Mathematical Problems in Engineering

Volume 2018, Article ID 1450683, 8 pages

https://doi.org/10.1155/2018/1450683

## Probabilistic Analysis of Tunnel Face Stability below River Using Bayesian Framework

^{1}School of Civil Engineering and Architecture, Nanchang University, Nanchang 330031, China^{2}Jiangxi Science and Technology Normal University, Nanchang 330013, China^{3}ARC Centre of Excellence for Geotechnical Science and Engineering, University of Newcastle, Callaghan, NSW 2308, Australia^{4}School of Civil Engineering and Architecture, Nanchang Institute of Technology, Nanchang 330099, China

Correspondence should be addressed to Lina Hu; nc.ude.ucn@aniluh

Received 28 February 2018; Accepted 10 May 2018; Published 6 June 2018

Academic Editor: Eric Feulvarch

Copyright © 2018 Weiping Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1], numerical methods [2, 3], and experimental methods [4]. The limit equilibrium method is commonly used for predicting limit support pressure, being relatively simple compared with finite element analysis. Investigations showed that soil parameters, such as cohesion and friction angle, are the most important soil properties for influencing the limit support pressure [5]. Therefore, accurate estimation of soil strength parameters is necessary for assessment of the tunnel face stability under tunnel excavation.

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 [6, 7] and can be modeled using stochastic analysis [8].

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 [9]. Mollon et al. [10, 11] use response surface method to perform probabilistic analysis of the face stability of tunnel. Li et al. [12] use the first-order reliability method and Monte Carlo simulation to carry out reliability analysis of a circular tunnel subjected to a hydrostatic stress field. Oreste [13] presents a probabilistic design approach for the tunnel taking into account the uncertainties of the rock mass quality index. Celestino et al. [14] evaluate the probability of failure according to load and resistance factor design principles. Langford et al. [15] use a modified point estimate method to design the tunnel. Lu et al. [16] evaluate failure probability of each failure mode of a rock tunnel by the first-order reliability method and the response surface method via an iterative procedure. Eshraghi et al. [17] study the face stability of TBD-driven tunnel in heterogeneous soil using probabilistic approach.

Bayesian approach can update the current state of knowledge about the model parameters based on the measurement data [18]. Many successful applications of Bayesian approach have been reported, e.g., estimation of the parameters of hydrological model [19], confidence interval of SWCC [20], and braced excavation [21]. Although previous studies have been done on the probability analysis of tunnel, a systematic Bayesian framework and Markov Chain Monte Carlo (MCMC) simulation for the probabilistic analysis has not yet been developed. The effect of uncertainties of soil parameters on prediction uncertainty of tunnel face stability below river using a Bayesian framework has not yet been investigated.

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 [22], with a delayed rejection adaptive metropolis (DRAM) [23] algorithm. The effects of uncertainty of strength parameters on the limit support pressure are discussed using the proposed Bayesian framework.

#### 2. Limit Support Pressure

Wedge analysis [24] based on the limit equilibrium theory is adopted the limit support pressure for the face stability of a tunnel below river. A simple failure mechanism which idealizes the three-dimensional mode of tunnel face is shown in Figure 1. The collapsing soil in front of the tunnel is schematized as a triangular wedge; the wedge is assumed as a rigid body. are the width, length, and height of the prism in Figure 1, respectively. The length of prism is , and the inclination of wedge is . This three-dimensional model is first proposed by Horn [25], and a right-angled prism extends from the tunnel crown to the surface. The Mohr-Coulomb failure condition with friction angle and cohesion is assumed. Terzaghi’s formula [26] has generally been adopted as vertical earth pressure acting the tunnel.