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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.

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 [

For the properties of the tunnel lining structure material, the tunnel lining structure is mainly made of reinforced concrete in China [

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 [

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 [

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) [

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 [

When the concrete is monotonically tensile, the stress-strain curve is as follows [

The related research [

Considering the effect of peak compressive stress

Similarly, when concrete is subjected to tensile loads, the formula can be proposed as follows:

According to the relevant fatigue test, Holmen [

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

The study [

The concrete uniaxial static load and fatigue stress-strain curves.

Uniaxial compression

Uniaxial tension

According to the softening section shape of the monotonic stress-strain relationship curves of the concrete, the envelope equation of the residual fatigue strength of the concrete can be obtained as follows: [

Considering the initial conditions of concrete residual strength and failure criterion, the boundary conditions were taken into account in the concrete compressive and tensile fatigue residual envelope equation. The concrete compressive and tensile dependent variables A and B were available. Substituting those variables to formulas (

Holmen obtained the formula of the fatigue residual strain of concrete by the curve fitting the experimental data, without considering the stress ratio. The formula is as follows: [

Life estimation was based on FE-SAFE fatigue analysis software. Firstly, the ABAQUS calculation was used to obtain the dynamic response of the cross structure in the process of the train operation in the tunnel. Secondly, the concrete stress-life curve (i.e., S-N curve) of concrete was determined by formulas (

The fitting formula for the tensile fatigue life and fatigue load curves of concrete [

The tensile fatigue life and fatigue load curves of concrete were obtained by the fitting formula based on concrete fatigue splitting test [

Plastic flow, microcracks, and microvoids are the fundamental reasons of nonlinearity of concrete. From the macroscopic performance, it shows the obvious difference of concrete tensile strength and compressive strength and the residual deformation of concrete [

According to the energy equivalent principle proposed by Sidoroff [

According to the plastic damage theory of concrete, when the concrete is tensile, the cracking strain is [

When the concrete is compressive, the inelastic strain is [

The complex cross structures of the tunnel with the designed service life up to 100 years are subjected to high cycle fatigue problems under the train caused vibration loads. It is uneconomical to calculate the dynamic impact of train on the tunnel every time. Petryna and Krätzig proposed the idea of high cycle structural fatigue [

Flowchart of structural damage analysis.

Guangzhou-Shenzhen-Hong Kong Railway Passenger Dedicated Line is a fast railway channel connecting Guangzhou, Shenzhen, and Hong Kong, which is an important part of the intercity railway network in the Pearl River Delta. The full length of the Shiziyang subsea tunnel is 10.8 km, and it is the longest and highest standard subsea railway tunnel in China. Guangzhou-Shenzhen-Hong Kong Passenger Dedicated Line Shiziyang Tunnel is located at Dongyong Station, Humen Station interval. The ground layers where the tunnel run through are mainly soil, mucky soil, silty clay and fine sand, coarse sand, weathered and weak weathered argillaceous siltstone, siltstone, and fine sandstone. The planar graph and the vertical sectional profile of the Shiziyang subsea tunnel are shown in Figures

The planar graph of the Shiziyang subsea tunnel.

The vertical profile of the Shiziyang subsea tunnel.

According to the actual situation of the project, the connecting transverse passage cross structure of Shiziyang railway shield tunnel was selected and studied here. The dynamic response of cross structure was simulated when the marshalling train is running in the A tunnel. It is assumed that the train is running in the main tunnel A. The clear distance between two tunnels is 5.0 m and the design speed is 300 km/h. The buried depth of the selected section of the tunnel is 19.0 m, located in the weak weathered muddy siltstone, topsoil layer covered with lighter silty clay layer, and fine sand layer. The outer and inner diameters of the shield tunnel are 10.8 m and 9.8 m. The lining is assembled in a 7 + 1 block way with a universal wedge ring reinforced concrete single segment. In order to consider the impact of the segment on the structure, the stiffness reduction ring is set at the main tunnel spacing, and the reduction factor is 0.8. The width and height of connecting transverse passage are 4.0 m and 5.0 m, respectively. The length, width, and height of the stratigraphic structure model are 800.0 m, 80.0 m, and 50.0 m, respectively. All the boundaries except for the upper boundary are simulated using a continuously distributed parallel spring-damper system. This boundary treatment can effectively solve the near field fluctuation problem at soil-structure dynamic interaction.

Marshalling train does not consider the connection between the carriages. The train contains 8 carriages with single carriage length of 25.0 m. Each carriage at the front and rear part has two pairs of axles, a total of 32 pairs of axles. The physical and mechanical parameters of surrounding rock, lining concrete, and track are shown in Table

Physical and mechanical parameters table.

Material | Density | Elastic modulus | Poisson’s ratio | Friction angle | Cohesion |
---|---|---|---|---|---|

(kg/m^{3}) | (GPa) | (°) | (MPa) | ||

Track | 7850 | 200 | 0.2 | - | - |

Lining | 2400 | 34.5 | 0.2 | 43.0 | 1.10 |

Surrounding rock | 2000 | 3.65 | 0.325 | 33.0 | 0.45 |

The model of the structure.

The surrounding rock

Cross structure

The train marshalling

The profile irregularity of a railway line is one of the essential vibration sources for vehicles and track [

Measured train vibration load curve (300 km/h).

In order to simulate the variation of the spatial position and the vibration load time in the upper tunnel where the high-speed train is running, the travel speed of 300 km/h was applied to the marshalling train, to simulate the space driving effect of the train, as shown in Figure

Schematic diagram of train vibration load.

Track and train were simulated using linear elastic materials, the surrounding rock was simulated by the elastic-plastic model with damping, and the vibration system damping used Rayleigh damping. According to the above analysis process and the dynamic response analysis model of the cross tunnel structure under the train vibration, the train vibration response and fatigue cumulative damage analysis of the shield tunnel cross structure were carried out.

Firstly, the fatigue life of the tunnel cross structure under the vibration load of the train was analyzed. The stress time-history of the tunnel cross structure obtained by the calculation during the train operation in the tunnel for the first time was introduced into the FE-SAFE software. The logarithmic life distribution nephogram of the cross tunnel structure was calculated as shown in Figure

Distribution nephogram of fatigue life of cross tunnel structure.

From Figure ^{6} times.

Based on the response law and fatigue life of the structure under the train vibration load for the first time, the fatigue constitutive model of the structure can be obtained. Then, the dynamic response and cumulative damage effect of the cross tunnel structure after operating a certain period of time can be calculated.

Four positions of the tunnel arch bottom were selected as the analysis points, and the analysis points layouts are shown in Figure

Diagram of tunnel analysis points.

The acceleration amplitude of the main tunnel analysis points A1, A2, A3, and B were extracted, after high-speed trains ran in the tunnel 1, 1 × 10^{3}, 1 × 10^{4}, 1 × 10^{5}, 2 × 10^{5}, 5× 10^{5}, 1 × 10^{6}, 1.5× 10^{6} times, as shown in Table

Acceleration amplitude of tunnel analysis points.

Number of runs | Running time | A1 | A2 | A3 | B |
---|---|---|---|---|---|

(Times) | (Years) | (m/s^{2}) | (m/s^{2}) | (m/s^{2}) | (m/s^{2}) |

1 | 0 | 0.89 | 2.28 | 0.73 | 0.35 |

1.0 × 10^{3} | 0.07 | 0.91 | 2.29 | 0.76 | 0.35 |

1.0 × 10^{4} | 0.69 | 0.93 | 2.30 | 0.77 | 0.35 |

1.0 × 10^{5} | 6.85 | 0.94 | 2.30 | 0.81 | 0.35 |

2.0 × 10^{5} | 13.70 | 0.94 | 2.33 | 0.87 | 0.36 |

5.0 × 10^{5} | 34.25 | 0.94 | 2.36 | 0.89 | 0.37 |

1.0 × 10^{6} | 68.49 | 0.95 | 2.43 | 0.90 | 0.39 |

1.5 × 10^{6} | 102.74 | 0.99 | 2.51 | 0.99 | 0.41 |

From Table ^{2}.

The maximum principal stress time-history curve of point A2 of the main tunnel A is shown in Figure

Time-history curve of maximum principal stress (A2).

Figure

The maximum principal stress curves of the analysis points are shown in Figure

The variation curves of the maximum principal stress of tunnel analysis points.

From Figure

Figure

Development trend of maximum principal stress of main tunnel segment.

The distribution nephogram of the cumulative tensile damage of the cross structure after tunnel operation 102.74 years is shown in Figure

Distribution nephogram of cumulative tensile damage of cross structure (102.74 a).

Figure

The development trend of the tensile damage of the analysis points C1, C2, and C3 is shown in Figure

The development trend of tensile damage of the main tunnel segment.

Figure

As can be seen from the above, the connection location is a weak part of the structure due to the stiffness singularity of main tunnel and connected transverse passage. Therefore, the interface of the connected transverse passage and the main tunnel was taken as the analysis section, and the maximum envelope of the acceleration in the time-history range was obtained by extracting the acceleration of the interface of the typical time point (the train first operation, operating for 0.07, 70.68, 49, and 102.44 years), as shown in Figure

Maximum envelope of acceleration at different operation times (MPa).

Frist

0.07 a

13.70 a

68.49 a

102.74 a

Figure ^{2}. The minimum value of acceleration is about 0.76 m/s^{2} in the vault of connected transverse passage. With the increase of the operation time, the acceleration of the middle of the connecting transverse passage floor, both sides of the wall corner and vault increase significantly.

The maximum principal stress values of the analysis point of the left side wall corner of the transverse passage were extracted, when the train is running for the first time, as shown in Figure

Figure

The maximum principal stress time-history curve of the left side wall for the first running.

The maximum envelope of maximum principal stress in time-history range when the train is running for the first time was obtained by extracting the maximum principal stress at the typical time point interface, as shown in Figure

Maximum envelope of maximum principal stress at different operation times (MPa).

Frist

0.07 a

13.70 a

68.49 a

102.74 a

Figure

Cumulative compressive and tensile damage of connecting transverse passage was extracted to further analyze damage situation of connecting transverse passage, as shown in Figures

Compressive damage nephogram of the connecting transverse passage at different operation times.

0.07 a

13.7 a

48.49 a

102.74 a

Tensile damage nephogram of the connecting transverse passage at different operation times.

0.07 a

13.7 a

48.49 a

102.74 a

Figures

Considering that the connecting transverse passage is mainly affected by the tensile damage, the cumulative maximum damage values of the transverse passage at the different operation times were extracted and the maximum development curve was shown in Figure

Maximum development curve of tensile damage of the transverse passage.

Figure

Considering the driving effects of high-speed train, the vibration fatigue life of tunnel cross structure was calculated using fatigue analysis software. The dynamic response and cumulative damage characteristics of cross tunnel structure of Shiziyang railway shield tunnel at various operation years were analyzed, which meet the requirements of the designed life of 100 years according to the Chinese standard. The following main conclusions are obtained:

(1) According to the latest concrete design code, the uniaxial cyclic loading constitutive model of concrete is proposed by taking into account the factors such as the stiffness degradation of concrete, the strength decrease of concrete, and the increase of fatigue residual strain. The proposed model is suitable for high cycle vibration fatigue analysis of train and can reflect the current commonly used concrete mechanical properties.

(2) The high-speed train ran in the cross tunnel structure and the middle area of the main tunnel in which the train runs were the most violent. The dynamic response of the arch bottom in the middle of the main tunnel opposite the transverse passage was the largest. The vibration response of the hance of the main tunnel near the transverse passage was larger than that of the opposite of the main tunnel hance.

(3) The stiffness singularity between the connecting transverse passage and the main tunnel caused large stress concentration phenomenon at the interface, and the stress and acceleration were relatively large at the interface. The maximum principal stress of the transverse passage mainly appeared near the side wall corner, while the maximum acceleration mainly appeared near the middle of the transverse passage floor.

(4) The maximum principal stress and acceleration of the connecting transverse passage increased with the increasing train operation years. The acceleration increased significantly in the middle of the transverse passage floor, both sides of side wall corner and vault, while the maximum principal stress developed obviously only in both sides of side wall corner.

(5) The cumulative damage of connecting transverse passage was mainly distributed in both sides of side wall corner. With the increasing operation years, the accumulative damage developed towards the side wall and floor of connecting transverse passage. The damage value and range of the left side wall corner were larger than the corresponding position of the right side wall.

(6) The cumulative compressive damage of connecting transverse passage was mainly distributed in the outer wall corner position, while cumulative tensile damage was mainly distributed in the inner wall corner, which was close to the linear development. The size and range of tensile damage of structure were larger than those of the compressive damage.

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

Financial support for this work provided by the National Key R&D Program of China (Grant no. 2016YFC0802205) and the National Science Foundation of China (Grants nos. 51178400, 51278425, and U1361210) is gratefully acknowledged.