Effects of Spatially Varying Seismic GroundMotions and Incident Angles on Behavior of Long Tunnels

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

Seismic behavior of tunnels has been widely studied by many researchers [7][8][9][10][11], and these researches have concentrated mostly on 2D analysis.For the analysis of axial and bending deformations of tunnels, it is most appropriate to utilize 3D models.
However, for the 3D analysis of seismic behavior of tunnel, soil-tunnel analyses in the past were typically limited to relatively small regions, which made it difficult to fully consider the complex spatial features involved in such large structures [12].Tunnels often had significant length and could be built on different strata foundations, which made the seismic analysis of tunnels a complex problem and was usually evaluated under idealized conditions by using numerical methods, such as the finite-element method (FEM).
For the seismic analysis of complex stress distribution and deformation of long-distance tunnels, it is often more reliable to adopt three-dimensional (3D) methods.With the rapid development of science and technology, it is now possible to use highperformance computers to conduct large-scale 3D FEM seismic analysis for tunnels.ere are three kinds of deformation, such as axial compression/extension, longitudinal bending, and ovalling/ racking, occurring in tunnels during earthquake [10].Particularly, the cross-sectional distortion of the tunnel can be related to seismic waves propagating along the tunnel.
Yu et al. [13,14] presented a multiscale 3D FEM analysis of long tunnels under seismic loads where the mechanical characteristics of the tunnel segments and joints under artificial or recorded earthquake loads were evaluated in detail.e model in this paper would take into account not only the motion distribution with time but also the spatial variability (incoherency effect), the wave-passage effect, and the site-response effect.
is paper, based on an earlier report by the authors [15], attempts to develop a new model for seismic analysis of long tunnels with multisupport excitations, which properly accounts for the spatial variability.e main focus of this paper is to assess the influence of multisupport input earthquake waves and uniform input earthquake waves as well as their incident angles on the diameter strain rate and tensive/compressive principal stresses under different strata.A full-scale 3D finiteelement model is built comprising geological data, tunnel geometry, and so on to farthest simulate actual situation.

Simulation of Seismic Ground Motions
2.1.Simulation of Uniform Seismic Ground Motion.Consider a zero mean Gaussian stationary seismic ground motion with a target autospectral density S(ω).e seismic ground motions can be generated through the following expression: where N is the number of frequency intervals, Δω � ω u /N is frequency increment with ω u as the cutoff frequency, ω l � lΔω, and the φ l 's are statistically independent random phase angles uniformly distributed between (0, 2π].Equation ( 1) is valid if there is an upper cutoff frequency ω u above which the contribution of the power spectral density (PSD) to the simulations is negligible for practical purposes.

Simulation of Spatially Varying Seismic Ground Motions.
e variations in ground motion are caused by the following four sources: (1) the "incoherence effect," (2) the "wave-passage effect," (3) the "site-response effect," and (4) the "attenuation effect."e spectral representation method is one of the most widely used methods in simulating the spatially varying seismic ground motions.
In practical application, spatially correlated ground motions can be considered as a one-dimensional, n-variety (1D-nV) stochastic vector process X(t) with components x j (t) • (j � 1, 2, . . ., n).Based on the spectral representation method, the jth component of the ground motions can be generated by [13] x where ω u is the upper cutoff frequency beyond which elements of the power spectral can be assumed to be zero for either mathematical or physical reasons and φ jk are independent random phase angles uniformly distributed over (0, 2π]. |H jk (ω)| and ϕ jk (ω) are the modulus and phase parts of H jk (ω), respectively, which can be obtained by the root decomposition of power spectral density matrix as follows: where the superscript T denotes conjugate transpose.
e power spectral density matrix is given as where S jj (ω) is the autopower spectral density function and S jk (ω, t) is the cross-power spectral density function, which can be expressed as where |c jk (ω)| is the lagged coherence function representing "incoherence" effect and θ jk (ω) is composed of wave passage.However, the ground motions simulated by the above method are stationary, while the actual seismic records are nonstationary.
erefore, to obtain nonstationary seismic ground motion, the way of multiplying an envelope function is applied.e envelope function is as follows: where t 1 , t 2 , and c are three parameters describing the shape of the envelope function.In this study, they are set to be t 1 � 6, t 2 � 10, and c � 0.5, respectively.

FE Model
e tunnel model adopted in this paper was built with circular appearance, whose outside diameter is 10 m, inside diameter is 9 m, and length is 1000 m.It had a buried depth of 30 m from the tunnel center to soil surface.e simulation setup is shown in Figure 1.It consists of a 1000 × 300 × 100 m box with the 1000 m length circular tunnel.e space coordinates were built by taking the length direction as the Z-axis, the width direction as the X-axis, and the height direction as the Y-direction.e soil profile was modeled as four layers of Mohr-Coulomb materials and tunnel as the elastic material.All the property parameters used can be found in Tables 1 and 2.An infinite domain by using the artificial boundary was adopted [12], where both borders of the tunnel were fixed in the Z-direction.
In this work, the constitutive model of the tunnel is an elastic model, which is given as follows: in which σ is the stress tensor, ε is the strain tensor, I 1 is the first strain invariant, λ is the Lame constant, and G is the shear modulus.e Mohr-Coulomb yield criterion for soil is expressed as follows: 2

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where and the parameters S yc and S yt are the yield stresses of the material in uniaxial compression and tension, respectively.On the interface, the deformation of the tunnel and soil is in concert and harmony.

Computation and Analysis
In computation, a regular 3D nite-element model was built based on geotechnical data, tunnel geometry, and so on.e traveling waves velocity motivated from bedrock surface was 500 m/s in the 3D spatially varying seismic ground motion eld.e schematic diagram of di erent incident angles used is shown in Figure 2. e simulated spatially varying ground motions are plotted in Figure 3.

E ect on Radial Deformations.
e diameter strain rate Δd is de ned as where d and d ′ are the tunnel diameters before and after deformation.e positive rate value denotes that the tunnel cross section undergoes tensile deformation in radial direction, and the negative value corresponds to compressive deformation.In this section, d max is de ned as the maximum of Δd in the earthquake time process.d max in the vertical direction is calculated from displacement time processes at the vertex and nadir of tunnel in the Y-axis direction while the one in the lateral direction is obtained from processes at the right and left sides of the tunnel in the X-axis direction.It was found from the simulation that lateral Δd and vertical Δd were equal and opposite in the direction at any arbitrary time, videlicet, the primary deformation of tunnel under seismic waves was ovalling.In addition, the partial feature results under uniform and spatially varying seismic ground motions, that is, the lateral d max at Z 100 m, 300 m, 500 m, 700 m, and  900 m with incident angles of 0 °, 30 °, 45 °, 60 °, and 90 °can be found in Tables 3 and 4.
It could be observed that the lateral d max increases with increasing incident angles for both seismic input forms, but the maximum positions di er from each other.For uniform seismic input simulations, the lateral d max position is at the middle point (Z 500 m) of the tunnel and the diameter strain rates are symmetrically distributed on the left and right sides of the tunnel.
is can be attributed to the balanced boundary conditions and uniform seismic waves.For spatially varying seismic ground motions, the lateral d max is located between Z 700 m and Z 900 m and exhibits asymmetric features due to the nonuniform waves.
e table also implies that the lateral d max for the whole tunnel under spatially varying seismic ground motions is greater than that under uniform input ground motions.

E ect on Stress Distributions.
e maximum principal shear (σ tmax ) and compressive (σ cmax ) stresses are commonly used to analyze the e ect of seismic input ground motions on tunnels.e peak stresses under di erent incident angles are illustrated in Table 5.
It is indicated that the maximum tensile stress is identical to the maximum compressive stress while the incident angle is 0 °for uniform seismic input tunnels. is is owing to the tunnel and soil pro le placed in pure shear stress state.When incident angles are equal to 30 °, 45 °, 60 °, and 90 °, the values of maximum tensile and compressive stresses are di erent because of the complex shear and normal stress states induced by seismic load.It can be thus concluded that the earthquake acceleration in the vertical direction changes not only the maximum stress but also the stress characteristics.In addition, the spatially varying seismic ground motion is another factor in uencing the stress state according to the di erent values between the maximum tensile stress and maximum compressive stress at an incident angle of 0 °under spatially varying seismic ground motions.Meanwhile, the value of maximum tensile/compressive stresses decreases with the increasing incident angles.e maximum tensile stress nephogram at an incident angle of 0 °for both uniform and spatially varying seismic ground motions is, respectively, shown in Figures 4 and 5.   4 is the maximum tensile stress, responding to the red lower portion near the tunnel end.It is suggested that the spatially varying seismic ground motion a ects the stress distribution.

Conclusions
Numerical simulation was an e ective method for studying the tunnel responses under seismic event.In this paper, a novel large-scale analytical method was established for estimating the seismic response of long tunnels within soil foundations.Based on the 3D FEM platform, this method could o er a reliable way for investigating the nonlinear seismic behavior of long tunnels.Main ndings are summarized as follows: (1) e spatially varying seismic ground motions induced larger diameter strain rates (radial deformation) than the uniform input seismic motion.e maximum radial strain rate increased with the increasing incident angles for both seismic input forms.
(2) For uniform seismic input simulations, the maximum radial strain rate occurred at the middle point of the tunnel and the radial strain rates were symmetrically distributed on the left and right sides of the tunnel.Moreover, the spatially varying seismic   Advances in Civil Engineering ground motions had an asymmetric effect on the radial strain rate distributions.
(3) e rising incident angles changed the pure shear stress state into a complex stress state for tunnels under specified input motion.e spatially varying seismic ground motions also influenced the stress state relative to the uniform seismic input.Meanwhile, the values of maximum tensile/compressive stresses decreased with the increasing incident angles.

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

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

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

Table 2 :
Properties of the tunnel.

Table 3 :
Lateral maximum diameter strain rate of tunnel under uniform seismic motion.

Table 4 :
Lateral maximum diameter strain rate of tunnel under multisupport seismic motion.

Table 5 :
Maximum stresses of tunnel body at di erent incident angles.