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System Reliability Assessment for a Rock Tunnel with Multiple Failure Modes

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Abstract

This paper presents a practical procedure for assessing the system reliability of a rock tunnel. Three failure modes, namely, inadequate support capacity, excessive tunnel convergence, and insufficient rockbolt length, are considered and investigated using a deterministic model of ground-support interaction analysis based on the convergence–confinement method (CCM). The failure probability of each failure mode is evaluated from the first-order reliability method (FORM) and the response surface method (RSM) via an iterative procedure. The system failure probability bounds are estimated using the bimodal bounds approach suggested by Ditlevsen (1979), based on the reliability index and design point inferred from the FORM. The proposed approach is illustrated with an example of a circular rock tunnel. The computed system failure probability bounds compare favorably with those generated from Monte Carlo simulations. The results show that the relative importance of different failure modes to the system reliability of the tunnel mainly depends on the timing of support installation relative to the advancing tunnel face. It is also shown that reliability indices based on the second-order reliability method (SORM) can be used to achieve more accurate bounds on the system failure probability for nonlinear limit state surfaces. The system reliability-based design for shotcrete thickness is also demonstrated.

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Abbreviations

D :

Disturbance factor

E i :

Young’s modulus of intact rock

E rm :

Young’s modulus of rock mass

g(x):

Performance function

L s :

Distance between support and tunnel face

N :

Simulation sample size

P f :

Probability of failure

\( p_{\text{s}}^{\text{D}} \) :

Final design pressure

\( p_{\text{s}}^{ \max } \) :

Support capacity

R :

Correlation matrix

R pl :

Plastic-zone radius

R t :

Radius of tunnel

u :

Vector of standard normal variables u i

u*:

Design point in U-space

\( u_{\text{r}}^{\text{D}} \) :

Final ground displacement

\( u_{\text{r}}^{\text{in}} \) :

Initial ground displacement

β :

Reliability index

\( \varepsilon_{u}^{\max } \) :

Maximum permissible tunnel convergence

κ :

Vector of principal curvatures κ i

ρ :

Vector of correlation coefficients

Φ(·):

CDF of the standard normal variable

CCM:

Convergence–confinement method

CDF:

Cumulative distribution function

COV:

Coefficient of variation

FORM:

First-order reliability method

GRC:

Ground reaction curve

GSI:

Geological strength index

LDP:

Longitudinal deformation profile

MCS:

Monte Carlo simulation

NATM:

New Austrian tunneling method

RSM:

Response surface method

SCC:

Support characteristic curve

SORM:

Second-order reliability method

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Acknowledgments

The authors would like to thank the Jurong Town Corporation of Singapore for providing research funding, which enable the first author (Lü Qing) to work as a post-doctoral research fellow and the second author (Chan Chin Loong) as a project officer at Nanyang Technological University. This research was also supported by Zhejiang Province Key Innovation Team Support Program (no. 2009R50050) and Zhejiang Province Special Scientific and Technological Project on “Methodology and modeling of tunnel excavation” (2010C13029).

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Authors and Affiliations

  1. College of Civil Engineering and Architecture, Zhejiang University, Hangzhou, 310058, People’s Republic of China

    Qing Lü

  2. School of Civil and Environmental Engineering, Nanyang Technological University, Singapore, 639798, Singapore

    Chin Loong Chan & Bak Kong Low

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  1. Qing Lü
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  2. Chin Loong Chan
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  3. Bak Kong Low
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Correspondence to Chin Loong Chan.

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Lü, Q., Chan, C.L. & Low, B.K. System Reliability Assessment for a Rock Tunnel with Multiple Failure Modes. Rock Mech Rock Eng 46, 821–833 (2013). https://doi.org/10.1007/s00603-012-0285-3

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