Vibration of layered saturated ground with a tunnel subjected to an underground moving load

Vibration of layered saturated ground with a tunnel subjected to an underground moving load

Computers and Geotechnics 119 (2020) 103342 Contents lists available at ScienceDirect Computers and Geotechnics journal homepage: www.elsevier.com/l...

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Computers and Geotechnics 119 (2020) 103342

Contents lists available at ScienceDirect

Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo

Research Paper

Vibration of layered saturated ground with a tunnel subjected to an underground moving load

T

Anfeng Hua,b, Yijun Lia,b, , Yuebao Dengc, Kanghe Xiea,b ⁎

a

Research Center of Coastal and Urban Geotechnical Engineering, Zhejiang University, 310058 Hangzhou, China MOE Key Laboratory of Soft Soils and Geoenvironmental Engineering, Zhejiang University, 310058 Hangzhou, China c Institute of Geotechnical Engineering, Ningbo University, Ningbo City, Zhejiang Province 315211, China b

ARTICLE INFO

ABSTRACT

Keywords: Layered saturated ground Tunnel Moving load TRM method Soft/Hard interlayer

The transmission of vibrations within multi-layered saturated ground subjected to an underground moving load in a tunnel is investigated theoretically. A two-dimensional model of a layered saturated half-space with an embedded beam is established first. The beam simulating the tunnel is located between two horizontal soil layers with a moving point load acting upon it. The half-space soil is assumed as Biot’s poroelastic medium and the transmission and reflection matrices (TRM) method is applied to consider the wave propagation in the layered soil. Combined with the surface boundary conditions and the continuity conditions between the soil layers and the beam, the governing equations of the ground-beam coupled system are solved by the Fourier transform. Finally, the dynamic response of the layered saturated ground in the time-space domain is obtained by using the fast Fourier transform (FFT) method. Three different soil cases are selected to study the influences of the soft/ hard interlayer on dynamic response. The displacement as well as the velocity and acceleration of the surface vibration are analyzed. The displacement amplitude spectra at different depths are presented to find the attenuation law of the ground vibration along depth. The numerical results show that both the stiffness and the embedded depth of the interlayer have impacts on the critical velocity of the coupled system and the surface vibration. When the vibration propagates from the tunnel to ground surface, the high-frequency components decrease and the low-frequency components increase.

1. Introduction The dynamic response of a half-space ground subjected to a moving load is a typical research subject in both geotechnical engineering and traffic engineering. Many studies concerning the ground vibration caused by surface moving loads have been carried out [1–8]. In these studies, soils were first simulated as homogeneous elastic or viscoelastic solid and then the layered property of soils and the porosity/multiphase of natural soils were also taken into consideration. The track system in most of these studies was simplified as the beam or slab structure. For the underground railways, the moving loads are applied on the tunnel buried inside the ground, resulting in the generation and transmission of the waves in the ground being different from those for surface moving loads. Various models have been established to estimate the vibrations induced by underground moving trains. By simplifying the tunnel as a Euler-Bernoulli beam embedded in a viscoelastic soil layer with a fixed bottom boundary, Metrikine and Vrouwenvelder [9] investigated the



surface vibration under three different types of moving loads. Making use of an efficient wavelet approach, Koziol et al. [10] theoretically analyzed the surface vibration of a viscoelastic half-plane induced by a harmonic moving point load along a buried beam. Taking into account the pore water in the soil medium, Yuan et al. [11] investigated the dynamic response of a saturated poroelastic soil layer subjected to an underground moving load. Considering the circular boundary of the tunnel, some analytical models were established in the cylindrical coordinate. Lu [12] investigated the dynamic response of a porous fullspace with a cylindrical hole subjected to a moving axisymmetric ring load. By treating the circular tunnel as a Flügge cylindrical shell in the homogeneous elastic medium, Forrest and Hunt [13] proposed a threedimensional “Pipe in Pipe (PiP)” model to calculate the train-induced vibration. Yuan and Boström [14] presented an analytical solution for the vibrations of a viscoelastic half-space due to a point load moving in an elastic hollow cylinder. Numerical approaches have also been widely applied to solve this vibration problem. For example, to address the infinite boundary of the ground in the wave propagation problems, the

Corresponding author. E-mail address: [email protected] (Y. Li).

https://doi.org/10.1016/j.compgeo.2019.103342 Received 10 June 2019; Received in revised form 2 November 2019; Accepted 10 November 2019 0266-352X/ © 2019 Published by Elsevier Ltd.

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boundary element method (BEM) and infinite element method (IEM) were usually coupled with the finite element method (FEM). Andersen and Jones [15] investigated the vibrations of two different tunnel designs using the coupled FE-BE method. Gupta et al. [16] investigated the mechanism of the generation and propagation of vibrations from underground railways by using the coupled periodic FE-BE model and the PiP model. A comparative work was conducted by Xu et al. [17] to study the characteristics and differences of the two-dimensional (2D) and three-dimensional (3D) numerical models for ground vibration considering two kinds of underground tracks. Assuming the invariance of the material and geometry of the tunnel-soil system in the longitudinal direction, the two-and-a-half-dimensional (2.5D) method has been utilized to simplify the 3D problem. Yang and Hung [18] applied the 2.5D FE/IE method to calculate the ground displacement induced by subway trains. Adopting a gradually damped artificial boundary, Bian et al. [19] used the 2.5D FE method to investigate the ground surface vibration generated by underground moving train loads with different vibration frequencies. Using the FE method to simulate the underground structures, a 2.5D coupled FE-BE model was established by He et al. [20] to calculate the dynamic response of the soil-structure system, and the effect of the vibration isolating screen was also investigated. By coupling the railway track with the tunnel-soil system, Di et al. [21] presented the dynamic solution of the poroelastic half-space under asymmetric loading using of the 2.5D boundary element equation and the 2.5D Green’s function for a poroelastic half-space. Yang et al. [22] compared the dynamic results calculated by the 2D and 2.5D FEBE models. For the complexity of dynamic problems, the soil mediums in the above literature were simplified to be uniform and homogeneous. Nevertheless, the natural ground usually consists of several different soil layers. The underground railway lines were inevitably designed to pass through ground with complex soil conditions. A number of analytical solutions have been proposed to study the vibration of layered ground induced by underground trains. Assuming that the tunnel was far away from the ground surface and the interfaces of the soil layer, Hussein et al. [23] incorporated the PiP model and Green’s functions to calculate the vibration from the tunnel embedded in the layered halfspace ground. By employing the transfer matrix method, He and Zhou [24] presented an analytical solution for the vibration of multi-layered ground with an elastic hollow cylinder. Based on Green’s functions for the layered half-space and the cylindrical tunnel, Sheng et al. [25] studied the vibration of the ground generated by a harmonic moving load mathematically. In the above layered ground models, the soil medium, however, was considered as a single-phased solid, and the effect of the pore fluid on the dynamic response of the saturated soil [3,26,27] cannot be ignored. Yuan et al. [28] established a multi-layered model with a dry layer overlying a poroelastic half-space containing a cylindrical tunnel to investigate the influence of the variation in the water table on the vibration. However, this model cannot be used to directly evaluate ground vibration with more complex soil conditions. He et al. [29] numerically calculated the dynamic response of a tunnel in a layered poroelastic half-space with the application of the dynamic 2.5D BEM and 2.5D Green’s function. The analytical solutions for the vibration of the layered poroelastic half-space under moving loads in the tunnel are still rarely reported yet. Therefore, based on Biot’s theory for the saturated poroelastic medium [30,31], both the layered property of the ground and the pore fluid in the soil will be taken into account in this paper. The tunnel is modelled as an infinite long Euler-Bernoulli beam embedded in layered saturated ground. To take into account the layering effects, Luco and Apsel [32–34] established the TRM method to investigate the response of the layered viscoelastic medium. Afterwards, the procedure to obtain the generalized reflection and transmission coefficients has been applied to the layered saturated poroelastic medium [29,35]. In this paper, the analytical solutions for the dynamic response of the multi-layered saturated ground subjected to an

Fig. 1. Geometry of layered saturated ground-beam coupled system subjected to a buried moving load.

underground moving point load along the beam are deduced by employing the TRM method. The motion of the soil is assumed to obey the wave equations of Biot’s poroelastic medium [30]. The governing equations of saturated layered soil in the frequency-wave number domain are solved using the continuity conditions and the boundary conditions of the coupled system. The results in the time-space domain are calculated using the FFT method. Three different cases for soil conditions are selected to investigate the influence of a soft/hard interlayer on the dynamic response of the layered ground-beam coupled system. The critical velocities of the three cases are evaluated, and the surface vibrations in the three cases under moving loads with different velocities are analyzed. In addition, the vertical displacement spectra at different depths are plotted, and the attenuation law of the vibration along the depth is discussed. 2. Physical model and basic assumptions Fig. 1 shows the physical model of this paper. The model consists of n horizontal soil layers overlying the half-plane/bedrock and the soil material is assumed to be a Biot’s poroelastic saturated medium [30]. The beam is located between the l th layer and the l + 1th layer with a moving load F (t) acting upon it. The variable Li (i = 1ñ) denotes the i th soil layer and Ln + 1 denotes the half-plane/bedrock; z i 1 and z i denote the depths of the upper boundary and lower boundary of the i th layer respectively; the thickness of the i th layer is denoted by hi (hi = z i z i 1); H represents the embedded depth of the beam, and l H = z l = i = 1 hi = h1 + h2 + ···+ hl ; F (t) denotes the moving point load with F (t)=P0 e i 0 t [x ct], in which P0 is the amplitude of load, (...) is the Dirac function, 0 ( 0 = 2 f0 ) is the circular frequency of the load, and c is the velocity of the moving load. In this model, the surface of the saturated layered ground is assumed to be permeable and the solid displacement and pore water displacement as well as the soil stress and pore pressure are continuous at the interface between two adjacent layers. The vertical displacement of the Euler-Bernoulli beam is equal to the solid vertical displacement of the soil at the interface of the beam and the surrounding soil. The model assumes that the beam does not move along the direction of the load movement. The beam model in the paper ignores the geometry and ring joint of the tunnel. According to previous studies [11,28], the dynamic response results calculated by the 2D model can be used as a reference for the qualitative analysis of the ground vibration, which is suitable for vibration prediction in the designing stage of a metro line.

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Fig. 2. Comparison of the present results with the results of Metrikine et al. [9]: (a) c = 30 m/ s ; (b) c = 75 m/ s .

Fig. 3. Calculation results of vertical displacements using the parameters from He et al. [24]: (a) under moving constant loads; (b) under moving harmonic loads.

3. Basic equations

expressed as follows [30]: ij

3.1. General solutions for the poroelastic saturated soil

2M

Muj, ji + Mwj, ji =

+ µ ) uj, ji + Mwj, ji = u¨i + f

u¨ i + mw¨ i + bwi

f

w¨ i

+ µ (ui, j + uj, i)

ij

p=

The equations of motion for the soil matrix and pore fluid according to Biot’s fluid-saturated poroelastic theory can be expressed as follows [30,31]:

µui, jj + ( +

=

(3)

ij p

(4)

M +M

where ij is the total stress of the porous saturated material and p is the excess pore fluid pressure; = ui, i is the dilatation of the solid skeleton; = wi, i is the increment of the pore fluid; the subscripts ( ), i denote spatial derivatives; ij is Kronecker delta. Based on the Helmholtz decomposition, four potentials are introduced to decompose the displacements ui and wi as follows:

(1) (2)

where ui and wi (i = x , z ) are the solid displacement and the fluid displacement relative to the solid matrix in the i -direction, respectively; u¨ i and w¨ i (wi ) represent partial derivatives of ui and wi with respect to time t ; and µ are Lame constants of the solid; and M are two Biot’s parameters; is the mass densities of the saturated soil with = n f + (1 n) s ; s and f are the densities of soil matrix and pore fluid, respectively; n is the porosity of porous material; m is the densitylike parameter with m = f / n ; and b is a parameter describing the internal viscous friction between the solid skeleton and the pore fluid. The material damping of the solid skeleton is taken into account by * = (1 + 2i ) and introducing the complex Lame constants: µ* = µ (1 + 2i ) , where denotes the hysteretic damping ratio. The constitutive equations for Biot’s poroelastic medium can be

ui = grad

1

+ curl

(5a)

wi = grad

2

+ curl

(5b)

The following equations of the potentials are derived by substituting Eq. (5) into Eqs. (1) and (2): 2M

[( + ( M

( 3

2

+ 2µ ) 2

2

f

2

µ t2

)

t2

)

=

2

2

1

f

t2 =

(M

]

1

2

= m

( M 2

t2

2

2

b

f

t

)

t2 2

)

2

(6a) (6b)

2

µ t2

(6c)

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Fig. 4. The geometry models of the three different soil cases: (a) case 1;(b) case 2; (c) case 3. 2

Table 1 Parameters of the soil and load.

f

Value

Young’s modulus for solid E

5.0 × 107N/m2 1/3 0.1

Water density

1000 kg/m3 0.388

f

Porosity n Internal viscous constants b

Bulk modulus of the fluid M

Compressibility of the solid particle Load amplitude P0 Load frequency f0

2

t2

+b

t

)

(6d)

109

kg·m

2

where is the Laplacian operator and 2 = x 2 + z2 . To simplify the above partial differential equations, the double Fourier transform and the corresponding inverse Fourier transform with respect to time t and horizontal coordinate x are introduced by the following integral transform equations:

2020 kg/m3

1.0 ×

= (m

2

Parameters Poisson’s ratio Hysteretic damping ratio Soil density

t2

f¯ ( , z, ) =

f (x , z, t )e

i x e i t dxdt

- 3s 1

105 MPa 1.0

f (x , z , t ) =

105N

4× 0Hz

1 (2 ) 2

f¯ ( , z, )e i x e i t d d

where represents the wave number in the x direction and

(7a) (7b) represents

the frequency; and f¯ ( , z, ) denotes the variables in the wave number-frequency domain. By performing the double Fourier transformation in Eq. (7a) to Eq.

Fig. 5. The maximum vertical displacements |uzmax | versus the load velocity in three cases: (a) Es = E/5; (b) Es = 5E .

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Fig. 6. Time histories of the vertical displacement at the ground surface under the sub-critical moving loads: (a–d) Es = E/5, with c = 20m/ s , 40m/ s , 60m/ s , and 70m/ s , respectively; (e–h) Es = 5E , with c = 20m/ s , 60m/ s , 80m/ s , and 90m/ s ,respectively.

(6), the potentials in the frequency-wavenumber domain can be solved. Substituting the potentials into Eqs. (3)–(5), the general solutions for the dynamic components of the poroelastic soil medium in the

transformed domain can be expressed as follows:

{iu¯ x u¯ z w¯ z i ¯xz ¯zz p¯ }T = {R (z )}{ } 5

(8)

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Fig. 7. Vertical displacement amplitude spectra with c = 20m/ s : (a) Es = E/5; (b) Es = 5E .

where { } = {A B C D E F }T , in which A ( , ), B ( , ) , C ( , ) , E ( , ) , D ( , ) and F ( , ) are the undetermined constants and {R (z )}= er 1z

e r1 z

ir3 er 3 z r 1 e r1 z

r 1 er 1 z

r 2 er 2 z

r 1 1 er1 z r 1 er 1z 2µ



er 2 z

e r2z

ir3 e r 3 z

r2 e r 2 z i er 3 z r 1 1 e r1 z r 2 2 er 2 z

r2 2 e r 2 z r 1 e r 1z 2µ

i a3 e r 3 z c1 er 1z c1 e r 1 z a1 er1 z a1 e r1 z

r3z 3 e r 2 e r 2 z 2µ

i

c2 e r 2 z a2 er 2 z

i

c2 e r 2 z a2 e r 2 z

i e r3 z

i a3 er 3 z r3 er 3 z 0

2 iµ

(x , z i ) = 0

(12a)

wz(i) (x , z i )

wz(i + 1) (x , z i) = 0

(12b)

uz(i)

uz(i + 1)

(x , z i ) = 0

(12c)

p(i + 1) (x , z i ) = 0

(12d)

(x , z i )

(x , z i )

(i ) xz

r3 e r 3 z 0

2 iµ

2 i ) MLi

( +

2 1

1

=

( + 2µ) Li2 +

=

f

4 2

2 (m 2

;

ib )

2

b i 2

(i = 1, 2) ; a3 = µ ( ( + ) 2 ; S2 = f 3 µ 2

( + 2µ) M

2 4 f

2

f

2+ m 2

Li2 (i = 1, 2) ;

2

ri =

2

i

r3 = 1

=

(i = 1, 2) ;

+ r32) ; ci =2µri2

S2 ;

2 (m 2

=

3

f

ib

Li2

L12 =

1+

ib )( + 2M + 2µ) +

2 1

2 2M

2

; m 2

(i ) zz

ai =

(i ) zz

2

; f

( + 2µ) M

; and

.

3.2. Motion equation of the Euler-Bernoulli beam The motion equation of the Euler-Bernoulli beam is as follows: 2W B

t2

+ EI

4W

x4

= F (t ) + a [

zz (x ,

H

0, t )

zz (x ,

H + 0, t )] (10)

2W

B

t2

+ EI

4W

t4

(x , z i ) = W (x , t )

(x , z i ) = 0(i

)

(i = l) (12f) (12g)

l)

u x(n) (x , zn) = 0

(13a)

uz(n) (x , zn) = 0

(13b)

wz(n) (x , zn) = 0

(13c)

4. Development of the TRM method and the solution

where W (x, t) denotes the vertical displacement of the beam; B and EI are the mass density and bending stiffness of the beam, respectively; a is a characteristic length associated with the length of the structure in the y direction; F (t ) denotes the moving point load; zz (x , H 0, t ) and zz (x , H + 0, t ) denote the vertical stress in the soil beneath and above the tunnel, respectively.

According to the general solutions for the dynamic response of the saturated soil in the transformed domain in Eq.(8), the dynamic responses of the i th layer in the layered soil are factorized by the TRM method as the following expression [29,35]: (i ) (

3.3. Boundary conditions and continuity conditions

, ,z)6 × 1 =

× [Wd(i) ( ,

According to the model assumptions, the boundary conditions and continuity conditions of the layered saturated ground-beam coupled system shown in Fig. 1 can be expressed as follows: When z = 0 ,

(i ) (

Wd(i) ( ,

Dd(i) ( , ) Du(i) ( , ) Sd(i) ( , ) Su(i) ( , ) T

, z )T Wu(i ) ( ,

, ,z)6 × 1 =

(i ) [iu¯ x

(i ) u¯ z

, z )T ] (i ) w¯ z

(i )

(i ) 1 (z z i 1)

, z )3 × 1 = [A(i ) e

(i ) 1 (z i z )

0) = 0

(11a)

zz (x ,

0) = 0

(11b)

Wu(i) ( ,

(11c)

where Wd(i)

(14) (i )

i ¯ xz ¯ zz p¯

, z )3 × 1 = [B (i ) e

xz (x ,

p (x , 0) = 0

uz(i + 1)

(

Moreover, when the n + 1th layer is a half-plane, the positive index term of the dynamic response solutions of the n + 1th layer should be omitted because of the radiation characteristics of the waves. When the n + 1th layer is considered as the rigid bedrock, the boundary conditions are given as follows:

L22 = 2M

(x , z i )

(x , z i ) = F (t )

(x , z i) = (i + 1) zz

(12e)

(x , z i) = 0

(i + 1) zz

(x , z i ) uz(i )

ai (i = 1, 2); 4 2

(i + 1) xz

(x , z j )

(9) where

u x(i + 1)

p (i ) ( x , z i )

r3z 3 e

r2 e r 2 z

1) ,

When z = z i (i = 1, 2, …, n

u x(i)

D (i ) e

(i ) T ]

(i ) 2 (z zi 1)

(15a)

F (i ) e

( i) 3 (z z i 1) ]T

(15b)

6

( ,

, z)

and Wu(i)

( ,

C(i ) e

(i ) 2 (z i z )

E (i ) e

(i ) 3 (zi z ) ]T

(15c)

, z ) are defined as the down-going and

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Fig. 8. The time histories of the surface vertical displacement under the super-critical moving loads: (a–d) Es = E/5, with c = 90m/ s , 100m/ s , and 150m/ s , respectively; (e–h) Es = 5E , with c = 100m/ s , 120m/ s , 130m/ s and 150m/ s , respectively.

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Fig. 9. Amplitude spectra of the vertical displacement with c = 120m/ s : (a) Es = E/5; (b) Es = 5E .

up-going wave vectors in the i th soil layer respectively; and the matrix Dd(i) ( , ) , Du(i) ( , ) , Sd(i) ( , ) , Su(i ) ( , ) can be deduced from the expression of R (z) in Eq. (9). The following equations can be easily deduced from Eqs. (15b) and (15c):

Wd(i) ( ,

, z i 1) = [B (i) ( , ) D (i) ( , ) F (i) ( , )]T

(16a)

Wu(i) ( ,

, z i ) = [A(i ) ( , ) C (i) ( , ) E (i) ( , )]T

(16b)

Wd(i) ( ,

, z ) = E (i ) ( z

Wu(i) ( ,

z i 1) W d(i ) ( ,

z ) Wu(i ) ( ,

, z ) = E (i ) ( z i

Tdeg(n Rdeg(n

E (i ) ( ) =

( i) 1

0

0

0

.

0

0

e

(i ) 3

(17)

Combining the continuity conditions between the i th layer and the i + 1th layer, the following equations can be obtained:

W d(i + 1) (z i ) Wu(i ) (z i )

=

Td(i)

Ru(i)

Rd(i)

Tu(i)

Wu(l ) (z l ) Wd(l + 1) (z l )

(n + Tue

1)

1)

Rdeg(n) Tdeg(n

1)

(Sd(1)) 1Su(1) Wu(1) (z 0) = Rueg(0) Wu(1) (z1) g (1)

Wu(1) (z1) = Tue Wu(2) (z2) g (1) (2) Wd(2) (z1) = Rue Wu (z2)

, where

(22)

g (1) Tue = (I

(1) g (0) 1 (1) Rde Rue ) Tue . g (1) (1) + T (1) R g (0) T g (1) Rue = Rue ue de ue

g (l + 1) (l + 1) (Du(l + 1) Rde E (hl + 1)

g (l 1) (l) = S (l ) Rue E (hl ) + Sul d

(18)

+ (EI 4

(l) g (l 1) (l ) 2 E (hl ) B )(Dd Rue

+ Dd(l + 1) g (l + 1) (l + 1) (Su(l + 1) Rde E (hl + 1) (l + 1) + Sd

1

O Q

+ Dul )

where

Td(i) Rd(i )

(n Rdeg(n) ) 1Tde

g (l 1) (l) Dd(l ) Rue E (hl ) + Dul

W d(i ) (z i ) Wu(i + 1) (z i )

1)

Similarly, the wave vectors of the i th layer above the buried beam (i.e. 0 < i < l ) can be deduced by the up-going wave vector Wu(l) (z l ) of the l th layer consecutively as well. When i = l , the up-going wave vector Wu(l) (z l ) and the down-going wave vector Wd(l + 1) (z l ) can be obtained according to the continuity conditions along the beam as follows:

0

(i ) 2

e

1)

When i = 1,

where

e

(n = Rde

Wd(1) (z 0) =

(16d)

, zi)

1)

(n Rue

= (I

Then, the wave vectors of the i th layer below the beam (i.e. l < i < n ) can be deduced by the down-going wave vector Wd(l + 1) (z l ) of the l + 1 th layer consecutively. According to the boundary conditions on the free surface, the following expressions can be obtained:

(16c)

, z i 1)

1)

(23)

Ru(i ) Tu(i)

=

Dd(i + 1) Sd(i + 1)

Du(i )

1

Su(i)

Dd(i) Sd(i)

T

where O = [0 0 0]T ; Q = 0 F¯ 0 , and F¯ = 2 P0 ( + c 0). Moreover, if the n + 1th layer is a half-plane, the up-going wave vector Wu(n + 1) (z ) (z z n ) will not exist and Rdeg(n + 1) = 0 . Thus, the wave vectors Wu(i) ( , , z ) and Wd(i) ( , , z ) . of arbitrary soil layer (0 < i n + 1) in the ground can be obtned by Eqs. (16c) and (16d). Therefore, the dynamic response of the ith layer in the transformed domain can also be calculated by substituting the wave vectors into Eq. (14). By applying the inverse Fourier transform, the dynamic response in time-space domain can be expressed as follows:

Du(i + 1) Su(i + 1)

(19)

( , ) and ( , ) in Eq. (19) denote the The 3 × 3 matrix reflection matrices for the down-going and up-going waves that incident on the interface z = z i , respectively; while Td(i) ( , ) and Tu(i) ( , ) denote the transmission matrices for the down-going and upgoing waves that are incident on the interface z = z i [32–34]. When the n + 1th layer is the overlaying bedrock, the following expressions can be obtained by the boundary conditions at z = z n : Rd(i)

Ru(i)

Wu(n) (z n) = Rdg (n) Wd(n) (zn ) = Rdeg(n) Wd(n) (zn 1)

(20)

(x , z , t ) =

= where When i = n 1, the following recursive equations can be obtained from the equations in Eq. (18): Rdg (n)

(Du(n) ) 1Dd(n) .

(zn 1) =

1) W d(n 1) (zn 2) g (n 1) Rde Wd(n 1) (z n 2)

(

0

+c )¯( ,

, z )e i x ei t d d (24)

Depending on the property of Dirac’s delta function, Eq.(24) can be reduced to a single inverse Fourier integral by replacing with 0 c , that is:

Wd(n) (z n 1) = Tdeg(n Wu(n 1)

P0 (2 ) 2

(21)

(x , z , t ) =

where 8

P0 (2 ) 2

¯( ,

0

c , z )e i

(x ct ) e i 0 t d

(25)

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Fig. 10. The effect of the interlayer on velocity and acceleration of the surface vibration under sub-critical moving loads: (a–d) c = 20m/ s ; (e–h) c = 40m/ s .

The integral shown in Eq. (25) is solved by the use of the FFT method.

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Fig. 11. The effect of the interlayer on velocity and acceleration of the surface vibration under super-critical moving loads: (a–d) c = 100m/ s ; (e–h) c = 120m/ s .

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Fig. 12. The vertical displacement amplitude spectra at different depths:(a) c = 20m/ s ; (b) c = 50m/ s ; (c) c = 60m/ s ; (d) c = 80m/ s ; (e) c = 100m/ s ; (f) c = 120m/ s ; (g) c = 130m/ s ; (h) c = 150m/ s .

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Fig. 13. The vertical displacement amplitude spectra under moving loads with different frequencies: (a) f0 = 20Hz ; (b) f0 = 50Hz .

5. Results and discussion

5.2.1. Effect of the soft/hard interlayer on the displacement It is assumed that the n + 1th layer is the rigid bedrock and that the n thickness of n overlying layers on the bedrock is 27 m ( i = 1 hi = 27m) . The beam is embedded at a depth of 12m (H = h1 + ···+ hl = 12m ). To investigate the effect of the soft/hard interlayer on the vibration, three different cases are considered. Case 1: As shown in Fig. 4(a), the beam is buried between two soil layers with the same soil parameters (n = 2 ; l = 1; h1 = 12m ; and h2 = 15m ) and the values of the soil parameters are listed in Table 1. This case is used as a reference case; in cases 2 and 3, there is an interlayer in the homogeneous soil layer of case 1. The thickness of the interlayer is denoted by hs (hs = 4m ) and the Young’s modulus of the interlayer is denoted by Es . Case 2: As shown in Fig. 4(b), there is an interlayer located in the upper soil layer of the beam (n = 3; l = 2 ; H = h1 + h2 = 12m ; h1 = 8m ; h2 = hs = 4m ; and h3 = 15m ); Case 3: As shown in Fig. 4(c), there is an interlayer located in the lower soil layer of the beam (n = 3; l = 1; H = h1 = 12m ; h2 = hs = 4m ; and h3 = 11m ). Except for Young’s modulus, the other parameters of the interlayer are assumed to be same as those in Table 1. The values of the mass density and bending stiffness of the beam are taken as B /a = 3 × 10 4kg/m2 and EI / a = 109Nm , respectively. The maximum vertical displacement at the ground surface versus the load velocity in three cases are plotted in Fig. 5 with Es = E/5 in Fig. 5(a) and Es = 5E in Fig. 5(b). It shows that the |u zmax| increases at first and then decreases as the load velocity increases. There is a critical velocity for each case and when the load velocity increases to the critical velocity, |u zmax| reaches its maximum. Furthermore, it can be observed that the relationships of the critical velocities of the three cases are case 3 < case 2 < case 1 in Fig. 5(a) and case 3 > case 2 > case 1 in Fig. 5(b). The results indicate that the existence of the soft interlayer reduces the critical velocity of the considered structure, whereas the hard interlayer has an opposite effect. In addition, the effect of the interlayer beneath the beam (case 3) is greater than that of the interlayer lying on the beam (case 2). According to the above analysis, four sub-critical velocities and four super-critical velocities are selected. The time histories of the vertical displacement at the ground surface (z = 0 ) and the amplitude spectra of the surface vertical displacement under the sub-critical moving loads are depicted in Fig. 6 and Fig. 7 respectively. The corresponding time histories and the amplitude spectra under the super-critical moving loads are given in Fig. 8 and Fig. 9, respectively. In Fig. 6(a)–(h), when the velocity of the moving load is slower than the critical velocity, the time histories of the vertical displacement are impulse-like curves. The impulse amplitude increases and the impulse length decreases with increasing load velocity. For case 3 in Fig. 6(d) and case 1 in Fig. 6(h), the corresponding time histories show a light fluctuation before and after the amplitude peak when the load

5.1. 5.1Comparison with the existing results The surface vibration of a viscoelastic soil layer subjected to a load moving along a buried beam was analyzed by Metrikine et al. [9]. As mentioned in Cai et al. [3], when the poroelastic parameters approach zero (i.e., M = f = m = b = = 0 ), the solution for the saturated poroelastic medium is degenerated to that for the single-phase elastic medium. Assuming that the n + 1th layer is the rigid bedrock and setting n = 2 and l = 1, the values of the other parameters of the soil are assumed to be the same as those in Metrikine et al. [9], i.e., 3 E1 = E2 = 3 × 107N ·m 2 , 1 = 2 = 0.3, 1 = 2 = 1700kg ·m , H = h1 = 12m/7m , h2 = 15m , B = 3 × 10 4kg · m 1, EI = 109N ·m2 , and P0 = 10 4N . The time history curves of the vertical displacement uz under the moving constant load with c = 30 m / s and c = 75 m / s are presented in Fig. 2(a) and Fig. 2(b), respectively. It can be seen from these figures that the results calculated by two different methods agree fairly well with each other. He et al. [24] proposed an analytical 3D solution for the vibration of a multi-layered ground with an elastic hollow cylinder in it. By ignoring the influence of pore water, the new developed solutions (2D beam model) are used here to calculate the dynamic response of the dry single-phase soil for comparison. The corresponding parameters are the same as those in Ref. [24]. Fig. 3(a) and (b) show the vertical displacement at the point (x = 0, z = 0) under moving constant loads and moving harmonic loads, respectively. It can be observed from these comparisons that the effects of the load velocity and frequency on the vibration results are generally consistent with that in Ref. [24], which confirms the reliability of the newly proposed method. 5.2. Numerical examples To investigate the vibration induced by the subway trains in the multi-layered ground, the following analysis focuses on the displacement responses of the coupled system. The vertical displacement of the solid can be obtained according to Eq. (25):

u z (x , z , t ) =

P0 (2 )2

u¯ z ( ,

0

c , z )e i

(x ct ) ei 0 t d

(26)

Consequently, the corresponding amplitude spectrum of the solid displacement can be derived as follows:

u z (f ) =

u z (x , z , t ) e

2 ift dt

=

f 2 f 1 u¯ z ( 0 , 2 f , z) 2 c

(27) 12

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velocities c = 70m /s and c = 90m /s approach the critical velocities of the considered cases. By comparing Fig. 6(a)–(d) with Fig. 6(e)–(h), it can be seen that the existence of the soft interlayer enhances the vibration amplitude while the existence of the hard interlayer reduces the vibration amplitude. The amplification/reduction effect of the interlayer becomes more significant with increasing load velocity. In addition, the orders of the peaks of the time histories are case 3 > case 2 > case 1 in Fig. 6(a)–(d) and case 3 < case 2 < case 1 in Fig. 6(e)–(h), which suggests that the effect of the interlayer beneath the beam (case 3) is greater than that of the interlayer on the beam (case 2). The effect of the interlayer on the displacement amplitude under sub-critical moving loads can also be proven by the sub-critical part in the curves in Fig. 5. Taking c = 20m /s as an example, the results about the influence of an interlayer on the displacement amplitude obtained from Fig. 6 are also evident in the amplitude spectra displayed in Fig. 7, and it can also be observed in Fig. 7 that the dominant frequencies of the vibration in the three cases are almost unaffected by the interlayer. The time histories of the surface vertical displacement under the super-critical moving loads are plotted in Fig. 8. In Fig. 8(a)–(h), the time history curves manifest obvious oscillations under the super-critical moving loads. The maximum displacement decreases and the duration time of the vibration increases as the load velocity increases. As shown in Fig. 8(a)–(d), the vibration frequencies of the time history curves for cases 2 and 3 are lower than those for case 1. In contrast, the vibration frequencies of the time history curves for case 2 and case 3 are higher than those for case 1 in Fig. 8(e)–(h). Moreover, this influence on the vibration frequency for the interlayer beneath the beam (case 3) is larger than that for the interlayer lying on the beam (case 2). To further illustrate the effect of the interlayer, the amplitude spectra of the three cases under the moving load with c = 120m / s are given in Fig. 9. The sequences of the dominant frequencies are case 3 > case 2 > case 1 in Fig. 9(a) and case 3 < case 2 < case 1 in Fig. 9(b), which is consistent with the results shown in the time history curves in Fig. 8.

surface (z = 0 ) and the depth of the beam (z = 12) are presented in Fig. 12(a)–(h). For the amplitude spectra under sub-critical moving loads in Fig. 12(a)–(d), there is only one maximum on the curve, and the frequency range increases with increasing load velocity. Notably, not only the dominant frequency but also the maximum displacement amplitude increases as the moving load velocity increases. The displacement amplitude spectra under the super-critical moving loads are presented in Fig. 12(e)–(h). It can be observed that there is more than one peak point on the spectrum curves and that the maximum displacement amplitude decreases with increasing load velocity. Although the frequency ranges in the super-critical cases are wider than those in the sub-critical cases, the vibration is still mainly distributed in the lowfrequency band. By comparing the displacement spectra at two different depths in each figure, it can be observed that the frequency range of the vertical displacement decreases when the vibration transmits from the beam (z = 12) to the ground surface (z = 0 ). In addition, the spectrum at the depth of z = 12 is higher than the spectrum at the depth of z = 0 in the frequency band on the right side of the red line, while in the frequency band on the left side of the red line, the spectrum at the depth of z = 12 is lower than the spectrum at the depth of z = 0 . It can be inferred that high-frequency components in the vibration decrease, whereas the lowfrequency components in the vibration increase during the propagation of the waves. 5.2.4. The vertical displacement under harmonic loads In practice, because of the roughness and irregularity of the railways, the range of the actual train loading spectrum is rather wide, which starts from zero frequency and lasts up to hundreds of Hz [7]. To investigate the surface vibrations under moving loads with different frequencies, the displacement spectra under two moving harmonic point loads with f0 = 20Hz and f0 = 50Hz are displayed as in Fig. 13. In these two spectra, the frequency of the displacement response distributes around the excitation frequency of the load f0 . And it is indicated that the vibration frequency is closely related with the excitation frequency of the applied load.

5.2.2. The velocity and acceleration of the vertical vibration Taking c = 20m /s and c = 40m /s as examples in the sub-critical domain, the time histories of the velocity and acceleration of the vibration as well as the velocity and acceleration spectra on the ground surface are displayed in Fig. 10(a)–(f). Compared with the corresponding figures in Fig. 6 and Fig. 7, it is obvious that the influences of the stiffness and embedded depth of the soft/hard interlayer on the vibration velocity and acceleration are similar to those on the displacement. In addition, the amplitude of the vibration velocity and acceleration increase when the load velocity c = 20m /s increases to c = 40m /s . Likewise, the time histories of the velocity and acceleration of the vertical vibration as well as the velocity and acceleration spectra on the ground surface under two super-critical moving loads (c = 100m / s and c = 120m / s ) are displayed in Fig. 11(a)–h). The velocity and acceleration of the vibration under the super-critical moving loads are much greater than those under the sub-critical moving loads, which proves that the vibration becomes more violent when the load velocity exceeds the critical velocity. Furthermore, as shown in the Fig. 11(a), (c), (e) and (g), the order of the dominant frequencies of the velocity in three cases is in accord with that of the displacement. However, the time histories of the acceleration combined with the acceleration spectra shown in Fig. 11(b), (d), (f) and (h) indicate that the vibration acceleration changes too fast and the effect of the interlayer on the vibration acceleration is irregular.

6. Conclusions In this paper, the solution for the dynamic response of the layered saturated ground-beam coupled system under a buried moving load is obtained, which can be used to investigate the vibration of the ground with a complex soil profile subjected to underground moving trains. By applying the TRM method and the Fourier transform, the motion equations of the coupled system are solved, and the dynamic response components in the time-space domain are calculated by the FFT algorithm. The influences of the interlayer in the ground on the surface vibration are investigated from the aspects of vibration displacement, velocity and acceleration. The displacement amplitude spectra are presented to investigate the changes in the frequency content when the waves propagate from the tunnel to the surface. The results can be concluded as follows: (1) The critical velocity of the ground with a soft interlayer is smaller than that of the homogeneous soil layer and the critical velocity of the ground with a hard interlayer is larger than that of the homogeneous soil layer. (2) The effects of the interlayer on the surface vibration manifest in different ways under different moving loads. When the moving constant load is in the sub-critical domain, the interlayer mainly affects the vibration amplitude. The amplitudes of the vibration displacement, velocity and acceleration increase when there is a soft interlayer in the ground, which is opposite for the hard interlayer. When the moving constant load is in the super-critical domain, the existence of the interlayer affects the vibration frequency

5.2.3. Vertical displacement at different depths To investigate the attenuation law of the vibration propagating from the tunnel to the surface under different moving loads, the vertical displacement amplitude spectra of the soil layer in case 1 at the free 13

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of the oscillation considerably. The frequency of the displacement as well as the frequency of the vibration velocity decrease/increase for the layered ground with a soft/hard interlayer. (3) The effect of the interlayer on the critical velocity and surface vibration depends not only on the soil stiffness, but also on the embedded depth of the interlayer. With the same thickness and stiffness, the interlayer located in the lower soil layer of the tunnel has a larger impact on the vibration than the interlayer located in the upper soil layer of the tunnel. (4) For the displacement amplitude spectra under the sub-critical moving loads, the frequency range becomes wider and the peak frequency tends to increase with the increase of the load velocity. For the displacement amplitude spectra under the super-critical moving loads, the frequency range first increases and then decreases as the load velocity increases. Moreover, when the waves propagate to the ground surface, the frequency range of the vibration decreases, and the high-frequency components decrease while the low-frequency components increase.

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Declaration of Competing Interest The authors declare that there are no known competing financial interests or personal relationships that influence the work reported in this paper. Acknowledgments Financial support from the National Natural Science Foundation of China (Grant No. 51778572) is gratefully acknowledged. References [1] Hung HH, Yang YB. Elastic waves in visco-elastic half-space generated by various vehicle loads. Soil Dyn Earthq Eng 2001;21(1):1–17. [2] Jones CJC, Sheng X, Petyt M. Simulations of ground vibration from a moving harmonic load on a railway track. J Sound Vib 2000;231(3):739–51. [3] Cai YQ, Sun HL, Xu CJ. Steady state responses of poroelastic half-space soil medium to a moving rectangular load. Int J Solids Struct 2007;44(22–23):7183–96. [4] Xu B, Lu JF, Wang JH. Dynamic response of a layered water-saturated half space to a moving load. Comput Geotech 2008;35(1):1–10. [5] Takemiya H, Bian XC. Substructure simulation of inhomogeneous track and layered ground dynamic interaction under train passage. J Eng Mech 2005;131(7):699–711. [6] Yao HL, Hu Z, Lu Z, et al. Analytical model to predict dynamic responses of railway subgrade due to high-speed trains considering wheel-track interaction. Int J Geomech 2015:04015061. [7] Lu Z, Fang R, Yao HL, Dong C, Xian SH. Dynamic responses of unsaturated halfspace soil to a moving harmonic rectangular load. Int J Numer Anal Methods Geomech 2018;42:1057–77. [8] Dong K, Connolly DP, Laghrouche O, et al. Non-linear soil behaviour on high speed rail lines. Comput Geotech 2019;112:302–18. [9] Metrikine AV, Vrouwenvelder A. Surface ground vibration due to a moving train in a tunnel: two-dimensional model. J Sound Vib 2000;234(1):43–66. [10] Koziol P, Mares C, Esat I. Wavelet approach to vibratory analysis of surface due to a load moving in the layer. Int J Solids Struct 2008;45(7–8):2140–59.

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