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In order to explore the influence of internal water on the seismic response of hydraulic tunnel, the combined mechanical analysis models of multimaterial including surrounding rock, lining structure, and internal water are built. Based on the explicit central difference method, the dynamic finite element analysis methods for rock, lining, and water are discussed, respectively. The dynamic contact force method is used to simulate the rock-lining contact interaction, and the arbitrary Lagrange-Euler (ALE) method is used to simulate the lining-water coupling interaction. Then a numerical simulation analysis method for combined seismic response of rock-lining-water system in hydraulic tunnel is proposed, and the detailed solving steps are given. This method is used to study the seismic stability characteristics of the water diversion tunnel in a hydropower station, and the displacement, stress, and damage failure characteristics of the lining structure under the conditions of no water, static water, and dynamic water are comparatively analyzed. The results show that the hydrostatic pressure restricts the seismic response of the lining, while the hydrodynamic pressure exacerbates its seismic response and leads to damage, separation, and slip failure appearing on the haunch, which can provide a scientific reference for the seismic design of hydraulic tunnel with high water head and large diameter.

In order to ensure the national energy supply and optimize the water resources allocation, a large number of long-distance and cross-basin water diversion projects have been built, under construction and planned in Southwest China, which is rich in water resources. Most of the water transmission lines of these projects have to pass through high mountains, and they mainly rely on hydraulic tunnel for water diversion. Hydraulic tunnel generally inevitably passes through the high seismic intensity area in the construction process due to its long extension, so its seismic safety needs detailed analysis and demonstration [

There are many research results on the seismic response of ordinary tunnel [

On the other hand, the violent shaking of fluid under seismic load belongs to a large deformation problem of mesh. The methods adopted to describe the large deformation of mesh mainly include Lagrange method and Euler method in the finite element calculation of continuous medium. The pure Lagrange method and the pure Euler method have their own advantages, but also have serious defects. For the Lagrange method, the solution of motion equation is relatively simple, but the large deformation may lead to mesh deformity. For the Euler method, the calculation accuracy will not decrease due to the material distortion, but it needs to introduce a very complex mathematical mapping when dealing with the boundary problem, which may make a large difference between the calculation results and the real solutions. With the continuous development of computer technology, the fluid-structure coupling analysis method based on the ALE description [

In this paper, the Lagrange method is used to describe the motions of surrounding rock and lining structure, and the ALE method is used to describe the motion of internal water. Based on the mechanical analysis models of multimaterial in hydraulic tunnel, the dynamic explicit finite element analysis methods for rock, lining, and water are discussed, respectively. The dynamic contact force method [

Hydraulic tunnel is a complex underground structure composed of multimaterial including surrounding rock, lining structure, and internal water, so the seismic response of hydraulic tunnel can be regarded as the conditional combined seismic response of the three materials [

Mechanical models of multimaterial in hydraulic tunnel. (a) Hydraulic tunnel. (b) Rock. (c) Lining. (d) Water.

For rock or lining, its momentum conservation equation based on the Lagrange description can be expressed as

After the rock or lining is discretized by finite element in space, the dynamic balanced equation of finite element based on Lagrange can be obtained:

The explicit center difference method is used here to solve (

For internal water, its mass and momentum conservation equations based on the ALE description can be obtained when it is assumed to be incompressible viscous fluid:

Similarly, the dynamic discrete equation of finite element based on ALE can be obtained by discretizing the water in space:

From the above analysis, it can be seen that, for each material in hydraulic tunnel, the motion state of nodes at time

Many researches [

Node 1 and node 2 are assumed to be a contact node pair of rock-lining system.

When

In order to solve the dynamic contact force

If

where

If

When

Node 3 and node 4 are assumed to be a coupling node pair of lining-water system.

The lining-water coupling process based on ALE can be briefly described as follows [

When the motion state of the coupling nodes at time

When

According to

According to

When

It must be noted that the lining-water coupling force is ignored in the above rock-lining contact analysis, and the rock-lining contact force is ignored in the above lining-water coupling analysis. Therefore, in order to correctly simulate the seismic response characteristics of each material in hydraulic tunnel, it is necessary to combine the rock-lining contact analysis and the lining-water coupling analysis. The combined seismic response analysis steps of rock-lining-water system at time

When

According to

According to

When

According to

The installed capacity of the expansion project of a hydropower station is 80 MW, and the water diversion and power generation system is arranged according to a water diversion tunnel, a surge shaft, and two units. The water intake is about 2 km away from the upstream left bank of the dam. The total length of the diversion tunnel from the intake to the surge shaft is 7126 m, whose center elevation is reduced from 80.7 m to 32.0 m, and the total length of the diversion tunnel from the surge shaft to the power house is 148.1 m. The water diversion tunnel is located in the eroded middle and low mountain landform with undulating ground along the route, and the geological conditions are relatively poor.

In this paper, a tunnel section within 80 m range downstream of the surge shaft is selected for the seismic response analysis, whose center elevation is 32.0 m. The excavation section is circular, and the tunnel diameter is 10.5 m. The lining adopts C25 concrete structure, whose thickness is 0.75 m. The maximum piezometric head of internal water is 123.0 m. A 3D finite element model of the tunnel section is built, as shown in Figure

3D finite element model of diversion tunnel. (a) Whole model. (b) Local model.

The lateral pressure coefficients of initial in-situ stress field are

Physical and mechanical parameters of rock, lining, and water.

Materials | Deformation modulus (GPa) | Poisson ratio | Cohesive force (MPa) | Internal friction angle (°) | Dynamic viscosity (10^{−3 }Pa∙s) | Density (g∙cm^{−3}) | Tensile strength (MPa) | Compressive strength (MPa) |
---|---|---|---|---|---|---|---|---|

Rock | 8.0 | 0.27 | 0.55 | 37.0 | — | 2.7 | 1.20 | 60.0 |

Lining | 28.0 | 0.17 | 1.80 | 40.0 | — | 2.5 | 1.27 | 11.9 |

Water | — | — | — | — | 1.308 | 1.0 | — | — |

The self-developed 3D dynamic finite element program for underground cavern [

Viscoelastic artificial boundary condition is applied at the bottom of the calculation model, and free field artificial boundary condition is applied at the four vertical boundaries. The top of the model is built to the free surface.

The representative Imperial-Valley wave is truncated for 20 s as the input load, and its peak acceleration is adjusted to 1.575 m/s^{2} to match the seismic intensity of the project area, as shown in Figure

Acceleration time-history curve of input wave.

In order to ensure the stability of dynamic calculation, the mesh size in the direction of wave propagation should be smaller than 1/12-1/8 of the shortest wavelength [

The calculation is divided into three working conditions: (1) no water condition; (2) static water condition (the hydrostatic pressure is considered, while the hydrodynamic pressure is not considered); and (3) dynamic water condition (both the hydrostatic and hydrodynamic pressures are considered). Three monitoring points are arranged on the top arch, haunch, and bottom arch at the middle section of the lining to monitor the hydrodynamic pressure exerted on the lining, the displacement, stress and damage characteristics of the lining, as shown in Figure

Layout of monitoring points.

When the dynamic coupling interaction between the internal water and the lining is considered under seismic load, the hydrodynamic pressure produced by the internal water exerted on the lining can be calculated. The hydrodynamic pressure time-history of the three monitoring points is shown in Figure

Hydrodynamic pressure time-history of monitoring points.

It can be seen from Figure

The peak displacements of the three monitoring points all appear at time 14.9 s both in

Maximum displacements of monitoring points. (a)

It can be seen from Figure

In order to explore the influence of hydrodynamic pressure on the mechanical characteristics of the lining, the maximum principal stress is taken as an example to illustrate. The maximum principal stress time-history of the three monitoring points under the three working conditions are shown in Figure

Maximum principal stress time-history of monitoring points. (a) Condition (1). (b) Condition (2). (c) Condition (3).

It can be seen from Figure

The maximum principal stress of the lining is mainly reflected as tensile stress. Under condition (1), the maximum tensile stresses of the top arch, haunch, and bottom arch are 0.65 MPa, 1.09 MPa, and 0.92 MPa, respectively. Under condition (2), the maximum tensile stresses of the three parts are 0.80 MPa, 0.88 MPa, and 0.79 MPa, respectively, most of which decrease due to the restriction of the hydrostatic pressure compared with condition (1). Under condition (3), the maximum tensile stresses of the three parts all reach the concrete tensile strength (1.27 MPa), which increase obviously compared with condition (1). These show that the hydrodynamic pressure exacerbates the stress response of the lining and may lead to the lining damage.

According to the above analysis, when there is no water or only the hydrostatic pressure is considered, no damage appears on the lining under seismic load. While when the hydrodynamic pressure is considered, the tensile damage appears on the lining. Based on the calculation formulas of damage coefficient (equations (

Damage coefficient time-history of monitoring points.

It can be seen from Figure

The damage coefficient distribution of the lining (taking the middle section of the lining) under the hydrodynamic pressure after the earthquake is shown in Figure

Damage coefficient distribution of lining after earthquake.

It can be seen from Figure

The rock-lining contact interface is basically in bond contact state before the earthquake. As the lining damage develops gradually under seismic load, the rock-lining contact interface is also gradually destructive. The separation and slip zone distributions of contact interface after the earthquake are shown in Figure

Separation and slip zone distributions of rock-lining contact interface after earthquake.

It can be seen from Figure

Combining the dynamic contact analysis of rock-lining interaction and the dynamic coupling analysis of lining-water interaction, a combined seismic response analysis method for rock-lining-water system in hydraulic tunnel is proposed. Taking the water diversion tunnel project in a hydropower station as an example, the dynamic response characteristics and failure mechanisms of the lining structure under seismic load are simulated and analyzed, and the research results can provide a scientific reference for the seismic design of hydraulic tunnel with high water head and large diameter.

The hydrodynamic pressure produced by the internal water increases with the increasement of seismic strength, and the hydrodynamic pressure of the haunch is larger than that of the other parts. Compared with the no water condition, both the maximum displacement and maximum tensile stress of the lining decrease under the static water condition, while they both increase obviously under the dynamic water condition. These show that the hydrostatic pressure restricts the deformation and stress response of the lining, while the hydrodynamic pressure exacerbates the deformation and stress response of the lining.

When the hydrodynamic pressure is considered, the tensile damage appears on the lining, and the damage coefficient increases obviously with time during the violent fluctuation of seismic wave. The lining part in direct contact with the internal water is seriously damaged, and the haunch is the mostly seriously damaged. The separation and slip zones of rock-lining contact interface are mainly distributed in the haunch and its sides under the combined actions of seismic load and hydrodynamic pressure. Therefore, the haunch is the seismic weak part of lining structure.

The data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This study was supported by the IWHR Research and Development Support Program (Grant no. EB0145B082020). This support is greatly acknowledged and appreciated.