Based on the existing numerical models of shield tunnel with double lining, an improved numerical model is developed and its rationality is verified by a similar model test. In the improved numerical model, lining, joint, and junction surface, respectively, are simulated by beam, spring, and a combination of compression bar and spring. Through the comparison of the numerical analysis results of the improved model and existing models, it turns out that the defects or problems in the existing numerical models are resolved; tension appearance on the contact surface and junction surface and the abrupt change of bending moment in linings are solved in the improved model because the compression bar element and the coupling technology of node displacement in the junction surface is applied. Therefore, the improved numerical model could be applied to analyze double lining with waterproof on the junction surface and separation of the junction surface under an unfavorable load. In this paper, the parameter formulas of element stiffness, applicable to the junction surface and contact surface of double lining, are given definitely, and the influence of the element amount of junction surface on the analysis results is discussed. Based on the improved numerical model, the mechanical behavior of the double lining of the Huangpu River Tunnel in China is studied, and some conclusions are obtained as follows. (1) The thickness increase of the double lining will clearly increase its bending moment, but it has little influence on its axial force. (2) The allocation proportion of the bending moment between the segment lining and secondary lining has no linear relationship with the ratio of the lining thickness.
Double lining is a kind of tunnel structure composed of an outer segment lining and an inner cast-in-place concrete secondary lining formed after tunnel boring machine installation of a segment lining, as shown in Figure
Sketch of shield tunnel with double lining.
The researches on shield tunnel mainly focus on single lining tunnel. Do et al. [
Some researchers also studied the behaviour of double lining structures. Murakami and Koizumi [
The existing double lining models have some defects: no analytic model for the above-mentioned design is so scientific that it is widely recognized and adopted. ITA’s (International Tunneling Association) Guidelines [
Present numerical models of shield tunnel with double linings.
The difference between these three models lies in the junction surface between the segment lining and the secondary lining (hereinafter referred to as the junction surface). In Model I, a radial beam element is used to simulate the transmission of the stress between the segment lining and secondary lining. Two nodes of the beam element are fixed in the beam element of the segment lining and secondary lining, respectively, which will lead to an abrupt change in the bending moment at the fixing point of the segment lining or secondary lining. Meanwhile, radial tension that does not conform to the fact will appear in this radial beam. In Model II, the radial spring was adopted in the junction surface, which could not avoid the appearance of radial tension in the junction surface as well. In Model III, a contact friction element was employed to simulate the force transfer on the junction surface. Its feature is that the tangential force value on junction surface is in direct proportion to the radial force and friction coefficient, which could not reflect the relationship between the tangential force and tangential relative displacement. Moreover, as there is no experimental data, the friction coefficient value of the junction surface is difficult to determine.
In conclusion, the defects of the three models mentioned above are mainly as follows: the appearance of radial tension on the junction surface and the contact surface, an abrupt change in the bending moment, discordance of the radial force, and radial displacement of junction surface, in addition to evidence deficiency of friction coefficient determination. These defects would greatly limit the application of the models mentioned above in specific engineering. Aiming at these aforementioned defects, some improvements of the numerical model are conducted. A model of lining, joint, and junction surface, respectively, simulated by beam, spring, and combination of compression bar and spring is put forward to solve problems. Furthermore, the number of junction surface elements and their stiffness are discussed. Secondly, model tests are conducted to verify the improved model, and the comparison between the improved model and existing numerical models reveals the advantages of the improved model. Finally, the mechanical behavior of the Qiantang River Shield Tunnel with double lining in the Hangzhou-Changsha Railway in China is studied in terms of the improved model.
To overcome the defects of the aforementioned models, an improved model, whose lining, joint, and junction surface are, respectively, simulated by beam, spring, and a combination of compression bar and spring, is put forward, as shown in Figure
Comparison of elements between improved model and existing models.
Items | Segment lining and secondary lining | Segment lining joint | Junction surface | Contact surface |
---|---|---|---|---|
Model I | Beam element | Spring elements with functions of tension and compression, shear, and bending resistance | Fixed radial beam element | Radial and tangential spring elements with functions of tension and compression resistance |
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Model II | Beam element | Spring elements with functions of tension and compression, shear, and bending resistance | Radial and tangential spring elements with function of tension and compression resistance | Radial and tangential spring elements with functions of tension and compression resistance |
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Model III | Beam element | Spring elements with functions of tension and compression, shear, and bending resistance | Friction elements | Radial and tangential spring elements with functions of tension and compression resistance |
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Improved model | Beam element | Spring elements of node displacement coupling with functions of tension and compression, shear, and bending resistance | Radial compression bar with functions of compression resistance and tangential spring with coupling functions of elements node displacement | Radial compression bar with functions of compression resistance and tangential spring element |
Improved model.
Junction surface and contact surface
Segmental joint
The problem of the parameter value to choose relates to the lack of theory evidence in Model III. The improved model learns from the existing research results [
(1) Stiffness of junction surface: after the segment lining is assembled, the secondary lining is manufactured by cast-in-place concrete directly. Double lining can be taken as a superimposition lining structure, and its radial compression can be simplified as the mechanical model shown as in Figure
Compression stiffness calculation schematic diagram of junction surface.
Cross section
Combine Formulas (
The junction surface tangential relationship is similar to the radial relationship, and the tangential stiffness of the double lining is as follows:
(2) Stiffness of contact surface: the calculating model of the contact surface is shown as in Figure
Schematic diagram of calculation model of contact face between segment and strata.
Mechanical model of contact surface
Calculation area
According to the experience of Japan, the contact surface tangential stiffness is approximately half the radial stiffness [
To verify the rationality of the improved model and compare the advantages and defects of the existing numerical models, the shield tunnel-ground complex simulation equipment is employed to conduct the model experiment on the Shiziyang Tunnel in China with double lining. Without water pressure, the test results with the numerical analysis results of the aforementioned Model I and Model II in the same load and same structure parameters are compared.
The double lining section of the Shiziyang Tunnel in Guangzhou-Shenzhen railway, which is buried at the depth of 35.35 m, is taken as the prototype structure. The external and internal radiuses of the tunnel structure are 10.8 m and 9.4 m, and the thicknesses of the segment lining and the secondary lining are 0.5 m and 0.2 m. The width of the single segment lining in the longitudinal direction is 2.0 m. The segment lining is assembled with a staggered joint, and “
Double lining of Shiziyang Tunnel.
The tunnel-ground complex simulation equipment is a plane strain loading device of a shield tunnel, which mainly includes three parts: a hydraulic loading unit, stratum and structural body, and signal collection transaction and analysis apparatus. The whole test platform’s length, height, and width are 6.0 m × 6.0 m × 2.55 m, and the tunnel model and model soil around it are put inside the device. The available filling range of the model soil is 3.0 m × 3.0 m × 0.2 m, as shown in Figure
Sketch of shield tunnel-ground complex simulation equipment (unit: mm).
Plan
Elevation
Tunnel-ground complex simulation equipment.
The test takes the 1 : 20 geometric similarity ratio and 1 : 1 bulk density similarity ratio as the basic similarity ratios. According to the elastic similarity theory, Poisson’s ratio and strain and the internal friction angle similarity ratio can be deduced as
The model stratum employs a mixture made of barite powder, fly ash, river sand, coarse quartz, fine quartz, rosin, and oil with the specific ratio, and the mass ratio is 1 : 0.341 : 0.676 : 0.334 : 0.334 : 0.08 : 0.14. The mechanical properties of the argillaceous siltstone stratum prototype and its model material can be seen in Table
Parameters of argillaceous siltstone strata in model and prototype.
Name | Cohesion |
Internal friction angle |
Model elastic quantity |
Bulk density |
---|---|---|---|---|
The prototype stratum | 22.0 | 23.5° | 28 | 22.3 |
The model stratum | 1.1 | 23.5° | 1.4 | 22.3 |
Similarity ratio | 1 : 20 | 1 : 1 | 1 : 20 | 1 : 1 |
The material of the segment lining is C50 reinforced concrete, and the concrete model adopts the composite material made of water, gypsum, and diatomite with the ratio of 1 : 1.40 : 0.1, and the ring direction main reinforcement employs the fine iron with the diameter of
Depths of slits in test models.
Bending stiffness/(N·m·rad−1) | Equivalent slit depth in prototype/mm | Equivalent slit depth in prototype of model/mm |
---|---|---|
2.0 × 107 (positive bending) | 240 | 12 (tunnel crown/tunnel bottom) |
2.6 × 107 (negative bending) | 200 | 10 (left/right side lumbers) |
Test model of double lining.
Segment lining
Segment lining and secondary lining
The electric resistance strain gauges are laid out in a transverse direction inside and outside the segment lining and secondary lining, and the test employs the electric resistance strain indicator to test the values of the gauges. The structural internal force of the segment lining and secondary lining could be calculated based on the values of gauges.
Based on the calculating results of the section, the test takes the vertical soil pressure of the section (the loading direction
As the friction coefficient value of the junction surface of Model III is difficult to determine, it will only compare the result of test with the results of Model I, Model II, and the improved model here. The segment lining joint stiffness of Model I, Model II, and the improved model is in accord with that of test prototype. The measured internal force of the model test and the calculating internal force of the numerical models are shown in Figure
Comparison on extreme internal forces between numerical and test results.
Subject | Model I | Mode II | The improved model | Similarity model test | ||||
---|---|---|---|---|---|---|---|---|
Maximum value | Minimum value | Maximum value | Minimum value | Maximum value | Minimum value | Maximum value | Minimum value | |
Bending moment (kN·m) | ||||||||
Segment lining | 585.65 | −639.55 | 603.44 | −636.72 | 627.35 | −641.52 | 663.74 | −635.67 |
Secondary lining | 163.47 | −144.32 | 195.42 | −192.54 | 178.48 | −162.88 | 188.54 | −148.87 |
Axial force (MN) | ||||||||
Segment lining | −10.98 | −4.65 | −12.03 | −5.97 | −11.67 | −4.75 | −10.28 | −5.24 |
Secondary lining | −2.89 | −1.45 | −2.32 | −0.92 | −2.78 | −1.22 | −2.44 | −1.07 |
Diagrams for tested values and calculated values of internal forces in lining.
Bending moment of segmental lining (unit: kN·m)
Bending moment of secondary lining (unit: kN·m)
Axial force of segmental lining (unit: MN)
Axial force of secondary lining (unit: MN)
From Figure
According to Figure
When there is a relatively large difference between the vertical and horizontal loads, the deformation of segment lining and secondary lining will no longer be synchronous in some parts, where the junction surface will separate inevitably. And laying the waterproof layer between the segment lining and the secondary lining, the junction surface could not transfer tension. Under these conditions, the values from Model I and Model II are removed from reality. At this time, using the improved model will avoid this phenomenon effectively and acquire the internal force distribution much closer to the model test. Therefore, the improved model should be selected as the numerical analysis model of the double lining because of its greater rationality.
To know more about the mechanical behavior of double lining, taking the Huangpu River Tunnel in the Shanghai-Nantong Railway in China as the example here, the improved numerical model is adopted to simulate and analyse. The total length of the Huangpu River Tunnel is 8.3 km, and the internal and external diameters are 8.74 m and 10.3 m. The segment lining employs a reinforced concrete structure of the “
The double lining structure schematic is shown as in Figure
Schematic diagram of double linings and its load case.
Sketch of shield tunnel with double lining
Load condition
The thickness of the segment lining and the secondary lining in the models is calculated according to its actual thickness. The amount of the junction surface compression bar, the amount of the junction surface tangential spring, the stratum compression bar amount, and the stratum tangential spring amount all are 400. Putting the parameters in Table
Material parameters of double linings.
Subject | Material | Elastic modulus (kPa) | Density (kg/m3) | Poisson’s ratio |
---|---|---|---|---|
Segment lining | C50 concrete | 3.45 × 107 | 2450 | 0.2 |
Secondary lining | C30 concrete | 3.00 × 107 | 2450 | 0.2 |
According to the relevant research results [
Theoretically, the radial compression bar and tangential spring amount of the junction surface (i.e., the element amount of the junction surface, the same below) influences the stiffness of the single radial compression bar and tangential spring. Obviously, the larger the amount, the smaller the area corresponding to the single radial compression bar and tangential spring, and the corresponding stiffness is smaller. Different junction surface quantities must be set in order to optimize and analyse, so as to determine the reasonable amount of tangential spring and radial compression bar. Setting the radial compression bar and tangential spring amount between the segment lining and the secondary lining as 20, 40, 100, 200, 400, 1000, and 2000, respectively, and also comparing the maximum deformation of the segment lining and the secondary lining, the extreme value of the bending moment (the bending moment is positive when traction is in the outside surface, the opposite is negative, the maximum is the positive peak, and the minimum is the negative peak, the same below) and the extreme value of the axial force (the maximum value of the axial force is the maximum pressure value, and the minimum value is the minimum pressure value, the same below) can be noted. The calculated results can be seen in Figure
The influence of element number at junction surface on inner forces of double lining.
The max deformation
The extreme bending moment
The extreme thrust force
Figure
As far as the double lining of the Huangpu River Shield Tunnel is concerned, comprehensively considering all the factors including the maximum deformation value, the bending moment extreme value, and axial force extreme value of the segment lining and the double lining, a conclusion can be drawn that it is appropriate to take 400 as the junction surface amount. The discussion below will be based on the condition of 400 radial compression bars and 400 tangential springs.
Based on the radial compression bar and tangential spring amount of the junction surface above, and with a focus on the Huangpu River Tunnel, the mechanical behavior analysis of the different segment lining thickness and different double lining thickness within the same external load is carried out.
(1) Different thickness of segment lining: the thickness of the secondary lining is 0.30 m, and the thicknesses of the segment lining are 0.30 m, 0.40 m, 0.50 m, 0.60 m, and 0.70 m, respectively, and the thicknesses of the corresponding double lining are 0.60 m, 0.70 m, 0.80 m, 0.90 m, and 1.00 m, respectively. The bending moment extreme values of the double lining (the sum of the bending moment of the segment lining and the secondary lining in the same section within a single ring breadth with 2 m, the same below) and the corresponding bending moment of the segment lining can be seen in Figure
The bending moments extreme values under different thicknesses of segment lining.
Thickness of segment lining (cm) | 30 | 40 | 50 | 60 | 70 |
Thickness of double lining (cm) | 60 | 70 | 80 | 90 | 100 |
The increase ratio of double lining thickness | 100% | 117% | 133% | 150% | 167% |
The section of maximum bending moment of the double lining | |||||
Bending moment of double lining (kN·m) | 131.047 | 142.037 | 148.896 | 151.945 | 152.454 |
Bending moment of segment lining at the corresponding section (kN·m) | 63.178 | 97.604 | 120.650 | 133.701 | 140.300 |
The allocation proportion of bending moment of segment lining | 48% | 69% | 81% | 88% | 92% |
The increase ratio of bending moment of double lining | 100% | 108% | 114% | 116% | 116% |
The section of minimum bending moment of the double lining | |||||
Bending moment of double lining (kN·m) | −158.954 | −171.705 | −179.542 | −182.704 | −183.285 |
Bending moment of segment lining at the corresponding section (kN·m) | −77.292 | −118.510 | −145.771 | −161.090 | −168.780 |
The allocation proportion of bending moment of segment lining | 49% | 69% | 81% | 88% | 92% |
The increase ratio of bending moment of double lining | 100% | 108% | 113% | 115% | 115% |
The axial force extreme values under different thicknesses of segment lining.
Thickness of segment lining (cm) | 30 | 40 | 50 | 60 | 70 |
Thickness of double lining (cm) | 60 | 70 | 80 | 90 | 100 |
The increase ratio of double lining thickness | 100% | 117% | 133% | 150% | 167% |
The section of maximum axial force of the double lining | |||||
Axial force of double lining (MN) | −1.144 | −1.142 | −1.135 | −1.124 | −1.120 |
Axial force of segment lining at the corresponding section (MN) | −0.744 | −0.789 | −0.819 | −0.845 | −0.875 |
The allocation proportion of axial force of segment lining | 65% | 69% | 72% | 75% | 78% |
The increase ratio of axial force of double lining | 100% | 99% | 99% | 98% | 98% |
The section of minimum axial force of the double lining | |||||
Axial force of double lining (MN) | −1.030 | −1.031 | −1.022 | −1.021 | −1.012 |
Axial force of segment lining at the corresponding section (MN) | −0.669 | −0.715 | −0.744 | −0.778 | −0.797 |
The allocation proportion of axial force of segment lining | 65% | 69% | 73% | 76% | 79% |
The increase ratio of axial force of double lining | 100% | 99% | 99% | 99% | 98% |
Numerical results of double lining under different thicknesses of segment lining.
The bending moment
The axial force
Figure
From Tables
(2) Different thickness of secondary lining: the thickness of segment lining is 0.50 m, and the thicknesses of the secondary lining are 0.10 m, 0.20 m, 0.30 m, 0.40 m, and 0.50 m, respectively, and the thicknesses of the corresponding double lining are 0.60 m, 0.70 m, 0.80 m, 0.90 m, and 1.00 m, respectively. Through computing, the maximum and minimum bending moment of double lining and the bending moment of the segment lining at the corresponding section can be seen in Figure
The bending moments extreme values under different thicknesses of secondary lining.
Thickness of secondary lining (cm) | 10 | 20 | 30 | 40 | 50 |
Thickness of double lining (cm) | 60 | 70 | 80 | 90 | 100 |
The increase ratio of double lining thickness | 100% | 117% | 133% | 150% | 167% |
The section of maximum bending moment of the double lining | |||||
Bending moment of double lining (kN·m) | 146.689 | 147.398 | 148.896 | 150.660 | 151.945 |
Bending moment of segment lining at the corresponding section (kN·m) | 145.480 | 138.001 | 120.650 | 94.474 | 72.271 |
The allocation proportion of bending moment of segment lining | 99% | 94% | 81% | 62% | 48% |
The increase ratio of bending moment of double lining | 100% | 100% | 102% | 103% | 104% |
The section of minimum bending moment of the double lining | |||||
Bending moment of double lining (kN·m) | −177.101 | −177.886 | −179.542 | −181.469 | −182.814 |
Bending moment of segment lining at the corresponding section (kN·m) | −175.650 | −166.640 | −145.771 | −116.662 | −87.462 |
The allocation proportion of bending moment of segment lining | 99% | 94% | 81% | 64% | 48% |
The increase ratio of bending moment of double lining | 100% | 100% | 101% | 102% | 103% |
The axial force extreme values under different thicknesses of secondary lining.
Thickness of secondary lining (cm) | 10 | 20 | 30 | 40 | 50 |
Thickness of double lining (cm) | 60 | 70 | 80 | 90 | 100 |
The increase ratio of double lining thickness | 100% | 117% | 133% | 150% | 167% |
The section of maximum axial force of the double lining | |||||
Axial force of double lining (MN) | −1.121 | −1.121 | −1.121 | −1.120 | −1.120 |
Axial force of segment lining at the corresponding section (MN) | −1.009 | −0.897 | −0.818 | −0.762 | −0.728 |
The allocation proportion of axial force of segment lining | 90% | 80% | 73% | 68% | 65% |
The increase ratio of axial force of double lining | 100% | 100% | 100% | 100% | 100% |
The section of minimum axial force of the double lining | |||||
Axial force of double lining (MN) | −1.058 | −1.057 | −1.057 | −1.056 | −1.056 |
Axial force of segment lining at the corresponding section (MN) | −0.963 | −0.856 | −0.772 | −0.718 | −0.686 |
The allocation proportion of axial force of segment lining | 91% | 81% | 73% | 68% | 65% |
The increase ratio of axial force of double lining | 100% | 100% | 100% | 100% | 100% |
Numerical results of double lining under different thicknesses of secondary lining.
The bending moment
The axial force
From Figure
From Tables
Double lining of an underwater shield tunnel has wide application in the future. Based on the comparative analysis of the existing numerical models of double lining, a new improved numerical model has been presented in this paper. The lining, joint, and junction surface are simulated by beam, spring, and a combination of compression bar and spring. On the basis of employing a similar model test to contrast the existing and the improved models, the advantages of the improved model are compared to the existing model. Eventually, adopting the improved numerical model, the reasonable element amounts of junction surface needed for implementing the accurate analysis of the double lining of the Huangpu River Tunnel are discussed and the mechanic behaviors are studied when changing the thickness of the segment lining and the secondary lining, respectively, at the same load. The main conclusions are listed as follows: The existing double lining models have the defects of an abrupt change of bending moment, the appearance of radial tensile stress on the junction surface and contact surface, and so forth, and they are inadequate for the task of setting the waterproof layer and the separation of the junction surface. Through the radial compression bar element, the improved numerical model simulates the radial interaction of the junction surface or the contact surface, and the link between the bar element and beam element is hinged. Therefore, it can solve the problems of bending moment abrupt change in the junction surface and contact surface and the appearance of radial tensile stress. Through the coupling of node displacement, it demonstrates that the shear force of the junction surface is directly relevant to the shear displacement. Meanwhile, there are definite calculating methods of the element stiffness parameters. Results of the improved numerical model have better consistency with results from the similar model test, and the element amount of the junction surface and contact surface should be discussed in a specific project, which is the precondition of using the improved model to carry out the double lining mechanical analysis. With other conditions unchanged, the increase in thickness of the segment lining or the secondary lining will cause the increase of the bending moment of double lining, but this has little influence on the axial force of the double lining. The allocation proportion of the bending moment also has no linear relationship with the ratio of the thickness between the segment lining and the secondary lining. With the condition that the thickness of the secondary lining remains unchanged, increasing the thickness of the segment lining, the bending moment and its proportion ratio of segment lining increases more at the corresponding section of the maximum and minimum bending moment of the double lining. It shows that the increased part of the bending moment of the double lining mainly depends on the thickened segment lining. And the axial force of segment lining and its proportion have only a small increase. With the condition that the thickness of the segment lining is unchanged, increasing the thickness of the secondary lining, the bending moment and its proportion ratio of segment lining decreases sharply at the corresponding section of the maximum and minimum bending moment of the double lining. It shows that the increased part of the bending moment of the double lining mainly transfers to the thickened secondary lining. The axial force of the segment lining and its proportion have a small decrease.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This study was supported by the National Natural Science Foundation of China (nos. 51178400, 51278425, and 51408511) and Program for New Century Excellent Talents in University (NCET-11-0713).