Advances in Civil Engineering

Volume 2018, Article ID 5791354, 11 pages

https://doi.org/10.1155/2018/5791354

## Parameter Sensitivity of Shallow-Bias Tunnel with a Clear Distance Located in Rock

Correspondence should be addressed to Feifei Wang; moc.621@1991wff

Received 25 October 2017; Revised 23 December 2017; Accepted 16 January 2018; Published 29 March 2018

Academic Editor: Dimitris Rizos

Copyright © 2018 Xueliang Jiang 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

In order to obtain the seismic internal force response laws of a shallow-bias tunnel with a small clear distance, the reliability of the numerical simulation is verified by the shaking table model test. The parameter sensitivity of the tunnel is studied by using MIDAS-NX finite element software. The effects of seismic wave peak (0.1 g, 0.2 g, 0.3 g, 0.4 g, 0.5 g, and 0.6 g), existing slope angle (30°, 45°, 60°, and 90°), clear distance (1.0 D, 1.5 D, 2.0 D, and 3.0 D), and excitation mode ( direction, direction, direction, and direction) on the internal force response law of the tunnel are studied, respectively. The results show that (1) the shear force gradually increases with the increasing of seismic peak. The amplification is different with different measuring points. (2) Under different existing slope-angle conditions, the variation trend of shear force of the tunnel is similar, but the shear force is different. The existing slope has significant effect on the shear force response of the tunnel, and the degree is different with different slope angles. (3) Under the conditions of 1.5 D and 2.0 D, the shear force response of the tunnel is stronger, but the response of other conditions is relatively weak. The tunnel with 1.5 D to 2.0 D clear distance should be avoided. Different excitation modes have a significant effect on the shear force response of the tunnel. (4) Under the same excitation mode, the different excitation directions also have a significant effect on the shear force response. (5) The shear force response of the tunnel crosssection shows nonlinear variation trend. The shear force response is strongest at the arch shoulder and arch foot of the tunnel. The research results provide a useful reference for the design of antishock and vibration resistance of the tunnel.

#### 1. Introduction

Tunnels have been important components of the transportation in mountain areas, which is beneficial to optimize the direction of the line and save construction cost. Previously, many scholars believed that the earthquake-resistant behavior of tunnels were stronger than that of aboveground buildings. After the earthquake observation, however, it is found that under the action of strong ground motion, tunnels may be subjected to extensive deformation or even collapse [1–4]. The surrounding rock has various pattern fractures, and a series of morphological changes occur in an earthquake [5–8]. Moreover, the dynamic response of tunnel structure is highly distinct from that of superstructure [9–11].

The dynamic response of underground structures (e.g., tunnels, culverts, underground stations, and underground reservoir structure) against earthquake has been a subject of intense study by the study methods of experiment [12–26] and numerical simulation [27–35] during recent years. From the above scholars’ results, the study on seismic response of underground has achieved rich results. The research, however, mainly focuses on the dynamic response of the tunnel conventional type and underground structures.

The study, currently, on seismic force response of the new type tunnel (shallow-bias tunnel with a small distance) has a few been studied by scholars. The seismic force response of the new type tunnel is significantly different from that of the tunnel conventional type. The numerical methods (e.g., FEM, FDM, and BEM), as the most popular approaches, have been used to study the dynamic response of tunnels. A few experimental studies have been reported the acceleration response of the tunnel. Based on the comparison between the shaking table test and the numerical method, the results obtained by the numerical method were credible and reasonable. This study aims to perform a systematic experimental work followed by means of an extended numerical parametric study to elaborate some dimensions of the problem. The effects of various parameters, including seismic wave peak (0.1 g, 0.2 g, 0.3 g, 0.4 g, 0.5 g, and 0.6 g), seismic wave excitation mode ( direction, direction, direction, and direction), clear distance (1.0 D, 1.5 D, 2.0 D, and 3.0 D), and existing slope angle (30°, 45°, 60°, and 90°) on the seismic force response, were investigated in the present study.

#### 2. Numerical Simulation

##### 2.1. Eigenvalue Analysis

In order to calculate the mode shape and the natural period of undamped free vibration in MIDAS-NX, the characteristic equation is as follows:where *K* is the stiffness matrix of the structure, *M* is the mass matrix of the structure, is the eigenvalue of the *n* mode, and is the mode vector of the *n* mode.

Eigenvalue analysis, also known as free vibration analysis, is used to analyze the inherent dynamic characteristics of structures. By eigenvalue analysis, the main dynamic characteristics, such as mode shape, natural vibration period, and vibration mode participation coefficient, are determined by the structure quality and stiffness.

In the calculation, the load and damping in the equation of motion of the SDOF system are assumed to be 0. Then the equation degenerates into free vibration equation, which is as follows:where *u* is the displacement induced by vibration. If (*A* is related to the initial displacement constant), (2) can be changed into the following formula:

The conditions of (2) are that the item in parentheses is 0. So the eigenvalues are as follows:where is the eigenvalue, is the proper circular frequencies, is the natural period, and *f* is the natural frequency.

The vibration mode participation coefficient is calculated by the ratio of response between the mode shape and all involved vibration modes. The calculation formula is as follows:where *M* is the order of the mode shapes, is the mode participation coefficient, is the quality of *i* nodes, and is the *m* order vibration vector of *i* node position.

In order to ensure that the earthquake mainly contains enough vibration, modal effective mass *M* is greater than the sum of all the effective mass of 90% general provisions in the seismic design code.where is the effective mass of various modes.

##### 2.2. Computation Damping

The Rayleigh damping type is used in the numerical simulation. In order to reduce the uncertainty of stiffness damping in the high modes, the sum of both mass proportional damping and stiffness damping is used as the damping matrix.

The proportional damping matrix *C* adopts the formula proposed by Caughey:where *j* and *N* are the nodal degrees of freedom. can be calculated by the free vibration formula of the undamped system:

Substituting formula (9) in (8), the following formula can be obtained:

In , is equivalent to the number of modes, and it is expressed by .

Substituting in formula (7) and multiplying (7) by and on both sides, the following formula can be obtained:

The damping constant of the *s* order vibration mode is expressed by the following formulas (12) and (13):

The damping constant of the mass ratio and stiffness type is shown in the following formulas (14) and (15), respectively. The Rayleigh damping matrix is shown in the following formula (16).

Mass proportion type:

Stiffness proportional type:

Rayleigh type:

##### 2.3. Time History Analysis

The structural dynamic time history analysis refers to the process of calculating the structural response (displacement, velocity, internal force, etc.) at any moment, when the structure is subjected to dynamic loads. The dynamic equilibrium equation is used in the dynamic time history analysis of MIDAS-NX, which is as follows:where is the mass matrix, is the damping matrix, is the stiffness matrix, is the dynamic loading, is the relative acceleration, is the relative velocity, and is the relative displacement.

The modal superposition method for structural dynamic analysis is used to carry out dynamic time history analysis in MIDAS-NX. The mode superposition method means that the displacement of the structure is solved by a linear combination of orthogonal displacement vectors. This method is better for linear dynamic analysis of large structures. The premise of using this method is that the damping matrix can be represented by a linear combination of the mass matrix and the stiffness matrix.

The mode superposition method is one of the most widely used methods of structural analysis program. But in nonlinear dynamic analysis, this method has some limitations. In order to make up for this shortcoming, the nonlinear characteristics of stiffness and damping can be taken into account in MIDAS-NX. In nonlinear dynamic time history analysis, the direct integration method is adopted. The direct integral method is a method of time as the integral parameter solution of the dynamic equilibrium equation. Analysis of dynamic time history using the Newmark method with better convergence in MIDAS-NX is performed. The basic integral method is as follows: can be obtained from (19). It is substituted in formula (18), and is calculated to obtain the relationship between displacement, velocity, and acceleration of the current stage and that of the last stage as follows:

Substituting formula (20) in formula (21), the present displacement can be obtained. The present velocity and acceleration can be obtained by present and last displacements:where and are the integral parameters of Newmark . is the integral time interval.

Midas-NX analysis software based on finite element theory can transform differential equations into linear algebraic equations to solve problems. It is applicable to anisotropic, nonlinear, and heterogeneous materials and has an effective applicability for complex boundary conditions. It can better reveal the dynamic response law of tunnel under earthquake loading. In this paper, Midas-NX finite element software is used to analyze the nonlinear dynamic response of the tunnel. In order to reduce the boundary effect in numerical simulation, the model size is more than 5 times the diameter of the tunnel. Therefore, the length, width, and height of the numerical model are 60 m, 40 m, and 55 m, respectively.

Lysmer and Kuhlemeyer [36] showed that for accurate representation of wave transmission through a model, the element size must be smaller than approximately 1/10 to 1/8 of the wavelength associated with the highest frequency component of the input wave:where *λ* is the wavelength of the propagated wave in the model.

According to the calculation results, the total number of nodes and units in the numerical calculation model are 9,548 and 42,539, respectively. In the numerical simulation, the surrounding rock and the lining are simulated by a solid element, and the elastic-plastic constitutive model and Mohr-Coulomb yield criterion are used. In the computational model, the free field boundary and Rayleigh damping are used, and the critical damping ratio of 5% is considered. The specific calculation model is shown in Figure 1.