[17] G P Kouretzis G D Bouckovalas and C J Gantes ldquo3-D shell analysis of cylindrical underground structures underseismic shear (S) wave actionrdquo Soil Dynamics and EarthquakeEngineering vol 26 no 10 pp 909ndash921 2006

[18] M Esmaeili S Vahdani andANoorzad ldquoDynamic response oflined circular tunnel to plane harmonic wavesrdquo Tunnelling andUnderground Space Technology vol 11 no 17 pp 1ndash9 2005

[19] H Xu T B Li and L Q Li ldquoResearch on dynamic responseof underground circular lining tunnel under the action of Pwavesrdquo Applied Mechanics and Materials vol 99-100 pp 181ndash189 2011

[20] H Xu T B Li DWang Y S Li and Z H Lin ldquoStudy of seismicresponses of mountain tunnels with 3D shaking table modeltestrdquo Chinese Journal of Rock Mechanics and Engineering vol32 no 9 pp 1762ndash1771 2013

[21] G Y Gao Y S Li and T B Li ldquoDynamic analysis of shallowburied circular lining tunnel with vertical incident P waverdquoCentral South Highway Engineering vol 32 no 2 pp 56ndash602007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0

3

3

6

9

6

9

0

30

6090

120

150

210

270300

330

180

240

120579120572 = 0∘

120579120572 = 30∘

120579120572 = 60∘

|120591lowast 120579120579|

(a) 120578 = 1

0

3

3

6

9

6

9

0

30

6090

120

150

210

270300

330

180

240

120579120572 = 0∘

120579120572 = 30∘

120579120572 = 60∘

|120591lowast 120579120579|

(b) 120578 = 2

Figure 3 Dynamic stress concentration coefficient of tunnel lining with different incident angles

of toroidal dynamic stress in the lining with different burieddepth in a range of 10mndash150m With the increase of theburied depths the dynamic stress concentration coefficientof the lining does not converge to the situation of singleround hole inwhole spaceThedistributions of lining toroidaldynamic stress with different values of buried depth are verycomplex and uneven and no directionality can be obviouslyobserved

As Figure 5 shows when the buried depth in the softsurrounding rock is within 100m the maximum dynamicstress response of the lining varies with buried depth andtwo peak values can be observed at the buried depth ofℎ1= 30m and 100m When the buried depth is greater

than 100m (ℎ1gt 100m) the coefficient of dynamic stress

concentration of lining does not vary significantly with a highlevel of about 65 It appears that with an increasing burieddepth the decreasing trend of dynamic stress response is notpronounced in the soft surrounding rock and the dynamicstress of tunnel lining may still be high in the case of alarge buried depth There are three main reasons (1) thedifference in stiffness is large between tunnel lining and softsurrounding rock (2) the degree of stress concentration oftunnel lining is high and dynamic stress of the lining doesnot decrease obviously with increase of the buried depth and(3) the pressure of surrounding rock increases gradually withincrease of the buried depth In this case the soft surroundingrock has a low strength large porosity many joints andfractures and low capability of self-arching and these rockproperties lead to large tunnel dynamic response

Case 2 (hard surrounding rock medium with elastic modulus119890119904= 20GPa) Regarding the case with hard surrounding

rock medium no obvious directionality can be observed in

terms of the distributions of toroidal dynamic stress causedby P waves but the dynamic stress response of lining ismore pronounced than the case of soft rock as illustrated inFigure 6

Figure 7 shows that when the tunnel buried depth isless than 100m the coefficient of maximum dynamic stressconcentration in the tunnel lining is generally high rangingfrom 60 to 85 which indicates that the response of dynamicstress of the lining is large at the portal segment of hard rocktunnel and the tunnel lining may be damaged easily duringa large earthquake When the tunnel buried depth is greaterthan 75m the coefficient of dynamic stress concentrationis reduced significantly The coefficient of dynamic stressconcentration starts to reach a constant level after the burieddepth is greater than 100m It appears that the lining dynamicstress is small and tends to stabilize at the large buried depthsegment of hard rock tunnel and the tunnel liningmay be lesssusceptible to damage during a large earthquakeTherefore itis concluded that under the hard rock conditions the tunnelburied depth (ie thickness of the overlying rock) is nota major factor that affects the dynamic stress and seismicdamages of tunnel lining when the tunnel buried depth isgreater than 100m This conclusion is consistent with thefindings of Gao et al [21] which states that the dynamicstress concentration coefficient tends to be constant when theburied depth is greater than 25119877

1and the buried depth of

251198771approaches 100m

34 Effect of Elastic Modulus on the Dynamic Stress Con-centration Coefficient of Lining The elastic modulus (119890

1)

of the lining can be considered an index for the elasticdeformation of tunnel lining and the stresses caused by

8 Mathematical Problems in Engineering

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 10mh1 = 20mh1 = 30m

|120591lowast 120579120579|

(a) ℎ1 = 10sim30m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 40mh1 = 50mh1 = 75m

|120591lowast 120579120579|

(b) ℎ1 = 40sim75m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 100mh1 = 125mh1 = 150m

|120591lowast 120579120579|

(c) ℎ1 = 100sim150m

Figure 4 Dynamic stress concentration coefficient of tunnel lining with different values of buried depth in the soft surrounding rock

0123456789

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150Buried depth h1 (m)

|120591lowast 120579120579|

Figure 5 The maximum dynamic stress concentration coefficientof tunnel lining with different buried depths in the soft surroundingrock

elastic deformation increase with increasing elastic modu-lus This also means that at a given level of stresses thelining deformation decreases with increasing lining stiffnessWe focus on the effect of lining elastic modulus on thedynamic stress concentration coefficient of the lining ina soft surrounding rock Figure 8 shows the coefficientsof dynamic stress concentration (120591lowast

120579120579) with different lining

elastic modulus in a soft surrounding rock (119890119904= 2GPa)

As shown in Figure 8 the coefficient of dynamicstress concentration of the lining increases gradually withincreasing lining elastic modulus and the greater the elasticmodulus is the more uneven the distribution of the dynamicstress coefficient is The different adaptabilities of lining

Mathematical Problems in Engineering 9

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 10mh1 = 20mh1 = 30m

|120591lowast 120579120579|

(a) ℎ1 = 10sim30m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 40mh1 = 50mh1 = 75m

|120591lowast 120579120579|

(b) ℎ1 = 40sim75m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 100mh1 = 125mh1 = 150m

|120591lowast 120579120579|

(c) ℎ1 = 100sim150m

Figure 6 Dynamic stress concentration coefficient of tunnel lining with different buried depths in hard surrounding rock

0123456789

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150Buried depth h1 (m)

|120591lowast 120579120579|

Figure 7Themaximumdynamic stress concentration coefficient oftunnel lining with different buried depths in hard surrounding rock

with different stiffness to dynamic stresses may lead to thevariation of the dynamic stress of lining Herein the followingexample is adopted to illustrate this aspect When the elasticmodulus of the lining is 119890

1= 10Gpa and soft surrounding

rock is 119890119904= 2Gpa (ie 119890

1= 5119890119904) the maximum value of

dynamic stress concentration coefficient of the lining is morethan 100 and the variation of dynamic stress of lining acrossall directions is very complexTherefore under the premise ofmeeting the bearing capacity and deformation of the liningappropriately soft lining is recommended to be adopted inthe tunnel structure to mitigate the seismic damage of tunnellining and the ratio between the elastic modulus of the liningand soft surrounding rock should be less than 5

10 Mathematical Problems in Engineering

0

10

20

30

10

20

30

0

30

6090

120

150

210

270300

330

180

240

e1 = 2Gpae1 = 4Gpa

|120591lowast 120579120579|

(a) 1198901 = 2sim4GPa

0

100

100

200

300

400

200

300

400

0

30

6090

120

150

180

210

240270

300

330

e1 = 6Gpae1 = 8Gpa

e1 = 10Gpae1 = 20Gpa

|120591lowast 120579120579|

(b) 1198901 = 6sim20GPa

Figure 8 Dynamic stress concentration coefficient of tunnel lining with different lining elastic modulus in the soft surrounding rock

4 Conclusions

A series solution of wave equation for dynamic response ofunderground circular lining tunnels subjected to incidentplane P waves is presented by Fourier-Bessel series expansionmethod in this paper The effects on circular lining rocktunnels of different factors including incident frequencyincident angle buried depth rock conditions and liningstiffness are studied through the proposed series solution Inthis study the following conclusions are drawn

(1) In circular lining rock tunnels the coefficient oftoroidal dynamic stress concentration and dynamicstress decrease gradually with the increase of incidentfrequency of P waves 120591lowast

120579120579for low-frequency P waves

is about 10 times larger than that for high-frequencyindicating that low-frequency contents of seismicwaves are more detrimental to the tunnel linings thanhigh-frequency contents

(2) The incident angle of P waves has a significantinfluence on dynamic response of the tunnel liningWhen seismic waves propagate vertically or with asmall incident angle (120579120572 = 0∘ sim30∘) the distributionof the coefficients of dynamic stress concentration inthe tunnel lining is very complex and uneven whichis detrimental to the safety of the tunnel lining

(3) When the buried depth of a tunnel in the soft sur-rounding rock is in a range of 0sim100m the decreasingtendency of dynamic stress response of the liningwith increasing tunnel buried depth is not obviouseven if the buried depth is greater than 100m thecoefficient of dynamic stress concentration of thelining still remains at a high level of about 65 In thissituation tunnels crossing the fault zone or the high

stress segment may get damaged seriously duringlarge earthquake

(4) When the tunnel buried depth in hard surroundingrock is less than 100m the coefficient of dynamicstress concentration in the tunnel lining is highranging from 60 to 85 While the buried depth isgreater than 100m the dynamic stress of the lining issmall and tends to stabilize When the tunnel burieddepth is more than 100m the tunnel buried depth isnot a major factor that affects the dynamic stress

(5) The coefficient of dynamic stress concentration inthe lining increases gradually with increasing elasticmodulus of the lining and the greater the elasticmodulus is the more uneven the dynamic stresscoefficient is Therefore under the premise of meet-ing the bearing capacity and deformation of liningappropriate soft lining is recommended to be adoptedin the tunnel structures to mitigate seismic damageof tunnel lining and the ratio between the elasticmodulus of the lining and soft surrounding rockshould be less than 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of China(Grant no 20110184120033) the Open Foundation of StateKey Laboratory of Geohazard Prevention and Geoenviron-ment Protection (Chengdu University of Technology Grant

Mathematical Problems in Engineering 11

no SKLGP2011K014) and the Science Foundation for Excel-lent Youth Scholars of Southwest JiaotongUniversity of China(Grant no SWJTU11BR003)

References

[1] T B Li ldquoDamage to mountain tunnels related to theWenchuanearthquake and some suggestions for aseismic tunnel construc-tionrdquo Bulletin of Engineering Geology and the Environment vol71 no 2 pp 297ndash308 2012

[2] I Towhata Geotechnical Earthquake Engineering SpringerBerlin Germany 2008

[3] V W Lee and M D Trifunac ldquoResponse of tunnels to incidentSH-wavesrdquo Journal of the Engineering Mechanics Division vol105 no 4 pp 643ndash659 1979

[4] Y L Zheng L D Yang and W Y Li Seismic Design of Under-ground Structure Tongji University Press Shanghai China2005

[5] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak and CompanyNew York NY USA 1973

[6] V W Lee and H Cao ldquoDiffraction of SV waves by circularcanyons of various depthsrdquo Journal of Engineering Mechanicsvol 115 no 9 pp 2035ndash2056 1989

[7] H Cao and V W Lee ldquoScattering and diffraction of planeP waves by circular cylindrical canyons with variable depth-to-width ratiordquo International Journal of Soil Dynamics andEarthquake Engineering vol 9 no 3 pp 141ndash150 1990

[8] VW Lee and J Karl ldquoDiffraction of SV waves by undergroundcircular cylindrical cavitiesrdquo Soil Dynamics and EarthquakeEngineering vol 11 no 8 pp 445ndash456 1992

[9] V W Lee and J Karl ldquoOn deformation of near a circularunderground cavity subjected to incident plane P wavesrdquoEuropean Journal of Earthquake Engineering vol 7 no 1 pp 29ndash35 1993

[10] M Manoogian and V W Lee ldquoDiffraction of SH-waves bysubsurface inclusions of arbitrary shaperdquo Journal of EngineeringMechanics vol 122 no 2 pp 123ndash129 1996

[11] C A Davis V W Lee and J P Bardet ldquoTransverse responceof underground cavities and pipes to incident SV wavesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 383ndash410 2001

[12] J W Liang H Zhang and V W Lee ldquoAnalytical solution fordynamic stress concentration of underground cavities underincident plane P wavesrdquo Chinese Journal of Geotechnical Engi-neering vol 26 no 6 pp 815ndash819 2004

[13] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (I) analytical solutionrdquoEarthquake Engineering and Engineering Vibration vol 27 no1 pp 1ndash6 2007

[14] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (II) numerical resultsrdquoEarthquake Engineering and Engineering Vibration vol 27 no2 pp 1ndash11 2007

[15] X D Ji J W Liang and J J Yang ldquoDynamic stress concen-tration of an underground cylindrical lined cavity subjected toincident plane P and SV wavesrdquo Journal of Tianjin UniversityScience and Technology vol 39 no 5 pp 511ndash517 2006

[16] H B You and J W Liang ldquoScattering of plane SV waves by acavity in a layered half-spacerdquo Rock and Soil Mechanics vol 27no 3 pp 383ndash388 2006

[17] G P Kouretzis G D Bouckovalas and C J Gantes ldquo3-D shell analysis of cylindrical underground structures underseismic shear (S) wave actionrdquo Soil Dynamics and EarthquakeEngineering vol 26 no 10 pp 909ndash921 2006

[18] M Esmaeili S Vahdani andANoorzad ldquoDynamic response oflined circular tunnel to plane harmonic wavesrdquo Tunnelling andUnderground Space Technology vol 11 no 17 pp 1ndash9 2005

[19] H Xu T B Li and L Q Li ldquoResearch on dynamic responseof underground circular lining tunnel under the action of Pwavesrdquo Applied Mechanics and Materials vol 99-100 pp 181ndash189 2011

[20] H Xu T B Li DWang Y S Li and Z H Lin ldquoStudy of seismicresponses of mountain tunnels with 3D shaking table modeltestrdquo Chinese Journal of Rock Mechanics and Engineering vol32 no 9 pp 1762ndash1771 2013

[21] G Y Gao Y S Li and T B Li ldquoDynamic analysis of shallowburied circular lining tunnel with vertical incident P waverdquoCentral South Highway Engineering vol 32 no 2 pp 56ndash602007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 10mh1 = 20mh1 = 30m

|120591lowast 120579120579|

(a) ℎ1 = 10sim30m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 40mh1 = 50mh1 = 75m

|120591lowast 120579120579|

(b) ℎ1 = 40sim75m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 100mh1 = 125mh1 = 150m

|120591lowast 120579120579|

(c) ℎ1 = 100sim150m

Figure 4 Dynamic stress concentration coefficient of tunnel lining with different values of buried depth in the soft surrounding rock

0123456789

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150Buried depth h1 (m)

|120591lowast 120579120579|

Figure 5 The maximum dynamic stress concentration coefficientof tunnel lining with different buried depths in the soft surroundingrock

elastic deformation increase with increasing elastic modu-lus This also means that at a given level of stresses thelining deformation decreases with increasing lining stiffnessWe focus on the effect of lining elastic modulus on thedynamic stress concentration coefficient of the lining ina soft surrounding rock Figure 8 shows the coefficientsof dynamic stress concentration (120591lowast

120579120579) with different lining

elastic modulus in a soft surrounding rock (119890119904= 2GPa)

As shown in Figure 8 the coefficient of dynamicstress concentration of the lining increases gradually withincreasing lining elastic modulus and the greater the elasticmodulus is the more uneven the distribution of the dynamicstress coefficient is The different adaptabilities of lining

Mathematical Problems in Engineering 9

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 10mh1 = 20mh1 = 30m

|120591lowast 120579120579|

(a) ℎ1 = 10sim30m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 40mh1 = 50mh1 = 75m

|120591lowast 120579120579|

(b) ℎ1 = 40sim75m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 100mh1 = 125mh1 = 150m

|120591lowast 120579120579|

(c) ℎ1 = 100sim150m

Figure 6 Dynamic stress concentration coefficient of tunnel lining with different buried depths in hard surrounding rock

0123456789

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150Buried depth h1 (m)

|120591lowast 120579120579|

Figure 7Themaximumdynamic stress concentration coefficient oftunnel lining with different buried depths in hard surrounding rock

with different stiffness to dynamic stresses may lead to thevariation of the dynamic stress of lining Herein the followingexample is adopted to illustrate this aspect When the elasticmodulus of the lining is 119890

1= 10Gpa and soft surrounding

rock is 119890119904= 2Gpa (ie 119890

1= 5119890119904) the maximum value of

dynamic stress concentration coefficient of the lining is morethan 100 and the variation of dynamic stress of lining acrossall directions is very complexTherefore under the premise ofmeeting the bearing capacity and deformation of the liningappropriately soft lining is recommended to be adopted inthe tunnel structure to mitigate the seismic damage of tunnellining and the ratio between the elastic modulus of the liningand soft surrounding rock should be less than 5

10 Mathematical Problems in Engineering

0

10

20

30

10

20

30

0

30

6090

120

150

210

270300

330

180

240

e1 = 2Gpae1 = 4Gpa

|120591lowast 120579120579|

(a) 1198901 = 2sim4GPa

0

100

100

200

300

400

200

300

400

0

30

6090

120

150

180

210

240270

300

330

e1 = 6Gpae1 = 8Gpa

e1 = 10Gpae1 = 20Gpa

|120591lowast 120579120579|

(b) 1198901 = 6sim20GPa

Figure 8 Dynamic stress concentration coefficient of tunnel lining with different lining elastic modulus in the soft surrounding rock

4 Conclusions

A series solution of wave equation for dynamic response ofunderground circular lining tunnels subjected to incidentplane P waves is presented by Fourier-Bessel series expansionmethod in this paper The effects on circular lining rocktunnels of different factors including incident frequencyincident angle buried depth rock conditions and liningstiffness are studied through the proposed series solution Inthis study the following conclusions are drawn

(1) In circular lining rock tunnels the coefficient oftoroidal dynamic stress concentration and dynamicstress decrease gradually with the increase of incidentfrequency of P waves 120591lowast

120579120579for low-frequency P waves

is about 10 times larger than that for high-frequencyindicating that low-frequency contents of seismicwaves are more detrimental to the tunnel linings thanhigh-frequency contents

(2) The incident angle of P waves has a significantinfluence on dynamic response of the tunnel liningWhen seismic waves propagate vertically or with asmall incident angle (120579120572 = 0∘ sim30∘) the distributionof the coefficients of dynamic stress concentration inthe tunnel lining is very complex and uneven whichis detrimental to the safety of the tunnel lining

(3) When the buried depth of a tunnel in the soft sur-rounding rock is in a range of 0sim100m the decreasingtendency of dynamic stress response of the liningwith increasing tunnel buried depth is not obviouseven if the buried depth is greater than 100m thecoefficient of dynamic stress concentration of thelining still remains at a high level of about 65 In thissituation tunnels crossing the fault zone or the high

stress segment may get damaged seriously duringlarge earthquake

(4) When the tunnel buried depth in hard surroundingrock is less than 100m the coefficient of dynamicstress concentration in the tunnel lining is highranging from 60 to 85 While the buried depth isgreater than 100m the dynamic stress of the lining issmall and tends to stabilize When the tunnel burieddepth is more than 100m the tunnel buried depth isnot a major factor that affects the dynamic stress

(5) The coefficient of dynamic stress concentration inthe lining increases gradually with increasing elasticmodulus of the lining and the greater the elasticmodulus is the more uneven the dynamic stresscoefficient is Therefore under the premise of meet-ing the bearing capacity and deformation of liningappropriate soft lining is recommended to be adoptedin the tunnel structures to mitigate seismic damageof tunnel lining and the ratio between the elasticmodulus of the lining and soft surrounding rockshould be less than 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of China(Grant no 20110184120033) the Open Foundation of StateKey Laboratory of Geohazard Prevention and Geoenviron-ment Protection (Chengdu University of Technology Grant

Mathematical Problems in Engineering 11

no SKLGP2011K014) and the Science Foundation for Excel-lent Youth Scholars of Southwest JiaotongUniversity of China(Grant no SWJTU11BR003)

References

[1] T B Li ldquoDamage to mountain tunnels related to theWenchuanearthquake and some suggestions for aseismic tunnel construc-tionrdquo Bulletin of Engineering Geology and the Environment vol71 no 2 pp 297ndash308 2012

[2] I Towhata Geotechnical Earthquake Engineering SpringerBerlin Germany 2008

[3] V W Lee and M D Trifunac ldquoResponse of tunnels to incidentSH-wavesrdquo Journal of the Engineering Mechanics Division vol105 no 4 pp 643ndash659 1979

[4] Y L Zheng L D Yang and W Y Li Seismic Design of Under-ground Structure Tongji University Press Shanghai China2005

[5] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak and CompanyNew York NY USA 1973

[6] V W Lee and H Cao ldquoDiffraction of SV waves by circularcanyons of various depthsrdquo Journal of Engineering Mechanicsvol 115 no 9 pp 2035ndash2056 1989

[7] H Cao and V W Lee ldquoScattering and diffraction of planeP waves by circular cylindrical canyons with variable depth-to-width ratiordquo International Journal of Soil Dynamics andEarthquake Engineering vol 9 no 3 pp 141ndash150 1990

[8] VW Lee and J Karl ldquoDiffraction of SV waves by undergroundcircular cylindrical cavitiesrdquo Soil Dynamics and EarthquakeEngineering vol 11 no 8 pp 445ndash456 1992

[9] V W Lee and J Karl ldquoOn deformation of near a circularunderground cavity subjected to incident plane P wavesrdquoEuropean Journal of Earthquake Engineering vol 7 no 1 pp 29ndash35 1993

[10] M Manoogian and V W Lee ldquoDiffraction of SH-waves bysubsurface inclusions of arbitrary shaperdquo Journal of EngineeringMechanics vol 122 no 2 pp 123ndash129 1996

[11] C A Davis V W Lee and J P Bardet ldquoTransverse responceof underground cavities and pipes to incident SV wavesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 383ndash410 2001

[12] J W Liang H Zhang and V W Lee ldquoAnalytical solution fordynamic stress concentration of underground cavities underincident plane P wavesrdquo Chinese Journal of Geotechnical Engi-neering vol 26 no 6 pp 815ndash819 2004

[13] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (I) analytical solutionrdquoEarthquake Engineering and Engineering Vibration vol 27 no1 pp 1ndash6 2007

[14] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (II) numerical resultsrdquoEarthquake Engineering and Engineering Vibration vol 27 no2 pp 1ndash11 2007

[15] X D Ji J W Liang and J J Yang ldquoDynamic stress concen-tration of an underground cylindrical lined cavity subjected toincident plane P and SV wavesrdquo Journal of Tianjin UniversityScience and Technology vol 39 no 5 pp 511ndash517 2006

[16] H B You and J W Liang ldquoScattering of plane SV waves by acavity in a layered half-spacerdquo Rock and Soil Mechanics vol 27no 3 pp 383ndash388 2006

[17] G P Kouretzis G D Bouckovalas and C J Gantes ldquo3-D shell analysis of cylindrical underground structures underseismic shear (S) wave actionrdquo Soil Dynamics and EarthquakeEngineering vol 26 no 10 pp 909ndash921 2006

[18] M Esmaeili S Vahdani andANoorzad ldquoDynamic response oflined circular tunnel to plane harmonic wavesrdquo Tunnelling andUnderground Space Technology vol 11 no 17 pp 1ndash9 2005

[19] H Xu T B Li and L Q Li ldquoResearch on dynamic responseof underground circular lining tunnel under the action of Pwavesrdquo Applied Mechanics and Materials vol 99-100 pp 181ndash189 2011

[20] H Xu T B Li DWang Y S Li and Z H Lin ldquoStudy of seismicresponses of mountain tunnels with 3D shaking table modeltestrdquo Chinese Journal of Rock Mechanics and Engineering vol32 no 9 pp 1762ndash1771 2013

[21] G Y Gao Y S Li and T B Li ldquoDynamic analysis of shallowburied circular lining tunnel with vertical incident P waverdquoCentral South Highway Engineering vol 32 no 2 pp 56ndash602007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 10mh1 = 20mh1 = 30m

|120591lowast 120579120579|

(a) ℎ1 = 10sim30m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 40mh1 = 50mh1 = 75m

|120591lowast 120579120579|

(b) ℎ1 = 40sim75m

0

6

3

6

3

9

9

0

30

6090

120

150

180

210

240270

300

330

h1 = 100mh1 = 125mh1 = 150m

|120591lowast 120579120579|

(c) ℎ1 = 100sim150m

Figure 6 Dynamic stress concentration coefficient of tunnel lining with different buried depths in hard surrounding rock

0123456789

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150Buried depth h1 (m)

|120591lowast 120579120579|

Figure 7Themaximumdynamic stress concentration coefficient oftunnel lining with different buried depths in hard surrounding rock

with different stiffness to dynamic stresses may lead to thevariation of the dynamic stress of lining Herein the followingexample is adopted to illustrate this aspect When the elasticmodulus of the lining is 119890

1= 10Gpa and soft surrounding

rock is 119890119904= 2Gpa (ie 119890

1= 5119890119904) the maximum value of

dynamic stress concentration coefficient of the lining is morethan 100 and the variation of dynamic stress of lining acrossall directions is very complexTherefore under the premise ofmeeting the bearing capacity and deformation of the liningappropriately soft lining is recommended to be adopted inthe tunnel structure to mitigate the seismic damage of tunnellining and the ratio between the elastic modulus of the liningand soft surrounding rock should be less than 5

10 Mathematical Problems in Engineering

0

10

20

30

10

20

30

0

30

6090

120

150

210

270300

330

180

240

e1 = 2Gpae1 = 4Gpa

|120591lowast 120579120579|

(a) 1198901 = 2sim4GPa

0

100

100

200

300

400

200

300

400

0

30

6090

120

150

180

210

240270

300

330

e1 = 6Gpae1 = 8Gpa

e1 = 10Gpae1 = 20Gpa

|120591lowast 120579120579|

(b) 1198901 = 6sim20GPa

Figure 8 Dynamic stress concentration coefficient of tunnel lining with different lining elastic modulus in the soft surrounding rock

4 Conclusions

A series solution of wave equation for dynamic response ofunderground circular lining tunnels subjected to incidentplane P waves is presented by Fourier-Bessel series expansionmethod in this paper The effects on circular lining rocktunnels of different factors including incident frequencyincident angle buried depth rock conditions and liningstiffness are studied through the proposed series solution Inthis study the following conclusions are drawn

(1) In circular lining rock tunnels the coefficient oftoroidal dynamic stress concentration and dynamicstress decrease gradually with the increase of incidentfrequency of P waves 120591lowast

120579120579for low-frequency P waves

is about 10 times larger than that for high-frequencyindicating that low-frequency contents of seismicwaves are more detrimental to the tunnel linings thanhigh-frequency contents

(2) The incident angle of P waves has a significantinfluence on dynamic response of the tunnel liningWhen seismic waves propagate vertically or with asmall incident angle (120579120572 = 0∘ sim30∘) the distributionof the coefficients of dynamic stress concentration inthe tunnel lining is very complex and uneven whichis detrimental to the safety of the tunnel lining

(3) When the buried depth of a tunnel in the soft sur-rounding rock is in a range of 0sim100m the decreasingtendency of dynamic stress response of the liningwith increasing tunnel buried depth is not obviouseven if the buried depth is greater than 100m thecoefficient of dynamic stress concentration of thelining still remains at a high level of about 65 In thissituation tunnels crossing the fault zone or the high

stress segment may get damaged seriously duringlarge earthquake

(4) When the tunnel buried depth in hard surroundingrock is less than 100m the coefficient of dynamicstress concentration in the tunnel lining is highranging from 60 to 85 While the buried depth isgreater than 100m the dynamic stress of the lining issmall and tends to stabilize When the tunnel burieddepth is more than 100m the tunnel buried depth isnot a major factor that affects the dynamic stress

(5) The coefficient of dynamic stress concentration inthe lining increases gradually with increasing elasticmodulus of the lining and the greater the elasticmodulus is the more uneven the dynamic stresscoefficient is Therefore under the premise of meet-ing the bearing capacity and deformation of liningappropriate soft lining is recommended to be adoptedin the tunnel structures to mitigate seismic damageof tunnel lining and the ratio between the elasticmodulus of the lining and soft surrounding rockshould be less than 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of China(Grant no 20110184120033) the Open Foundation of StateKey Laboratory of Geohazard Prevention and Geoenviron-ment Protection (Chengdu University of Technology Grant

Mathematical Problems in Engineering 11

no SKLGP2011K014) and the Science Foundation for Excel-lent Youth Scholars of Southwest JiaotongUniversity of China(Grant no SWJTU11BR003)

References

[1] T B Li ldquoDamage to mountain tunnels related to theWenchuanearthquake and some suggestions for aseismic tunnel construc-tionrdquo Bulletin of Engineering Geology and the Environment vol71 no 2 pp 297ndash308 2012

[2] I Towhata Geotechnical Earthquake Engineering SpringerBerlin Germany 2008

[3] V W Lee and M D Trifunac ldquoResponse of tunnels to incidentSH-wavesrdquo Journal of the Engineering Mechanics Division vol105 no 4 pp 643ndash659 1979

[4] Y L Zheng L D Yang and W Y Li Seismic Design of Under-ground Structure Tongji University Press Shanghai China2005

[5] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak and CompanyNew York NY USA 1973

[6] V W Lee and H Cao ldquoDiffraction of SV waves by circularcanyons of various depthsrdquo Journal of Engineering Mechanicsvol 115 no 9 pp 2035ndash2056 1989

[7] H Cao and V W Lee ldquoScattering and diffraction of planeP waves by circular cylindrical canyons with variable depth-to-width ratiordquo International Journal of Soil Dynamics andEarthquake Engineering vol 9 no 3 pp 141ndash150 1990

[8] VW Lee and J Karl ldquoDiffraction of SV waves by undergroundcircular cylindrical cavitiesrdquo Soil Dynamics and EarthquakeEngineering vol 11 no 8 pp 445ndash456 1992

[9] V W Lee and J Karl ldquoOn deformation of near a circularunderground cavity subjected to incident plane P wavesrdquoEuropean Journal of Earthquake Engineering vol 7 no 1 pp 29ndash35 1993

[10] M Manoogian and V W Lee ldquoDiffraction of SH-waves bysubsurface inclusions of arbitrary shaperdquo Journal of EngineeringMechanics vol 122 no 2 pp 123ndash129 1996

[11] C A Davis V W Lee and J P Bardet ldquoTransverse responceof underground cavities and pipes to incident SV wavesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 383ndash410 2001

[12] J W Liang H Zhang and V W Lee ldquoAnalytical solution fordynamic stress concentration of underground cavities underincident plane P wavesrdquo Chinese Journal of Geotechnical Engi-neering vol 26 no 6 pp 815ndash819 2004

[13] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (I) analytical solutionrdquoEarthquake Engineering and Engineering Vibration vol 27 no1 pp 1ndash6 2007

[14] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (II) numerical resultsrdquoEarthquake Engineering and Engineering Vibration vol 27 no2 pp 1ndash11 2007

[15] X D Ji J W Liang and J J Yang ldquoDynamic stress concen-tration of an underground cylindrical lined cavity subjected toincident plane P and SV wavesrdquo Journal of Tianjin UniversityScience and Technology vol 39 no 5 pp 511ndash517 2006

[16] H B You and J W Liang ldquoScattering of plane SV waves by acavity in a layered half-spacerdquo Rock and Soil Mechanics vol 27no 3 pp 383ndash388 2006

[17] G P Kouretzis G D Bouckovalas and C J Gantes ldquo3-D shell analysis of cylindrical underground structures underseismic shear (S) wave actionrdquo Soil Dynamics and EarthquakeEngineering vol 26 no 10 pp 909ndash921 2006

[18] M Esmaeili S Vahdani andANoorzad ldquoDynamic response oflined circular tunnel to plane harmonic wavesrdquo Tunnelling andUnderground Space Technology vol 11 no 17 pp 1ndash9 2005

[19] H Xu T B Li and L Q Li ldquoResearch on dynamic responseof underground circular lining tunnel under the action of Pwavesrdquo Applied Mechanics and Materials vol 99-100 pp 181ndash189 2011

[20] H Xu T B Li DWang Y S Li and Z H Lin ldquoStudy of seismicresponses of mountain tunnels with 3D shaking table modeltestrdquo Chinese Journal of Rock Mechanics and Engineering vol32 no 9 pp 1762ndash1771 2013

[21] G Y Gao Y S Li and T B Li ldquoDynamic analysis of shallowburied circular lining tunnel with vertical incident P waverdquoCentral South Highway Engineering vol 32 no 2 pp 56ndash602007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0

10

20

30

10

20

30

0

30

6090

120

150

210

270300

330

180

240

e1 = 2Gpae1 = 4Gpa

|120591lowast 120579120579|

(a) 1198901 = 2sim4GPa

0

100

100

200

300

400

200

300

400

0

30

6090

120

150

180

210

240270

300

330

e1 = 6Gpae1 = 8Gpa

e1 = 10Gpae1 = 20Gpa

|120591lowast 120579120579|

(b) 1198901 = 6sim20GPa

Figure 8 Dynamic stress concentration coefficient of tunnel lining with different lining elastic modulus in the soft surrounding rock

4 Conclusions

A series solution of wave equation for dynamic response ofunderground circular lining tunnels subjected to incidentplane P waves is presented by Fourier-Bessel series expansionmethod in this paper The effects on circular lining rocktunnels of different factors including incident frequencyincident angle buried depth rock conditions and liningstiffness are studied through the proposed series solution Inthis study the following conclusions are drawn

(1) In circular lining rock tunnels the coefficient oftoroidal dynamic stress concentration and dynamicstress decrease gradually with the increase of incidentfrequency of P waves 120591lowast

120579120579for low-frequency P waves

is about 10 times larger than that for high-frequencyindicating that low-frequency contents of seismicwaves are more detrimental to the tunnel linings thanhigh-frequency contents

(2) The incident angle of P waves has a significantinfluence on dynamic response of the tunnel liningWhen seismic waves propagate vertically or with asmall incident angle (120579120572 = 0∘ sim30∘) the distributionof the coefficients of dynamic stress concentration inthe tunnel lining is very complex and uneven whichis detrimental to the safety of the tunnel lining

(3) When the buried depth of a tunnel in the soft sur-rounding rock is in a range of 0sim100m the decreasingtendency of dynamic stress response of the liningwith increasing tunnel buried depth is not obviouseven if the buried depth is greater than 100m thecoefficient of dynamic stress concentration of thelining still remains at a high level of about 65 In thissituation tunnels crossing the fault zone or the high

stress segment may get damaged seriously duringlarge earthquake

(4) When the tunnel buried depth in hard surroundingrock is less than 100m the coefficient of dynamicstress concentration in the tunnel lining is highranging from 60 to 85 While the buried depth isgreater than 100m the dynamic stress of the lining issmall and tends to stabilize When the tunnel burieddepth is more than 100m the tunnel buried depth isnot a major factor that affects the dynamic stress

(5) The coefficient of dynamic stress concentration inthe lining increases gradually with increasing elasticmodulus of the lining and the greater the elasticmodulus is the more uneven the dynamic stresscoefficient is Therefore under the premise of meet-ing the bearing capacity and deformation of liningappropriate soft lining is recommended to be adoptedin the tunnel structures to mitigate seismic damageof tunnel lining and the ratio between the elasticmodulus of the lining and soft surrounding rockshould be less than 5

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Specialized Research Fundfor the Doctoral Program of Higher Education of China(Grant no 20110184120033) the Open Foundation of StateKey Laboratory of Geohazard Prevention and Geoenviron-ment Protection (Chengdu University of Technology Grant

Mathematical Problems in Engineering 11

no SKLGP2011K014) and the Science Foundation for Excel-lent Youth Scholars of Southwest JiaotongUniversity of China(Grant no SWJTU11BR003)

References

[1] T B Li ldquoDamage to mountain tunnels related to theWenchuanearthquake and some suggestions for aseismic tunnel construc-tionrdquo Bulletin of Engineering Geology and the Environment vol71 no 2 pp 297ndash308 2012

[2] I Towhata Geotechnical Earthquake Engineering SpringerBerlin Germany 2008

[3] V W Lee and M D Trifunac ldquoResponse of tunnels to incidentSH-wavesrdquo Journal of the Engineering Mechanics Division vol105 no 4 pp 643ndash659 1979

[4] Y L Zheng L D Yang and W Y Li Seismic Design of Under-ground Structure Tongji University Press Shanghai China2005

[5] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak and CompanyNew York NY USA 1973

[6] V W Lee and H Cao ldquoDiffraction of SV waves by circularcanyons of various depthsrdquo Journal of Engineering Mechanicsvol 115 no 9 pp 2035ndash2056 1989

[7] H Cao and V W Lee ldquoScattering and diffraction of planeP waves by circular cylindrical canyons with variable depth-to-width ratiordquo International Journal of Soil Dynamics andEarthquake Engineering vol 9 no 3 pp 141ndash150 1990

[8] VW Lee and J Karl ldquoDiffraction of SV waves by undergroundcircular cylindrical cavitiesrdquo Soil Dynamics and EarthquakeEngineering vol 11 no 8 pp 445ndash456 1992

[9] V W Lee and J Karl ldquoOn deformation of near a circularunderground cavity subjected to incident plane P wavesrdquoEuropean Journal of Earthquake Engineering vol 7 no 1 pp 29ndash35 1993

[10] M Manoogian and V W Lee ldquoDiffraction of SH-waves bysubsurface inclusions of arbitrary shaperdquo Journal of EngineeringMechanics vol 122 no 2 pp 123ndash129 1996

[11] C A Davis V W Lee and J P Bardet ldquoTransverse responceof underground cavities and pipes to incident SV wavesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 383ndash410 2001

[12] J W Liang H Zhang and V W Lee ldquoAnalytical solution fordynamic stress concentration of underground cavities underincident plane P wavesrdquo Chinese Journal of Geotechnical Engi-neering vol 26 no 6 pp 815ndash819 2004

[13] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (I) analytical solutionrdquoEarthquake Engineering and Engineering Vibration vol 27 no1 pp 1ndash6 2007

[14] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (II) numerical resultsrdquoEarthquake Engineering and Engineering Vibration vol 27 no2 pp 1ndash11 2007

[15] X D Ji J W Liang and J J Yang ldquoDynamic stress concen-tration of an underground cylindrical lined cavity subjected toincident plane P and SV wavesrdquo Journal of Tianjin UniversityScience and Technology vol 39 no 5 pp 511ndash517 2006

[16] H B You and J W Liang ldquoScattering of plane SV waves by acavity in a layered half-spacerdquo Rock and Soil Mechanics vol 27no 3 pp 383ndash388 2006

[17] G P Kouretzis G D Bouckovalas and C J Gantes ldquo3-D shell analysis of cylindrical underground structures underseismic shear (S) wave actionrdquo Soil Dynamics and EarthquakeEngineering vol 26 no 10 pp 909ndash921 2006

[18] M Esmaeili S Vahdani andANoorzad ldquoDynamic response oflined circular tunnel to plane harmonic wavesrdquo Tunnelling andUnderground Space Technology vol 11 no 17 pp 1ndash9 2005

[19] H Xu T B Li and L Q Li ldquoResearch on dynamic responseof underground circular lining tunnel under the action of Pwavesrdquo Applied Mechanics and Materials vol 99-100 pp 181ndash189 2011

[20] H Xu T B Li DWang Y S Li and Z H Lin ldquoStudy of seismicresponses of mountain tunnels with 3D shaking table modeltestrdquo Chinese Journal of Rock Mechanics and Engineering vol32 no 9 pp 1762ndash1771 2013

[21] G Y Gao Y S Li and T B Li ldquoDynamic analysis of shallowburied circular lining tunnel with vertical incident P waverdquoCentral South Highway Engineering vol 32 no 2 pp 56ndash602007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

no SKLGP2011K014) and the Science Foundation for Excel-lent Youth Scholars of Southwest JiaotongUniversity of China(Grant no SWJTU11BR003)

References

[1] T B Li ldquoDamage to mountain tunnels related to theWenchuanearthquake and some suggestions for aseismic tunnel construc-tionrdquo Bulletin of Engineering Geology and the Environment vol71 no 2 pp 297ndash308 2012

[2] I Towhata Geotechnical Earthquake Engineering SpringerBerlin Germany 2008

[3] V W Lee and M D Trifunac ldquoResponse of tunnels to incidentSH-wavesrdquo Journal of the Engineering Mechanics Division vol105 no 4 pp 643ndash659 1979

[4] Y L Zheng L D Yang and W Y Li Seismic Design of Under-ground Structure Tongji University Press Shanghai China2005

[5] Y H Pao and C C Mow Diffraction of Elastic Waves andDynamic Stress Concentrations Crane Russak and CompanyNew York NY USA 1973

[6] V W Lee and H Cao ldquoDiffraction of SV waves by circularcanyons of various depthsrdquo Journal of Engineering Mechanicsvol 115 no 9 pp 2035ndash2056 1989

[7] H Cao and V W Lee ldquoScattering and diffraction of planeP waves by circular cylindrical canyons with variable depth-to-width ratiordquo International Journal of Soil Dynamics andEarthquake Engineering vol 9 no 3 pp 141ndash150 1990

[8] VW Lee and J Karl ldquoDiffraction of SV waves by undergroundcircular cylindrical cavitiesrdquo Soil Dynamics and EarthquakeEngineering vol 11 no 8 pp 445ndash456 1992

[9] V W Lee and J Karl ldquoOn deformation of near a circularunderground cavity subjected to incident plane P wavesrdquoEuropean Journal of Earthquake Engineering vol 7 no 1 pp 29ndash35 1993

[10] M Manoogian and V W Lee ldquoDiffraction of SH-waves bysubsurface inclusions of arbitrary shaperdquo Journal of EngineeringMechanics vol 122 no 2 pp 123ndash129 1996

[11] C A Davis V W Lee and J P Bardet ldquoTransverse responceof underground cavities and pipes to incident SV wavesrdquoEarthquake Engineering and Structural Dynamics vol 30 no3 pp 383ndash410 2001

[12] J W Liang H Zhang and V W Lee ldquoAnalytical solution fordynamic stress concentration of underground cavities underincident plane P wavesrdquo Chinese Journal of Geotechnical Engi-neering vol 26 no 6 pp 815ndash819 2004

[13] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (I) analytical solutionrdquoEarthquake Engineering and Engineering Vibration vol 27 no1 pp 1ndash6 2007

[14] J W Liang Z L Ba and VW Lee ldquoScattering of plane P wavesaround a cavity in poroelastic half-space (II) numerical resultsrdquoEarthquake Engineering and Engineering Vibration vol 27 no2 pp 1ndash11 2007

[15] X D Ji J W Liang and J J Yang ldquoDynamic stress concen-tration of an underground cylindrical lined cavity subjected toincident plane P and SV wavesrdquo Journal of Tianjin UniversityScience and Technology vol 39 no 5 pp 511ndash517 2006

[16] H B You and J W Liang ldquoScattering of plane SV waves by acavity in a layered half-spacerdquo Rock and Soil Mechanics vol 27no 3 pp 383ndash388 2006

[17] G P Kouretzis G D Bouckovalas and C J Gantes ldquo3-D shell analysis of cylindrical underground structures underseismic shear (S) wave actionrdquo Soil Dynamics and EarthquakeEngineering vol 26 no 10 pp 909ndash921 2006

[18] M Esmaeili S Vahdani andANoorzad ldquoDynamic response oflined circular tunnel to plane harmonic wavesrdquo Tunnelling andUnderground Space Technology vol 11 no 17 pp 1ndash9 2005

[19] H Xu T B Li and L Q Li ldquoResearch on dynamic responseof underground circular lining tunnel under the action of Pwavesrdquo Applied Mechanics and Materials vol 99-100 pp 181ndash189 2011

[20] H Xu T B Li DWang Y S Li and Z H Lin ldquoStudy of seismicresponses of mountain tunnels with 3D shaking table modeltestrdquo Chinese Journal of Rock Mechanics and Engineering vol32 no 9 pp 1762ndash1771 2013

[21] G Y Gao Y S Li and T B Li ldquoDynamic analysis of shallowburied circular lining tunnel with vertical incident P waverdquoCentral South Highway Engineering vol 32 no 2 pp 56ndash602007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of