Probabilistic dislocation capacity model for mountain tunnel crossing strike–slip creeping fault

Fault dislocations significantly affect the performances of cross–fault tunnels. However, there is a lack of rapid and accurate methods for estimating the maximum fault dislocation that tunnels can withstand (i.e., the dislocation capacity of tunnels). This study systematically investigates the dislocation capacity of circular tunnels crossing strike–slip creeping faults while considering uncertainties in the lining parameters, rock strength, and crossing angles. Through pseudostatic numerical simulations, the dislocation capacities of tunnels under various working conditions were determined, and a probabilistic dislocation capacity model was developed using a support vector machine. The proposed model can estimate the probability distribution of the dislocation capacity of a tunnel, with its median closely aligned with the numerical result. Another advantage lies in its direct correlation with the lining parameters, rock strength, and crossing angles. This facilitates a rapid assessment of the tunnel dislocation capacity based on existing parameters. The dislocation capacity and vulnerability curves of a case tunnel are estimated using our probabilistic model, which serves as a demonstration. The results demonstrate that the proposed model is convenient for engineering applications and can assist in the design and optimization of cross–fault tunnels.


Introduction
The permanent deformation of strata caused by fault dislocation leads to cracking, misalignment, and even the collapse of the tunnel lining, which seriously threatens the safety of cross-fault tunnel structures [1][2][3].As a rule of thumb, the tunnel route should avoid active fault areas.However, owing to the complexity of the terrain, the uncertainty of fault location, and the impact of construction costs, tunnels inevitably cross fault.Therefore, the potential impact of fault dislocations on tunnel structural safety should be considered before the tunnel is constructed near active fault areas.
Experimental [4,5] and numerical [6][7][8][9][10] studies have been conducted to investigate the damage characteristics of a tunnel under fault dislocations.Several prevention and control measures have been proposed to mitigate the risk of damage to cross-fault tunnels [10,11].A recent literature review can be found in [11].These studies revealed the failure mechanisms of cross-fault tunnels and identified various factors that directly affect their performance.However, there is currently a lack of probabilistic models to quickly estimate the maximum fault dislocation that a tunnel can withstand, which is required for fragility analysis and optimal design of cross-fault tunnels.
This study aimed to establish a probabilistic dislocation capability model for circular mountain tunnels across strike-slip creeping faults.Uncertainties in the lining parameters, rock strength, and intersection angle between the tunnel and the fault were considered.Parameter-capacity sample pairs are obtained through quasi-static numerical simulations, and a probabilistic model is developed using a IOP Publishing doi:10.1088/1755-1315/1333/1/012004 2 support vector machine.Finally, a case study is conducted to demonstrate the accuracy and convenience of the proposed probabilistic model.The results demonstrate that the proposed probabilistic model can quickly estimate the probability distribution of the tunnel dislocation capacity, whose mean value is close to the result of the numerical simulation.This can facilitate the fragility analysis and optimal design of cross-fault tunnels.

Uncertainties of the design parameters considered in this study
Strike-slip faults typically exhibit relatively steep dip angles.Therefore, in this study, the dip angles of strike-slip faults were uniformly assumed to be 90°.To enhance the applicability of the capacity model, the uncertainties associated with various design parameters were considered, including the intersection angle between the tunnel and fault, lining dimensions, rock mass strength, lining concrete grade, and reinforcement strength and rate.The number of levels and value ranges of the design parameters are listed in Table 1.
Table 1.Level numbers and value range for the considered design parameters.The mechanical behavior of the tunnel is affected by the angle between the tunnel and fault.Figure 1 illustrates the distinct forces formed in the tunnels at different angles.The angle between the tunnel axis and the direction of fault displacement is noted as θ.It can be observed that when θ is less than 90°, fault dislocation will stretch the tunnel, resulting primarily in tensile-shear failure, whereas when θ is greater than 90°, fault dislocation will compress the tunnel, leading to predominantly compressive-shear failure.The angle θ is considered to range from 10° to 170° with an interval of 10° to investigate how it affects the capacities of tunnels in resisting fault dislocation． dislocation compresses the tunnel.For the uncertainty of the tunnel dimensions, three different lining diameters were chosen, and three lining thicknesses were considered for each.Consequently, nine combinations of lining sizes were obtained ( Table 1).
Geological survey reports provide annual fault dislocations and maximum fault dislocations at the ground surface for tunnel design.However, there remains a lack of information regarding the attenuation relationship between bedrock and surface dislocations.When the rock mass is sufficiently hard, fault dislocations at different depths are essentially the same (an extreme case is when the rock mass is a rigid body).Therefore, by incorporating the estimated fault dislocation, the developed capacity model can directly assess the safety of tunnels in hard rock.In this study, the elastic modulus of the rock mass 3 ranges from 20217 to 39000 MPa, which corresponds to the Class Ⅰ and Class Ⅱ rock mass defined in the Chinese Code [12].The density ρs, Poisson's ratio νs, internal friction angle φs, dilatancy angle ψs, and cohesion force cs all use fixed values, that is, ρs = 2650 kg/m 3 , νs = 0.2, φs = 60°, ψs =8°, cs =2.1 MPa, corresponding to the critical values of Class Ⅰ and Class Ⅱ rock mass, respectively.Nine lining concrete grades were considered, from C30 to C70.The concrete density ρc = 2460 kg/m 3 , Poisson's ratio νc = 0.2, and the strength parameters corresponding to different grades of concrete are listed in Table 2. Similarly, nine levels of reinforcement ratios and yield strengths of the reinforced steel bars were considered, as listed in Table 1.The density of the steel bar was 7800 kg/m 3 , and the elasticity modulus was 20 GPa.
A mixed uniform design was employed based on the influencing factors and level numbers provided in Table 1 to generate representative parameter combinations using data processing system (DPS) software.Subsequently, a quasistatic numerical simulation was conducted for each set of parameter combinations to determine the capacity of the tunnel to resist fault dislocations.This process resulted in a total of 17 × 9 = 153 pairs of parameter-capacity samples for developing the capacity model.

Details of the numerical model used in this paper
To evaluate the capacity of the tunnel to resist fault dislocation, a series of quasi-static analyses of the rock-tunnel-fault system was performed using the finite element software ABAQUS [13].Figure 2 illustrates a representative 3D quasi-static numerical model used in this study.The length of the model was 200 m.The width and height of the model were set to 100 m.The rock mass consisted of a moving block and a fixed block.The "surface-to-surface contact" interaction between the moving block and the fixed block is established to simulate the fault plane, which is commonly utilized in numerous studies [5,14].The lining was composed of concrete and an equivalent reinforced steel cage.The steel bar was modeled using linear truss elements (T2D2), an elastic-perfectly plastic constitutive model, and embedded in the lining concrete.The concrete lining and rock mass were meshed by eight-node reduced-integrated brick elements (C3D8R).The Mohr-Coulomb constitutive model was used to simulate the rock mass.The nonlinear behavior of concrete was simulated using the concrete damaged plasticity (CDP) constitutive model.The correspondence among the damage factors, stress, and strain of the CDP model for C30 concrete is illustrated in Figure 3, where dt is the tensile damage factor and dc is the compressive damage factor.
The interactions between the moving and fixed blocks and between the lining and rock mass were considered using surface-to-surface contact behaviors.The normal behavior was set as a hard contact, and the tangential behavior was simulated using a penalty function with a friction coefficient of 0.7.A finer mesh was used for parts near the contact surface.According to [15], when the same fault dislocation value is applied, using fixed boundaries at both ends of the pipeline structure results in a greater structural response than using equivalent spring and infinite element boundaries.Therefore, in this study, fixed boundaries were used at both ends of the lining to obtain a relatively conservative tunnel capacity for resisting fault dislocations.This analysis involved three steps.First, considering that the tunnel is in place, the geostatic stresses are introduced to establish a reasonable "reference" initial stress state [16].The bottom surface of the model was fixed, and the vertical displacement of the side surfaces was constrained.Upon balancing the geostatic stresses, the vertical reaction forces extracted from the side surfaces were applied to the corresponding nodes to release the constraint in the horizontal direction of the moving block while maintaining the same initial stress state.Finally, a displacement load parallel to the fault plane was applied at the bottom of the driving disk to conduct a quasi-static analysis.Following the completion of the calculations, the value of the fault dislocation when the tunnel broke down was determined.When the damage factor of the lining concrete exceeds 0.9 and the steel bar yields, the tunnel is considered damaged; this criterion usually corresponds to a severely damaged tunnel [5].

Development of the probabilistic dislocation capacity model
Based on the parameter-capability sample pairs obtained through numerical simulation, the support vector machine (SVM) method was used to develop a probabilistic dislocation capacity model for tunnels.A detailed explanation of the SVM can be found in the literature [17].There are several types of kernel functions for support vector machines.Considering its good performance in nonlinear regression, the radial basis function was chosen as the kernel function in this study.
The dataset used in this study comprised 153 samples, among which, 120 were randomly selected as the training dataset, and the remaining 33 were used as the test dataset.The optimal hyperparameters, c = 8 and g = 0.3536 were determined through grid optimization, and an ideal support vector regression model was obtained.The predictive effect of the model is illustrated in Figure 4.The model demonstrated excellent prediction accuracy for both the training and test datasets, indicating that it not only achieved high prediction accuracy but also good generalization ability.
The uncertainty of the probabilistic model was reflected by the standard deviation of the regression residuals for all data with reference to the least-squares regression and Bayesian regression methods.Figure 5 illustrates the probability distribution of the regression residuals obtained using all data in this study.Finally, the probabilistic capability model is expressed as follows: where, x is a set of input variables required by the support vector regression model, is the output result of the support vector regression model, representing the logarithmic mean of the fault dislocation capacity of the tunnel, 0.091ε is the error correction term representing the logarithmic standard deviation of the capacity, and ε is a random variable that follows the standard normal distribution [18].Further, t, r, and CFD are in meters.fc, fy, and Es are expressed in Pa.The probabilistic model enables rapid estimation of the capacity probability distribution for a tunnel in resisting fault dislocations based on existing design parameters.

Case study
In this section, a circular tunnel example is used to validate the proposed probabilistic capacity model and demonstrate its application to fragility analysis.6 illustrates the dislocation capacity of the target obtained from the quasi-static analysis estimated using the probabilistic model.It can be observed that the probabilistic model can provide the probability density function of the lining's dislocation capacity, the median value of which is similar to the result of numerical simulation.
Based on the estimated probability distribution of the tunnel dislocation capability, the exceedance probability of the lining failure at a given dislocation value d can be calculated using Equation (2) [19].
where, Φ is the standard cumulative probability function, βC is the lognormal standard deviation of the dislocation capacity, and the value of βC is 0.091.Thus, the fragility curve of the target tunnel (Figure 7) was derived.
The proposed probabilistic model can update the dislocation capacity and fragility curve of the lining without necessitating re-simulation when the design parameters are changed, thereby facilitating rapid optimization of the tunnel lining design.

Conclusions
This study presents a simple yet comprehensive approach to developing a probabilistic dislocation capacity model for circular mountain tunnels across strike-slip creeping faults.The uncertainties in the lining parameters, rock mass strength, and crossing angle were considered.Through quasistatic numerical simulations, the capacities of the tunnels to withstand fault dislocations under different parameter combinations were determined, and a probabilistic dislocation capacity model was established using the support vector regression method.A significant advantage of the proposed probabilistic model is that it can provide a probability distribution of the dislocation capacity rather than a fixed value.In addition, the proposed probabilistic model can be used to quickly and accurately estimate the dislocation capacity of a tunnel based on the available design parameters without re-numerical simulation.In general, the proposed probabilistic model can facilitate the fragility analysis and optimal design of cross-fault tunnels.
This study has certain limitations.For example, the faults were considered to be fault planes.Further research should incorporate a deeper understanding of the fault rupture mechanism and consider the impact of the uncertainty of fault rupture morphology.

Figure 1 .
Figure 1.Aerial perspective of strike-slip faults: (a) fault dislocation stretches the tunnel and (b) faultdislocation compresses the tunnel.For the uncertainty of the tunnel dimensions, three different lining diameters were chosen, and three lining thicknesses were considered for each.Consequently, nine combinations of lining sizes were obtained ( Table1).Geological survey reports provide annual fault dislocations and maximum fault dislocations at the ground surface for tunnel design.However, there remains a lack of information regarding the attenuation relationship between bedrock and surface dislocations.When the rock mass is sufficiently hard, fault dislocations at different depths are essentially the same (an extreme case is when the rock mass is a rigid body).Therefore, by incorporating the estimated fault dislocation, the developed capacity model can directly assess the safety of tunnels in hard rock.In this study, the elastic modulus of the rock mass

Figure 3 .
Figure 3. Damage factor and (a) tensile and (b) compressive stress-strain relationships for the CDP model.

Figure 4 .
Figure 4. Comparison between the estimated mean dislocation capacities and the calculated results obtained from ABAQUS.

Figure 5 .
Figure 5. Probability distribution of the regression residuals for all the data used in this paper.

Figure 6 .
Figure 6.Comparison of the probability density curve for the dislocation capacity of the target tunnel estimated by the proposed probabilistic models with those calculated by ABAQUS.

Figure 7 .
Figure 7. Fragility curve of the target tunnel.

Table 2 .
Strength parameters corresponding to different grades of concrete.

Table 3 .
Design parameters of the example tunnel.