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Portal section is the weak link of seismic fortification for tunnel structure. Assuming that seismic wave is the vertical incident elastic plane wave, the plane wave input method for the portal section was discussed in this paper; that is, the wave input problem can be converted to the problem of calculating equivalent nodal force at artificial boundaries. Based on different damage evolution processes of concrete under tension and compression conditions, the tension and compression damage variables were defined and solved, respectively. And then a simple elastic dynamic damaged constitutive model for concrete lining was built. According to the characteristics of dynamic interaction between the lining and rock, and based on the dynamic contact force algorithm, an analytical model for joint loading between the lining and rock was built. This model can simulate lining features such as bond, separation, and slip under seismic load. The dynamic response characteristics of lining structure for the portal section under seismic load were analyzed by taking example for an exit section of Dianzhong diversion project in strong earthquake area. The results show that the relative displacement magnitudes of the lining parts are related to the vibration direction of the seismic wave, and the peak displacements decrease gradually to the fixed values from the portal to the interior. The damage coefficients of the lining parts accumulate gradually over time, and the farther the lining is away from the portal, the less serious the seismic damage is. The separation and slip zone distributions of the lining are basically consistent with its severe seismic damage area, which are mainly at haunch, spandrel, and arch foot within a certain range of distance from the portal. The seismic fortified length and key fortified parts of tunnel structure for the portal section can be determined according to the calculation results.

It was found that the portal section is destroyed easily when an earthquake occurs according to the seismic damage investigation of highway tunnels in epicentral area after Wenchuan earthquake. The portal section is the seismic weak section of tunnel structure, second only to the fault fracture zone [

There have been some research results for the seismic response analysis of the portal section. The main research methods are model test and numerical simulation. In the model test, Shen et al. [

This paper assumed that the seismic wave is the elastic plane wave of vertical incidence. The input methods of plane P wave and S wave for the portal section were discussed. A simple dynamic damage model for the concrete lining was built according to different damage evolution processes of the concrete under tension and compression conditions. An analytical model for joint loading between the lining and rock was built based on the dynamic contact force algorithm. The seismic response characteristics of the lining structure were simulated combining with a case study for an exit section of Dianzhong diversion project, which is expected to provide a useful reference for seismic design of the tunnel portal section.

The total wave field can be decomposed into internal and external wave field based on the wave field decomposition principle. There is no need to deal with the external wave field specifically, because it can be transmitted at artificial boundaries and has no influence on the finite element calculation. Therefore, the key to realizing the accurate input of seismic load is to obtain the internal wave field at artificial boundaries. The internal wave field at the bottom boundary of the model is equal to the incident wave field considering the vertical incidence of the seismic wave. Because the incident field of the lateral boundary and the wave field reflected by ground surface are both parallel to the boundary, whereas the local artificial boundary cannot simulate the wave field parallel to the lateral boundary, the internal field of the lateral boundary can be calculated as the free field at this point. When the incident wave field is known, the free field of the lateral boundary can be obtained by superimposing the incident wave field and reflected wave field. The wave input problem can be transformed into the problem of calculating equivalent nodal force acting on artificial boundaries based on the viscoelastic artificial boundary [

Assuming that the seismic wave is the elastic plane wave of vertical incidence, the wave front is parallel to the bottom of the tunnel model, as shown in Figure

Input model for seismic wave.

Assuming that

Given the displacement field, the corresponding velocity field can be solved by derivation or difference. The corresponding stress field can be calculated by the internal wave field at artificial boundaries according to the generalized Hook’s law, and then the equivalent nodal force at artificial boundaries can be obtained [

For the bottom boundary:

For the left boundary:

For the right boundary:

For the front and back boundaries:

S wave is divided into SH wave and SV wave. The calculation methods of seismic load of the two under vertical incidence condition are similar. In this paper, SH wave is taken as an example for illustration. Assuming that

The equivalent nodal force at artificial boundaries is as follows.

For the bottom boundary:

For the left boundary:

For the right boundary:

For the front and back boundaries:

Concrete is a kind of heterogeneous man-made stone. Its material composition determines natural micro cracks existing in the concrete. It is easy to produce damage characteristics under seismic load. Therefore, establishing a reasonable dynamic damage model for the concrete is the key to simulating the seismic response of the lining structure.

Assuming that the concrete is an isotropic medium, and based on the basic theory of continuum damage mechanics, the stress-strain relationship of the concrete can be expressed as

The concrete shows different strength and stiffness characteristics under tension and compression conditions, which is called unilateral effect [

It is necessary to define and solve the damage variables

Dynamic damage model for concrete.

The damage evolution process of the concrete can be divided into three stages under uniaxial tension condition: elastic stage, damage softening stage, and complete damage stage. Correspondingly, the damage variable

The damage evolution process of the concrete can be also divided into three stages under uniaxial compression condition: elastic stage, damage softening stage, and residual strength stage. Correspondingly, the damage variable

Assuming that the concrete damage is also isotropic in three-dimensional stress state, when an element meets the maximum tensile strain criterion and produces tension damage, the one-dimensional damage model can be extended to three-dimensional state. At this point,

Similarly, when an element meets the Mohr-Coulomb yield criterion and produces compression damage, the three-dimensional dynamic compressive damage model for the concrete can be obtained by replacing

The concrete material generally shows obvious rate correlation property under dynamic load, and the mechanical parameters increase with the increase of the strain rate. To describe this growth property, DIF is adopted to express the dynamic increase factor of the concrete parameters. Research shows that the elastic modulus changing law of the concrete is relatively discrete as the strain rate. Therefore, the dynamic elastic modulus can be replaced by the static elastic modulus from the bias safety perspective of engineering project. However, the concrete strength shows some rules as the strain rate changes. The European International Concrete Committee gave the calculation formulas of the dynamic tensile strength increase factor DIFt and compressive strength increase factor DIFc of the concrete under different strain rates based on a large number of experimental data [

There are complex dynamic contact behaviors in the joint seismic response process between the lining and rock. The earthquake may lead to lining damage, crack, separation, or slip. Engineering practice shows that the separation or slip failure of the lining structure is mainly partial in the tunnel without the fault fracture zone, and the large slip failure is rare. Therefore, it is assumed that the contact interface nodes between the lining and rock are always or approximately in point to point contact state during the earthquake and that the contact node pairs are in bond contact state before the earthquake [

For a contact node pair

Point to point contact.

Thus, the relative normal displacement

Assuming that the contact node pair is still in bond contact state at

After each time step is calculated, the contact state of the contact interface needs to be checked, and the dynamic contact force needs to be corrected. The detailed method is as follows.

(1) If

(2) If

Dianzhong diversion project is mainly composed of water source project, water transmission project, auxiliary project, and so on. It is a water conservancy project with comprehensive utilization of water resources. The total length of the water transmission trunk line is 848.18 km. The project scale is huge, and the geological conditions along the route are complex. The seismic fortified intensity is high, and the tunnel stability problem is salient.

An exit section along the route is selected to analyze its seismic response in this paper. The maximum buried depth of the section is 35 m, and the slope angle is about 50°. The design depth of water is 7.51 m, and the surrounding rock type is mainly IV. The excavation section is in horseshoe shape with a maximum excavation size of 8.3 m × 9.87 m. C25 reinforced concrete lining structure is adopted, and the lining thickness is 0.5 m.

A three-dimensional finite element model for the exit section is built. This model is discretized totally by 8-node hexahedron elements, and 110,532 elements and 117,280 nodes are divided, 5,760 of which are lining elements.

Calculation model for portal section.

According to the geostress test results, the lateral pressure coefficients are valued as

Mechanical parameters of materials.

Parameters | Rock | Lining |
---|---|---|

Deformation modulus (GPa) | 3.0 | 28.0 |

Poisson ratio | 0.30 | 0.17 |

Cohesion force (MPa) | 0.5 | 1.8 |

Internal friction angle (°) | 35.0 | 40.0 |

Tensile strength (MPa) | 1.0 | 1.27 |

Compressive strength (MPa) | 20.0 | 11.9 |

The calculation program adopts the three-dimensional elastoplastic dynamic damaged displayed finite element platform [

The viscous-elastic artificial boundary is applied at the bottom of the model, and the free field artificial boundary is applied at the four sides. American strong earthquake record, El-Centro wave, is selected as the seismic wave, whose peak acceleration is 3.417 m/s^{2}. A section of 20 s is intercepted as the incident wave, as shown in Figure

Acceleration time-history curve of El-Centro wave.

A monitoring section is arranged every 6 m from the portal along the tunnel axis, and a total of 12 monitoring sections are arranged. Five monitoring points are arranged on the lining structure in each monitoring section, as shown in Figure

Monitoring points’ layout of lining.

Different parts of the lining are in synchronous vibration state during the earthquake. The maximum positive displacement is defined as the peak displacement; then the variation laws of

As seen in Figure

The peak displacements of the spandrel and haunch of the lining are larger than those of the other parts, and the peak displacement of the bottom arch is the smallest under SH wave action. The maximum relative displacement reaches 0.98 cm. The peak displacement of the bottom arch of the lining is larger than that of the other parts, and the peak displacement of the top arch is the smallest under P wave action. The maximum relative displacement is 0.55 cm. These indicate that the relative displacement magnitudes of the lining parts are related to the vibration direction of the seismic wave.

The seismic damage failure of the lining is mainly caused by tensile damage, because the compressive strength of the concrete is much greater than its tensile strength. For that reason, this paper mainly analyzes the maximum principal stress variation law of the lining. Taking the sixth monitoring section as an example, the maximum principal stress time-history curves of monitoring points are plotted in Figure

Maximum principal stress time-history of monitoring points.

As seen in Figure

Based on the dynamic damage model for the concrete in this paper, the time-history curves of the damage coefficients of monitoring points are obtained, as plotted in Figure

Damage coefficients variation time-history of monitoring points.

As seen in Figure

The damage coefficients’ distribution of the lining structure (taking 60 m range of distance from the portal) after the earthquake is shown in Figure

Damage coefficients’ distribution of lining after the earthquake.

As seen in Figure

The lining and rock are in bond contact state before the earthquake. Once the dynamic contact force of the contact nodes between the lining and rock breaks through the cohesive force during the earthquake, the contact interface will separate or slip easily. And the process is irreversible. The separation and slip zone distributions of the lining structure (taking 60 m range of distance from the portal) after the earthquake are shown in Figures

Separation zone distribution of lining after the earthquake.

Slip zone distribution of lining after the earthquake.

As seen in Figure

The displacement response of the lining is large within 48 m range of distance from the portal, where the lining damage is relatively serious according to its displacement and damage analysis. Therefore, the seismic fortified length of tunnel structure for the portal section can be set to 48 m. In addition, the separation and slip zones of the lining are mainly distributed within 21 m range of distance from the portal, which can be served as the key fortified length.

Based on the dynamic algorithm of the concrete lining, the dynamic response characteristics of the lining structure under seismic load were analyzed by taking as an example an exit section of Dianzhong diversion project. The following conclusions are drawn.

(1) The relative displacement magnitudes of the lining parts are related to the vibration direction of the seismic wave under seismic load. The farther the lining is away from the portal, the smaller the peak displacements are, and they tend gradually to the fixed values.

(2) The maximum principal stresses of the lining parts can easily reach the tensile strength. The violent fluctuation of the maximum principal stresses may aggravate the lining damage and cause the lining fatigue failure.

(3) The damage coefficients of the lining parts accumulate gradually over time. The seismic damage area of the lining is mainly distributed at the haunch, spandrel, and arch foot within a certain range of distance from the portal. The farther the lining is away from the portal, the less serious the seismic damage is.

(4) The separation and slip zones of the lining are mainly distributed at the haunch, spandrel, and arch foot within a certain range of distance from the portal, which are basically consistent with the distribution law of the severe seismic damage area of the lining.

(5) The seismic fortified length and key fortified parts of tunnel structure for the portal section can be determined according to the displacement, damage, separation, and slip analysis of the lining.

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

This study is supported by the National Key Basic Research Program of China (2015CB057904) and the National Natural Science Foundation of China (51579191). These supports are greatly acknowledged and appreciated.