Journal of Geological Research

Volume 2015, Article ID 394761, 9 pages

http://dx.doi.org/10.1155/2015/394761

## Tunnel Probabilistic Structural Analysis Using the FORM

Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15875-4413, Iran

Received 26 January 2015; Accepted 22 July 2015

Academic Editor: Umberta Tinivella

Copyright © 2015 Yousef Mirzaeian 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

In this paper tunnel probabilistic structural analysis (TuPSA) was performed using the first order reliability method (FORM). In TuPSA, a tunnel performance function is defined according to the boundary between the structural stability and instability. Then the performance function is transformed from original space into the standard normal variable space to obtain the design point, reliability index, and also the probability of tunnel failure. In this method, it is possible to consider the design factors as the dependent or independent random parameters with arbitrary probability distributions. A software code is developed to perform the tunnel probabilistic structural analysis (TuPSA) using the FORM. For validation and verification of TuPSA, a typical tunnel example with random joints orientations as well as mechanical properties has been studied. The results of TuPSA were compared with those obtained from Monte-Carlo simulation. The results show, in spite of deterministic analysis which indicates that the rock blocks are stable, that TuPSA resulted in key-blocks failure with certain probabilities. Comparison between probabilistic and deterministic analyses results indicates that probabilistic results, including the design point and probability of failure, are more rational than deterministic factor of safety.

#### 1. Introduction

Structural instability is the most dominant failure mechanism of underground openings in moderately jointed rock masses. Traditional analyses in such openings have been largely based on rock mass classification methods. Rock load [1], RQD [2], RSR [3], RMR [4], and rock tunneling quality index () [5] are the most practical rock mass classifications. Despite the empirical effectiveness of these classifications, they largely ignored particular stability problem due to the formation of removable rock blocks around the tunnel walls.

Block instability is not necessarily dependent on any rock mass classifications. Block theory [6] provides a mathematical and geometrical procedure to define the stability of rock structures triggered by discontinuities geometry. For a number of joint intersections behind an excavation free face, block theory determines the combinations of joints half spaces which create a removable rock block, block geometry, probable sliding or falling direction, and the safety factor of block using a rigorous approach. Block theory method considers the rock mass and support properties as deterministic parameters. However, they improved our understanding of structural instability of tunnels, in spite of their assumptions, most of the geomechanical characteristics of rock masses such as discontinuities orientation and mechanical properties and also support properties vary in wide ranges.

Unlike the probabilistic stress-controlled instability analysis, few researches have been performed on probabilistic instability analysis of block failures in underground openings: [7–9]. On the other hand, in these cases, the analysis is limited to the specific case studies with limited random dimension or specific probability distributions. So far, there is no direct study on the probabilistic analysis of structural instability in underground openings. Therefore, in this research a tunnel probabilistic structural stability analysis (TuPSA) was developed using first order reliability method.

For this purpose, at first, the reliability problem and first order reliability method (FORM) are explained, and then the performance function for tunnel structural stability is presented. By combining these two definitions, tunnel probabilistic structural stability analysis (TuPSA) will be explained. A computer code is developed for reliability analysis of structural failures and performing the TuPSA method. The reliability problem of a typical tunnel example is solved by developing computer code assuming different scenarios. Finally, the results are compared to those obtained from Monte-Carlo simulations.

#### 2. First Order Reliability Method (FORM)

For a tunnel system, reliability could be defined as the ability of the tunnel or its components to perform the required functions under the specific uncertain conditions. A random uncertain parameters vector such as (), which could include rock material, support, and joints geometrical parameters, follows a multivariate probability density function . For a specific designed tunnel system, there is a performance function that is defined to be zero at a limit state, less than zero at system failure domain, and greater than zero for safe state. The objective of reliability analysis is to calculate the probability of failure which can be expressed as [10, 11]where is failure probability and () is uncertain parameters vector. The concepts of performance limit state, failure and safe regions, and probability of failure (domain of integration that is illustrated by shaded region) are illustrated in Figure 1. In Figure 1, shaded area indicates the failure domain and dashed lines are the contours that indicate the probability distributions of two variables (for sake of simplicity).