Shock and Vibration

Volume 2018, Article ID 9525680, 14 pages

https://doi.org/10.1155/2018/9525680

## Dynamic Characteristic and Fatigue Accumulative Damage of a Cross Shield Tunnel Structure under Vibration Load

^{1}Key Laboratory of Transportation Tunnel Engineering, Ministry of Education, Southwest Jiaotong University, Chengdu 610031, China^{2}Chengdu Survey, Design and Research Co. Ltd., Chengdu 610031, China

Correspondence should be addressed to Hang Chen; moc.361@dssgnahnehc and Shuqi Ma; moc.qq@amiquhs

Received 19 September 2017; Accepted 9 January 2018; Published 13 March 2018

Academic Editor: Sandris Ručevskis

Copyright © 2018 Qixiang Yan 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

This study presents an improved constitutive model for concrete under uniaxial cyclic loading which considers the fatigue stiffness degradation, fatigue strength degradation, and fatigue residual strain increment of concrete fatigue damage. According to the constitutive model, the dynamic response and cumulative damage of the tunnel cross structure under various train operation years were analyzed. The results show that the vibration in the middle of the main tunnel is most violent. With the increase of train operation period, the acceleration in the middle of the transverse passage floor, both sides of the wall corner and the vault increase significantly, and the maximum principal stress increases significantly only in both sides of the wall corner. The compressive damage is mainly distributed at both sides of the wall corner, while tensile damage is distributed in both sides of the inner wall corner. The accumulative damage of the cross structure exhibits a two-stage profile. The size and range of accumulative tensile damage of the connecting transverse passage are greater than those of accumulative compressive damage.

#### 1. Introduction

For long and large tunnels, a certain number of connecting transverse passages are normally set up to meet the needs of the operating ventilation, accident evacuation, fire rescue, and other functions. The use of these passages forms the cross tunnel structures. This kind of structure is complex, which lead to uneven distributed forces for the whole structure. The stress concentration most often appears at the intersection due to the train vibration loads [1–4]. In addition, during the service life of the tunnel, the cross tunnel structure is subjected to the vibration loads caused by the train for a long time. The concrete material deteriorates continuously, and the structure is damaged and cracked all the time, which ultimately results in structural damage and poses serious threat to the operational safety of the structure [5–10]. Therefore, it is very important to study the dynamic response and fatigue cumulative damage law of the special structure of the cross tunnel under long-term train which caused loads for the safety of long and large tunnels.

For the properties of the tunnel lining structure material, the tunnel lining structure is mainly made of reinforced concrete in China [11–15]. The fatigue of concrete structures under cyclic loading causes the damage and cracking of concrete lining. At present, the fatigue performance of concrete structure is mainly studied using indoor fatigue tests. Aas-Jakobsen [16] proposed a general formula for logarithmic lifetime and cyclic stress; Tepfer and Kutti [17] identified the general formula of the basic parameters by the fatigue test; Holmen [18] studied compressive fatigue performance of concrete cylindrical specimens under fatigue load; Huang et al. [19] studied the propagation law of the main crack of reinforced concrete beams strengthened with prestressed CFRP sheet under the fatigue load; Cao et al. [20] studied the fractal characterization in the evolving damage of concrete structures based on physical model experiments and found the surface-crack distribution of the damaged concrete structures.

The fatigue test can accurately describe the fatigue performance of the material, but the test is very time-consuming, and it is difficult to conduct full-scale tests. Researchers started to use numerical methods to study the fatigue damage of the complex structure. Teng and Wang [21] proposed a two-dimensional damage constitutive model of a reinforced concrete structure; Petryna and Krätzig [22] proposed a calculation method for long-term performance evaluation of reinforced concrete structures considering the accumulated damage; Zhang and Shi [23] used the finite element method to study the interface peel stress and its influencing factors on reinforcement and concrete under the fatigue load; J.-S. Zhu and X.-C. Zhu [24] proposed a simplified method for numerical simulation of the fatigue failure process of reinforced concrete bridge structures under operating loads; Wang [25] established a stochastic damage constitutive model based on modified elastomeric Helmholtz free energy under tension and compressive conditions; Wang [26] proposed an equivalent static analysis method for the fatigue cumulative damage process of concrete components.

For the vibration effects caused by train, studies have been conducted on the dynamic response characteristics of tunnels under the train vibration. Gharehdash and Barzegar [27] used a complex elastoplastic 3D dynamic finite difference model by fully considering the joints to study the dynamic response of the shield tunnel buried in soft soil under the vibration loads; Gupta et al. [28] presented the experimental validation of a numerical model for the prediction of subway induced vibrations; Gupta et al. [29] used a coupled periodic finite element-boundary element model to study the vibration response from a Thalys high-speed train in the Groene Hart tunnel; Lin [30] studied the dynamic response of the tunnel under different conditions, such as the preconstruction of the train vibration load.

However, most of the current methods are complex in theory and cannot simulate the fatigue damage behavior of concrete structures under high cyclic loads. Most of the studies only focus on the fatigue analysis of concrete beams, aiming at the dynamic response of the tunnel under the train causing vibration loads. There is a lack of research on dynamic responses of the tunnel structure under the high cyclic loads and lack of the fatigue damage analysis of the cross tunnel structure, formed by the main tunnel and the transverse passage.

The improved uniaxial cyclic loading constitutive model for concrete is proposed based on the latest concrete uniaxial monotone load constitutive model given by “Code for Design of Concrete Structures” (GB50010-2010) [31], together with the concrete fatigue constitutive relation proposed by J. S. Zhu and X. C. Zhu [24]. This model is able to more accurately simulate the mechanical behavior of the commonly used concrete. The formulas for calculating the concrete fatigue stiffness variables, fatigue residual strength variables, and fatigue residual strain variables are included in the cyclic loading constitutive model. Based on the actual situation of the Shiziyang tunnel project of Guangzhou-Shenzhen-Hongkong Railway Passenger Dedicated Line, numerical analysis models were established, and the dynamic response and cumulative damage characteristics of the tunnel cross structures under train vibration load were analyzed.

#### 2. Constitutive Model of Concrete Uniaxial Monotone Loading

The stress-strain curves of concrete under monotonic compression were obtained according to the test data fitting in the “Code for Design of Concrete Structures” (GB50010-2010) and are as follows [31]: wherewhere is concrete strain; is nondestructive elastic modulus of concrete; is compressive stress of concrete; is peak compressive stress; is peak compressive strain corresponding to the peak compressive stress and can be taken as .

When the concrete is monotonically tensile, the stress-strain curve is as follows [32]:wherewhere is concrete tensile stress; is peak tensile stress; is the peak tensile strain corresponding to the peak tensile stress and can be taken as .

#### 3. Constitutive Model of Concrete under Uniaxial Cyclic Loading

The related research [33] shows that the fatigue damage of concrete structures under cyclic loading is mainly demonstrated in three aspects: stiffness decrease, strength degradation, and residual strain increase. Therefore, according to concrete uniaxial constitutive model of the above specification and the concrete fatigue constitutive relation proposed by J. S. Zhu and X. C. Zhu [24], the fatigue constitutive model of concrete under uniaxial compression can be proposed: that is, the stress-strain curve is as follows:where the residual strain of concrete after times fatigue loads, the peak compressive strain , and the modulus of elasticity after times fatigue loads are considered, and the relevant revised parameters for the constitutive model are as follows:

Considering the effect of peak compressive stress after times fatigue loads, the peak compressive strain can be obtained after times fatigue loads:

Similarly, when concrete is subjected to tensile loads, the formula can be proposed as follows:where the residual strain of concrete , peak tensile strain , and elastic modulus are considered, and the parameters are revised as follows:where is the structural concrete peak tensile stress after loading the times fatigue load and is the peak tensile strain after loading times fatigue load.

##### 3.1. Concrete Fatigue Stiffness Related Variable

According to the relevant fatigue test, Holmen [18] proposed the degradation formula for the concrete elastic modulus:where is the concrete fatigue life.

##### 3.2. Concrete Fatigue Residual Strength Variables and

The residual fatigue strength of concrete is related to the number of fatigue load cycles and the maximum and minimum stresses of the load [21].

The study [32] shows that the maximum total strain when concrete is broken under tensile and compressive fatigue loads is equivalent to the strain corresponding to the maximum stress of fatigue load in monotonic loading softening section, as shown in Figure 1, the point B in the stress-strain curve of concrete under uniaxial static load and fatigue process. It is assumed that the concrete fatigue residual strength envelope [34] is represented by the shape of softening section of monotonic loading stress-strain curve of the concrete. Therefore, the concrete residual strength envelope can be obtained by the softening section shape of monotonic loading stress-strain curve of the concrete.