# Experimental and Numerical Research on Defective Shield Segment under Cyclic Loading

Authors: Long-gang Tian; Zhi-qiang Hu; Jie Chen
Journal: Shock and Vibration
 (2021)

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## Abstract

The retaining structure of a shield tunnel is usually subjected to static loading; however, cyclic loading such as the vibration loading of the train will act on the lining structure year after year that cannot be neglected. Due to the complex manufacturing and assembling process of the lining segment, initial defects in the segment are inevitable. Such defects will propagate under the cyclic loading such as the train vibration, which may cause a large threat to the shield tunnel system. In this paper, a defective shield tunnel lining segment under cyclic loading is studied by both a full-scale experiment and three-dimensional numerical simulations to investigate its fracture properties and failure mechanism and make a rational estimate of its fatigue life. Results show that crack propagation of the defective shield tunnel segment can be identified as several different stages based on its deformation characteristics, and the failure pattern of the segment is determined by its initial defects. The results by the experiment and numerical investigation are in good agreement for each other.


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## Body

## 1. Introduction

Shield tunnel has been extensively adopted as the main form of the urban subway system during the past several decades. Usually, a shield tunnel is supported by the precast reinforced concrete lining segment to resist the substantial soil and water pressures surrounding the tunnel. These concrete segments are precast in factory and assembled in situ, which inevitably bring about initial defects such as small cracks in the lining segment. What is more, the surrounding pressures of shield tunnel lining such as the hydraulic pressure and temperature pressure change seasonally, and the vibration loading of the train acts on the lining structure day after day. These initial defects propagate stealthily under such cyclic loadings, which will weaken the safety level of the tunnel lining, and can even lead to total collapse for the tunnel in the worst situation.With the increasing demand for shield tunnels in recent years, lots of experimental and numerical studies have been conducted to investigate the mechanical behaviour and failure mechanism of shield tunnel segments. Chen and Mo [1] adopted the 3D finite element method to analyse crack problems of shield tunnel at both the construction stage and service stage and testified that crack and breakage can be effectively reduced by improving the anticrack performance of the segment. Cividini et al. [2] studied on the reasons of a deep tunnel by both the in situ investigation and nonlinear finite element calculation, showing that the damages of the tunnel are related to incomplete filling of the gap. Some full-scale tests have been conducted by Ding et al. [3] to study the mechanical behaviours of the shield tunnel segment joints and proposed a mechanical model of the segment joints. Yang et al. [4] used the dynamic finite element method to analyse the segment opening distribution subjected to seismic loading. A new longitudinal model for shield tunnel was proposed in [5], which could consider the shearing dislocation between the lining rings of the tunnel structure. Li et al. [6] employed both analytical and experimental methods to study joint opening of a shield tunnel under different bending moments and under the ultimate limit state. Several model tests were conducted to study the failure mechanism of shield segments in sandy ground in [7], showing that removal of the segment may result in a sudden bending moment growth of the adjacent segments. Kavvadas et al. [8] investigated and compared three shield lining models consisting of continuous shell model, shell model with aligned joints, and shell model with staggered joints. Yang et al. [9] investigated the fracture of shield segments through a case study of the shield tunnel in Shanghai. A cyclic four-point bending test was conducted on a reinforced concrete beam and a smart shield tunnel lining segment using the optical fibre sensor to investigate the bending behaviour of reinforced concrete structures [10]. Based on an unloading situation, the bearing capacity of the shield tunnel structure was studied by performing some full-scale experiments [11]. Tian et al. [12, 13] conducted a series of prototype load tests and three-dimensional numerical experiments to investigate the failure mechanisms of cracked shield segments. Yan et al. [14] used the extended finite element method (XFEM) to explore the fracture properties of an underwater shield segment, and the impacting load of high-speed train was considered in this study. Through data analyses of acoustic emission, Wang et al. [15, 16] conducted some experimental studies on the damage evolution characteristics of shield tunnel segment with cracks. A novel multiscale modelling method was proposed by Wang et al. [17] to analyse the random propagation of multiple cracks and its mechanical behaviours of shield segments.Performing fracture and fatigue analyses efficiently and precisely for three-dimensional structures with cracks is always a challenging subject in engineering. Plenty of effort has been made to obtain the advanced method for analysing three-dimensional crack problems, such as the embedded-singularity elements [18, 19], the singular quarter-point elements [20, 21], the XFEM [22–27], and the symmetric Galerkin boundary element method-finite element method (SGBEM-FEM) alternating method [28–31]. Recently, Tian et al. [32–35] have made a series of two-dimensional and three-dimensional fatigue analyses on several kinds of typical structures in mechanical and civil engineering and obtained some useful results for the practical engineering.In this paper, the defective shield segments under cyclic loading are studied by both a full-scale experiment and the three-dimensional numerical investigations. Three-dimensional numerical models of the defective lining segments are built using the SGBEM-FEM alternating method, and the stress intensity factors of the crack-front are captured by this method. The fatigue-crack propagation subjected to cyclic loading is depicted by the Paris exponential law. The results by the experiment and numerical simulation are compared and analysed, showing quite good agreement with each other.


## 2. The Full-Scale Experiment

### 2.1. Specimen Design and Equipment

The full-scale experiment of the defective shield tunnel segment is conducted with the real shield segment of Nanjing Metro Line 5. The outer diameter of the concrete lining ring is 6.2 m, and the thickness of the segment is 0.35 m. The length of each lining ring is 1.2 m. The grade of the concrete strength is C55, with an elastic modulus of 3.55 × 104 MPa, and Poisson’s ratio of 0.2. The initial defects are two cut-through cracks along the length of the lining segment on its inner surface, with the depth of about 0.05 m, which is shown in Figure 1.Figure 1
The defective shield tunnel segment.The loading device shown in Figure2 is used to carry out the experiment. This specimen is subjected to the four-point bending, as illustrated in Figure 3, in which P represents the vertical load and N represents the axial load. The amplitude of the cyclic loading is set to 70% of the bearing capacity for the shield segment, and the stress ratio is set to 0.1.Figure 2
Loading system of the full-scale experiment.Figure 3
Sketch of the defective shield segment (unit: mm).Only half of the shield segment is analysed due to the symmetrical property of the specimen. The moment in the middle of the specimen is illustrated in Figure4 and expressed as follows:(1)M=P+WL1−PL2−WL3−Nh1,where N denotes the axial force and M denotes the bending moment; W represents the self-weight of the specimen and P represents the prescribed vertical load.Figure 4
Half of the specimen is used for the force analysis.According to (1), the vertical load can be expressed as follows:(2)P=Ne+h1−WL1−L3L1−L2,where e=M/N, which is defined as the loading eccentricity. For this full-scale experiment, e is set to 0.24 m.In this experiment, the parameters to be measured include the load, displacement, surface strain of the concrete, and crack mouth opening displacement (CMOD). The load is directly captured with the controlling computer and the concrete strain is measured with resistance strain gauge. CMOD and displacement of the segment are captured by the displacement meter. Figures5–7 show the related devices used in this experiment.Figure 5
Measuring concrete strain using strain gauge.Figure 6
Measuring CMOD of the segment with displacement meter.Figure 7
Measuring vertical displacement of the segment with displacement meter.


### 2.2. Observed Phenomena of the Experiment

For this initially defective shield segment, the initial cracks start to grow as their stress intensity factors reach the concrete’s fracture toughness and show very typical I-type fracture characteristics. The cracking stages of the specimen during the entire loading process can be identified as several different phases based on its deformation characteristics, which include (a) the initial crack stage; (b) the crack developing stage; (c) the stable crack growth; (d) the unstable crack growth; and (e) the final collapse. Figure8 shows the captured pictures of the crack propagation and final failure mode for the specimen. The defective shield segment demonstrates some I-type fracture properties, and the collapse pattern of the segment is determined by its initial defects.Figure 8
The deformation and failure process of the defective specimen.
(a)(b)(c)(d)(e)Both the specimen (including the defects) and the load are symmetrical, so the deformation and crack propagation will show some symmetrical properties, theoretically. However, it is very hard, sometimes even impossible, to make the material homogeneous of the specimen in practical engineering. It is also difficulty to set the loads absolutely symmetrical for the specimen. Hence, the actual deformation characteristic and collapse pattern of the specimen demonstrate some unsymmetrical and unbalanced properties to a certain extent.


### 2.3. Experiment Results and Analysis

The double-K fracture criterion [36] is used to analyse fatigue and fracture characteristics of the specimen. As with the propagating process of the crack, the load which impels the initial crack to begin growing is defined as cracking load, and the load which the specimen can finally endure is defined as ultimate load. The variation of CMOD and the vertical displacement of the specimen is illustrated in Figure 9, and the load versus stain near the crack-front is demonstrated in Figure 10. In Figure 9, we can obtain the cracking load by searching for the first turning point on CMOD curve. For the curves in Figure 10, the cracking load can also be acquired by searching for these turning points. The results of cracking load for the specimen are listed in Table 1 and good agreement can be reached.Figure 9
CMOD and vertical displacement of the specimen.Figure 10
Load versus strain at the crack-front of the specimen.Table 1
The cracking loads of the shield segment by the experiment.
MethodThe cracking loadsAxial load (kN)Vertical load (kN)CMOD-time curve269.22143.60Load-strain curve281.25139.48Error (%)4.282.87


## 2.1. Specimen Design and Equipment

The full-scale experiment of the defective shield tunnel segment is conducted with the real shield segment of Nanjing Metro Line 5. The outer diameter of the concrete lining ring is 6.2 m, and the thickness of the segment is 0.35 m. The length of each lining ring is 1.2 m. The grade of the concrete strength is C55, with an elastic modulus of 3.55 × 104 MPa, and Poisson’s ratio of 0.2. The initial defects are two cut-through cracks along the length of the lining segment on its inner surface, with the depth of about 0.05 m, which is shown in Figure 1.Figure 1
The defective shield tunnel segment.The loading device shown in Figure2 is used to carry out the experiment. This specimen is subjected to the four-point bending, as illustrated in Figure 3, in which P represents the vertical load and N represents the axial load. The amplitude of the cyclic loading is set to 70% of the bearing capacity for the shield segment, and the stress ratio is set to 0.1.Figure 2
Loading system of the full-scale experiment.Figure 3
Sketch of the defective shield segment (unit: mm).Only half of the shield segment is analysed due to the symmetrical property of the specimen. The moment in the middle of the specimen is illustrated in Figure4 and expressed as follows:(1)M=P+WL1−PL2−WL3−Nh1,where N denotes the axial force and M denotes the bending moment; W represents the self-weight of the specimen and P represents the prescribed vertical load.Figure 4
Half of the specimen is used for the force analysis.According to (1), the vertical load can be expressed as follows:(2)P=Ne+h1−WL1−L3L1−L2,where e=M/N, which is defined as the loading eccentricity. For this full-scale experiment, e is set to 0.24 m.In this experiment, the parameters to be measured include the load, displacement, surface strain of the concrete, and crack mouth opening displacement (CMOD). The load is directly captured with the controlling computer and the concrete strain is measured with resistance strain gauge. CMOD and displacement of the segment are captured by the displacement meter. Figures5–7 show the related devices used in this experiment.Figure 5
Measuring concrete strain using strain gauge.Figure 6
Measuring CMOD of the segment with displacement meter.Figure 7
Measuring vertical displacement of the segment with displacement meter.


## 2.2. Observed Phenomena of the Experiment

For this initially defective shield segment, the initial cracks start to grow as their stress intensity factors reach the concrete’s fracture toughness and show very typical I-type fracture characteristics. The cracking stages of the specimen during the entire loading process can be identified as several different phases based on its deformation characteristics, which include (a) the initial crack stage; (b) the crack developing stage; (c) the stable crack growth; (d) the unstable crack growth; and (e) the final collapse. Figure8 shows the captured pictures of the crack propagation and final failure mode for the specimen. The defective shield segment demonstrates some I-type fracture properties, and the collapse pattern of the segment is determined by its initial defects.Figure 8
The deformation and failure process of the defective specimen.
(a)(b)(c)(d)(e)Both the specimen (including the defects) and the load are symmetrical, so the deformation and crack propagation will show some symmetrical properties, theoretically. However, it is very hard, sometimes even impossible, to make the material homogeneous of the specimen in practical engineering. It is also difficulty to set the loads absolutely symmetrical for the specimen. Hence, the actual deformation characteristic and collapse pattern of the specimen demonstrate some unsymmetrical and unbalanced properties to a certain extent.


## 2.3. Experiment Results and Analysis

The double-K fracture criterion [36] is used to analyse fatigue and fracture characteristics of the specimen. As with the propagating process of the crack, the load which impels the initial crack to begin growing is defined as cracking load, and the load which the specimen can finally endure is defined as ultimate load. The variation of CMOD and the vertical displacement of the specimen is illustrated in Figure 9, and the load versus stain near the crack-front is demonstrated in Figure 10. In Figure 9, we can obtain the cracking load by searching for the first turning point on CMOD curve. For the curves in Figure 10, the cracking load can also be acquired by searching for these turning points. The results of cracking load for the specimen are listed in Table 1 and good agreement can be reached.Figure 9
CMOD and vertical displacement of the specimen.Figure 10
Load versus strain at the crack-front of the specimen.Table 1
The cracking loads of the shield segment by the experiment.
MethodThe cracking loadsAxial load (kN)Vertical load (kN)CMOD-time curve269.22143.60Load-strain curve281.25139.48Error (%)4.282.87


## 3. The Numerical Investigation

### 3.1. Numerical Model

The numerical model for the defective shield segment is built according to the full-scale experiment as described in Section2. The global structure with no crack is modelled with the 10-noded tetrahedron element, and the initial crack surface is modelled with the 8-noded quadrilateral elements, independently. The numerical model is shown in Figure 11 and quite coarse meshes are used for the simulation.Figure 11
Numerical model for the defective shield segment.


### 3.2. Segment Bearing Capacity

The fracture toughness for concrete C55 isKIcini=0.53×106PamandKIcun=1.42×106Pam based on the double-K fracture criterion of concrete. Three-dimensional SGBEM-FEM alternating method is employed to capture the stress intensity factors at the crack-front.The efficient stress intensity factor for this problem is defined as follows:(3)Keff=KI2+KII2+KIII21−ν,where KI, KII, and KIII are the stress intensity factors for fracture mode I, II, and III, respectively.The initial cracking loadNini and Pini, and the ultimate load Nun and Pun can be obtained by constantly changing their values, until the computed stress intensity factors approach the fracture toughness of concrete C55. Computed cracking loads for this numerical model are(4)Nini=290kN,Pini=147.725kN,Keff=0.5304×106PamNmax=700kN,Pmax=370.296kN,Keff=1.436×106Pam.


### 3.3. Structure Reaction under Cyclic Load

The surrounding pressures of shield tunnel lining segment, such as the water pressure and temperature pressure, usually change seasonally. Moreover, the vibration loading of the train will act on the lining structure in the long term that cannot be neglected. These various kinds of load will cause the fatigue-related problems, which, on one hand, may initiate microcrack inside the concrete segment of the shield tunnel and propagate the existing cracks on the other hand. Therefore, it has significant meaning to investigate the fracture and fatigue mechanism of defective shield tunnel segment under cyclic loading.For the concrete shield tunnel segment with initial defects, its Paris fatigue law is expressed as follows [37]:(5)dadN=3.43×10−3ΔK−17.393a0/D+12.844,where da/dN denotes the crack growth rate, ∆K is the range of stress intensity factors, a0 denotes the initial crack depth, and D denotes the thickness of the specimen.The amplitude of the cyclic loading is 70% of the bearing capacity for the shield segment, and the stress ratio is 0.1. Numerical model mentioned above is studied under such cyclic loading. The crack growth versus load cycles is illustrated in Figure12. The stress intensity factor at the crack-front approaches the fracture toughness of concrete C55 after about 1.51 × 107 load cycles, thus the segment is broken. The total fatigue-crack increment is 0.067 m.Figure 12
The crack growth versus number of load cycles.The final form of the crack in shield segment in a two-dimensional view is illustrated in Figure13, and the three-dimensional fatigue-crack propagation is demonstrated in Figure 14, which approximatively demonstrates the I-type fracture of the defective shield tunnel segment. The initial defect does not strictly extend along its initial path; in fact, it slightly propagates to the middle of the segment.Figure 13
The final crack in the segment, a 2D view.Figure 14
Fatigue-crack propagation of the segment under cyclic loading, a 3D view.
(a)(b)(c)(d)(e)(f)


### 3.4. Results and Discussion

The cracking loads for the defective tunnel segment by numerical simulation in Section3.2 are compared with those by the experiment in Section 2.3, which are listed in Table 2; also, good agreement can be reached for each other.Table 2
Comparison of the cracking loads between numerical and experimental results.
MethodCracking loadsAxial loadNini (kN)Vertical loadPini (kN)By the load-strain curve281.25139.48By numerical simulation290.00147.73Error (%)3.115.91By the CMOD-time curve269.22143.60By numerical simulation290.00147.73Error (%)7.162.80


## 3.1. Numerical Model

The numerical model for the defective shield segment is built according to the full-scale experiment as described in Section2. The global structure with no crack is modelled with the 10-noded tetrahedron element, and the initial crack surface is modelled with the 8-noded quadrilateral elements, independently. The numerical model is shown in Figure 11 and quite coarse meshes are used for the simulation.Figure 11
Numerical model for the defective shield segment.


## 3.2. Segment Bearing Capacity

The fracture toughness for concrete C55 isKIcini=0.53×106PamandKIcun=1.42×106Pam based on the double-K fracture criterion of concrete. Three-dimensional SGBEM-FEM alternating method is employed to capture the stress intensity factors at the crack-front.The efficient stress intensity factor for this problem is defined as follows:(3)Keff=KI2+KII2+KIII21−ν,where KI, KII, and KIII are the stress intensity factors for fracture mode I, II, and III, respectively.The initial cracking loadNini and Pini, and the ultimate load Nun and Pun can be obtained by constantly changing their values, until the computed stress intensity factors approach the fracture toughness of concrete C55. Computed cracking loads for this numerical model are(4)Nini=290kN,Pini=147.725kN,Keff=0.5304×106PamNmax=700kN,Pmax=370.296kN,Keff=1.436×106Pam.


## 3.3. Structure Reaction under Cyclic Load

The surrounding pressures of shield tunnel lining segment, such as the water pressure and temperature pressure, usually change seasonally. Moreover, the vibration loading of the train will act on the lining structure in the long term that cannot be neglected. These various kinds of load will cause the fatigue-related problems, which, on one hand, may initiate microcrack inside the concrete segment of the shield tunnel and propagate the existing cracks on the other hand. Therefore, it has significant meaning to investigate the fracture and fatigue mechanism of defective shield tunnel segment under cyclic loading.For the concrete shield tunnel segment with initial defects, its Paris fatigue law is expressed as follows [37]:(5)dadN=3.43×10−3ΔK−17.393a0/D+12.844,where da/dN denotes the crack growth rate, ∆K is the range of stress intensity factors, a0 denotes the initial crack depth, and D denotes the thickness of the specimen.The amplitude of the cyclic loading is 70% of the bearing capacity for the shield segment, and the stress ratio is 0.1. Numerical model mentioned above is studied under such cyclic loading. The crack growth versus load cycles is illustrated in Figure12. The stress intensity factor at the crack-front approaches the fracture toughness of concrete C55 after about 1.51 × 107 load cycles, thus the segment is broken. The total fatigue-crack increment is 0.067 m.Figure 12
The crack growth versus number of load cycles.The final form of the crack in shield segment in a two-dimensional view is illustrated in Figure13, and the three-dimensional fatigue-crack propagation is demonstrated in Figure 14, which approximatively demonstrates the I-type fracture of the defective shield tunnel segment. The initial defect does not strictly extend along its initial path; in fact, it slightly propagates to the middle of the segment.Figure 13
The final crack in the segment, a 2D view.Figure 14
Fatigue-crack propagation of the segment under cyclic loading, a 3D view.
(a)(b)(c)(d)(e)(f)


## 3.4. Results and Discussion

The cracking loads for the defective tunnel segment by numerical simulation in Section3.2 are compared with those by the experiment in Section 2.3, which are listed in Table 2; also, good agreement can be reached for each other.Table 2
Comparison of the cracking loads between numerical and experimental results.
MethodCracking loadsAxial loadNini (kN)Vertical loadPini (kN)By the load-strain curve281.25139.48By numerical simulation290.00147.73Error (%)3.115.91By the CMOD-time curve269.22143.60By numerical simulation290.00147.73Error (%)7.162.80


## 4. Conclusions

In this paper, a full-scale experiment of the defective shield tunnel segment is conducted, and its corresponding three-dimensional numerical model is built by using the three-dimensional SGBEM-FEM alternating method. The fracture properties and bearing capacity of the initially defective shield segment under cyclic loading are studied by both the experiment and the numerical simulation. The following conclusions can be drawn:(1)
Crack propagation of the defective shield segment can be identified as several different stages based on its deformation characteristics, which include the initial crack stage, the crack developing stage, the stable crack growth, the unstable crack growth, and the final collapse.(2)
The complete crack propagation up to collapse of the shield segment can be conveniently and efficiently simulated by the SGBEM-FEM alternating method, which is similar with that by the experiment.(3)
The shield segment with initial defects approximatively demonstrates the I-type fracture properties under the cyclic loading, and the final failure mode of the segment is mainly determined by its initial defects.(4)
The cracking loads for the defective shield segment are acquired by both the experiment and numerical investigation, and good agreement can be reached for each other.Some three-dimensional effects are neglected in this study, such as the three-dimensional corner singularity, existence of a critical angle at which crack-front intersects the free surface, and coupling of mode I, II, and III, since they have insignificant effects for fracture and fatigue analysis of the cracked structures. Also, crack closure induced by plasticity can be helpful to consider the influence of thickness of the structure on the crack propagation rates. Very simple Paris fatigue law is employed to predict the fatigue propagation rate in this study; however, other complex models which consider the effect of structure thickness can also be incorporated in the current framework of SGBEM-FEM alternating method, which will be our study in the future.



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*Source: 8567899-2021-07-05.xml*