Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 4106970, 14 pages

https://doi.org/10.1155/2017/4106970

## Seismic Response Analysis of Concrete Lining Structure in Large Underground Powerhouse

^{1}State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China^{2}Key Laboratory of Rock Mechanics in Hydraulic Structural Engineering, Ministry of Education, Wuhan University, Wuhan, China

Correspondence should be addressed to Juntao Chen

Received 24 March 2017; Accepted 7 June 2017; Published 17 July 2017

Academic Editor: Fabrizio Greco

Copyright © 2017 Xiaowei Wang 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

Based on the dynamic damage constitutive model of concrete material and seismic rock-lining structure interaction analysis method, the seismic response of lining structure in large underground powerhouse is studied in this paper. In order to describe strain rate dependence and fatigue damage of concrete material under cyclic loading, a dynamic constitutive model for concrete lining considering tension and shear anisotropic damage is presented, and the evolution equations of damage variables are derived. The proposed model is of simple form and can be programmed into finite element procedure easily. In order to describe seismic interaction characteristics of the surrounding rock and lining, an explicit dynamic contact analysis method considering bond and damage characteristics of contact face between the surrounding rock and lining is proposed, and this method can integrate directly without iteration. The proposed method is applied to seismic stability calculation of Yingxiuwan Underground Powerhouse, results reveal that the amplitude and duration of input seismic wave determine the damage degree of lining structure, the damage zone of lining structure is mainly distributed in its arch, and the contact face damage has great influence on the stability of the lining structure.

#### 1. Introduction

More than a dozen large-scale underground powerhouses of hydropower stations are located in the high seismic area of Southwest China, and their stability under seismic action is essential for the safe operation of hydropower stations. Many scholars have done a lot of researches on the dynamic response of the underground powerhouse [1–4] and have made many great achievements. However, there are few studies on the dynamic response characteristics of the lining structure in underground powerhouse so far. The “Wenchuan” earthquake disaster investigation [5, 6] shows that lining structure is the weak part of the underground powerhouse, so it is significant to study its seismic response.

The seismic response analysis of the concrete lining structure in underground powerhouse mainly includes two aspects: one is the dynamic response of the concrete material and the other is seismic interaction of surrounding rock and lining structure. Dynamic constitutive model of concrete material is the basis for dynamic analysis of lining structure in underground powerhouse. Based on laboratory experiments or theoretical analysis, many scholars have established a variety of concrete dynamic constitutive models to study the dynamic response process of concrete material, mainly including dynamic elastoplastic constitutive model [7–9], nonlinear elastic dynamic constitutive model [10, 11], viscoplastic dynamic constitutive model [12, 13], and dynamic damage constitutive model [11, 14, 15], wherein the dynamic damage constitutive model is the most widely used. Because of the structural features of concrete material, minor defects are distributed inside concrete structure before loading and continue to develop under earthquake cyclic loading, so the failure process of concrete material is the process of damage accumulation. It is necessary to study the evolution process of internal defects of concrete material in order to fully understand its dynamic response characteristics [16], and this is why damage theory is widely used in the dynamic response analysis of concrete structures [17, 18].

Because the seismic interaction of surrounding rock and lining structure is of complex nonlinear characteristics, a large number of nonlinear calculations, as well as higher demands for both the seismic contact analysis model and solving algorithm, are needed. Therefore, it has been a difficult problem in the dynamic response analysis of lining structure in underground powerhouse. At present, the numerical method is mainly used to solve the dynamic contact problem, such as penalty function method [19], Lagrange multiplier method [20], numerical programming method [21], and contact element method [22]. However, these methods tend to have a large amount of computation, and the iteration does not usually converge. Therefore, it is difficult to be applied to the calculation of dynamic contact problems of large underground structures. Liu and Sharan [23] proposed a dynamic contact force method, which combines the contact constraints with the explicit center difference method. This method is widely used in large complex practical engineering calculation with better convergence and no iteration [2, 24] but it does not consider the bonding effects of the contact face itself. In fact, due to grouting effects, the lining structure is in strong bonding with the surrounding rock before the contact face cracking and the cohesion of contact face between lining structure and surrounding rock cannot be ignored.

Based on the above ideas, a dynamic constitutive model for concrete lining considering tension and shear anisotropic damage is presented in this paper, and the evolution equations of damage variables are derived. Combined with the proposed explicit dynamic contact analysis method considering bond and damage characteristics of contact face between lining and surrounding rock, a dynamic response analysis method of concrete lining structure in underground powerhouse is constructed. The method is applied to seismic stability calculation of Yingxiuwan Underground Powerhouse and used to study the seismic damage evolution process of concrete lining structure of underground powerhouse. The results provide references for seismic design of concrete lining in large underground powerhouse.

#### 2. Dynamic Damage Constitutive Model for Concrete Lining

Based on a large number of experimental studies [25–28], it is found that the concrete material shows significant strain rate dependence and fatigue damage characteristics under dynamic cyclic loading. Its strain rate dependence is mainly reflected by the fact that its mechanical parameters increase with the increase of the strain rate, which has been proved by numerous experiments and theoretical studies [29–32]. Because concrete is a kind of heterogeneous material, its inherent micro cracks and voids continually develop and accumulate under seismic cyclic loading and finally cause damage on a macro level. In this process, the strength and stiffness of the concrete materials are continuously degraded with fatigue damage characteristics emerging. Based on the above facts, these two features of concrete material should be considered to build a dynamic constitutive model for concrete lining. Therefore, on the basis of damage theory, a concrete dynamic damage constitutive model considering the effects of strain rate is established in this section.

##### 2.1. Dynamic Damage Model for Concrete Based on the Separation of Tension and Shear

The dynamic damage constitutive model for concrete needs to reflect the strain rate sensitivity as well as the generation and evolution of the damage; its general form can be written aswhere means strain rate and means dynamic damage variable.

The rationality of the damage variable largely determines the accuracy of the damage model. The damage variable used in current engineering calculations is a scalar, which is simple and convenient for theoretical derivation and large-scale numerical calculation. However, as a kind of heterogeneous material, the damage evolution of concrete is directional under the action of load, especially the seismic cyclic load, so the measurement of damage should take the form of tensor [16]. Damage forms of concrete are mainly tensile damage and shear damage, and both are of different generation mechanisms, and so are their effects on strength and stiffness. Therefore, the dynamic damage variable can be divided into tensile damage and shear damage [33]: where and are, respectively, tensile and shear damage variables, which are of the scalar form, and their specific evolution equations will be given in Section 2.2 and and are of fourth-order tensor form and are expressed as functions of the principal value of the stress tensor and the principal direction :

Combined with (2) and (3), the dynamic damage variable , which is of fourth-order tensor form, can be derived. According to the static constitutive relation of concrete, the dynamic form of concrete can bewhere is dynamic Young’s modulus of concrete. In the absence of experimentation, it is advisable to take the static Young’s modulus; is fourth-order consistency tensor.

##### 2.2. Evolution Equation of Tensile and Shear Damage Variables of Concrete

According to the derivation process above, in order to establish a complete dynamic damage model, it is essential to set up evolution equation of tensile and shear damage variables of concrete in dynamic action. There is difference between the tensile and shear damage evolution process under dynamic action and under static load, because damage evolution process under dynamic action relates not only to the strain but also to the strain rate. Therefore, the key point of the paper lies in setting up rate-dependent damage evolution equation.

Assuming the maximum tensile strain rule applying to the tensile damage of element in 3D (three-dimensional) stress state, article [34] substitutes the maximum tensile strain in three-dimensional stress state by equivalent strain, so the tensile damage evolution equation can be expressed as where is the threshold of tensile strain damage and is the limit tensile strain, both considering strain rate, and is the residual strength coefficient, while is equivalent strain, which can be defined aswhere , and are three principal strains, respectively. The function can be defined as

It is difficult to acquire the relation between the strain rate and the threshold of tensile strain damage and the limit tensile strain under dynamic action directly. According to a number of experiments [35], there is a high similarity between the static and the dynamic stress strain curves of concrete. According to the similarity,where and are threshold of tensile strain damage and the limit tensile strain in static state respectively, while is dynamic strain amplification factor of concrete. Based on results of series of dynamic experiments on concrete, Euro-International Committee for Concrete proposed an equation: where is static strain rate whose value is and is dynamic strain rate ranging from to .

It is obvious that (5) to (9) compose a complete evolution equation of tensile damage variable. Similarly, evolution equation of shear damage variable can be defined aswhere is the maximum compressive principal strain and is limit compressive strain of element. When Mohr-Coulomb criterion applies to the stress state of the element, then we havewhere and are dynamic friction angle and Poisson’s ratio, which are usually assigned with static value, and is dynamic uniaxial compressive strength of concrete. According to the similarity of the static and the dynamic stress strain curves of concrete, we havewhere is static compressive strength of concrete material and is amplification coefficient of dynamic compressive strength. The proposed equation by Euro-International Committee for Concrete iswhere .

Similarly, (10) to (13) compose a complete evolution equation of dynamic shear damage variable. Obviously, tensile and shear damage variables in this paper are set on the basis of maximum strain criterion and Mohr-Coulomb criterion considering strain rate effect, which is brief and easy for calculation in finite element method.

##### 2.3. Finite Element Simulation of Lining Element

Steel bars are distributed in the concrete lining of underground powerhouse, so their reinforcement effects on the lining structure should be considered in the process of finite element calculation of concrete lining. In this paper, it is assumed that the steel bars are uniformly distributed in all of the lining elements, and the stiffness of the lining element is the superposition of the stiffness of the concrete element and the stiffness of the steel element; its element stiffness matrix can be expressed aswhere is stress transformation matrix and is equivalent elastic matrix for steel bar element; its expression can be seen in literature [36]. is equivalent elastic matrix for concrete element, and, based on the derivation of Sections 2.1 and 2.2, it can be expressed aswhere .

#### 3. Dynamic Contact Analysis Model of Lining Structure and Surrounding Rock

The dynamic contact analysis between large underground structure and surrounding rock is of numerous contact elements as well as complex contact states, and, due to the consolidation grouting and contact grouting, the contact face between the underground structure and the surrounding rock is in bonding state before the bonding failure with certain tensile strength and shear strength. Therefore, the suitable dynamic contact model of underground structure and surrounding rock should not only be able to support large-scale nonlinear calculation, but also reflect their dynamic contact characteristics. Based on the above facts as well as time-domain explicit integration method, a dynamic contact analysis model of surrounding rock and lining is established in this section, which combines the contact conditions and bonding properties.

##### 3.1. The Explicit Time-Domain Integration Method for the Motion Equations of Contact System

The contact system model of surrounding rock and lining structure and the dynamic contact forces on their contact face is shown in Figure 1. According to the idea of Shu et al. [37], the two sides of the contact face are divided into surface of the surrounding rock and surface of the lining structure, the correspondent nodes (as and in Figure 1) on surfaces and are, namely, node pairs (pairs of nodes), the two nodes of each node pair occupy a common local coordinate system and the same global coordinates, and dynamic forces on the two nodes satisfy the interaction relationship; that is, . The main content of this subsection is to derive the time-domain progressive integration scheme for the nodes of the contact system according to the equation of motion.