Advances in Civil Engineering

Volume 2018, Article ID 8195396, 6 pages

https://doi.org/10.1155/2018/8195396

## Effects of Spatially Varying Seismic Ground Motions and Incident Angles on Behavior of Long Tunnels

Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, Hohai University, Nanjing 210098, China

Correspondence should be addressed to Yongxin Wu; moc.361@uhhuwxy

Received 14 March 2018; Accepted 7 May 2018; Published 11 June 2018

Academic Editor: Yongfeng Deng

Copyright © 2018 Yundong Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Seismic behavior of long circle tunnels is significantly influenced by the nature of input motion. This study, based on the 3D finite-element 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 ground motions 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.

#### 1. Introduction

Tunnels, as an integral part of the infrastructure of modern society, would suffer damages when subjected to dynamic loadings during earthquake activity, for example, in Kobe earthquake (1995) [1], Chi-Chi earthquake (1999) [2], Duzce earthquake (1999) [3], Mid-Niigata Prefecture earthquake (2004) [4], Wenchuan earthquake (2008) [5], and Tohoku earthquake (2011) [6]. Nevertheless, the number of large tunnels and underground structures had grown significantly in recent decades. Therefore, seismic evaluation of tunnels in seismically active areas is critical during engineering design.

Seismic behavior of tunnels has been widely studied by many researchers [7–11], 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 high-performance computers to conduct large-scale 3D FEM seismic analysis for tunnels. There 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. The 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.

This 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. The 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 finite-element model is built comprising geological data, tunnel geometry, and so on to farthest simulate actual situation.

#### 2. 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 . The seismic ground motions can be generated through the following expression:where is the number of frequency intervals, is frequency increment with as the cutoff frequency, , and the ’s are statistically independent random phase angles uniformly distributed between (0, 2*π*]. Equation (1) is valid if there is an upper cutoff frequency above which the contribution of the power spectral density (PSD) to the simulations is negligible for practical purposes.

##### 2.2. Simulation of Spatially Varying Seismic Ground Motions

The 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.” The 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-*n*V) stochastic vector process with components . Based on the spectral representation method, the component of the ground motions can be generated by [13]where 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 are independent random phase angles uniformly distributed over (0, 2π]. and are the modulus and phase parts of , respectively, which can be obtained by the root decomposition of power spectral density matrix as follows:where the superscript denotes conjugate transpose.

The power spectral density matrix is given aswhere is the autopower spectral density function and is the cross-power spectral density function, which can be expressed aswhere is the lagged coherence function representing “incoherence” effect and is composed of wave passage.

However, the ground motions simulated by the above method are stationary, while the actual seismic records are nonstationary. Therefore, to obtain nonstationary seismic ground motion, the way of multiplying an envelope function is applied. The envelope function is as follows:where , and are three parameters describing the shape of the envelope function. In this study, they are set to be , , and , respectively.

#### 3. FE Model

The 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. The simulation setup is shown in Figure 1. It consists of a 1000 × 300 × 100 m box with the 1000 m length circular tunnel. The 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. The 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.