A simplified approach to reliability evaluation of deep rock tunnel deformation using First-Order Reliability Method and Monte Carlo simulations

Caio Cesar Cardoso da Silva

Universidade Federal do Rio Grande do Sul, Programa de Pós-graduação em Engenharia Civil, Porto Alegre, RS, Brasil

Mauro de Vasconcellos Real

Universidade Federal do Rio Grande, Programa de Pós-graduação em Engenharia Oceânica, Rio Grande, RS, Brasil

Samir Maghous

Universidade Federal do Rio Grande do Sul, Programa de Pós-graduação em Engenharia Civil, Porto Alegre, RS, Brasil

Corresponding author: Caio Cesar Cardoso da Silva. E-mail: caiocesarcardosodasilva@gmail.com

Conflict of interest: We wish to confirm that there are no known conflicts of interest associated with this publication.

Author contributions: CCCS: conceptualization, investigation, data curation, methodology, validation, writing, review, editing; MVR: conceptualization, data curation, formal analysis, project administration, supervision, review, editing; SM: conceptualization, formal analysis, project administration, supervision, review, editing.

Geotechnical parameter	Mean ( $μ$ )	Standard Deviation ( $σ$ )	Coefficient of Variation ( $δ$ )
Friction angle ( $φ$ )	22.85°	1.31°	0.06
Cohesion (c)	0.230 MPa	0.068 MPa	0.30
Deformation modulus (E)	373 MPa	48 MPa	0.13
Far field stress ( $P_{\infty}$ )	2.50 MPa	0.25 MPa	0.10

Geotechnical parameter	Mean ( $μ$ )	Standard Deviation ( $σ$ )	Coefficient of Variation ( $δ$ )
Friction angle ( $φ$ )	35.00°	0.50°	0.01
Cohesion (c)	1.30 MPa	0.13 MPa	0.10
Deformation modulus (E)	5000 MPa	500 MPa	0.10
Far field stress ( $P_{\infty}$ )	8.00 MPa	0.80 MPa	0.10

Shotcrete parameter	Mean ( $μ$ )	Standard Deviation ( $σ$ )	Coefficient of Variation ( $δ$ )
Deformation modulus ( $E_{s}$ )	16,000 MPa	800 MPa	0.05
Uniaxial compressive strength ( $σ_{s}$ )	30 MPa	1.5 MPa	0.05

	Results of this work		Li and Low [⁹9 H. Li and B. K. Low, "Reliability analysis of circular tunnel under hydrostatic stress field," Comput. Geotech., vol. 37, no. 1-2, pp. 50–58, 2010.]
$P_{i} (MPa)$	FORM Low and Tang	MCS	FORM Low and Tang
0 (unsupported tunnel)	24.30%	24.68%	24.40%
0.10	6.26%	6.42%	6.31%
0.20	0.87%	0.89%	0.88%
0.30	0.06%	0.05%	0.06%

	P_eq (MPa)		P_max (MPa)		U_eq (%)		U_max (%)		u_eq (mm)		u_max (mm)
	μ	σ	μ	σ	μ	σ	μ	σ	μ	σ	μ	σ
Tunnel A	0.53	0.11	0.87	0.04	0.11	0.02	0.18	0.01	4.70	1.00	7.93	0.56
Tunnel B	0.91	0.11	1.33	0.07	0.12	0.01	0.18	0.01	5.40	0.64	7.93	0.56

[R] = [\begin{matrix} 1 & ρ_{1,2} & \dots & ρ_{1,n} \\ ρ_{2,1} & 1 & \dots & ρ_{2,n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ ρ_{n,1} & ρ_{n,2} & \dots & 1 \end{matrix}]

\{\begin{matrix} K_{s} = \frac{E_{s}}{(1 - ν_{s})} \frac{t_{s}}{R_{i}} for \frac{R_{i}}{t_{s}} > 10 \\ K_{s} = \frac{E_{s} [R_{i}^{2} - {(R_{i} - t_{s})}^{2}]}{(1 - ν_{s}) [(1 - ν_{s}) R_{i}^{2} + {(R_{i} - t_{s})}^{2}]} for \frac{R_{i}}{t_{s}} \leq 10 \end{matrix}

\{\begin{matrix} P_{max} = \frac{σ_{s} t_{s}}{R_{i}} for \frac{R_{i}}{t_{s}} > 10 \\ P_{max} = \frac{σ_{s}}{2} [1 - \frac{{(R_{i} - t_{s})}^{2}}{R_{i}^{2}}] for \frac{R_{i}}{t_{s}} \leq 10 \end{matrix}

P_{eq} - K_{c} \{(\frac{1 + ν}{E}) [A + B {(\frac{R_{i}}{R_{p}})}^{K_{p} - 1} + C {(\frac{R_{p}}{R_{i}})}^{K_{b} + 1}] - U_{0}\} = 0

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