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Elastic-slip interface effect on dynamic response of a lined tunnel in a semi-infinite alluvial valley under SH waves
Tunnelling and Underground Space Technology 74 (2018) 96–106
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Elastic-slip interface effect on dynamic response of a lined tunnel in a semiinfinite alluvial valley under SH waves
T
⁎
Xue-Qian Fang , Teng-Fei Zhang, Hai-Yan Li Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Tunnel Alluvial valley Elastic-slip interface Dynamic response
A lined tunnel in a semi-infinite alluvial valley with an elastic-slip interface is constructed, and the dynamic stress distribution around the circular tunnel subjected to SH waves is analyzed. The elastic-slip interface model is introduced to simulate the actual interface condition. The Green's function is used to derive the wave fields of incident, scattered, and refracted waves. The elastic and slip coefficients are used to describe the interface properties. The displacement contours and dynamic stresses with different interface coefficients are analyzed. The interface effects under different wave frequencies and embedded depths are also examined in detail.
1. Introduction
alluvial valley under SH waves was studied using the indirect-integral BEM, and the effects of wave frequency on the surface displacement and dynamic stress were discussed (Zhao et al., 2016). Based on the finite element method with a viscous-spring artificial boundary, formulas of equivalent nodal forces for plane P wave scattering with arbitrary incident angles were deduced and implemented into Abaqus (Huang et al., 2017). Alielahi and Adampira investigated the effects of an unlined tunnel (Alielahi et al., 2015) or two long unsupported parallel tunnels (Alielahi and Adampira, 2016) on the seismic response of the ground surface using the BEM in the time-domain. In their work, a linear elastic medium subjected to vertical SV and P waves was assumed. The time-domain BEM was also employed to predict site-effects of hill-cavity interaction subjected to SV and P waves. Significant effects of underground cavity and hill topography on the surface ground motion were found (Alielahi and Adampira, 2017). In addition, a saturated porous medium was also introduced to simulate the surrounding medium, and Biot's dynamic theory was proposed. Based on Biot's dynamic theory, the diffraction of plane P-waves by a hemispherical alluvial valley was studied, and the effects of incident P-waves on the surface displacement amplitudes were discussed (Zhao et al., 2006). By introducing three potentials of Helmholtz equations and employing Biot's dynamic theory, the scattering of plane P waves by a circular-arc alluvial valley embedded in a poroelastic halfspace was solved, and the stresses and pore pressures were analyzed (Zhou et al., 2008). By combining the method of fundamental solutions and Biot's dynamic theory, the diffraction of Rayleigh waves by a fluidsaturated poroelastic alluvial valley of arbitrary shape was investigated. The effects of alluvium porosity, valley shape, and incident frequency on the dynamic response were discussed in numerical examples (Liu
Many irregular geological conditions such as alluvial valleys may significantly amplify ground movement resulting from the earthquakes. Consequently, concentrated damage to industrial and civil structures often occurs during earthquakes. Various numerical methods have been developed to study the response of alluvial valleys and underground structures, including the finite difference method (FDM) Chaillat et al., 2009, the finite element method (FEM) Najafizadeh et al., 2014; Bielak et al., 1991 and the boundary element method (BEM) Ba and Liang, 2017; Kawase and Aki, 1989. In an alluvial valley, lined tunnels are usually built to pass through the valley. The irregular local geological topography may have a significant effect on the deformation and stress, and even initiate earthquakes (Bard and Bouchon, 1985; Dravinski, 1982; Luco and De Barros, 1994). Therefore, it is of great importance to predict the dynamic response of a field with special local geological conditions in earthquake resistance design. By employing Fourier–Bessel series expansion technique, the scattering and reflection of plane P waves by circular-arc alluvial valleys were described and the surface displacement was analyzed (Li et al., 2005). The boundary integral equation method was used to solve the site response of an alluvial valley or canyon under SH waves, and the half-plane radiation and scattering problems with circular boundaries were considered (Chen et al., 2008). A multi-domain indirect BEM was used to investigate SH wave scattering from a complex local site in a layered half-space (Ba and Yin, 2016). An indirect boundary integral equation method was introduced to solve the scattering of seismic waves by a three-dimensional layered alluvial basin (Liu et al., 2016). The seismic response of tunnel passing through an ⁎
Corresponding author. E-mail address: [email protected] (X.-Q. Fang).
https://doi.org/10.1016/j.tust.2018.01.008 Received 3 September 2017; Received in revised form 25 December 2017; Accepted 5 January 2018 0886-7798/ © 2018 Elsevier Ltd. All rights reserved.
Tunnelling and Underground Space Technology 74 (2018) 96–106
X.-Q. Fang et al.
et al., 2016). The scattering of plane SV waves around a canyon embedded in a saturated poroelastic half-space was investigated by using the indirect boundary integrate equation method. The effects of incident frequency, soil porosity, boundary drainage condition, and canyon shape on the amplification effect of displacement were discussed (Liu et al., 2015). The scattering of plane fast compressional waves by a shallowly embedded tunnel in a poroelastic half-space was solved by using the indirect boundary integration equation method, and the amplification effect on the surface ground motion and the hoop stress in the tunnel was observed (Liu et al., 2017). However, the above research mainly concerns the scattering of elastic waves around the local geological topography with perfect interfaces. Continuous displacements and stresses between the scattering bodies and the surrounding medium are assumed. This assumption simplifies the solution procedure. To simulate the actual interface, imperfect interface models such as a spring-type interface were proposed (Valier-Brasier et al., 2012; Fang et al., 2015). The imperfect interface effect on the strength was also found. In this interface model, the slip of the interface and the coupling of slip and elastic coefficients are ignored. The purpose of this paper is to develop a new interface model to investigate the displacements and stresses around a circular tunnel embedded in a semi-infinite alluvial valley under SH waves. The elasticslip interface model is used to simulate the actual interface condition. The boundary conditions around the tunnel become more complicated. A closed-form solution of SH wave scattering in the semi-infinite alluvial valley is presented using the Green's functions of the semi-infinite medium. Numerical solutions are obtained by discretising the boundaries of the tunnel and the alluvial valley. The displacement contour and dynamic stress distributions around the tunnel under different interface conditions are discussed.
Fig. 2. Five scattering sources in the tunnel and alluvial valley.
Ω1,Ω2 and Ω3 . The governing equation for the displacement W is expressed as (Pao and Mow, 1973) ∂ 2W ∂ 2W + + k 2W = 0, ∂x 2 ∂y 2
(1)
where k = ω/ cSH is the wave number of the anti-plane wave with ω being the incident frequency. cSH = μ/ ρ is the wave speed. ρ and μ are, respectively, the mass density and shear modulus of the medium. The stresses resulting for the anti-plane displacement can be written as (Pao and Mow, 1973)
τxz = μ
∂W ∂x
τyz = μ
∂W , ∂y
(2)
where τxz and τyz are the shear stresses in the medium. 3. Wave fields in the regions To obtain the total wave fields in the regions, the incident, scattered, and refracted waves should be given. Because of the existence of different regions, five scattering sources come into being, as depicted in Fig. 2. The outer and inner scattering waves at the boundary of the alluvial valley are denoted by SC1 and SC2, respectively. The outer and inner scattering waves at the interface between the tunnel and alluvial valley are denoted by SC3 and SC4, respectively. The scattering wave at the inner boundary of the tunnel is SC5. The displacement Win (x ,y ) of incident waves can be expressed as
2. Governing equations Consider an alluvial valley existing in a semi-infinite space and a circular tunnel with infinite length embedded in the alluvial valley, as depicted in Fig. 1. The isotropic property of materials is assumed. The depth of the tunnel is d. The inner and outer radii of the tunnel are, respectively, denoted by a and b. The origin of the tunnel is o, and the coordinate system is shown in Fig. 1. 0An anti-plane wave with incident angle α propagates in the alluvial valley, as depicted in Fig. 2. The traction-free boundary condition at the semi-infinite surface is assumed. In this paper, the type of incident wave is oversimplified as SH waves. The incident wave field may be more complex than SH waves. The actual ground may behave as a three-dimensional structure, rather than a two-dimensional one. However, the closed-form solution of the twodimensional structure can be derived in the following sections. If significant damping exists, surface waves cannot propagate over a large distance, so the damping of materials is neglected. To simulate the imperfect interface around the tunnel, an elastic-slip interface model is developed, as shown in Fig. 1. In this model, an elastic spring in the radial direction and slip in the circumferential direction are introduced. The system is divided into three regions, i.e.,
Win (x ,y ) = exp[i(k x x + k y y )] + exp[i(k x x −k y y )],
(3)
where k x = k sinα , k y = k cosα , and α is the incident angle of the SH waves. Subscript in denotes the incident waves and i= −1 . To satisfy the traction-free boundary condition at the semi-infinite surface, the Green's functions of the semi-infinite medium for the displacements and stresses are expressed as (Kawase and Aki, 1989; Chen et al., 2008)
G (r ,r1) =
i (H0(2) (kr ) + H0(2) (kr ′)), 4
(4)
∂G (r ,r1) ∂G (r ,r1) ⎞ + ⎟, ∂n x ∂n y ⎠
(5)
T (r ,r1) = μ ⎛⎜ ⎝
H0(2)
(•) is the zero-th Hankel function of the where r ∈ Ω1 and r1 ∈ SC1 . second kind. n x and n y are the normal unit vectors corresponding to the r point in the scattering wave field. r ′ is the imaging point of r about the semi-infinite boundary. The larger argument in the complex Hankel function is introduced to ensure H0(2) (•) singularity and series convergence. To avoid the strong singularity at r ′, the image wave sources are deviated a certain distance. Then, the displacement and stresses in the semi-infinite space (Ω1) are expressed as
Fig. 1. A circular tunnel in semi-infinite alluvial valley and interface model.
W1 (r ) =
∫SC
a (r1) G (r ,r1) dSC1,
(6)
σ1 (r ) =
∫SC
a (r1) T (r ,r1) dSC1,
(7)
1
1
where a (r1) is the density of the scattering wave source SC1 at the point 97
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ks = 0.5 N m3 Ds = 0.1m3 N
ks = 0.5 N m3 Ds = 1m3 N
0
0
3
-5
3
-5
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2.5
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ks = 2 N m 3 Ds = 0.1m3 N
1.5 -20
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1.5 20
2
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-20
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ks = 5 N m3 Ds = 3m3 N
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ks = 5 N m3 Ds = 1m3 N
-5
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ks = 5 N m3 Ds = 0.1m3 N
-30 -30
1.5 -20
-10
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-25
ks = 2 N m3 Ds = 3m3 N
2.5 -15
0
2
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-10
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ks = 2 N m 3 Ds = 1m3 N
-10
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-5 -10 -15
2
-20
0
ks = 0.5 N m3 Ds = 3m3 N
0
1.5
1.5 -20
-10
0
10
20
-30 -30
30
-20
-10
0
10
20
30
Fig. 3. The displacement contour under different interface coefficients ( f = 0.5 , R = 30 m, d = 12 m, μ1: μ 2 : μ3 = 4: 1: 25 ).
r1 .
introduced, i.e.,
In the alluvial valley region (Ω2 ), the total scattering wave is a superposition of the scattering waves SC2 and SC3. The displacement and stresses in this region are expressed as
W2−W3 = −
W2 (r ) =
∫SC
a (r2) G (r ,r2) dSC2 +
σ3 (r ) =
∫SC
a (r2) T (r ,r2) dSC2 +
2
2
∫SC
3
∫SC
3
a (r3) G (r ,r3) dSC3, a (r3) T (r ,r3) dSC3,
(
σ2 = σ3
∫S
a (x 4 ) Gi (r ,r4 ) dSC4 +
σ3 (r ) =
∫S
a (r4 ) Tij (r ,r4 ) dSC4 +
4
4
∫S
5
∫S
5
a (r5) Gi (r ,r5) dSC5,
a (r5) Tij (r ,r5) dSC5,
(9)
σ3 = 0.
(13)
(14)
5. Solving for the densities of scattering wave sources
(10)
To obtain the wave fields in the regions, the densities of scattering wave sources should be solved. By substituting Eqs. (3)–(11) into Eqs. (12)–(14), a system of linear algebraic equations can be obtained. After rearrangement, these equations are written as
(11)
where r ∈ ΩIII ,r4 ∈ SC4,r5 ∈ SC5, a (r4 ) and a (r5) correspond to the densities of scattering wave sources (SC4 and SC5 ) at the points r4 and r5 .
[G]{A} = {B},
(15)
where [G] is a (2N1 + 2N2 + N3) × (M1 + M2 + M3 + M4 + M5) matrix and represents the influence matrix of displacements and stresses. {B} is a vector related to the incident waves, and {A} = [a (r1) a (r2) a (r3) a (r4 ) a (r5)]. Ni (i = 1,2,3) are the discrete points at the boundaries of Γi (i = 1,2,3) . Mi (i = 1−5) are the discrete points around the boundaries of the five scattering sources. Therefore, the densities of scattering wave sources are obtained as
4. Boundary conditions with elastic-slip interface effect Two different kinds of materials are combined at the interface; the perfect interface is an ideal condition. The perfect interface may bring large stresses around the tunnel. To simulate the actual interface condition and find a more optimized interface model, an elastic-slip interface is proposed. At the interface of the alluvial valley (Γ1), the boundary conditions are written as
W1 + Win = W2 + W3 ⎫ . σ1 + σn = σ2 + σ3 ⎬ ⎭
)
+ Ds σ2zθ ⎫ , ⎬ ⎭
where ks and Ds are, respectively, the elastic and slip coefficients of the interface. These two coefficients are introduced to describe the interface effect. At the inter-surface of the tunnel, the traction-free boundary condition is considered, i.e.,
(8)
where r ∈ Ω2,r2 ∈ SC2,r3 ∈ SC3, and a (r2) a (r3) correspond to the densities of scattering wave sources (SC2 and SC3 ) at the points r2 and r3 . Similarly, the total scattering wave in the alluvial valley region (Ω3 ) is a superposition of the scattering waves SC4 and SC5. The displacement and stresses in this region are expressed as
W3 (r ) =
σ2zr ks
A = (G T G )−1G T B.
(16)
To solve Eq. (16), an analytical discrete method is introduced. In this method, the residual error E is defined as
(12)
E 2 = [B−A·G]T ·[B−A·G].
At the interface of the tunnel (Γ2 ), the elastic-slip interface model is 98
(17)
Tunnelling and Underground Space Technology 74 (2018) 96–106
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ks = 0.5 N m3 Ds = 0.1m3 N
ks = 0.5 N m3 Ds = 1m3 N
ks = 2 N m 3 Ds = 0.1m3 N
ks = 2 N m3 Ds = 1m3 N
ks = 2 N m3 Ds = 3m3 N
ks = 5 N m3 Ds = 0.1m3 N
ks = 5 N m3 Ds = 1m3 N
ks = 5 N m3 Ds = 3m3 N
ks = 0.5 N m3 Ds = 3m3 N
Fig. 4. The displacement contour under different interface coefficients ( f =1.0, R = 30 m, d = 12 m, μ1: μ 2 : μ3 = 4: 1: 25 ).
ks = 0.5 N m 3 Ds = 0.1m3 N
ks = 0.5 N m3 Ds = 1m3 N
ks = 0.5 N m3 Ds = 3m3 N
ks = 2 N m3 Ds = 0.1m3 N
ks = 2 N m3 Ds = 1m3 N
ks = 2 N m3 Ds = 3m3 N
ks = 5 N m3 Ds = 0.1m3 N
ks = 5 N m3 Ds = 1m3 N
ks = 5 N m3 Ds = 3m3 N
Fig. 5. The displacement contour under different interface coefficients ( f = 2.0, R = 30 m, d = 12 m, μ1: μ 2 : μ3 = 4: 1: 25 ).
6. Numerical examples and discussion
paper is to analyze the effect of the elastic-slip interface; the effect of incident angle has been studied in Li et al. (2005). The influence of the elastic-slip interface is unrelated to the incident angle, so the oblique propagation is ignored in this paper. The SH wave propagates along the y-axis. The dimensionless wave frequency is f = 2R/ λ , where λ (λ = 2π / k ) is the wavelength of the SH wave in the semi-infinite space
The dynamic response of tunnels in a semi-infinite alluvial valley is very significant for their safe operation. In the following numerical examples, a lined tunnel in a semi-circular alluvial valley is considered, and the incident angle of the SH wave is zero. The main objective of this 99
Tunnelling and Underground Space Technology 74 (2018) 96–106
X.-Q. Fang et al. 90
Obtained from this paper
15 60
120
and R is the radius of the semi-circular alluvial valley. The ratio of wave speeds in the three regions (semi-infinite medium, alluvial valley and 1 2 3 : cSH : cSH = 2: 1: 5. To discuss the interface eftunnel) is defined as cSH fect under different impedance ratios between the alluvial valley and the tunnel lining, different shear modulus ratios of the three regions (semi-infinite medium, alluvial valley, and tunnel), namely, μ1 : μ2 : μ3 , are selected. The values of ks and Ds are, respectively, selected as ks = 0.5−5.0N/m3 , and Ds = 0.1−3.0m3/N. The number of discrete points around the boundary depend on the incident frequency. Through computation, it is found that the convergence of the numerical results to the exact boundary condition depends on the number of discrete points. The number of discrete points increases with the wave frequency. The optimal numbers of discrete points Ni (i = 1,2,3) are 110, 50 and 50. The optimal number of discrete points Mi (i = 1−5) are 110, 30, 30, 30 and 20.
Obtained from Ref. [26]
10 150
30 5
180
0
330
210
240
300 270
Fig. 6. DSCF without interface effect ( f = 0.5 , d = 24m , μ1: μ 2 : μ3 = 4: 1: 25 ).
ks = 0.5 N m 3 Ds = 0.1m3 N 90
1 2 3
40
120
ks = 0.5 N m3 Ds = 1m3 N
60
90
20
150
1 2 3
30
120
60
30
90 120
10
10
3
1
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20 60
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1
0
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330
210
90
20 60
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ks = 5 N m3 Ds = 3m3 N
1 2 3
90
10
150
60
1
0
3 330
300
10
150
30
2 1
0
1
180
210
330
300
210
330
240
300 270
Fig. 7. Dynamic stress distribution around the tunnel with different interface coefficients ( f = 0.5, R = 30 m, μ1: μ 2 : μ3 = 4: 1: 25 ).
100
0
3
3
270
30
2
5
180
240
1 2 3
15
5
5
210
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15 30
2
300 270
ks = 5 N m3 Ds = 1m3 N
1 2 3
0
330
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15 10
1
180
270
ks = 5 N m3 Ds = 0.1m3 N 60
0
330
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2
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3
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10 5
1
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3
90
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20 60
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90 120
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300
ks = 2 N m3 Ds = 3m3 N
1 2 3
20
5
270
330
270
15
240
0
300
ks =2 N m3 Ds = 1m3 N
ks = 2 N m3 Ds = 0.1m3 N
180
180
270
270
30
3
210
330
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240
1
5
210
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30
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90
60 20 15
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1
1 2 3
25
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ks = 0.5 N m3 Ds = 3m3 N
Tunnelling and Underground Space Technology 74 (2018) 96–106
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90
1 2 3
150 60
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30
1
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20 10 0
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ks = 5 N m3 Ds = 0.1m3 N
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25 60
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30 5
3
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5
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1 0
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90 120
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ks = 5 N m3 Ds = 3m3 N
1 2 3
25
120
300 270
ks = 5 N m3 Ds = 1m3 N
1 2 3
0
2
270
270
30
3 0
330
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300
1 2 3
1
10
1 3
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ks = 2 N m3 Ds = 3m3 N
1 2 3
30
150
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300 270
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30
0
330
20
15
30
3
210
ks =2 N m3 Ds = 1m3 N
1 2 3
25
120
180
0
270
ks = 2 N m3 Ds = 0.1m3 N
1
20
150
330
240
300
90
60
3
2
270
1 2 3
40 30
30
210
330
240
1
90 120
10
3 2
180
150
30
sp=0.5 D=3
1 2 3
50
1 3
180
0
2
210
330
240
300 270
270
Fig. 8. Dynamic stress distribution around the tunnel with different interface coefficients ( f = 1.0, R = 30 m, μ1: μ 2 : μ3 = 4: 1: 25 ).
with those in Fig. 3, it can be seen that the maximum displacement increases significantly. The distribution of maximum displacement shows great variation with the interface coefficients. The maximum displacement shifts towards the lower part of the semi-circular alluvial valley because of the elastic coefficient of the interface. This phenomenon results from the stronger scattering waves when the elastic interface exists. The greater the elastic coefficient is, the greater the shift of maximum displacement is. The displacements around the tunnel decrease because of the slip at the interface, and the decrease becomes more evident when the elastic coefficient is greater. Therefore, it can be concluded that the slip interface weakens the incident waves. Fig. 5 shows the displacement contours in the high frequency region ( f = 2.0 and μ1 : μ2 : μ3 = 4: 1: 25). By comparing the results with those in Figs. 3 and 4, it can be seen that more displacements peaks occur owing to the more concentrated areas of high frequency seismic waves. The displacements in the semi-circular alluvial valley increase significantly because of the elastic interface, especially in the lower part. The large displacement results from the strong incident and scattering effect at the illuminated side of the tunnel. When the wavelength is small, the incident wave cannot easily pass through the tunnel section
6.1. Displacement contours To display the dynamic response clearly, the displacement contours under different interface coefficients are illustrated. Fig. 3 shows the displacement contours in the low frequency region ( f = 0.5 and μ1 : μ2 : μ3 = 4: 1: 25). The outer and inner radii of the tunnel are defined as b = 3.4 m and a = 3.0 m. The radius of the semi-circular alluvial valley is R = 30 m, and the embedded depth is d = 12 m. It can be seen that the maximum displacement occurs at the surface of the semi-circle alluvial valley, and the displacement at the bottom is a minimum. This is because the waves with greater wavelengths can easily pass through the tunnel section to impact the ground surface. Due to the existence of the elastic-slip interface, the displacement distribution shows significant variation. The displacement at the upper points of the tunnel increases with increasing values for the elastic and slip coefficients of the interface. It can be concluded that smaller values of elastic and slip coefficients of the interface are preferred to reduce the displacements around the tunnel. Fig. 4 shows the displacement contours in the medium frequency region ( f = 1.0 and μ1 : μ2 : μ3 = 4: 1: 25). By comparing the results 101
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k s = 0.5 N m 3 D s = 0.1m 3 N 90
1 2 3
50
120
ks = 0.5 N m3 Ds = 1m3 N
60 40
90
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150 20
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2 3
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2 0
ks = 2 N m3 Ds = 0.1m3 N
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ks = 5 N m3 Ds = 3m3 N
ks = 5 N m3 Ds = 1m3 N
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ks = 5 N m3 Ds = 0.1m3 N
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ks = 2 N m3 Ds = 3m3 N
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ks = 2 N m 3 Ds = 1m3 N
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ks = 0.5 N m3 Ds = 3m3 N
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Fig. 9. Dynamic stress distribution around the tunnel with different interface coefficients ( f = 2.0, R = 30 m, μ1: μ 2 : μ3 = 4: 1: 25 ).
tunnel. To validate the present dynamic model, a comparison with existing results is given in Fig. 6. In this figure, ks = 200 N/m3 and Ds = 0 mean that the interface effect is ignored. d = 24 m denotes that the semi-infinite body reduces to an infinite one. The DSCF of a circular tunnel without interface effect is studied in Pao and Mow (1973). Excellent agreement with reference Pao and Mow (1973) can be seen. The DSCFs around the outer boundary of the tunnel with different interface properties are illustrated in Figs. 7–9. In Fig. 7, the wave frequency is low ( f = 0.5, and μ1 : μ2 : μ3 = 4: 1: 25). It can be seen that the maximum dynamic stress decreases with increasing slip coefficients of the interface when a smaller elastic coefficient is selected. This occurs because the circumferential slip of the interface weakens the incident and scattering waves around the tunnel. However, the maximum dynamic stress shows little variation with the slip coefficients of the interface when a larger elastic coefficient is selected. Therefore, to
to impact the ground surface, and the tunnel acts as a barrier. If the slip coefficient is smaller, the effect of the elastic coefficient of the interface on the displacement distribution will become more significant. 6.2. Dynamic stress distribution around the tunnel Fatigue failures often occur in the regions with high stress concentration. Evaluating the strength of the tunnel requires knowledge of the dynamic stress distribution. According to the work of Pao and Mao (Pao and Mow, 1973), the dynamic stress concentration factor (DSCF) is defined as the ratio of the hoop stress to the maximum amplitude of the incident stress, i.e.,
DSCF = σθz /(k1 μ1),
(18)
where k1 and μ1 are, respectively, the wave number and shear modulus of the semi-infinite medium, and σθz is the shear stress around the 102
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Fig. 10. Dynamic stress distribution around the tunnel with different interface coefficients ( f = 1.0, R = 30 m, μ1: μ 2 : μ3 = 4: 1: 15 ).
In Fig. 9, the high wave frequency is defined ( f = 2.0, and μ1 : μ2 : μ3 = 4: 1: 25). It can be seen that the maximum dynamic stress occurs at the lower part of the tunnel when a smaller elastic coefficient of the interface is selected. When the embedded depth is smaller, the maximum dynamic stress occurs near the upper and lower positions of the tunnels. If a large elastic coefficient is selected, the maximum dynamic stress increases with decreasing depth of the tunnel, and the effect of the slip coefficient becomes smaller. Compared with the results in Fig. 7, it can be seen that the stresses at the illuminated side increase significantly. This phenomenon is explained by the fact that a wave with smaller wave-length cannot easily pass through the tunnel section to impact the ground surface. However, the elastic and slip properties of the interface diffuse the incident and scattering waves, and shift the maximum dynamic stress towards the propagation direction. To discuss the dynamic responses with different impedance ratios between the alluvial valley and the tunnel lining, Figs. 10 and 11 are presented. By comparing with the results in Fig. 8, it can be seen that the dynamic stress increases with increasing elastic modulus of the tunnel lining. This phenomenon results from the strong scattering of SH
reduce the maximum dynamic stress, large elastic coefficients and small slip coefficients of interfaces are preferred. It is clear that the dynamic stress with the depth of d = 18 m is the greatest. This phenomenon results from the radius of the semi-circular alluvial valley. Multiple scattering between the tunnel and the alluvial valley is strongest when the depth is d = 18 m. If the embedded depth is d = 12 m, the tunnel is near the position of increasing displacements of the semi-circular alluvial valley, so, the relative displacement between the tunnel and the alluvial valley occurs, which results in small stresses around the tunnel. In Fig. 8, the medium wave frequency is defined ( f = 1.0, and μ1 : μ2 : μ3 = 4: 1: 25). It can be seen that the maximum dynamic stress decreases with increasing slip coefficients of the interface when a smaller elastic coefficient is selected. The maximum dynamic stress with D = 0.1 m3/N will be a factor of 3 higher than that with D = 3.0 m3/N. The imperfect interface effect increases with decreasing depth of the tunnel. For the medium incident frequency, the dynamic stress with a depth of d = 18 m is the minimum. This phenomenon results from the relative displacement between the tunnel and the alluvial valley. 103
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Fig. 11. Dynamic stress distribution around the tunnel with different interface coefficients ( f = 1.0, R = 30 m, μ1: μ 2 : μ3 = 4: 1: 35 ).
becomes shorter, the tunnel will gradually become a barrier and stronger scattering at the interface will arise. The effect of the slip coefficient becomes weaker when the wavelength is smaller.
waves around the stiffer tunnel lining. However, if an elastic-slip interface is introduced, the dynamic stress decreases significantly. If the elastic coefficient of the interface is greater that 5 N/m3, the interface effect becomes minor. A large elastic coefficient of the interface weakens the incident waves and makes the scattering effect weaker. To find the elastic-slip interface effect on the dynamic stress under larger tunnel dimension, Fig. 12 is illustrated ( f = 2.0, and μ1 : μ2 : μ3 = 4: 1: 15). The outer and inner radii of the tunnel are defined as b = 10 m and a = 8.0 m. By comparing with the results in Fig. 9, it can be seen that the interface effect decreases if larger dimension tunnels are designed. Owing to the large dimension of tunnels, the dynamic stresses at the illuminated side become much larger than that at the positions near the ground surface. This phenomenon results from the waves with greater wavelengths more easily passing through the tunnel section to impact the ground surface. However, when the wavelength
7. Conclusions An elastic-slip interface model is developed to study the dynamic response of tunnels in a semi-infinite alluvial valley under SH waves. By introducing the Green's functions of a semi-infinite medium, the displacements and stresses in the whole region are expressed, and the densities of scattering waves are solved by satisfying the boundary conditions with the interface effect. Through numerical examples, it is found that the elastic and slip coefficients of the interface can change the distributions of displacement and dynamic stress in the alluvial valley. The main conclusions drawn from this study are as follows: 104
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Fig. 12. Dynamic stress distribution around the tunnel with different interface coefficients ( f = 2.0, R = 30 m, μ1: μ 2 : μ3 = 4: 1: 15 ).
is selected. e. The dynamic stress increases with increasing elastic modulus of the tunnel lining, and the interface effect decreases when larger dimension tunnels are employed.
a. In the low frequency region, smaller values of interfacial elastic and slip coefficients are preferred to reduce the displacements and dynamic stress around the tunnel. b. The slip interface weakens the incident waves, and the displacement and maximum dynamic stress decrease with increasing slip coefficients. The effect of slip coefficients becomes smaller when a larger elastic coefficient is selected. c. In the high frequency region, the displacement and maximum dynamic stress in the semi-circular alluvial valley increase significantly owing to the effect of the elastic interface. d. The effect of the interface coefficients on the displacement and dynamic stress becomes more evident when a smaller embedded depth
Acknowledgements This work is supported by the National Natural Science Foundations of China (No. 11472181) and the Natural Science Foundation in Hebei Province of China (No. E2015210020).
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Alielahi, H., Adampira, M., 2016. Effect of twin-parallel tunnels on seismic ground response due to vertically in-plane waves. Int. J. Rock. Mech. Mining. Sci. 85, 67–83. Alielahi, H., Adampira, M., 2017. Evaluation of 2D seismic site response due to hill-cavity interaction using boundary element technique. J. Earthquake Eng. http://dx.doi.org/ 10.1080/13632469.2016.1277437. Alielahi, H., Kamalian, M., Adampira, M., 2015. Seismic ground amplification by unlined tunnels subjected to vertically propagating SV and P waves using BEM. Soil Dyn. Earthquake Eng. 71, 63–79. Ba, Z., Liang, J., 2017. Dynamic response analysis of periodic alluvial valleys under incident plane SH-waves. J. Earthquake Eng. 21, 531–550. Ba, Z.N., Yin, X., 2016. Wave scattering of complex local site in a layered half-space by using a multidomain IBEM: incident plane SH waves. Geophys. J. Int. 205, 1382–1405. Bard, P.Y., Bouchon, M., 1985. The two dimensional resonance of sediment filled valleys. Bull. Seism. Soc. Am. 75, 519–541. Bielak, J., MacCamy, R.C., McGhee, D.S., Barry, A., 1991. Unified symmetric BEM-FEM for site effects on ground motion-SH wave. J. Eng. Mech. 117, 2265–2285. Chaillat, S., Bonnet, M., Semblat, J.F., 2009. A new fast multi-domain BEM to model seismic wave propagation and amplification in 3-D geological structures. Geophys. J. Int. 177, 509–531. Chen, J.T., Chen, P.Y., Chen, C.T., 2008. Surface motion of multiple alluvial valleys for incident plane SH-waves by using a semi-analytical approach. Soil. Dyn. Earthq. Eng. 28, 58–72. Dravinski, M., 1982. Scattering of elastic waves by an alluvial valley. J. Eng. Mech. Div. 108, 19–31. Fang, X.Q., Jin, H.X., Wang, B.L., 2015. Dynamic interaction of two circular lined tunnels with imperfect interfaces under cylindrical P-waves. Int. J. Rock. Mech. Mining. Sci. 79, 172–182. Huang, J., Zhao, M., Du, X., 2017. Non-linear seismic responses of tunnels within normal fault ground under obliquely incident P waves. Tunn. Undergr. Space Technol. 61, 26–39. Kawase, H., Aki, K., 1989. A study on the response of a soft basin for incident S, P and
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Elastic-slip interface effect on dynamic response of a lined tunnel in a semiinfinite alluvial valley under SH waves
T
⁎
Xue-Qian Fang , Teng-Fei Zhang, Hai-Yan Li Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Tunnel Alluvial valley Elastic-slip interface Dynamic response
A lined tunnel in a semi-infinite alluvial valley with an elastic-slip interface is constructed, and the dynamic stress distribution around the circular tunnel subjected to SH waves is analyzed. The elastic-slip interface model is introduced to simulate the actual interface condition. The Green's function is used to derive the wave fields of incident, scattered, and refracted waves. The elastic and slip coefficients are used to describe the interface properties. The displacement contours and dynamic stresses with different interface coefficients are analyzed. The interface effects under different wave frequencies and embedded depths are also examined in detail.
1. Introduction
alluvial valley under SH waves was studied using the indirect-integral BEM, and the effects of wave frequency on the surface displacement and dynamic stress were discussed (Zhao et al., 2016). Based on the finite element method with a viscous-spring artificial boundary, formulas of equivalent nodal forces for plane P wave scattering with arbitrary incident angles were deduced and implemented into Abaqus (Huang et al., 2017). Alielahi and Adampira investigated the effects of an unlined tunnel (Alielahi et al., 2015) or two long unsupported parallel tunnels (Alielahi and Adampira, 2016) on the seismic response of the ground surface using the BEM in the time-domain. In their work, a linear elastic medium subjected to vertical SV and P waves was assumed. The time-domain BEM was also employed to predict site-effects of hill-cavity interaction subjected to SV and P waves. Significant effects of underground cavity and hill topography on the surface ground motion were found (Alielahi and Adampira, 2017). In addition, a saturated porous medium was also introduced to simulate the surrounding medium, and Biot's dynamic theory was proposed. Based on Biot's dynamic theory, the diffraction of plane P-waves by a hemispherical alluvial valley was studied, and the effects of incident P-waves on the surface displacement amplitudes were discussed (Zhao et al., 2006). By introducing three potentials of Helmholtz equations and employing Biot's dynamic theory, the scattering of plane P waves by a circular-arc alluvial valley embedded in a poroelastic halfspace was solved, and the stresses and pore pressures were analyzed (Zhou et al., 2008). By combining the method of fundamental solutions and Biot's dynamic theory, the diffraction of Rayleigh waves by a fluidsaturated poroelastic alluvial valley of arbitrary shape was investigated. The effects of alluvium porosity, valley shape, and incident frequency on the dynamic response were discussed in numerical examples (Liu
Many irregular geological conditions such as alluvial valleys may significantly amplify ground movement resulting from the earthquakes. Consequently, concentrated damage to industrial and civil structures often occurs during earthquakes. Various numerical methods have been developed to study the response of alluvial valleys and underground structures, including the finite difference method (FDM) Chaillat et al., 2009, the finite element method (FEM) Najafizadeh et al., 2014; Bielak et al., 1991 and the boundary element method (BEM) Ba and Liang, 2017; Kawase and Aki, 1989. In an alluvial valley, lined tunnels are usually built to pass through the valley. The irregular local geological topography may have a significant effect on the deformation and stress, and even initiate earthquakes (Bard and Bouchon, 1985; Dravinski, 1982; Luco and De Barros, 1994). Therefore, it is of great importance to predict the dynamic response of a field with special local geological conditions in earthquake resistance design. By employing Fourier–Bessel series expansion technique, the scattering and reflection of plane P waves by circular-arc alluvial valleys were described and the surface displacement was analyzed (Li et al., 2005). The boundary integral equation method was used to solve the site response of an alluvial valley or canyon under SH waves, and the half-plane radiation and scattering problems with circular boundaries were considered (Chen et al., 2008). A multi-domain indirect BEM was used to investigate SH wave scattering from a complex local site in a layered half-space (Ba and Yin, 2016). An indirect boundary integral equation method was introduced to solve the scattering of seismic waves by a three-dimensional layered alluvial basin (Liu et al., 2016). The seismic response of tunnel passing through an ⁎
Corresponding author. E-mail address: [email protected] (X.-Q. Fang).
https://doi.org/10.1016/j.tust.2018.01.008 Received 3 September 2017; Received in revised form 25 December 2017; Accepted 5 January 2018 0886-7798/ © 2018 Elsevier Ltd. All rights reserved.
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et al., 2016). The scattering of plane SV waves around a canyon embedded in a saturated poroelastic half-space was investigated by using the indirect boundary integrate equation method. The effects of incident frequency, soil porosity, boundary drainage condition, and canyon shape on the amplification effect of displacement were discussed (Liu et al., 2015). The scattering of plane fast compressional waves by a shallowly embedded tunnel in a poroelastic half-space was solved by using the indirect boundary integration equation method, and the amplification effect on the surface ground motion and the hoop stress in the tunnel was observed (Liu et al., 2017). However, the above research mainly concerns the scattering of elastic waves around the local geological topography with perfect interfaces. Continuous displacements and stresses between the scattering bodies and the surrounding medium are assumed. This assumption simplifies the solution procedure. To simulate the actual interface, imperfect interface models such as a spring-type interface were proposed (Valier-Brasier et al., 2012; Fang et al., 2015). The imperfect interface effect on the strength was also found. In this interface model, the slip of the interface and the coupling of slip and elastic coefficients are ignored. The purpose of this paper is to develop a new interface model to investigate the displacements and stresses around a circular tunnel embedded in a semi-infinite alluvial valley under SH waves. The elasticslip interface model is used to simulate the actual interface condition. The boundary conditions around the tunnel become more complicated. A closed-form solution of SH wave scattering in the semi-infinite alluvial valley is presented using the Green's functions of the semi-infinite medium. Numerical solutions are obtained by discretising the boundaries of the tunnel and the alluvial valley. The displacement contour and dynamic stress distributions around the tunnel under different interface conditions are discussed.
Fig. 2. Five scattering sources in the tunnel and alluvial valley.
Ω1,Ω2 and Ω3 . The governing equation for the displacement W is expressed as (Pao and Mow, 1973) ∂ 2W ∂ 2W + + k 2W = 0, ∂x 2 ∂y 2
(1)
where k = ω/ cSH is the wave number of the anti-plane wave with ω being the incident frequency. cSH = μ/ ρ is the wave speed. ρ and μ are, respectively, the mass density and shear modulus of the medium. The stresses resulting for the anti-plane displacement can be written as (Pao and Mow, 1973)
τxz = μ
∂W ∂x
τyz = μ
∂W , ∂y
(2)
where τxz and τyz are the shear stresses in the medium. 3. Wave fields in the regions To obtain the total wave fields in the regions, the incident, scattered, and refracted waves should be given. Because of the existence of different regions, five scattering sources come into being, as depicted in Fig. 2. The outer and inner scattering waves at the boundary of the alluvial valley are denoted by SC1 and SC2, respectively. The outer and inner scattering waves at the interface between the tunnel and alluvial valley are denoted by SC3 and SC4, respectively. The scattering wave at the inner boundary of the tunnel is SC5. The displacement Win (x ,y ) of incident waves can be expressed as
2. Governing equations Consider an alluvial valley existing in a semi-infinite space and a circular tunnel with infinite length embedded in the alluvial valley, as depicted in Fig. 1. The isotropic property of materials is assumed. The depth of the tunnel is d. The inner and outer radii of the tunnel are, respectively, denoted by a and b. The origin of the tunnel is o, and the coordinate system is shown in Fig. 1. 0An anti-plane wave with incident angle α propagates in the alluvial valley, as depicted in Fig. 2. The traction-free boundary condition at the semi-infinite surface is assumed. In this paper, the type of incident wave is oversimplified as SH waves. The incident wave field may be more complex than SH waves. The actual ground may behave as a three-dimensional structure, rather than a two-dimensional one. However, the closed-form solution of the twodimensional structure can be derived in the following sections. If significant damping exists, surface waves cannot propagate over a large distance, so the damping of materials is neglected. To simulate the imperfect interface around the tunnel, an elastic-slip interface model is developed, as shown in Fig. 1. In this model, an elastic spring in the radial direction and slip in the circumferential direction are introduced. The system is divided into three regions, i.e.,
Win (x ,y ) = exp[i(k x x + k y y )] + exp[i(k x x −k y y )],
(3)
where k x = k sinα , k y = k cosα , and α is the incident angle of the SH waves. Subscript in denotes the incident waves and i= −1 . To satisfy the traction-free boundary condition at the semi-infinite surface, the Green's functions of the semi-infinite medium for the displacements and stresses are expressed as (Kawase and Aki, 1989; Chen et al., 2008)
G (r ,r1) =
i (H0(2) (kr ) + H0(2) (kr ′)), 4
(4)
∂G (r ,r1) ∂G (r ,r1) ⎞ + ⎟, ∂n x ∂n y ⎠
(5)
T (r ,r1) = μ ⎛⎜ ⎝
H0(2)
(•) is the zero-th Hankel function of the where r ∈ Ω1 and r1 ∈ SC1 . second kind. n x and n y are the normal unit vectors corresponding to the r point in the scattering wave field. r ′ is the imaging point of r about the semi-infinite boundary. The larger argument in the complex Hankel function is introduced to ensure H0(2) (•) singularity and series convergence. To avoid the strong singularity at r ′, the image wave sources are deviated a certain distance. Then, the displacement and stresses in the semi-infinite space (Ω1) are expressed as
Fig. 1. A circular tunnel in semi-infinite alluvial valley and interface model.
W1 (r ) =
∫SC
a (r1) G (r ,r1) dSC1,
(6)
σ1 (r ) =
∫SC
a (r1) T (r ,r1) dSC1,
(7)
1
1
where a (r1) is the density of the scattering wave source SC1 at the point 97
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X.-Q. Fang et al.
ks = 0.5 N m3 Ds = 0.1m3 N
ks = 0.5 N m3 Ds = 1m3 N
0
0
3
-5
3
-5
-10
2.5
-15
-10
2.5
-15 2
-20
-25
1.5
-30 -30
-20
-10
0
10
20
-25 -30 -30
30
ks = 2 N m 3 Ds = 0.1m3 N
1.5 -20
-10
0
10
20
-5
3
-5
3
2
-20 -25
1.5 20
2
-25
1.5 -20
-10
0
10
20
3
-5
3
2
-25
1.5 -10
0
10
20
2
-20 -25
1.5
2.5
30
2
-20
-10
0
10
20
30
-5
3 2.5
-15 2
-20 -25
-25 -30 -30
-20
-10
-15
-20
2.5
ks = 5 N m3 Ds = 3m3 N
2.5
-20
30
0
-10
-15
20
3
ks = 5 N m3 Ds = 1m3 N
-5
10
-5
-30 -30
30
0
0
-10
0
-15
-20
-30 -30
30
-10
2.5
ks = 5 N m3 Ds = 0.1m3 N
-30 -30
1.5 -20
-10
-15
10
-25
ks = 2 N m3 Ds = 3m3 N
2.5 -15
0
2
-20
0
-10
-10
2.5
ks = 2 N m 3 Ds = 1m3 N
-10
-20
3
-30 -30
30
0
-30 -30
-5 -10 -15
2
-20
0
ks = 0.5 N m3 Ds = 3m3 N
0
1.5
1.5 -20
-10
0
10
20
-30 -30
30
-20
-10
0
10
20
30
Fig. 3. The displacement contour under different interface coefficients ( f = 0.5 , R = 30 m, d = 12 m, μ1: μ 2 : μ3 = 4: 1: 25 ).
r1 .
introduced, i.e.,
In the alluvial valley region (Ω2 ), the total scattering wave is a superposition of the scattering waves SC2 and SC3. The displacement and stresses in this region are expressed as
W2−W3 = −
W2 (r ) =
∫SC
a (r2) G (r ,r2) dSC2 +
σ3 (r ) =
∫SC
a (r2) T (r ,r2) dSC2 +
2
2
∫SC
3
∫SC
3
a (r3) G (r ,r3) dSC3, a (r3) T (r ,r3) dSC3,
(
σ2 = σ3
∫S
a (x 4 ) Gi (r ,r4 ) dSC4 +
σ3 (r ) =
∫S
a (r4 ) Tij (r ,r4 ) dSC4 +
4
4
∫S
5
∫S
5
a (r5) Gi (r ,r5) dSC5,
a (r5) Tij (r ,r5) dSC5,
(9)
σ3 = 0.
(13)
(14)
5. Solving for the densities of scattering wave sources
(10)
To obtain the wave fields in the regions, the densities of scattering wave sources should be solved. By substituting Eqs. (3)–(11) into Eqs. (12)–(14), a system of linear algebraic equations can be obtained. After rearrangement, these equations are written as
(11)
where r ∈ ΩIII ,r4 ∈ SC4,r5 ∈ SC5, a (r4 ) and a (r5) correspond to the densities of scattering wave sources (SC4 and SC5 ) at the points r4 and r5 .
[G]{A} = {B},
(15)
where [G] is a (2N1 + 2N2 + N3) × (M1 + M2 + M3 + M4 + M5) matrix and represents the influence matrix of displacements and stresses. {B} is a vector related to the incident waves, and {A} = [a (r1) a (r2) a (r3) a (r4 ) a (r5)]. Ni (i = 1,2,3) are the discrete points at the boundaries of Γi (i = 1,2,3) . Mi (i = 1−5) are the discrete points around the boundaries of the five scattering sources. Therefore, the densities of scattering wave sources are obtained as
4. Boundary conditions with elastic-slip interface effect Two different kinds of materials are combined at the interface; the perfect interface is an ideal condition. The perfect interface may bring large stresses around the tunnel. To simulate the actual interface condition and find a more optimized interface model, an elastic-slip interface is proposed. At the interface of the alluvial valley (Γ1), the boundary conditions are written as
W1 + Win = W2 + W3 ⎫ . σ1 + σn = σ2 + σ3 ⎬ ⎭
)
+ Ds σ2zθ ⎫ , ⎬ ⎭
where ks and Ds are, respectively, the elastic and slip coefficients of the interface. These two coefficients are introduced to describe the interface effect. At the inter-surface of the tunnel, the traction-free boundary condition is considered, i.e.,
(8)
where r ∈ Ω2,r2 ∈ SC2,r3 ∈ SC3, and a (r2) a (r3) correspond to the densities of scattering wave sources (SC2 and SC3 ) at the points r2 and r3 . Similarly, the total scattering wave in the alluvial valley region (Ω3 ) is a superposition of the scattering waves SC4 and SC5. The displacement and stresses in this region are expressed as
W3 (r ) =
σ2zr ks
A = (G T G )−1G T B.
(16)
To solve Eq. (16), an analytical discrete method is introduced. In this method, the residual error E is defined as
(12)
E 2 = [B−A·G]T ·[B−A·G].
At the interface of the tunnel (Γ2 ), the elastic-slip interface model is 98
(17)
Tunnelling and Underground Space Technology 74 (2018) 96–106
X.-Q. Fang et al.
ks = 0.5 N m3 Ds = 0.1m3 N
ks = 0.5 N m3 Ds = 1m3 N
ks = 2 N m 3 Ds = 0.1m3 N
ks = 2 N m3 Ds = 1m3 N
ks = 2 N m3 Ds = 3m3 N
ks = 5 N m3 Ds = 0.1m3 N
ks = 5 N m3 Ds = 1m3 N
ks = 5 N m3 Ds = 3m3 N
ks = 0.5 N m3 Ds = 3m3 N
Fig. 4. The displacement contour under different interface coefficients ( f =1.0, R = 30 m, d = 12 m, μ1: μ 2 : μ3 = 4: 1: 25 ).
ks = 0.5 N m 3 Ds = 0.1m3 N
ks = 0.5 N m3 Ds = 1m3 N
ks = 0.5 N m3 Ds = 3m3 N
ks = 2 N m3 Ds = 0.1m3 N
ks = 2 N m3 Ds = 1m3 N
ks = 2 N m3 Ds = 3m3 N
ks = 5 N m3 Ds = 0.1m3 N
ks = 5 N m3 Ds = 1m3 N
ks = 5 N m3 Ds = 3m3 N
Fig. 5. The displacement contour under different interface coefficients ( f = 2.0, R = 30 m, d = 12 m, μ1: μ 2 : μ3 = 4: 1: 25 ).
6. Numerical examples and discussion
paper is to analyze the effect of the elastic-slip interface; the effect of incident angle has been studied in Li et al. (2005). The influence of the elastic-slip interface is unrelated to the incident angle, so the oblique propagation is ignored in this paper. The SH wave propagates along the y-axis. The dimensionless wave frequency is f = 2R/ λ , where λ (λ = 2π / k ) is the wavelength of the SH wave in the semi-infinite space
The dynamic response of tunnels in a semi-infinite alluvial valley is very significant for their safe operation. In the following numerical examples, a lined tunnel in a semi-circular alluvial valley is considered, and the incident angle of the SH wave is zero. The main objective of this 99
Tunnelling and Underground Space Technology 74 (2018) 96–106
X.-Q. Fang et al. 90
Obtained from this paper
15 60
120
and R is the radius of the semi-circular alluvial valley. The ratio of wave speeds in the three regions (semi-infinite medium, alluvial valley and 1 2 3 : cSH : cSH = 2: 1: 5. To discuss the interface eftunnel) is defined as cSH fect under different impedance ratios between the alluvial valley and the tunnel lining, different shear modulus ratios of the three regions (semi-infinite medium, alluvial valley, and tunnel), namely, μ1 : μ2 : μ3 , are selected. The values of ks and Ds are, respectively, selected as ks = 0.5−5.0N/m3 , and Ds = 0.1−3.0m3/N. The number of discrete points around the boundary depend on the incident frequency. Through computation, it is found that the convergence of the numerical results to the exact boundary condition depends on the number of discrete points. The number of discrete points increases with the wave frequency. The optimal numbers of discrete points Ni (i = 1,2,3) are 110, 50 and 50. The optimal number of discrete points Mi (i = 1−5) are 110, 30, 30, 30 and 20.
Obtained from Ref. [26]
10 150
30 5
180
0
330
210
240
300 270
Fig. 6. DSCF without interface effect ( f = 0.5 , d = 24m , μ1: μ 2 : μ3 = 4: 1: 25 ).
ks = 0.5 N m 3 Ds = 0.1m3 N 90
1 2 3
40
120
ks = 0.5 N m3 Ds = 1m3 N
60
90
20
150
1 2 3
30
120
60
30
90 120
10
10
3
1
180
0
2
180
0
1 2 3
20 60
120
10
150
90
240
120
60
10
150
1
0
210
330
210
90
20 60
120
ks = 5 N m3 Ds = 3m3 N
1 2 3
90
10
150
60
1
0
3 330
300
10
150
30
2 1
0
1
180
210
330
300
210
330
240
300 270
Fig. 7. Dynamic stress distribution around the tunnel with different interface coefficients ( f = 0.5, R = 30 m, μ1: μ 2 : μ3 = 4: 1: 25 ).
100
0
3
3
270
30
2
5
180
240
1 2 3
15
5
5
210
20
120
15 30
2
300 270
ks = 5 N m3 Ds = 1m3 N
1 2 3
0
330
240
300
15 10
1
180
270
ks = 5 N m3 Ds = 0.1m3 N 60
0
330
240
30
2
3
210
300
120
150
3
270
20
10 5
1
180
3
90
15
30
2
1 2 3
20 60
5
180
240
90 120
15
30
2
300
ks = 2 N m3 Ds = 3m3 N
1 2 3
20
5
270
330
270
15
240
0
300
ks =2 N m3 Ds = 1m3 N
ks = 2 N m3 Ds = 0.1m3 N
180
180
270
270
30
3
210
330
240
300
240
1
5
210
330
210
2
150
30
3
90
60 20 15
2
150
30
1
1 2 3
25
20
10
150
ks = 0.5 N m3 Ds = 3m3 N
Tunnelling and Underground Space Technology 74 (2018) 96–106
X.-Q. Fang et al.
90
1 2 3
150 60
120
90
60
120 40
100 150
30
1
50
20 10 0
210
180
60 20
2
180
1
150 10
240
300
5
90
60
90
0
210
2
180
ks = 5 N m3 Ds = 0.1m3 N
210
330
240
300
25 60
120 20
90
60
150
30 5
3
10
1
5
0
2
210
330
300
60
180
3
10
1 0
2
210
330
240
300
30
150
30
10
1 2 3
30
20
15
150
270
90 120
20
15
240
ks = 5 N m3 Ds = 3m3 N
1 2 3
25
120
300 270
ks = 5 N m3 Ds = 1m3 N
1 2 3
0
2
270
270
30
3 0
330
240
300
1 2 3
1
10
1 3
210
330
90
60
150
30
180
30
120 20
10
2
240
ks = 2 N m3 Ds = 3m3 N
1 2 3
30
150
3
180
300 270
120
30
0
330
20
15
30
3
210
ks =2 N m3 Ds = 1m3 N
1 2 3
25
120
180
0
270
ks = 2 N m3 Ds = 0.1m3 N
1
20
150
330
240
300
90
60
3
2
270
1 2 3
40 30
30
210
330
240
1
90 120
10
3 2
180
150
30
sp=0.5 D=3
1 2 3
50
1 3
180
0
2
210
330
240
300 270
270
Fig. 8. Dynamic stress distribution around the tunnel with different interface coefficients ( f = 1.0, R = 30 m, μ1: μ 2 : μ3 = 4: 1: 25 ).
with those in Fig. 3, it can be seen that the maximum displacement increases significantly. The distribution of maximum displacement shows great variation with the interface coefficients. The maximum displacement shifts towards the lower part of the semi-circular alluvial valley because of the elastic coefficient of the interface. This phenomenon results from the stronger scattering waves when the elastic interface exists. The greater the elastic coefficient is, the greater the shift of maximum displacement is. The displacements around the tunnel decrease because of the slip at the interface, and the decrease becomes more evident when the elastic coefficient is greater. Therefore, it can be concluded that the slip interface weakens the incident waves. Fig. 5 shows the displacement contours in the high frequency region ( f = 2.0 and μ1 : μ2 : μ3 = 4: 1: 25). By comparing the results with those in Figs. 3 and 4, it can be seen that more displacements peaks occur owing to the more concentrated areas of high frequency seismic waves. The displacements in the semi-circular alluvial valley increase significantly because of the elastic interface, especially in the lower part. The large displacement results from the strong incident and scattering effect at the illuminated side of the tunnel. When the wavelength is small, the incident wave cannot easily pass through the tunnel section
6.1. Displacement contours To display the dynamic response clearly, the displacement contours under different interface coefficients are illustrated. Fig. 3 shows the displacement contours in the low frequency region ( f = 0.5 and μ1 : μ2 : μ3 = 4: 1: 25). The outer and inner radii of the tunnel are defined as b = 3.4 m and a = 3.0 m. The radius of the semi-circular alluvial valley is R = 30 m, and the embedded depth is d = 12 m. It can be seen that the maximum displacement occurs at the surface of the semi-circle alluvial valley, and the displacement at the bottom is a minimum. This is because the waves with greater wavelengths can easily pass through the tunnel section to impact the ground surface. Due to the existence of the elastic-slip interface, the displacement distribution shows significant variation. The displacement at the upper points of the tunnel increases with increasing values for the elastic and slip coefficients of the interface. It can be concluded that smaller values of elastic and slip coefficients of the interface are preferred to reduce the displacements around the tunnel. Fig. 4 shows the displacement contours in the medium frequency region ( f = 1.0 and μ1 : μ2 : μ3 = 4: 1: 25). By comparing the results 101
Tunnelling and Underground Space Technology 74 (2018) 96–106
X.-Q. Fang et al.
k s = 0.5 N m 3 D s = 0.1m 3 N 90
1 2 3
50
120
ks = 0.5 N m3 Ds = 1m3 N
60 40
90
60
120
60
150
90
150 20
10
1
180
0
2 3
210
240
330
2 0
ks = 2 N m3 Ds = 0.1m3 N
330
240
300
40
120
60
90
40
120
60
1
90
60
0
210
180
20
150
30
0
3
240
300
180
330
240
300
60
300 270
ks = 5 N m3 Ds = 3m3 N
ks = 5 N m3 Ds = 1m3 N
1 2 3
30
90
40 60
120
1 2 3
90
60
0
3 330
210
300 270
20
150
30
1
10
2
240
20
150
2 0
180
210
330
300 270
0
3
3
240
30
1
10
2
180
1 2 3
30
30 30
40
120
20
1
0
3
270
ks = 5 N m3 Ds = 0.1m3 N
180
2
210
330
270
10
30
1
10
210
330
1 2 3
30
2
3
150
40
120
1
10
2
120
1 2 3
20
150
30
10
ks = 2 N m3 Ds = 3m3 N
30
30 20
300 270
ks = 2 N m 3 Ds = 1m3 N
1 2 3
0
210
330
240
30
2
3
180
270
240
1 2 3
1
10
210
300
20
1 3
180
270
180
60
150
30
20
90
40
120 30
30
90
1 2 3
40
30
150
ks = 0.5 N m3 Ds = 3m3 N
210
330
240
300 270
Fig. 9. Dynamic stress distribution around the tunnel with different interface coefficients ( f = 2.0, R = 30 m, μ1: μ 2 : μ3 = 4: 1: 25 ).
tunnel. To validate the present dynamic model, a comparison with existing results is given in Fig. 6. In this figure, ks = 200 N/m3 and Ds = 0 mean that the interface effect is ignored. d = 24 m denotes that the semi-infinite body reduces to an infinite one. The DSCF of a circular tunnel without interface effect is studied in Pao and Mow (1973). Excellent agreement with reference Pao and Mow (1973) can be seen. The DSCFs around the outer boundary of the tunnel with different interface properties are illustrated in Figs. 7–9. In Fig. 7, the wave frequency is low ( f = 0.5, and μ1 : μ2 : μ3 = 4: 1: 25). It can be seen that the maximum dynamic stress decreases with increasing slip coefficients of the interface when a smaller elastic coefficient is selected. This occurs because the circumferential slip of the interface weakens the incident and scattering waves around the tunnel. However, the maximum dynamic stress shows little variation with the slip coefficients of the interface when a larger elastic coefficient is selected. Therefore, to
to impact the ground surface, and the tunnel acts as a barrier. If the slip coefficient is smaller, the effect of the elastic coefficient of the interface on the displacement distribution will become more significant. 6.2. Dynamic stress distribution around the tunnel Fatigue failures often occur in the regions with high stress concentration. Evaluating the strength of the tunnel requires knowledge of the dynamic stress distribution. According to the work of Pao and Mao (Pao and Mow, 1973), the dynamic stress concentration factor (DSCF) is defined as the ratio of the hoop stress to the maximum amplitude of the incident stress, i.e.,
DSCF = σθz /(k1 μ1),
(18)
where k1 and μ1 are, respectively, the wave number and shear modulus of the semi-infinite medium, and σθz is the shear stress around the 102
Tunnelling and Underground Space Technology 74 (2018) 96–106
X.-Q. Fang et al.
ks = 0.5 N m3 Ds = 0.1m3 N 90
40
120 30 20
150
90
d=12m d=18m d=24m
1 2 3
60
30 20
150
20 15
15
0
d=12m d=18m d=24m
1 2 3
60 20
30
150
30
3
10
180
330
3
180
0
210
300
0
2
210
330
240
1
5
1
2
210
240
25
120 15
10
150
5
2
d=12m d=18m d=24m
1 2 3
60
1
180
k s = 2 N m 3 Ds = 3m 3 N 90
20
120
30
3
300 270
90
d=12m d=18m d=24m
1 2 3
60
5
240
300
k s = 2 N m 3 Ds = 1m3 N
k s = 2 N m 3 Ds = 0.1m 3 N
10
330
270
270
0
2
210
330
240
300
180
0
2
210
330
210
150
3
3 180
0
90
30
1 10
2
120
20
150
d=12m d=18m d=24m
1 2 3
60 30
30
1
40
120
10
180
240
d=12m d=18m d=24m
1 2 3
60
3
10
90
40
120
30
1
k s = 0.5N m 3 Ds = 3m 3 N
k s = 0.5N m 3 Ds = 1m 3 N
330
240
300
300
270
270
270
k s = 5N m 3 Ds = 0.1m3 N
k s = 5N m 3 Ds = 1m3 N
k s = 5N m 3 Ds = 3m3 N
90
90
20
120
60 15 10
150
3
0
330
210
300 270
15
d=12m d=18m d=24m
1 2 3
25
120
d=12m d=18m d=24m
1 2 3
60 20 15
10
3
180
0
330
210
300 270
30 10
1
5
2
240
150
30
1
5
2
240
60
150
30
90
20
120
1
5 180
d=12m d=18m d=24m
1 2 3
180
3 0
2
330
210
240
300 270
Fig. 10. Dynamic stress distribution around the tunnel with different interface coefficients ( f = 1.0, R = 30 m, μ1: μ 2 : μ3 = 4: 1: 15 ).
In Fig. 9, the high wave frequency is defined ( f = 2.0, and μ1 : μ2 : μ3 = 4: 1: 25). It can be seen that the maximum dynamic stress occurs at the lower part of the tunnel when a smaller elastic coefficient of the interface is selected. When the embedded depth is smaller, the maximum dynamic stress occurs near the upper and lower positions of the tunnels. If a large elastic coefficient is selected, the maximum dynamic stress increases with decreasing depth of the tunnel, and the effect of the slip coefficient becomes smaller. Compared with the results in Fig. 7, it can be seen that the stresses at the illuminated side increase significantly. This phenomenon is explained by the fact that a wave with smaller wave-length cannot easily pass through the tunnel section to impact the ground surface. However, the elastic and slip properties of the interface diffuse the incident and scattering waves, and shift the maximum dynamic stress towards the propagation direction. To discuss the dynamic responses with different impedance ratios between the alluvial valley and the tunnel lining, Figs. 10 and 11 are presented. By comparing with the results in Fig. 8, it can be seen that the dynamic stress increases with increasing elastic modulus of the tunnel lining. This phenomenon results from the strong scattering of SH
reduce the maximum dynamic stress, large elastic coefficients and small slip coefficients of interfaces are preferred. It is clear that the dynamic stress with the depth of d = 18 m is the greatest. This phenomenon results from the radius of the semi-circular alluvial valley. Multiple scattering between the tunnel and the alluvial valley is strongest when the depth is d = 18 m. If the embedded depth is d = 12 m, the tunnel is near the position of increasing displacements of the semi-circular alluvial valley, so, the relative displacement between the tunnel and the alluvial valley occurs, which results in small stresses around the tunnel. In Fig. 8, the medium wave frequency is defined ( f = 1.0, and μ1 : μ2 : μ3 = 4: 1: 25). It can be seen that the maximum dynamic stress decreases with increasing slip coefficients of the interface when a smaller elastic coefficient is selected. The maximum dynamic stress with D = 0.1 m3/N will be a factor of 3 higher than that with D = 3.0 m3/N. The imperfect interface effect increases with decreasing depth of the tunnel. For the medium incident frequency, the dynamic stress with a depth of d = 18 m is the minimum. This phenomenon results from the relative displacement between the tunnel and the alluvial valley. 103
Tunnelling and Underground Space Technology 74 (2018) 96–106
X.-Q. Fang et al.
ks = 0.5 N m3 Ds = 0.1m3 N
90
90
90
150 60
120 100
d=12m d=18m d=24m
1 2 3
150
80
120
60 60
2
180
0
2
210
330
60 20 1
k s = 2 N m 3 Ds = 1m3 N
k s = 2 N m 3 Ds = 3m 3 N 90
30
120
60 20
180
0
210
2
270
60 20
d=12m d=18m d=24m
1 2 3
15 1
150
90 30
120
60 20
10
20 150
1
0
330
180
0 2
2
330
240
30
3 0
210
d=12m d=18m d=24m
1 2 3
10
180
2
300
60
3
180
210
30
120
1
10 3
5
d=12m d=18m d=24m
1 2 3 30
150
30
k s = 5N m 3 Ds = 3m3 N
k s = 5N m 3 Ds = 1m3 N 90
25
300
240
270
k s = 5N m 3 Ds = 0.1m 3 N
270
330
210
300
240
300
0 2
330
270
120
30
180
0
210
330
240
20
3
2
90
d=12m d=18m d=24m
1 2 3
10
3
3
240
60 30
150
30
40
120
1
10
10
d=12m d=18m d=24m
1 2 3
1
150
30
300 270
d=12m d=18m d=24m
1 2 3
240
300
90 30
330
270
270
k s = 2 N m 3 Ds = 0.1m 3 N
0
2
210
330
240
300
150
30
3
10
1 210
180
1 20
3 180
0
120
d=12m d=18m d=24m
1 2 3
30 150
30
3
90
60 40
20
240
50
120
1
50
180
d=12m d=18m d=24m
1 2 3
40
150
30
k s = 0.5N m 3 Ds = 3m3 N
k s = 0.5N m 3 Ds = 1m 3 N
300
210
330
240
300 270
270
Fig. 11. Dynamic stress distribution around the tunnel with different interface coefficients ( f = 1.0, R = 30 m, μ1: μ 2 : μ3 = 4: 1: 35 ).
becomes shorter, the tunnel will gradually become a barrier and stronger scattering at the interface will arise. The effect of the slip coefficient becomes weaker when the wavelength is smaller.
waves around the stiffer tunnel lining. However, if an elastic-slip interface is introduced, the dynamic stress decreases significantly. If the elastic coefficient of the interface is greater that 5 N/m3, the interface effect becomes minor. A large elastic coefficient of the interface weakens the incident waves and makes the scattering effect weaker. To find the elastic-slip interface effect on the dynamic stress under larger tunnel dimension, Fig. 12 is illustrated ( f = 2.0, and μ1 : μ2 : μ3 = 4: 1: 15). The outer and inner radii of the tunnel are defined as b = 10 m and a = 8.0 m. By comparing with the results in Fig. 9, it can be seen that the interface effect decreases if larger dimension tunnels are designed. Owing to the large dimension of tunnels, the dynamic stresses at the illuminated side become much larger than that at the positions near the ground surface. This phenomenon results from the waves with greater wavelengths more easily passing through the tunnel section to impact the ground surface. However, when the wavelength
7. Conclusions An elastic-slip interface model is developed to study the dynamic response of tunnels in a semi-infinite alluvial valley under SH waves. By introducing the Green's functions of a semi-infinite medium, the displacements and stresses in the whole region are expressed, and the densities of scattering waves are solved by satisfying the boundary conditions with the interface effect. Through numerical examples, it is found that the elastic and slip coefficients of the interface can change the distributions of displacement and dynamic stress in the alluvial valley. The main conclusions drawn from this study are as follows: 104
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90
90
15
120
k s = 0.5N m 3 Ds = 3m3 N
k s = 0.5N m 3 Ds = 1m 3 N
ks = 0.5 N m3 Ds = 0.1m3 N 60
90
25 60
120
40 60
120
20 10
15
150
30
30
150
20
150
30
10
5
3
180
2 180
0
210
330
k s = 2 N m 3 Ds = 0.1m 3 N 8 6 4
330
240
300
300
270
270
k s = 2 N m 3 Ds = 1m3 N
k s = 2 N m 3 Ds = 3m 3 N
90
d=12m d=18m d=24m
1 2 3
60
210
330
240
0 1
210
300
90
2 180
0
1
270
120
10
5
1
240
3
3
2
150
30
8
120 6
90
d=12m d=18m d=24m
1 2 3
60
60 8 6
30
4
150
30
2
150
30 4
2
2
2 1 0
210
240
1
180
330
3
2
2
1
180
10
120
180
0
210
330
0
210
330
3
300
240
3
300
240
300
270
270
270
k s = 5N m 3 Ds = 0.1m 3 N
k s = 5N m 3 Ds = 1m3 N
k s = 5N m3 Ds = 3m 3 N
90
8 6 4
150
2
90
d=12m d=18m d=24m
1 2 3
60
120
8
120 6
30
4
150
2
2
4
150
2
2
330
30 2 1
180
0
210
3
d=12m d=18m d=24m
1 2 3
60 6
1 0
3
8
120
30
1 180
210
90
d=12m d=18m d=24m
1 2 3
60
330
180
0
210
330 3
300
240 270
300
240 270
240
300 270
Fig. 12. Dynamic stress distribution around the tunnel with different interface coefficients ( f = 2.0, R = 30 m, μ1: μ 2 : μ3 = 4: 1: 15 ).
is selected. e. The dynamic stress increases with increasing elastic modulus of the tunnel lining, and the interface effect decreases when larger dimension tunnels are employed.
a. In the low frequency region, smaller values of interfacial elastic and slip coefficients are preferred to reduce the displacements and dynamic stress around the tunnel. b. The slip interface weakens the incident waves, and the displacement and maximum dynamic stress decrease with increasing slip coefficients. The effect of slip coefficients becomes smaller when a larger elastic coefficient is selected. c. In the high frequency region, the displacement and maximum dynamic stress in the semi-circular alluvial valley increase significantly owing to the effect of the elastic interface. d. The effect of the interface coefficients on the displacement and dynamic stress becomes more evident when a smaller embedded depth
Acknowledgements This work is supported by the National Natural Science Foundations of China (No. 11472181) and the Natural Science Foundation in Hebei Province of China (No. E2015210020).
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