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A key issue in assessment on tunnel face stability is a reliable evaluation of required support pressure on the tunnel face and its variations during tunnel excavation. In this paper, a Bayesian framework involving Markov Chain Monte Carlo (MCMC) simulation is implemented to estimate the uncertainties of limit support pressure. The probabilistic analysis for the three-dimensional face stability of tunnel below river is presented. The friction angle and cohesion are considered as random variables. The uncertainties of friction angle and cohesion and their effects on tunnel face stability prediction are evaluated using the Bayesian method. The three-dimensional model of tunnel face stability below river is based on the limit equilibrium theory and is adopted for the probabilistic analysis. The results show that the posterior uncertainty bounds of friction angle and cohesion are much narrower than the prior ones, implying that the reduction of uncertainty in cohesion and friction significantly reduces the uncertainty of limit support pressure. The uncertainty encompassed in strength parameters are greatly reduced by the MCMC simulation. By conducting uncertainty analysis, MCMC simulation exhibits powerful capability for improving the reliability and accuracy of computational time and calculations.

Valid estimation of the tunnel face stability under excavation requires a reliable evaluation of limit support pressure which prevents soil collapse. This issue has been extensively studied with limit equilibrium method [

Normally the mean values of soil parameters are used in deterministic analysis. Although the method can deliver accurate analytical results of support pressure, it requires the input parameters for every calculation point in situ and cannot control the spatial variability of the input parameters. In fact, geotechnical materials are natural materials, and their properties are affected by various spatially variable factors during their formation processes. The inherent spatial variability has been considered as one of the major sources of uncertainties [

A major difficulty in estimating accurate limit support pressure arises from the uncertainties incorporated in the input parameters of the computer model. The input strength parameters, such as cohesion and friction angle, are usually determined by direct measurements in laboratory. These samples will be disturbed in the process of testing. Meanwhile, the test conditions in laboratory cannot be exactly the same as in situ. This also leads to a significant uncertainty in predicting the limit support pressure for keeping tunnel face stability. This uncertainty poses challenges for obtaining reliable design of tunnel excavation. The stochastic approach can improve the traditional deterministic methods for taking the uncertainties of the parameters into account.

In practice, tunnel face collapse prediction can be formulated as a classification. Many studies have been performed to analyze the stability of tunnel face [

Bayesian approach can update the current state of knowledge about the model parameters based on the measurement data [

In this paper, the probability is associated with the different parameters which are governing the tunnel face stability and furthermore detailed in the following. The soil strength parameters, such as friction angle and cohesion, are assumed as random variables. The tunnel face below river is in the homogeneous soils. The three-dimensional model of tunnel face stability below river is based on the limit equilibrium theory and is adopted for the probabilistic analysis. The probabilities analysis and parameters uncertainty estimation are performed using the Markov Chain Monte Carlo (MCMC) simulation method which is good efficiency for highly nonlinear problem [

Wedge analysis [

Wedge stability model.

The circular cross-section of the tunnel is approximated by a square whose sides are as long as the circular tunnel diameter

Considering an element has dimension

Vertical friction applied to lateral of the element is

The vertical equilibrium equation of

From the equilibrium equation (

For the sake of simplicity, the soil of wedge is considered to be homogeneous. Assuming the failure criterion holds along the failure face, a static equilibrium equation can be set up. When a shield tunnel is located below the river, the stability analysis of the shield tunnel face needs to consider the influence of pore water pressure. The pore water pressure usually is considered an external force [

Forces acting upon the wedge.

The vertical mean stress

The shear forces

The shear force

Not taking the infiltration in excavation face into account, the overburden strata are assumed to be permeable with high permeability, such as sand and gravel. So a complete hydraulic connection exists between the river water and groundwater. In this sense, the pore water pressure generated by river water can be expressed as

By considering the water pressure and equating force in the vertical and horizontal direction, the limit support force on the tunnel face is

The support pressure is simplified and considered uniform, and the minimum support pressure termed as limit support pressure which keeps the tunnel face stable is expressed as

This would provide a simple design method for limit support pressure on the tunnel face. Effect of a multilayered overburden also can be taken into account. In this paper, this mechanism will be used for the probabilistic analysis.

Within a Bayesian framework, inferences are made about the parameters of interest by a probability distribution given the data [

The error or difference between the actual performance and the model prediction is defined as the model correction factor

The likelihood function gives a measure of the agreement between the available data and the corresponding model output. Assuming that error

Bayesian updating can be achieved using Bayesian framework, when conjugate priors are given. The posterior distribution function of input parameters and model predicted response cannot be derived through analytical means. An alternative approach is to use a MCMC method to obtain the numerical summarization of the posterior distribution. Therefore, random sampling methods are needed to generate samples from the posterior distribution function. These posterior distributions are obtained using the MCMC simulation, which is an effective random sampling method. The MCMC simulation can maintain adequate sampling density as the number of parameter increases and compute efficiently which has gained popularity in recent years to sample the posterior probability density function [

The definition of the empirical covariance matrix determined

For

In the delayed rejection algorithm, the proposed sample

The second stage proposal is accepted with probability if the

This process of delaying rejection can be iterated.

Numerous test investigations and numerical studies of tunnel face stability have illustrated that the friction angle

The friction angel and cohesion are considered as random input parameters with normal variables. In this study, the respective means of friction angle and cohesion are denoted as

Prior distributions.

Parameter | Mean | Std. dev. | COV |
---|---|---|---|

| 30 | 4.5 | 0.15 |

| 5 | 1 | 0.20 |

For probabilistic analysis of support stress on tunnel face, variation of friction angle and cohesion are considered. Other parameters come from the crossing Ganjiang River tunnel and are used as follows: saturated unit weight ^{3}, the tunnel diameter

The chains with total number of evaluations equal to 30,000 are used in MCMC simulation with a DEAM algorithm, and the posterior samples generated are shown in Figure

Plot of samples with MCMC simulation steps.

Since the samples are generated by MCMC method, the posterior distributions of the parameters are computed using the chains of the two MCMC simulations. Figures

Statistic of posterior distribution parameters.

Parameter | Mean | Std. dev. | COV |
---|---|---|---|

| 27.816 | 3.181 | 0.114 |

| 4.268 | 0.697 | 0.163 |

Prior and posterior distribution of friction angle.

Prior and posterior distribution of cohesion.

According to Table

In this section, the probability distribution of the limit support stress is determined for a given condition. Using updated soil parameters of friction angle

Posterior distribution of the limit support stress.

Cumulative distribution of the limit support stress.

In this paper, a Bayesian framework for updating soil strength parameters in tunnel excavation below river is presented. The method employs a MCMC simulation based approach to derive the posterior distribution of the parameters. The posterior distribution can be used for probabilistic analysis of the limit support stress on the circular tunnel face. The 3D analysis model with the limit equilibrium method is used as deterministic model. The uncertain parameters considered in the analysis are the soil strength parameters as the friction angle and cohesion. Comparing with the prior distribution, the means of prediction improve and the variation of prediction reduces by the proposed Bayesian framework. Therefore, the proposed method is effective in reducing the uncertainty of soil strength parameters, demonstrating its potential as a practical geotechnical engineering tool. The Bayesian framework with MCMC method might be more favorable in the uncertainty analysis and risk management.

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

The authors declare that they have no conflicts of interest.

The authors sincerely acknowledge the support from the National Natural Science Foundation of China (51468041), the Specialized Research Fund for the Doctoral Program of Higher Education (20123601110001), and Jiangxi Science Foundation (20161BAB203078).