Acta Mech DOI 10.1007/s00707-017-1893-5

O R I G I NA L PA P E R

Jian-Fei Lu · Jiang-Jiang Ding · Zeng Fan · Min-Hao Li

Response of a circular tunnel embedded in saturated soil to a series of equidistant moving loads

Received: 4 October 2016 / Revised: 7 May 2017 © Springer-Verlag GmbH Austria 2017

Abstract The dynamic response of an infinitely long circular tunnel embedded in the saturated soil to a series of equidistant moving loads (SEML) is investigated in this study. The saturated soil surrounding the tunnel is described by Biot’s theory. Two scalar potentials and one vector potential are introduced to represent the displacement of the solid skeleton and the pore pressure. Based on Biot’s theory, the Helmholtz equations for the potentials are derived using the Fourier transform method. Performing the Fourier transform with respect to the longitudinal coordinate on the Helmholtz equations gives the frequency–wavenumber domain general solutions for the potentials. With the general solutions and the boundary conditions along the tunnel surface, the solution for the circular tunnel subjected to a single moving load (SML) is obtained. Superposition of the solutions for SMLs gives the representation for the response of the tunnel to a SEML, with which the resonance and cancelation conditions for the SEML are derived. Numerical simulations suggest that for the circular infinite tunnel there is a critical velocity. When the load moves with the critical velocity, the frequency domain response of the tunnel exhibits some dominant peak. Further, for a SEML moving with the critical velocity, the resonance and cancelation phenomena in the time domain may occur when the aforementioned resonance and cancelation conditions are fulfilled.

1 Introduction The vibration generated by a train running inside a tunnel is crucial for the design of a tunnel. Hence, the dynamic behavior of a tunnel has drawn much attention in the transportation engineering and environmental engineering [1]. Different approaches have been developed to investigate the response of a tunnel to dynamic loads. Based on the 2.5D finite/infinite element approach, Yang and Hung [2] investigated the ground vibration induced by moving loads in the tunnel. Bian et al. [3] used a 2.5D finite element model to investigate the interaction between an underground tunnel and the surrounding soil. They found that the wave propagation zone at the ground surface becomes narrower with increasing vibration frequency. Gupta and Degrande [4] J.-F. Lu (B) · J.-J. Ding · Z. Fan Department of Civil Engineering, Jiangsu University, Zhenjiang 212013, Jiangsu, People’s Republic of China E-mail: [email protected] J.-J. Ding E-mail: [email protected] Z. Fan E-mail: [email protected] M.-H. Li Department of Photonics Engineering, Yuan Ze University, Taoyuan 320, Taiwan, People’s Republic of China E-mail: [email protected]

J.-F. Lu et al.

developed a coupled model for the periodic track and tunnel–soil system using the Floquet transform method and investigated the vibration isolation effect of continuous and discontinuous floating slab tracks. Metrikine and Vrouwenvelder [5] obtained an analytical solution for the tunnel subjected to the moving load embedded in a two-dimensional viscoelastic soil layer, wherein the tunnel is simplified as an Euler–Bernoulli beam. In order to examine the influence of the liner on the response of the tunnel, by using the discrete wavenumber fictitious force method, a numerical model for the lined and unlined tunnels subjected to moving loads was developed by Sheng et al. [6]. Compared with the models based on the boundary element method, the model by Sheng et al. [6] has the advantage that only the Green’s function of the displacement is required. Using elasticity theory for the soil, shell theory for the tunnel, and beam theory for the track and rail, Forrest and Hunt [7,8] developed an analytical 3D model for the dynamic response of a deep circular tunnel considering the coupling between the soil, tunnel, track, and rail. By using the elasticity theory and shell theory to describe the soil and tunnel, respectively, Alekseeva and Ukrainets [9] derived an analytical solution for the dynamic response of a shallow-buried lined tunnel in an elastic half-space to moving loads. Recently, Hussein et al. [10] used the fictitious force method to establish a model for a tunnel embedded in a layered half-space subjected to moving loads, extending the pipe-in-pipe model [7,8] for the homogeneous soil to the stratified case. Very recently, Yuan et al. [11] proposed a semi-analytical model for a tunnel embedded in a half-space soil by using the transformation properties between the plane and cylindrical wave functions to handle the boundary conditions along the surface of the half-space soil. It is worth noting that all aforementioned research treats the surrounding medium of the tunnel as a singlephase elastic or viscoelastic medium. However, in the coastal region, the soil is often saturated by water, and hence the saturated soil model is more realistic than the single-phase soil model. To date, some research has been conducted to study the dynamic response of the structure embedded in the saturated soil subjected to dynamic loads. Kumar et al. [12] used the Laplace transform method to derive the closed form solution for the saturated porous medium with a cylindrical cavity subjected to an arbitrary time-dependent force. Based on Biot’s theory, Senjuntichai and Rajapakse [13] investigated the transient response of a pressurized long cylindrical cavity embedded in an infinite poroelastic medium and subjected to axially symmetric transient radial tractions. They found that the maximum radial displacement and hoop stress at the cavity surface are substantially higher than the classical static solutions and different considerably from the transient elastic solutions. Lu and Jeng [14] initiated the research for a circular tunnel embedded in a full-space saturated soil and subjected to moving loads by using Biot’s theory and the Fourier transform method. Based on Biot’s theory and double Fourier transform method, Hasheminejad and Komeili [15] studied the dynamic response of a circular lined tunnel to an axially moving ring load. Their attention was focused on the influence of the bonding condition at the liner/soil interface and the load velocity on the dynamic response of the system. A 2D model for the dynamic response of a tunnel embedded in a half-space saturated soil to a moving point load was presented by Yuan et al. [16], wherein the tunnel is simplified as an infinite Euler–Bernoulli beam, and the critical velocity of the tunnel–soil system is analyzed via the frequency–wavenumber spectrum method. Recently, based on Biot’s theory, Yuan et al. [17] presented an analytical model for a lined circular tunnel embedded in the saturated soil and subjected to moving loads by using the wave function expansion method. It is noted that although some research has been conducted to investigate the dynamic response of the tunnel embedded in the saturated soil to moving loads, no research has been carried out to study the dynamic response of the tunnel to series equidistant moving loads (SEML) so far. As the real train can usually be simplified as equidistant moving loads, it is thus crucial to examine the dynamic response of the tunnel embedded in the saturated soil to a SEML and the relevant new phenomena. In this study, a 3D analytical solution for the dynamic response of an infinitely long circular tunnel embedded in a saturated soil to a SEML is developed. The saturated soil is described by Biot’s theory [18,19]. For convenience, two scalar potentials and one vector potential are introduced to represent the displacement of the solid skeleton and the pore pressure. Based on Biot’s theory and the Fourier transform method, the Helmholtz equations for the potentials are obtained. Performing the Fourier transform with respect to the longitudinal coordinate on the Helmholtz equations yields the general solutions for the potentials in the cylindrical coordinate system. With the general solutions and the boundary conditions of the tunnel surface, the frequency–wavenumber domain solution for the tunnel subjected to an SML is obtained. Superposition of the solutions for the SMLs gives the representation for the response of the tunnel to a SEML, by which the resonance and cancelation conditions for the SEML are established. Analytical inversion of the Fourier transform with respect to the axial wavenumber recovers the frequency domain response of the tunnel. Inversion of the Fourier transform with respect to the angular frequency further via the FFT method retrieves the time domain response of the tunnel. With the proposed model, numerical simulations are conducted. The presented

Response of a circular tunnel embedded in saturated soil

numerical results show that for the circular tunnel there exists a critical velocity. When the velocity of the load is equal to the critical velocity, the frequency domain response of the tunnel displays some dominant sharp peaks. Further, at the critical velocity, the predicted resonance and cancelation phenomena in the time domain occur when the resonance and cancelation conditions are fulfilled.

2 Governing equations and the corresponding general solutions 2.1 Biot’s theory for the saturated soil When dealing with the problem of the tunnel subjected to moving loads, two kinds of Fourier transforms are involved: the Fourier transform with respect to time and that with respect to a spatial coordinate. In this study, the two Fourier transforms are defined as follows [20]: f˜(ω) =

f¯(ξ ) =

+∞ f (t)e

−iωt

dt,

−∞ +∞

f (z)eiξ z dz,

1 f (t) = 2π f (z) =

−∞

1 2π

+∞ −∞ +∞

f˜(ω)eiωt dω,

f¯(ξ )e−iξ z dξ

(1)

−∞

where t and ω denote time and angular frequency, respectively; the superimposed tilde denotes the Fourier transform with respect to time; z and ξ represent the coordinate√and wavenumber, respectively; the bar denotes the Fourier transform with respect to the coordinate, and i = −1. The constitutive equations for the homogeneous saturated soil have the form [19] σij = 2μεij + λδij e − αδij p,

p = −α Me + Mϑ

(2)

where σij and εij are the stress and strain tensors of the saturated soil; p is the excess pore pressure, and δij is the Kronecker delta; λ and μ are Lame constants of the solid skeleton; α and M are the Biot effective stress coefficient and the Biot modulus, respectively; e and ϑ are the dilatation of the solid skeleton and the increment of water content, which are defined as follows: e = u i,i , ϑ = −wi,i , wi = ϕ(Ui − u i )

(3)

where u i and Ui denote the average displacements of the solid skeleton and pore fluid; wi describes the flow of the water relative to the solid skeleton, and ϕ is the porosity of the saturated soil. The equations of motion for the saturated soil and the pore fluid are expressed in terms of the displacements u i and wi as follows [19]: μu i,jj + (λ + α 2 M + μ)u j,ji + α Mw j,ji = ρb u¨ i + ρ f w¨ i , η α Mu j,ji + Mw j,ji = ρ f u¨ i + m w¨ i + w˙ i k

(4)

where ρb = (1 − ϕ)ρs + ϕρ f denotes the bulk density of the saturated soil, and ρs as well as ρ f denote the densities of the solid skeleton and pore water, respectively; ϕ is the porosity of the saturated soil; m = a∞ ρ f /ϕ and a∞ is the tortuosity of the soil; η and k account for the viscosity of the pore water and the permeability of the saturated soil, respectively; a superimposed dot on a variable denotes time derivative.

2.2 The potentials for Biot’s theory In this section, for simplicity, the potentials for the two P waves and one shear wave of the saturated soil are introduced in the frequency domain. Since there are two kinds of P waves (P1 and P2 waves) occurring in the saturated soil, according to the Helmholtz decomposition method [21], the displacement of the solid skeleton can be represented as follows: u˜ i = φ˜ f,i + φ˜ s,i + eijk ψ˜ k, j (5)

J.-F. Lu et al.

˜ is where φ˜ f and φ˜ s denote scalar potentials corresponding to the P1 and P2 waves of the saturated soil; ψ the vector potential associated with the shear wave of the saturated soil, and eijk is the Levi–Civita symbol. ˜ satisfies the generalized gauge condition of the form ψˆ i,i = 0. As the pore Moreover, the vector potential ψ pressure of the saturated soil is only relevant to the two P waves of the saturated soil, it thus has the following expression [22,23]: p˜ = A f φ˜ f,ii + As φ˜ s,ii (6) where A f and As are two frequency-dependent constants determined by the following equation [23]: A2f,s +

β3 + β2 β5 − (λ + 2μ)β4 (λ + 2μ)β5 A f,s − =0 β2 β4 β2 β4

(7)

where β1 = mω2 − iηω/kβ2 = α − ρ f ω2 /β1 , β3 = ρb ω2 − ρ 2f ω4 /β1 , β4 = β1 /M, and β5 = αβ1 − ρ f ω2 . Employing Eqs. (2) and (4), the relative displacement of the pore water to the solid skeleton has the form w˜ i =

1 ( p,i − ρ f ω2 u˜ i ). β1

(8)

Due to the independent propagation of the P1 and P2 waves, the potentials φ˜ f and φ˜ s of the P1 and P2 waves fulfill the following Helmholtz equations independently [22]: ∇ 2 φ˜ f + k 2f φ˜ f = 0, ∇ 2 φ˜ s + ks2 φ˜ s = 0

(9)

where ∇ 2 is the Laplacian and k f and ks are the complex wavenumbers for the P1 and P2 waves, which are determined by [23] k 2f =

β4 A f + β5 β3 β3 β4 A s + β5 = , ks2 = = . λ + 2μ − β2 A f Af λ + 2μ − β2 As As

(10)

Besides, the vector potential of the shear wave of the saturated soil satisfies the following Helmholtz equation: ∇ 2 ψ˜ + kt2 ψ˜ = 0, kt2 = β3 /μ.

(11)

Once the potentials for the saturated soil are determined, the displacements and pore pressure are given by Eqs. (5) and (6), while the relative displacement of the pore fluid to the solid skeleton is given by Eq. (8) and the stresses of the saturated soil are determined by Eq. (2). 2.3 General solutions in the cylindrical coordinate system for the potentials For the problem considered in this study, it is more convenient to work in the cylindrical coordinate system as shown in Fig. 1. In the cylindrical coordinate system, the vector potential ψ˜ of the saturated soil can be represented by [21] ψ˜ = χ˜ ez + ∇ × (ηe ˜ z) (12) where ez is the unit vector along the longitudinal direction of the cylindrical coordinate system; χ˜ and η˜ are ˜ The potentials χ˜ and η˜ in Eq. (12) satisfy the two scalar potentials used to represent the vector potential ψ. the following Helmholtz equations: ∇ 2 χ˜ + kt2 χ˜ = 0, ∇ 2 η˜ + kt2 η˜ = 0.

(13)

By using the potentials φ˜ f , φ˜ s , χ˜ and η, ˜ the displacement of the solid skeleton in the cylindrical coordinate system as shown in Fig. 1 can be represented by   ∂ φ˜ f ∂ φ˜ s 1 ∂ χ˜ ∂ 2 η˜ ∂ φ˜ s ∂ χ˜ ∂ 2 η˜ 1 ∂ φ˜ f u˜ ρ = + + + , u˜ θ = + − + , ∂ρ ∂ρ ρ ∂θ ∂z∂ρ ρ ∂θ ∂θ ∂ρ ρ∂z∂θ   ∂ φ˜ f ∂ φ˜ s 1 ∂ ∂ η˜ 1 ∂ 2 η˜ u˜ z = (14) + − ρ − 2 2 ∂z ∂z ρ ∂ρ ∂ρ ρ ∂θ

Response of a circular tunnel embedded in saturated soil

where u˜ ρ , u˜ θ and u˜ z are the displacements of the solid skeleton in the ρ, θ , and z direction, respectively. Performing the Fourier transform with respect to the longitudinal coordinate z on Eqs. (9) and (13), the Helmholtz equations for the potential are reduced to ∂ 2 φˆ f 1 ∂ 2 φˆ f 1 ∂ φˆ f + + + (k 2f − ξ 2 )φˆ f ∂ρ 2 ρ ∂ρ ρ 2 ∂θ 2 ∂ 2 φˆ s 1 ∂ φˆ s 1 ∂ 2 φˆ s + + (ks2 − ξ 2 )φˆ s + ∂ρ 2 ρ ∂ρ ρ 2 ∂θ 2 1 ∂ χˆ 1 ∂ 2 χˆ ∂ 2 χˆ + + (kt2 − ξ 2 )χˆ + ∂ρ 2 ρ ∂ρ ρ 2 ∂θ 2 1 ∂ ηˆ 1 ∂ 2 ηˆ ∂ 2 ηˆ + + 2 2 + (kt2 − ξ 2 )ηˆ 2 ∂ρ ρ ∂ρ ρ ∂θ

= 0, = 0, = 0, =0

(15)

where the superimposed caret denotes the combination of the Fourier transforms with respect to time and the longitudinal coordinate. Solving the differential equations in Eq. (15) by the separation variable method, the following frequency–wavenumber domain general solutions for the potentials are obtained: φˆ f = φˆ s = ηˆ = χˆ =

+∞ 

Am (ω, ξ )Hm(2) (γ f ρ) cos(mθ ),

m=0 +∞  m=0 +∞ 

Bm (ω, ξ )Hm(2) (γs ρ) cos(mθ ), Cm (ω, ξ )Hm(2) (γt ρ) cos(mθ ),

m=0 +∞ 

Dm (ω, ξ )Hm(2) (γt ρ) sin(mθ )

(16)

m=1

where Am (ω, ξ )− Dm (ω, ξ ) are the mth order arbitrary constants to be determined by the boundary conditions (2) along the tunnel surface; Hm (∗) denotes the second kind of the mth order Hankel function; γ 2f = k 2f − ξ 2 , γs2 = ks2 − ξ 2 and γt2 = kt2 − ξ 2 , and the branches of γ f , γs , and γt should be chosen such that Re(γ f ) ≥ 0, Re(γs ) ≥ 0, and Re(γt ) ≥ 0. It is noted that since the tunnel is only subjected to moving loads at the tunnel surface, only the outward waves are taken into account in Eq. (16). Also, because of the symmetry of the moving load with respect to the x-axis (Fig. 1), the trigonometric terms associated with the ring modes are chosen such that the resulting solution is also symmetric about the x-axis. With the displacement–potential relations as given in Eq. (14) and the constitutive relation of the saturated soil, the displacement, stress, pore pressure, and relative displacement of the pore water can be obtained using the potential as given by Eq. (16) as follows:

the full-space saturated soil

Fn …

v

y R

o

z

…θ ρ

the circular tunnel

x Fig. 1 An infinite circular tunnel embedded in saturated soil and subjected to a series of equidistant moving loads with a uniform velocity

J.-F. Lu et al.

{uˆ i , wˆ i } = {uˆ i , wˆ i } = σˆ ij = σˆ ij = pˆ =

+∞  m=0 +∞  m=1 +∞  m=0 +∞  m=1 +∞ 

{u im , wim }(ρ, ξ, ω, Am , Bm , Cm , Dm ) cos(mθ ), i = ρ, z, {u im , wim }(ρ, ξ, ω, Am , Bm , Cm , Dm ) sin(mθ ), i = θ, σijm (ρ, ξ, ω, Am , Bm , Cm , Dm ) cos(mθ ), ij = ρρ, θ θ, zz, zρ, σijm (ρ, ξ, ω, Am , Bm , Cm , Dm ) sin(mθ ), ij = ρθ, zθ, pm (ρ, ξ, ω, Am , Bm ) cos(mθ ),

(17)

m=0

in which the expressions for uˆ im , wˆ im , σˆ ijm , pˆ i are given in “Appendix A.” 3 Formulation of the boundary-value problem and the solution for an SML As noted above, a circular tunnel with radius R embedded in the saturated soil subjected to moving loads in uniform motion is investigated in this study (Figure 1). The moving load may be a concentrated point load or a uniformly distributed load. Using the general solutions derived in the above section and the boundary conditions along the tunnel surface, the solution for the tunnel subjected to an SML is derived in this section. If a point SML with a velocity v is acting on the tunnel surface, the boundary conditions for the stresses have the following forms: Fn δ(θ )δ(z − vt), σρθ (R, θ, z, t) = 0, R Ft (18) σzρ (R, θ, z, t) = − δ(θ )δ(z − vt) R where Fn and Ft are the magnitudes of the normal and tangent components of the point load, respectively; δ(∗) is the Dirac-δ function, and R is the radius of the tunnel. If the moving load is a distributed load over a length 2d, the boundary conditions are as follows: σρρ (R, θ, z, t) = −

fn δ(θ ) [H (z − vt + d) − H (z − vt − d)] , σρθ (R, θ, z, t) = 0, R ft σzρ (R, θ, z, t) = − δ(θ ) [H (z − vt + d) − H (z − vt − d)] (19) R where f n and f t are the intensities of the distributed load in the normal and tangent directions; H (∗) is the Heaviside function. If the tunnel surface is completely permeable or impermeable, the pore pressure or the relative displacement of the pore water to the solid skeleton at the tunnel surface should vanish, namely [24] σρρ (R, θ, z, t) = −

p(R, θ, z, t) = 0, wρ (R, θ, z, t) = 0.

(20)

Performing the double Fourier transform with respect to t and z on Eq. (18), the following frequency– wavenumber domain boundary conditions are obtained: 2π Fn δ(θ )δ(ω − ξ v), σˆ ρθ (R, θ, ξ, ω) = 0, R 2π Ft δ(θ )δ(ω − ξ v). (21) σˆ zρ (R, θ, ξ, ω) = − R Similarly, applying the double Fourier transform to Eq. (19), one has the following frequency–wavenumber domain boundary conditions for the distributed load: σˆ ρρ (R, θ, ξ, ω) = −

σˆ ρρ (R, θ, ξ, ω) = −

4π f n sin(ξ d) δ(θ )δ(ω − ξ v), σˆ ρθ (R, θ, ξ, ω) = 0, Rξ

Response of a circular tunnel embedded in saturated soil

σˆ zρ (R, θ, ξ, ω) = −

4π f t sin(ξ d) δ(θ )δ(ω − ξ v). Rξ

(22)

Likewise, the hydraulic boundary conditions for the permeable and impermeable tunnel surface in the frequency–wavenumber domain can also be obtained via Eq. (20) as follows: p(R, ˆ θ, ξ, ω) = 0, wˆ ρ (R, θ, ξ, ω) = 0.

(23.1,2)

To determine the arbitrary constants of the potentials via the boundary conditions, δ(θ ) in boundary conditions (21) and (22) is expanded into the following Fourier series: δ(θ ) =

+∞ 

1

m=0

αm π

 cos(mθ ),

αm = 2, m = 0 . αm = 1, m > 0

(24)

In the remainder of this section, we use the tunnel with a permeable surface subjected to a moving point load only with a normal component as an example to show the determination of the arbitrary constants Am (ω, ξ ) − Dm (ω, ξ ) of the potentials. For the other cases, the four arbitrary constants of the potentials can be determined in a similar manner. using Eq. (24) at the right-hand sides of Eq. (21), and using Eq. (17) and Eq. (21), the following three equations for the arbitrary constants of the potentials are obtained: 2Fn δ(ω − ξ v), σˆ zρm (R, ξ, ω, Am ∼ Dm ) = 0, m = 0 ∼ ∞, αm R σˆ ρθ m (R, ξ, ω, Am ∼ Dm ) = 0, m = 1 ∼ ∞. (25)

σˆ ρρm (R, ξ, ω, Am ∼ Dm ) = −

Besides, for the tunnel with a permeable surface, substituting Eq. (17) into Eq. (23.1,2), one has the following: pˆ m (R, ξ, ω, Am , Bm ) = 0, m = 0 ∼ ∞.

(26)

By solving Eqs. (25) and (26), the arbitrary constants Am (ω, ξ ) − Dm (ω, ξ ) in Eq. (16) can be determined. The expressions for the mth order displacement, stress, pore pressure, and relative displacement of the pore water in the frequency–wavenumber domain can then be determined using the expressions given in “Appendix A.” The total frequency–wavenumber domain displacement, stress, pore pressure, and relative displacement of the pore water with respect to the solid skeleton are the sums of the contributions from all orders of m. In numerical simulation, the infinite series in Eq. (16) should be truncated in terms of the desired accuracy requirement. The time-space domain solution can be retrieved by applying the double inverse Fourier transform with respect to angular frequency and wavenumber to the frequency-wavenumber domain solution. To show ˆ is used to represent the variables in the frequency– the inversion of the double inverse Fourier transform,  wavenumber domain due to the aforementioned moving point load only with a normal component. Hence, the frequency–wavenumber domain response of the tunnel can be represented by ˆ ˆ ∗ (ρ, θ, ξ, ω). (ρ, θ, ξ, ω) = Fn δ(ω − ξ v)

(27)

By inversion of the double Fourier transform, the time–space domain solution of the tunnel has the following form:  2 +∞ +∞ 1 ˆ ∗ (ρ, θ, ξ, ω)ei(ωt−ξ z) dωdξ . Fn δ(ω − ξ v) (28) (ρ, θ, z, t) = 2π −∞ −∞

Using the property of the Dirac-δ function, the inversion of the Fourier transform with respect to the wavenumber can be performed analytically, namely  +∞ 1 Fn ∗ ω ˜ ˆ (ρ, θ, , ω)e−iωz/v . ˜ (ρ, θ, z, t) = θ, z, ω) = (29)  (ρ, θ, z, ω)eiωt dω, (ρ, 2π −∞ 2πv v Equation (29) suggests that the time domain solution can be obtained simply by performing the inverse Fourier transform on the corresponding frequency domain solution, which can be accomplished by the FFT method.

J.-F. Lu et al.

4 The solution for the tunnel subjected to a SEML and the corresponding resonance and cancelation conditions In this section, the solution for the tunnel subjected to a SEML is derived, and corresponding resonance and cancelation conditions for the SEML are proposed. The dynamic response of the circular tunnel to a SEML can be obtained by superposing all the responses of the tunnel to the SMLs constituting the SEML. Suppose that a SEML is moving over the tunnel surface with a constant velocity v and the number of the loads constituting the SEML is N L . The distance between the neighboring loads of the SEML is l. Supposing that the response of the circular tunnel due to the jth loading is  j (ρ, θ, z, t), the total response of the tunnel (ρ, θ, z, t) due to the SEML in the time domain is thus equal to  (ρ, θ, z, t) =

N L −1

 j (ρ, θ, z, t)

j=0



 l l + · · · + 0 ρ, θ, z, t − (N L − 1) . = 0 (ρ, θ, z, t) + 0 ρ, θ, z, t − v v

(30)

Applying the Fourier transform with respect to time to Eq. (30), the total frequency domain response of the tunnel is obtained as follows: ˜ (ρ, θ, z, ω) =

N L −1

˜ j (ρ, θ, z, ω) 

j=0

 ˜ 0 (ρ, θ, z, ω) × 1 + e−iω(l/v) + · · · + e−iω(N L −1)(l/v) . =

(31)

If the time interval between two neighboring loadings satisfies the following condition: l 2π =n , v ω in which n is an arbitrary integer, Eq. (31) is reduced to ˜ 0 (ρ, θ, z, ω). ˜ (ρ, θ, z, ω) = N L 

(32)

(33)

On the other hand, if the time interval fulfills the following condition:

Eq. (31) is reduced to ˜ (ρ, θ, z, ω) =



π l = (2n − 1) , v ω

(34)

0, Mod(N L , 2) = 0 ˜ 0 (ρ, θ, z, ω), Mod(N L , 2) = 1 , 

(35)

in which Mod denotes the Mod function. Equations (32) and (34) are the conditions for amplifying and repressing the SEML response of the tunnel for a specific frequency. If the frequency domain SML response of the tunnel is smooth and without obvious peaks, fulfillment of Eqs. (32) or (34) only enhances or suppresses the response of the tunnel at a common frequency, and hence it will not entail obvious resonance or cancelation. However, if the frequency domain SML response of the tunnel has obvious peaks, fulfillment of Eqs. (32) or (34) at an obvious peak will result in obvious resonance or cancelation. Therefore, the conditions for resonance or cancelation to occur when the circular tunnel is subjected to a SEML are as follows: l 2π = n (D) , v ωi (D)

l π = (2n − 1) (D) , i = 1, . . . , N D v ω

(36)

i

where ωi denotes the angular frequency of the ith resonance peak of the SML response of the tunnel; N D is the number of the resonance peaks appearing in the SML response of the tunnel. Equation (36) suggests that when the time interval between two neighboring moving loadings is equal to the integer multiples of the period of one of the resonance peaks of the SML response of the tunnel, resonance will occur; when the time interval is equal to the half-integer multiples of the period of one of the resonance peaks of the SML response, cancelation will take place.

Response of a circular tunnel embedded in saturated soil

y

o

ζc

q σ*

x Fig. 2 A circular tunnel subjected to a static antiplane point shear force Table 1 The material and geometrical parameters for the saturated soil and circular tunnel The shear modulus of the solid skeleton μ The Poisson’s ratio of the solid skeleton ν The density of the solid skeleton ρs The density of the pore fluid ρ f The porosity of the soil ϕ The Biot effective stress coefficient of the soil α The Biot modulus of the soil M The viscosity of the pore fluid η The permeability of the soil k The tortuosity of the soil α∞ The radius of the circular tunnel R

3.0 × 107 Pa 0.35 1.9 × 103 kg/m3 1.0 × 103 kg/m3 0.3 0.95 6.96 × 107 Pa 1.0 × 10−3 Pa s 2.0 × 10−13 m2 3.0 3.0 m

5 Numerical results and corresponding discussions In this section, based on the proposed model, numerical results for the frequency and time domain responses of the tunnel to an SML and SEML are presented. Before presenting numerical results, to validate the proposed model, results of this paper are compared with those due to an existing solution for a circular tunnel subjected to a static antiplane shear force. 5.1 Comparisons of present results with a static solution for a circular tunnel To validate the proposed method, our results are compared with the solution for a circular tunnel subjected to a static tangent distributed load over −∞ < z < +∞ at the tunnel surface. As the static tangent load is distributed over −∞ < z < +∞ along the tunnel surface uniformly, the problem can thus be treated as an antiplane static problem (Fig. 2), the solution of which can thus be derived via the complex variable method (“Appendix B”). If the velocity of the moving distributed load is small, the dynamic effect can be ignored and the role of the pore water is hence negligible. In this case, the dynamic problem of the moving load is reduced to a quasi-static problem. Further, if the moving load is distributed over a length much larger than the radius of the tunnel, then the response of the tunnel near the center of the distributed load should be equal to that of the corresponding antiplane static problem approximately. In this example, the geometric and material parameters for the circular tunnel and the saturated soil are given in Table 1. The tangent load is distributed over a segment of length 2d √ = 100 m, and the velocity of the load is v/vsh = 0.013, in which vsh is a wave velocity defined by vsh = μ/ρb . The time domain response of the tunnel to the distributed moving load is calculated using the proposed method. The parameters used to implement the FFT for the inversion of the Fourier transform in Eq. (29) are given in Table 2. According to Eq. (19), as the instant when the center of the distributed load passes z = 0 is chosen as the zero time, to have a full view about the response of an observation point, the observation point should be far from z = 0. In this example, the coordinate z of the observation point is taken as 1035.15 m. To validate the proposed method, the time domain stresses σzρ and σzθ of the points with ρ = 5.0 m, −π < θ < π and z = 1035.15 m at the time when the center of the distributed load passes z = 1035.15 m are compared with the antiplane static solution for the tunnel (see “Appendix B”). When evaluating the response of the tunnel to the moving distributed load, 30 terms, namely, m = 0 to m = 29, are used to approximate the infinite series in Eq. (16). Figure 3 shows the comparison of the shear stresses σzρ and σzθ between the two solutions. It follows

J.-F. Lu et al.

Table 2 Parameters for implementing the FFT for different velocities of the load The calculated variables

The velocity v/vsh

σzρ σzθ uρ uρ uρ uρ p p p p

0.013 0.013 0.5 0.92 0.98 1.3 0.5 0.92 0.98 1.3

The number of sample points N 401 401 2001 4801 24,001 24,001 4001 3201 20,001 8001

(a)

The sample spacing in the frequency domain ω (rad/s)

The sample spacing in the time domain t (s)

2.7810−3 3.0410−3 0.19 0.41 0.14 0.75 0.1 0.43 0.095 0.27

5.63 5.15 1.63 × 3.21 × 1.85 × 3.49 × 1.51 × 4.61 × 3.29 × 2.89 ×

10−2 10−3 10−3 10−4 10−2 10−3 10−3 10−3

0.14 0.12

σz /ft (Pa/(N/m))

0.10 0.08 0.06 0.04 0.02 0.00

The present solution The antiplane static solution

-3.140 -2.355 -1.570 -0.785 0.000

0.785

1.570

2.355

3.140

0.785

1.570

2.355

3.140

θ (rad)

(b)

0.06

σz /ft (Pa/(N/m))

0.04

The present solution The antiplane static solution

0.02 0.00 -0.02 -0.04 -0.06 -3.140 -2.355 -1.570 -0.785 0.000

θ (rad)

Fig. 3 The comparison of the shear stresses between the present results and the results due to the antiplane static solution: a the comparison of the shear stress σzρ ; b the comparison of the shear stress σzθ

from Fig. 3 that the results due to the solution for the moving distributed load agree very well with those of the antiplane static solution, validating the proposed method. 5.2 Dynamic response of the circular tunnel to an SML 5.2.1 Frequency domain response of the circular tunnel to an SML In this section, the frequency domain response of the tunnel to an SML is calculated. The geometric and material parameters for the circular tunnel are given in Table 1. The surface of the saturated soil is assumed to be permeable. The moving load is a point load only with the normal component Fn . Four cases corresponding to v/vsh = 0.5, v/vsh = 0.92, v/vsh = 0.98, and v/vsh = 1.3, respectively, are considered. The observation

Response of a circular tunnel embedded in saturated soil

point is located at ρ = 5.0 m and θ = π/4. In calculation, the partial sum of the first thirty terms is used to approximate the infinite series in Eq. (16). The amplitudes of the frequency domain radial displacement u˜ ρ and the pore pressure p˜ for the four velocities are shown in Fig. 4. Figure 4 shows that the radial displacement and the pore pressure for the velocity v/vsh = 0.98 are much larger than those for other velocities. For velocities different from the velocity v/vsh = 0.98, the curves for the displacement and pore pressure are smooth, and no obvious peaks appear. However, Fig. 4c, g indicates that for the velocity v/vsh = 0.98, when the angular frequency is around 171.97 rad/s, two dominant peaks occur and the responses at other frequencies are almost negligible. This resonance type response only occurs at the velocity v/vsh = 0.98, implying that the velocity v/vsh = 0.98 is a critical velocity of the tunnel. 5.2.2 Time domain response of the circular tunnel to an SML In this section, the time domain response of the tunnel to the moving load is investigated. The tunnel and the load are the same as those for Sect. 5.2.1. The observation point is located at ρ = 5.0 m, θ = π/4 and z = 1000 m. The parameters used to implement the FFT for the inversion of the Fourier transform in Eq. (29) for different velocities are given in Table 2. As with Sect. 5.2.1, the infinite series of the potentials in Eq. (16) are approximated by the first thirty terms. Figure 5 plots the time domain radial displacement and pore pressure at the observation point. Figure 5 illustrates that the values of u ρ and p for the velocity v/vsh = 0.5 are the smallest among the four, while those for the velocity v/vsh = 0.98 are the largest among the four, indicating again that the velocity v/vsh = 0.98 is a critical velocity for the tunnel. As shown in Fig. 5a, b, e, f, since the velocities v/vsh = 0.5 and v/vsh = 0.92 are smaller than the body wave velocities of the saturated soil, the response at the observation point is nearly symmetric with respect to the arrival time. However, it follows from Fig. 5c, d, g, h that the responses at the observation point become asymmetric with respect to the arrival time for the velocity v/vsh = 0.98 and v/vsh = 1.3. Figure 5c, g show that for v/vsh = 0.98, shockwave type response at the observation point occurs. Also, it is interesting to note that the shockwave type response phenomenon at the observation point for the velocity v/vsh = 0.98 is more pronounced than that for the velocity v/vsh = 1.3. 5.3 Dynamic response of the circular tunnel to a SEML 5.3.1 Frequency domain response of the circular tunnel to a SEML In this section, the frequency domain response of the tunnel to a SEML is investigated. The circular tunnel is the same as that in Sect. 5.2. The SEML consists of ten point loads (N L = 10) with only the normal component. The coordinates for the observation point are ρ = 5.0 m and θ = π/4. The velocity of the SEML is equal to the critical velocity of the tunnel, namely, v/vsh = 0.98. All the infinite series of the potentials in Eq. (16) are truncated at the thirtieth term. As noted above, the largest resonance peaks of the SML radial displacement and pore pressure occur at the angular frequency 171.97 rad/s. Hence, according to Eq. (36), for the angular frequency 171.97 rad/s and velocity v/vsh = 0.98, the loading distances l = 14.87 m and l = 12.39 m satisfy the resonance and cancelation conditions, respectively (n = 3). Hence, in this section, the frequency domain radial displacement u˜ ρ and pore pressure p˜ due to the SEML with the loading distance l = 14.87 m and l = 12.39 m are calculated. The radial displacement u˜ ρ and pore pressure p˜ versus the angular frequency for l = 14.87 m and l = 12.39 m are illustrated in Fig. 6. Figure 6a indicates that compared with the SML response, the radial displacement due to the SEML at the angular frequency 171.97 rad/s for l = 14.87 m increases tenfold. On the contrary, Fig. 6b shows that when l = 12.39 m, the SEML radial displacement at the angular frequency 171.97 rad/s is almost negligible. Figure 6c, d indicates that with the radial displacement the pore pressure exhibits the similar resonance and cancelation phenomena at the angular frequency 171.97 rad/s when the loading distance equals l = 14.87 m and l = 12.39 m, fulfilling the resonance and cancelation conditions, respectively. 5.3.2 Time domain response of the circular tunnel to a SEML In this section, the time domain response of the tunnel to the SEML considered in Sect. 5.3.1 is investigated. The circular tunnel and the SEML are the same as those in Sect. 5.3.1. As with Sect. 5.3.1, only the response of the tunnel to the SEML with the velocity v/vsh = 0.98 and loading distance l = 14.87 m and l = 12.39 m

J.-F. Lu et al.

(a)

(e) 2.50E-010

0.00010 v/vsh=0.50

v/vsh=0.50

2.00E-010

Abs(p/F n) (Pa/N)

0.00008

1.50E-010 1.00E-010 5.00E-011 0.00E+000 0.00

(b)

0.00006 0.00004 0.00002 0.00000

48.16

96.32

144.48

0.00

192.64

32.34

64.68

97.02

129.36

161.70

194.04

(f)

1.60E-010

0.00005 v/vsh=0.92

v/vsh=0.92

0.00004

Abs(p/F n) (Pa/N)

1.20E-010

8.00E-011

4.00E-011

0.00003 0.00002 0.00001

0.00E+000 0.00

0.00000 244.61

489.22

733.83

978.44

(c)

0.00

103.53

207.06

310.59

414.12

517.65

621.18

(g) 1.60E-008

0.020

v/vsh=0.98

v/vsh=0.98

Abs(p/F n) (Pa/N)

1.20E-008

8.00E-009

4.00E-009

0.00E+000 0.00

0.015

0.010

0.005

0.000 343.94

687.88

1031.82

0.00

1375.76

(d)

(h) 8.00E-011

2.00E-011

515.91

687.88

859.85

0.00006 0.00005

Abs(p/F n) (Pa/N)

4.00E-011

343.94

v/vsh=1.30

v/vsh=1.30

6.00E-011

171.97

0.00004 0.00003 0.00002 0.00001

0.00E+000 0.00

0.00000 1800.19

3600.38

5400.57

7200.76

9000.95

0.00

167.92

335.84

503.76

671.68

839.60

1007.52

Fig. 4 The frequency domain response at the observation point with the coordinates ρ = 5.0 m and θ = π/4 to a moving point load with the normal component Fn for the velocity v/vsh = 0.5, 0.92, 0.98, and 1.3: a the radial displacement u˜ ρ for the velocity v/vsh = 0.5; b the radial displacement u˜ ρ for the velocity v/vsh = 0.92; c the radial displacement u˜ ρ for the velocity v/vsh = 0.98; d the radial displacement u˜ ρ for the velocity v/vsh = 1.3; e the pore pressure p˜ for the velocity v/vsh = 0.5; f the pore pressure p˜ for the velocity v/vsh = 0.92; g the pore pressure p˜ for the velocity v/vsh = 0.98; h the pore pressure p˜ for the velocity v/vsh = 1.3

Response of a circular tunnel embedded in saturated soil

(a)

(e) 0.00E+000

0.00030

-1.00E-010

v/vsh=0.50

0.00025 v/vsh=0.50

-2.00E-010

p/F n (Pa/N)

u /Fn (m/N)

0.00035

-3.00E-010 -4.00E-010

0.00020 0.00015 0.00010 0.00005 0.00000

-5.00E-010 12.529

13.266

14.003

14.740

15.477

16.214

-0.00005 10.318

16.951

11.792

13.266

t (s)

14.740

16.214

17.688

19.162

t (s)

(b)

(f)

0.0012

0.00E+000

v/vsh=0.92

0.0006

u /Fn (m/N)

p/F n (Pa/N)

v/vsh=0.92

-6.00E-010

-1.20E-009

0.0000

-0.0006

-1.80E-009 -0.0012

-2.40E-009 4.006

6.009

8.012

10.015

7.4111

12.018

7.6114

7.8117

(c)

8.0120

8.2123

8.4126

8.6129

t (s)

t (s)

(g)

2.25E-008

0.02

v/vsh=0.98

v/vsh=0.98

1.50E-008 0.01

p/F n (Pa/N)

u /Fn (m/N)

7.50E-009 0.00E+000 -7.50E-009

0.00

-0.01

-1.50E-008 -0.02 -2.25E-008 0.00

7.49

14.98

22.47

29.96

0.00

37.45

14.98

(d)

(h)

3.00E-009 v/vsh=1.30

2.00E-009

44.94

59.92

0.004 0.003 v/vsh=1.30

p/F n (Pa/N)

1.00E-009

u /Fn (m/N)

29.96

t (s)

t (s)

0.00E+000 -1.00E-009 -2.00E-009 -3.00E-009

0.002 0.001 0.000 -0.001

-4.00E-009 3.983

4.552

5.121

5.690

t (s)

6.259

6.828

7.397

-0.002 3.983

4.552

5.121

5.690

6.259

6.828

7.397

t (s)

Fig. 5 The time domain response at the observation point with the coordinates ρ = 5.0 m, θ = π/4, and z = 1000 m to a moving point load with the normal component Fn for the velocity v/vsh = 0.5, 0.92, 0.98, and 1.3: a the radial displacement u ρ for the velocity v/vsh = 0.5; b the radial displacement u ρ for the velocity v/vsh = 0.92; c the radial displacement u ρ for the velocity v/vsh = 0.98; d the radial displacement u ρ for the velocity v/vsh = 1.3; e the pore pressure p for the velocity v/vsh = 0.5; f the pore pressure p for the velocity v/vsh = 0.92; g the pore pressure p for the velocity v/vsh = 0.98; h the pore pressure p for the velocity v/vsh = 1.3

J.-F. Lu et al.

(a) 1.60E-007 l=14.87 m

1.20E-007

8.00E-008

4.00E-008

0.00E+000 0.00

(b)

343.94

687.88

1031.82

1375.76

1.00E-008 l=12.39 m

7.50E-009

5.00E-009

2.50E-009

0.00E+000 0.00

343.94

687.88

1031.82

1375.76

(c) 0.20 l=14.87 m

0.15

0.10

0.05

0.00 0.00

(d)

171.97

343.94

515.91

687.88

859.85

0.012 0.010 l=12.39 m

0.008 0.006 0.004 0.002 0.000 0.00

171.97

343.94

515.91

687.88

859.85

Fig. 6 The frequency domain response at the observation point with the coordinates ρ = 5.0 m and θ = π/4 to a SEML with N L = 10, velocity v/vsh = 0.98 and l = 14.87 m and 12.39 m: a the radial displacement u˜ ρ for l = 14.87 m; b the radial displacement u˜ ρ for l = 12.39 m; c the pore pressure p˜ for l = 14.87 m; d the pore pressure p˜ for l = 12.39 m

Response of a circular tunnel embedded in saturated soil

(a) 1.00E-007 l=14.87 m

u /Fn (m/N)

5.00E-008 0.00E+000 -5.00E-008 -1.00E-007 0.00

7.49

14.98

22.47

29.96

37.45

t (s)

(b)

3.00E-008 l=12.39 m

u /Fn (m/N)

2.00E-008 1.00E-008 0.00E+000 -1.00E-008 -2.00E-008 -3.00E-008 0.00

7.49

14.98

22.47

29.96

37.45

t (s)

(c) 0.10

l=14.87 m

p/F n (Pa/N)

0.05 0.00 -0.05 -0.10 0.00

14.98

29.96

44.94

59.92

t (s)

(d) 0.02 l=12.39 m

p/F n (Pa/N)

0.01 0.00 -0.01 -0.02 0.00

14.98

29.96

44.94

59.92

t (s)

Fig. 7 The time domain response of the observation point with the coordinates ρ = 5.0 m, θ = π/4, and z = 1000 m to a SEML with N L = 10, velocity v/vsh = 0.98, and l = 14.87 m and 12.39 m: a the radial displacement u ρ for l = 14.87 m; b the radial displacement u ρ for l = 12.39 m; c the pore pressure p for l = 14.87 m; d the pore pressure p for l = 12.39 m

J.-F. Lu et al.

are considered. The coordinates for the observation point are ρ = 5.0 m, θ = π/4, and z = 1000 m. The parameters used to implement the FFT for the inversion of the Fourier transform in Eq. (29) are shown in Table 2. Also, when calculating the response of the tunnel, the number of the terms used to approximate the infinite series of the potentials in Eq. (16) is the same as that of Sect. 5.3.1. The time domain radial displacement u ρ and pore pressure p due to the SEML are shown in Fig. 7. Figure 7a indicates that compared with the SML displacement the maximum radial displacement due to the SEML when l = 14.87 m increases by several times, exhibiting obvious resonance phenomenon. On the other hand, Fig. 7b shows that when l = 12.39 m the SEML maximum radial displacement is close to that due to the SML response, displaying the cancelation phenomenon. As with the radial displacement u ρ , Fig. 7c, d indicates that the pore pressure also exhibits obvious resonance and cancelation phenomena in the time domain when l = 14.87 m and l = 12.39 m. 6 Conclusions An analytical solution for the dynamic response of a circular tunnel embedded in a saturated soil and subjected to moving loads has been developed in this study. To decouple the displacement and pore pressure in Biot’s theory, two scalar potentials and one vector potential are introduced. Using the Fourier transform method, the frequency–wavenumber domain solution for the tunnel subjected to an SML is obtained. The frequency and time domain solutions for an SML are obtained by inversion of the Fourier transforms. With the SML solution, the SEML solution and the resonance and cancelation conditions for the SEML are derived. Based on the researches conducted in this study, the following conclusions can be drawn: • At low velocity, the response of the tunnel is almost symmetric with respect to the arrival time of the load; with increasing velocity, the response of the tunnel becomes asymmetric. For the velocity of the moving load larger than the critical velocity, the tunnel will show shockwave type response, but this phenomenon is also very pronounced at the critical velocity. • When the velocity of the moving load approaches the critical velocity, the response of the tunnel increases significantly. Also, in this case, the response of the tunnel is only determined by several dominant frequencies associated with several resonant peaks, while the influence of the responses at other frequencies is almost negligible. • At the critical velocity, as the response of the tunnel is dictated by several dominant frequencies, enhancement and suppression of the dominant frequencies will hence entail resonance or cancelation phenomenon. Therefore, when the time interval between two neighboring moving loads is equal to the integer or halfinteger multiples of the period of one of the dominant peaks of the corresponding SML response of the tunnel, resonance or cancelation will occur. The predicted resonance and cancelation phenomena are confirmed by the presented numerical examples. • This study provides a benchmark solution for various numerical methods addressing tunnels subjected to moving loads. However, as many real tunnels usually contain liners as well as tracks or rails, in order to apply the proposed model to more realistic tunnels subjected to a SEML, it is necessary to incorporate the aforementioned factors into the proposed model. Acknowledgements Financial support received from the national Natural Science Foundation of China (No. 1272137) is highly acknowledged by the authors. Also, constructive comments from the two referees which help to improve the quality of the paper substantially are highly appreciated by the authors.

Appendix A The expressions for uˆ ρm , uˆ θ m , uˆ zm , σˆ ρρm , σˆ θ θ m , σˆ zzm , σˆ ρθ m , σˆ zρm , σˆ zθ m , pˆ m , wˆ ρm , wˆ θ m and wˆ zm in Eq. (17) are given as follows: uˆ ρm =

1 (2) (2) (2) (2) (2) mAm H(m) f + mBm H(m)s + mDm H(m)t − ργ f Am H(m+1) f − ργs Bm H(m+1)s ρ

 (2) (2) +iξ Cm m H(m)t − ργt H(m+1)t ,

(A.1)

Response of a circular tunnel embedded in saturated soil

 1 (2) (2) (2) (2) (2) mAm H(m) f + mBm H(m)s + mDm H(m)t − ργt Dm H(m+1)t + imξ Cm H(m)t ρ × sin(mθ ), (A.2)  

(2) (2) (2) 2 (A.3) = mγt Cm H(m)t + iξ Am H(m) f + Bm H(m)s cos(mθ ),   1  (2) (2) (2) = 2 2μmDm (m − 1) H(m)t − γt ρ H(m+1)t + 2iμCm m (m − 1) H(m)t ρ  (2) (2) (2) (2) (2) + ργt H(m+1)t − ρ 2 γt2 H(m)t ξ + Am 2μ m 2 H(m) f − m H(m) f + ργ f H(m+1) f 

   (2) (2) (2) (2) − ρ 2 γ 2f H(m) f + ρ 2 γ 2f + ξ 2 α A f − λ H(m) f + Bm 2μ m 2 H(m)s − m H(m)s     (2) (2) (2) (A.4) + ργs H(m+1)s − ρ 2 γs2 H(m)s + ρ 2 γs2 + ξ 2 (α As − λ) H(m)s cos(mθ ),    1 (2) (2) (2) = − 2 2μmDm (m − 1) H(m)t − ργt H(m+1)t + 2iμCm m (m − 1) H(m)t ρ     (2) (2) (2) (2) +ρ γt H(m+1)t ξ + Am 2μ m 2 H(m) f − m H(m) f + ργ f H(m+1) f − α A f − λ

   (2) (2) (2) 2 (2) × ρ 2 γ 2f + ξ 2 H(m) f + Bm 2μ m H(m)s − m H(m)s + ργs H(m+1)s    (2)  − ρ 2 α A f − λ γs2 + ξ 2 H(m)s cos(mθ ), (A.5)  1 2 (2) (2) (2) = −λ γ f Am H(m) f + γs2 Bm H(m)s + 2iμξ γt2 Cm H(m)t − (2μ + λ) ξ 2 ρ  



  (2) (2) (2) (2)  × Am H(m) f + Bm H(m)s + α A f Am H(m) f γ 2f + ξ 2 + As Bm H(m)s γs2 + ξ 2 cos(mθ ),

uˆ θ m = − uˆ zm σˆ ρρm

σˆ θ θ m

σˆ zzm



1  (2) (2) (2) (2) (2) + 2mB H(m)s σˆ ρθ m = 2 μ 2mAm H(m) − m H + ργ H − m H(m)s f m f (m) f (m+1) f ρ   (2) (2) (2) (2) +ρ γs H(m+1)s + Dm −2H(m)t (m 2 − m) − 2ργt H(m+1)t + ρ 2 γt2 H(m)t + 2imCm 

(2) (2) (2) ×ξ H(m)t − m H(m)t + ργt H(m+1)t sin(mθ ), 

1   (2) (2) (2) (2) σˆ zρm = μ im 2 Am H(m) f + 2Bm H(m)s + Dm H(m)t − 2 γ f Am H(m+1) f + γs Bm ρ 

  (2) (2) (2) × H(m+1)s ρ ξ + Cm m H(m)t − ργt H(m+1)t γt2 − ξ 2 cos(mθ ),  1  (2) (2) (2) (2) σˆ zθ m = − iμ 2mAm H(m) f + 2mBm H(m)s + mDm H(m)t − ργt Dm H(m+1)t ξ ρ (2) − i mCm H(m)t (γt2 − ξ 2 ) sin(mθ ),

   (2) (2)  pˆ m = − A f Am H(m) f γ 2f + ξ 2 + As Am H(m)s γs2 + ξ 2 cos(mθ ), 1  2 (2) (2) (2) (2) ω ρ f mAm H(m) f + mBm H(m)s + mDm H(m)t − ργ f Am H(m+1) f wˆ ρm = − β1 ρ



 (2) (2) (2) (2) (2) + ργs Bm H(m+1)s + iξ Cm m H(m)t − ργt H(m+1)t + A f Am m H(m) f − ργ f H(m+1) f



  (2) (2) × γ 2f + ξ 2 + As Bm m H(m)s γs2 + ξ 2 cos(mθ ), − ργs H(m+1)s 1  2 (2) (2) (2) (2) ω ρ f mAm H(m) f + mBm H(m)s + mDm H(m)t − ργt Dm H(m+1)t + iCm wˆ θ m = β1 ρ 

  (2) (2) (2)  × mξ H(m)t + mρ A f Am H(m) f γ 2f + ξ 2 + mρ As Bm H(m)s γs2 + ξ 2 sin(mθ ),

(A.6)

(A.7)

(A.8)

(A.9) (A.10)

(A.11)

(A.12)

J.-F. Lu et al.

  i  (2) (2) (2) (2) −iω2 ρ f γt2 Cm H(m)t + ξ ω2 ρ f Am H(m) f + Bm H(m)s + A f Am H(m) f β1 

  (2)  × γ 2f + ξ 2 + As Bm H(m)s γs2 + ξ 2 cos(mθ ),

wˆ zm = −

(2)

(2)

(2)

(2)

(2)

(A.13)

(2)

where H(m) f ≡ Hm (γ f ρ), H(m)s ≡ Hm (γs ρ) and H(m)t ≡ Hm (γt ρ). Appendix B As the solution for the tunnel subjected to a moving load is validated by comparing its results with a static antiplane solution for a circular tunnel subjected to an antiplane shear force, this section will outline basic notations of the complex variable method for the static antiplane elasticity and the solution for a circular tunnel subjected to a static antiplane shear force. The complex potential in antiplane elasticity can be expressed in the following form [25]: φ(ς ) = μw(x, y) + i f (x, y), ς = x + iy,   ∂w ∂w ∂f ∂f φ (ς ) = σzx − iσzy = μ −i = +i ∂x ∂y ∂y ∂x

(B.1) (B.2)

where w is the antiplane displacement and f is the resultant force function, which is defined as follows:  ς f (x, y) = σzx dy − σzy dx. (B.3) ς0

In Eq. (B.3), the integration path is a curve linking the fixed point ς0 = x0 + iy0 and the generic point ς . Since the stress components σzx and σzy fulfill the following equilibrium equation: ∂σzx ∂σzx + = 0, ∂x ∂y

(B.4)

the integral in Eq. (B.3) is hence path independent. For a circular tunnel subjected to an antiplane shear force q at the point σ∗ (Fig. 2), the corresponding solution is determined by the following complex potential [26]: φ(ς ) = −

q q log(ς − ςc ) + log(σ∗ − ς ), 2π π

(B.5)

in which ςc is the center of the tunnel and σ∗ is the complex coordinate of the location of the shear force. Calculating the derivative of Eq. (B.5) with respect to ς , one has φ (ς ) = σzx − iσzy = −

q q + . 2π(ς − ςc ) π(ς − σ∗ )

(B.6)

By using Eq. (B.6), the stress components σzx and σzy can be obtained, and the shear stresses σzρ and σzθ in the cylindrical coordinate system as shown in Fig. 1 have the form σzρ = σzx cos θ + σzy sin θ, σzθ = −σzx sin θ + σzy cos θ.

(B.7)

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Response of a circular tunnel embedded in saturated soil

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