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Uncertainty is critical to the tunnel design. In this study, a novel reliability-based design (RBD) method was developed by integrating the rock-support interaction and its uncertainties for rock tunnels. In this method, the rock-support interactions were analyzed based on a convergence-confinement method. Uncertainties were estimated by reliability analysis using Excel Solver. Chaotic particle swarm optimization (CPSO) was adopted to search the optimal tunnel design parameters based on the rock-support interaction analysis. The proposed method for estimating the reliability index and determining the tunnel support parameters was introduced in detail. To illustrate the proposed method, it was applied to two tunnels with rock-bolt and combined support systems. The results indicated that the method could obtain accurate solutions with a dramatically improved computing efficiency, guaranteeing the tunnel stability at the same time. The developed method provides an excellent way to deal with the uncertainty in tunnel design. It was proved to be an efficient and effective method for the support designs of rock tunnels.

There are lots of uncertainty problems existing in various engineering systems. To improve the performance of engineering systems, reliability-based design (RBD) has been widely used in various engineering fields [

The optimization of engineering structure is essential to structure design in various fields. The support design for rock tunnels involves a complex problem and a tradeoff between safety and economy because of various uncertainties present in rock mass properties [

Many researchers have developed various methods to estimate the rock-support interaction for determining the support system [

The objective of this study was to present a practical approach to performing reliability-based designs of rock-support systems. In this study, an RBD was applied to a rock-support interaction design in a tunnel. First, the rock-support interaction analysis was reviewed. Second, a first-order reliability method (FORM) and an RBD were presented. Then, the procedure reliability-based design of the tunnel was presented in detail. Finally, the proposed method was applied to two tunnels related to the rock-bolt system, the combined support system of rock-bolt and shotcrete, and some conclusions were drawn.

The potential for instability in the surrounding rock in tunnels is an ever-present threat to both the safety of the workers and equipment in the tunnel. Support structures (rock-bolt, shotcrete, steel set, and so on) are significant to maintaining the stability of the surrounding rock in tunnels. Specific support designs for tunnels may be required in order to reduce the potential instability to an absolute minimum [

The rock-support interaction analyses, which are based on CCM, require the rock response curves and the support reaction curves, as illustrated in Figure

Rock-support interaction curve for a tunnel.

In this study, in order to present the concepts of the rock-support interactions, a circular tunnel was analyzed in a hydrostatic stress field, in which the horizontal and vertical stresses were equal. The surrounding rock was assumed to behave as an elastic-perfectly plastic material. The failure was assumed to occur with zero plastic volume change. Since the support was modeled as an equivalent internal pressure, the reinforcement provided by the support could not be taken into account in this type of simple model. A typical plot of the rock response curve is presented in Figure _{i} in Figure _{i} equaled the in situ stress _{o}, which corresponded to point A on the rock response curve. As the tunnel advanced, the support provided by the rock mass diminished, and the rock mass responded elastically up to point B, at which point a plastic failure of the rock mass was initiated. The radius _{p} of the plastic zone and the radial convergence _{i} provided by the face had decreased to zero, and the radial convergence

Once the support had been installed, and in full and effective contact with the rock, the support started to deform elastically, as shown in Figure _{o} by which the support is installed is shown in Figure

The rock response curve is defined as the relationship between the fictitious internal pressure _{i} and the radial displacement of the wall _{r}, as shown in Figure

For the support of the reaction line, the support stiffness and maximum support pressure need to be determined. The support stiffness and maximum support pressure of anchored rock-bolt can be determined according to the following equations [_{b} is the diameter of the rock-bolt; _{b} is the elastic modulus of the rock-bolt; _{bf} represents the ultimate failure load from pull-out test; and _{c} and _{L} are the circumferential and longitudinal spacing of the rock-bolt, respectively. Figure

Support analysis of the rock-bolt.

The support stiffness and maximum support pressure of the shotcrete are given by the following equations [_{c} is the elastic modulus of the concrete or shotcrete; _{c} denotes the thickness of the shotcrete; and _{c} represents the uniaxial compressive strength of the shotcrete.

For the combined support system, the stiffness of the support system _{total} was determined by the following equation:_{1} and _{2} are the stiffness of the support systems 1 and 2, respectively.

The maximum support pressure of the support system _{totmax} was determined by the following equations [

Once the rock response curve and support reaction curve were determined, a rock-support interaction analysis was completed to determine the equilibrium point (point C in Figure

Rock-support interaction analysis based on an empirical yield criterion.

For reliability-based design, the optimization and determination of the reliability index require a great deal of repeated computation. This computation time decreases the efficiency and limits the application of RBDs in practice. In this study, Low and Tang introduced the algorithm to compute the reliability index using Excel Solver software [

The determination of the reliability index is important to a reliability-based design (RBD). In this study, a first-order reliability analysis method (FORM) was used to calculate the reliability index. The Hasofer–Lind index has been previously widely used [_{i};

Based on the Hasofer–Lind index, Low and Tang presented a practical and transparent FORM procedure, in which the constrained optimization was based on the perspective of an expanding ellipsoid in the original space of the basic random variables [_{i}. Then, equation (_{i} varied during the strained optimization, the corresponding value of _{i} was calculated as follows:

In this study, equation (

Worksheet of the reliability analysis and code.

Kennedy and Eberhart originally designed particle swarm optimization (PSO) for large search spaces [

In regard to the minimum problem, by supposing that _{i}= (_{i1}, _{i2}, …, _{in}) will be the current position of the particle; _{i} = (_{i} = (_{i1}, _{i2}, …, _{in}) is its best position. Then, the best position of particle

If the population is

According to the PSO theory, the following equations represent the process of the evolution:_{ij} is its position; _{ij} is its best previous position; and _{g} is the best position among all the particles in the population; _{1} and _{2} represent two independently uniformly distributed random variables with range [0, 1]; _{1} and _{2} denote the positive constant parameters called the acceleration coefficients and control the maximum step size, respectively; and

In a standard PSO, equation (

In PSO, the parameters (for example, the inertia weight factor) are crucial for the efficient identification of the optimum solution. Many researchers believe that the parameters _{1}, and _{2} in equation (_{1} and _{2} cannot ensure the entire ergodicity within the search space. Therefore, to enrich the search behavior and avoid entrapment at a local optimum, Zhao et al. proposed a novel approach that combined chaotic mapping with certainty, ergodicity, and stochastic property into the PSO [_{0} was not 0, 0.25, 0.5, 0.75, and 1. In this study, the parameters _{1}, and _{2} of equation (_{i}(0) were not 0, 0.25, 0.5, 0.75, and 1.

In this study, in order to investigate the reliability analysis of a tunnel and its support system, two failure modes of a tunnel (for example, the deformation exceedance criterion and the support system failure) were considered. Then, the performance function was obtained by the displacement solutions of the tunnel wall, and the support pressure of the support system is as follows:_{r} is the inward displacement of the tunnel wall; _{max} denotes the maximum deformation of the tunnel; _{i} represents the support pressure of the support system; and _{max} is the maximum capacity of the support system. The performance function became negative when the inward displacement of the tunnel wall was _{r} ≥ _{max}, or the support pressure was _{i} ≥ _{max}. This indicated that the tunnel or support system would experience failure. _{max} and _{max} were calculated by using equations (_{r} and _{i} could also be calculated using the equation in this study’s Appendix section.

In this study, _{i}, and represented the _{i}, respectively. The purpose of the RBD was to determine a set of design variables that could minimize the cost function (_{i}(d) and _{i}^{T} are the reliability index and reliability constraint of the _{i} represents the design variables; and are the lower and upper limitation of the _{m} and _{d} denote the number of reliability constraints and design variables, respectively. An RBD differs from the determination optimization designs and involves reliability constraints. In this study, FORM (equation (

An objective function is essential to any optimization problem. Therefore, in order to address the problems of a reliability-based design, the design variables need to search for the optimal value based on the objective function using CPSO. Then, the problems can be converted into the following nonconstrained optimization form using a penalty method:_{i}(_{i}(

The procedure of an RBD includes two loops. The inner loop is used to determine the reliability index based on FORM, and the outer loop is used to search the design variables. In this study, Excel Solver software was adopted to calculate the reliability index. Then, the CPSO was used to search the design variables. The flowchart of this study’s CPSO-based RBD is shown in Figure

Flowchart of the RBD using CPSO.

In this study, Excel Solver software was used to compute the reliability index using a FORM method in the inner loop. Then, CPSO was adopted to search the optimal design variables in the outer loop. A rock-support interaction model was used to determine the equilibrium point, and the support pressure and deformation of the rock mass were calculated to determine the performance function in the reliability analysis. The stepwise procedure was as follows:

This study illustrated and analyzed the performance of the approach mentioned above, with one tunnel characterized by a rock-bolt support system and one tunnel containing a combined support system (rock-bolt and shotcrete). The optimal design variables of the supports were determined, and the rock-support interaction of the tunnels was evaluated.

A six-meter diameter tunnel (_{o} = 10 MPa. The rock mass properties were uncertain and therefore were regarded as random variables with normal distributions. The statistics of the random variables are listed in Table _{b}) was 25 mm. The Young modulus (_{b}) was 200 GPa. The ultimate failure load (_{bf}) and load-deformation constant (

Statistical parameters of the random variables in example 1.

Distribution type | Mean value | Standard deviation | |
---|---|---|---|

Young’s modulus | Normal | 1000 | 100 |

Poisson’s ratio | Normal | 0.25 | 0.025 |

Cohesion | Normal | 2.6 | 0.26 |

Friction angle ^{o}) | Normal | 30 | 3 |

The spreadsheet of the reliability-based design is shown in Figure

Reliability-based support optimization design of the tunnel using CPSO.

Figure

Rock-support interaction curve for the tunnel using optimal support parameters.

Figures

Relationship between the objective function value and the cycle of the CPSO.

Distribution of the objective function value in each cycle.

A 10.7 m diameter highway tunnel was excavated in frail-quality gneiss, at a depth of 122 m below the surface. The in situ stress was _{o} = 3.31 MPa. The Poisson ratio and the unit weight of the rock were 0.2 and 0.02 MN/m^{3}, respectively. The Young modulus (_{r}, _{r}) were found to be normal distributions of the random variables. The statistical parameters of the random variables are listed in Table _{b}) was 25 mm. The Young modulus (_{b}) was 207 GPa. The ultimate failure load (_{bf}) and load-deformation constant (_{c}), uniaxial compressive strength (_{c}), and Poisson’s ratio (

Statistical parameters of the random variables in example 2.

Young’s modulus | Uniaxial compression strength _{c} (MPa) | Material constraints of the rock mass | ||||
---|---|---|---|---|---|---|

Original | Broken | |||||

_{r} | _{r} | |||||

Distribution type | Normal | Normal | Normal | Normal | Normal | Normal |

Mean value | 1380 | 69 | 0.5 | 0.001 | 0.1 | 0 |

Standard deviation | 138 | 6.9 | 0.05 | 0.0001 | 0.01 | 0.001 |

The spreadsheet of the reliability-based design is shown in Figure

Reliability-based combined support system design of the tunnel using CPSO.

Rock-support interaction curve for the tunnel using the optimal combined support system.

Variation process of the design variables.

In this study, a novel reliability-based design approach that took the uncertainties into account was proposed to design support systems in tunnels. This new design was applied to two circular tunnels, one with a rock-bolt support system and one with a combined support system. The results showed that the proposed method was able to obtain accurate solutions with a dramatically improved computing efficiency. The proposed method could also be generalized and used for any shape tunnel with a numerical solution. The reliability index was calculated using various reliability methods, with the exception of FORM. A CPSO method was utilized in the demonstration cases in this study. However, any search optimization method could be applied, such as a genetic algorithm, gradient-based methods, and so on. The reliability-based design of the rock-support interaction design was found to conform to the practical requirements of tunnels. It provided a rational and reliable way to conduct the analyses of tunnel stability and support designs in tunnel projects.

In the solution based on Mohr–Coulomb (M-C) yield criterion, it was assumed that the circular tunnel was excavated in a continuous, homogeneous, isotropic, initially elastic rock mass, which had been subjected to hydrostatic far-field stress _{0} and uniform support pressure _{i}. Therefore, if _{i} was less than the critical pressure _{cr}, a plastic zone existed. According to the Mohr–Coulomb criterion, the plastic zone radius _{p} and the inward displacement of the tunnel wall _{ip} could be obtained by the following equations [_{cr},

For the solution based on the empirical yield criterion, if the plastic failure occurred around the tunnel, then the radius of the plastic zone could be obtained by the following equations:

The deformation around the tunnel was obtained by the following equation:_{c} represents the uniaxial compressive strength of the intact rock; _{r} and _{r} are the material constants for the broken rock mass.

If the deformation around the tunnel was elastic, then the deformation around the tunnel could be obtained by the following equation:

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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

This research was funded by the Open Research Fund of the State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, China (Grant no. Z020006).