Experimental and Numerical Research on Defective Shield Segment under Cyclic Loading

(e 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. (e results by the experiment and numerical investigation are in good agreement for each other.


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. ese 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. ese 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 fullscale 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. rough 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 embeddedsingularity elements [18,19], the singular quarter-point elements [20,21], the XFEM [22][23][24][25][26][27], and the symmetric Galerkin boundary element method-finite element method (SGBEM-FEM) alternating method [28][29][30][31]. Recently, Tian et al. [32][33][34][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. ree-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. e fatigue-crack propagation subjected to cyclic loading is depicted by the Paris exponential law. e results by the experiment and numerical simulation are compared and analysed, showing quite good agreement with each other.

Specimen Design and Equipment.
e full-scale experiment of the defective shield tunnel segment is conducted with the real shield segment of Nanjing Metro Line 5. e outer diameter of the concrete lining ring is 6.2 m, and the thickness of the segment is 0.35 m. e length of each lining ring is 1.2 m. e grade of the concrete strength is C55, with an elastic modulus of 3.55 × 10 4 MPa, and Poisson's ratio of 0.2. e 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.
e loading device shown in Figure 2 is used to carry out the experiment. is 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. e 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.
Only half of the shield segment is analysed due to the symmetrical property of the specimen. e moment in the middle of the specimen is illustrated in Figure 4 and expressed as follows: 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. According to (1), the vertical load can be expressed as follows: 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). e 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. Figures 5-7 show the related devices used in this experiment.

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. e 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. Figure 8 shows the captured pictures of the crack propagation and final failure mode for the specimen. e defective shield segment demonstrates some I-type fracture properties, and the collapse pattern of the segment is determined by its initial defects.
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.

Experiment Results and Analysis.
e 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. e 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. e results of cracking load for the specimen are listed in Table 1 and good agreement can be reached.

Numerical
Model. e numerical model for the defective shield segment is built according to the full-scale experiment as described in Section 2. e 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. e numerical model is shown in Figure 11 and quite coarse meshes are used for the simulation. ree-dimensional SGBEM-FEM alternating method is employed to capture the stress intensity factors at the crack-front. e efficient stress intensity factor for this problem is defined as follows:

Segment
where K I , K II, and K III are the stress intensity factors for fracture mode I, II, and III, respectively. e initial cracking load N ini and P ini , and the ultimate load N un and P un 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

Structure Reaction under Cyclic Load.
e 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. ese various kinds of load will cause the fatiguerelated 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. erefore, 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]: where da/dN denotes the crack growth rate, ∆K is the range of stress intensity factors, a 0 denotes the initial crack depth, and D denotes the thickness of the specimen. e 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. e crack growth versus load cycles is  Shock and Vibration illustrated in Figure 12.
e stress intensity factor at the crack-front approaches the fracture toughness of concrete C55 after about 1.51 × 10 7 load cycles, thus the segment is broken. e total fatigue-crack increment is 0.067 m. e final form of the crack in shield segment in a twodimensional view is illustrated in Figure 13, and the threedimensional fatigue-crack propagation is demonstrated in Figure 14, which approximatively demonstrates the I-type fracture of the defective shield tunnel segment. e initial defect does not strictly extend along its initial path; in fact, it slightly propagates to the middle of the segment.

Results and Discussion.
e cracking loads for the defective tunnel segment by numerical simulation in Section 3.2 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.

Conclusions
In this paper, a full-scale experiment of the defective shield tunnel segment is conducted, and its corresponding threedimensional numerical model is built by using the threedimensional SGBEM-FEM alternating method. e fracture properties and bearing capacity of the initially defective shield segment under cyclic loading are studied by both the experiment and the numerical simulation. e 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) e 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) e 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) e 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,  Shock and Vibration 7 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.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare no conflicts of interest.