The determination of the reliability of tunnel linings is an important problem but also a very difficult task. The variability of the soil properties involves using random fields. In addition the problem is time dependent, requiring the use of stochastic processes, and three-dimensional, leading to very complex models which adds to the generally complex nature of soil-structure interaction problems.

The aim of the paper is to continue previous studies developed by the working group [3,5] and give an overview of the resulting error which may be expected when applying simplified methods to this type of soil-structure interaction problem.

The soil is modelled using the description proposed by Hoek in 2002 [4], where the mechanical characteristics of the rock are defined by several parameters (GSI: geological resistance index, D: damage index and : resistance in compression) while the tunnel concrete lining is considered circular and governed by an elasto-plastic material law. The Hoek's model is based on the New Austrian Tunnelling Method (NATM) and basically consists of finding a state of equilibrium described by a pair pressure-displacement at the interface between the soil and the lining.

In the present paper several methods are employed to calculate the probability of failure: Monte Carlo Simulation (MCS) [1], the Fast Approximation Method (FAM) based on MCS [6], the Point Estimate Method (PEM) [7] proposed by Rosenblueth, a modified version of the former one (MPEM), the Response Surface Method (RSM) [2] and Level II Methods (FORM). The results obtained by MCS have been considered as reference values for the evaluation of the performance of the other probabilistic techniques. Some methods (FAM, MPEM and FORM) evaluate the probability of failure using the definition of the reliability index, i.e: the smallest distance between the failure surface and the origin of the uncorrelated standard normal space. The point of minimum distance over the failure surface is named design point and its search allows calculating the weighting factors and the sensitivity coefficients that have been compared.

The random variables are related to the excavation (geometry, lining characteristics and Panet coefficient), and are related to the soil (mechanical characteristics of the rock and density). The properties of five types of rock have been considered resulting from an in situ study at the tunnels of Padrún in Northern Spain. Three limit states have been considered, the convergence (service limit), the lining resistance and the plastification radius of the soil (ultimate limits).

A total of three studies have been carried out considering uncorrelated Gaussian variables, correlated Gaussian variables and correlated non-Gaussian variables. As the tendencies in the three studies are similar only the last one is presented in the paper. A comparison of the results obtained in terms of the reliability index permits the identification of some of the methods used as inadequate for the particular problem under consideration while others reveal an acceptable performance, i.e. the relative error does not exceed 5% with respect to the reference value obtained by Monte Carlo simulation. Also the weighting factors and sensitivity coefficients are compared.

1

Cruse T. "Reliability based mechanical design". Dekker, 1997.

2

Faravelli L. "Response-surface approach for reliability analysis". Jour. Eng. Mech. ASCE 115, 12, 1989. doi:10.1061/(ASCE)0733-9399(1989)115:12(2763)

3

Fraile A.; del Rey I.; Gómez M.S.; Alarcón E. "Reliability of tunnel liners". Progress in structural mechanics. Morton J. & París F. Editores. Sevilla, 2000.

4

Hoek, E.; Carranza-Torres, C.T.; Corkum, B. "Hoek-Brown failure criterion - 2002 edition". Proc. North American Rock Mechanics Society meeting in Toronto in July 2002.

5

Laso E.; Gómez M.S.; Alarcón E. "A level II reliability approach to tunnel support design". Appl. Mathem. Modelling. Vol. 19, 1995. doi:10.1016/0307-904X(95)00019-G

6

Melchers R.E.; Ahammed M. "A fast approximate method for parameter sensitivity estimation in Monte Carlo structural reliability" Computers & Structures no. 82 pp. 55-61, 2004. doi:10.1016/j.compstruc.2003.08.003

7

Rosenblueth E. "Two-point estimates in probabilities". Jour. Applied. Math. Modelling no 5 pp 329-335, 1981. doi:10.1016/S0307-904X(81)80054-6

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