This study presents a probabilistic elasto-plastic analysis of a circular tunnel in a rock mass using Monte Carlo Simulations (MCS). Ladanyi’s closed-form solutions as modified by Hoek and Brown, is used as the deterministic model. Uniaxial compressive strength of intact rock (σci), Hoek–Brown material constants for original rock mass (mb, sb), and broken rock mass (mr, sr) are treated as random variables. σci, mb, and mr are assumed to follow the normal distribution, while sb and sr are treated as log-normal distributed random variables. The realizations of these random variables are generated using MATLAB’s inbuilt functions and then fed to the deterministic model for analysis. The number of simulations is determined to achieve a 95% confidence level with a 0.5% error margin in the mean of random variables. The study evaluates the radial stress (σr), tangential stress (σθ), critical stress concentration factor <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3272_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">$\left( {\sigma_{\theta ,\max } /p_{0} } \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mrow> <msub> <mi>σ</mi> <mrow> <mi>θ</mi> <mo>,</mo> <mo movablelimits="true">max</mo> </mrow> </msub> <mo stretchy="false">/</mo> <msub> <mi>p</mi> <mn>0</mn> </msub> </mrow> </mfenced> </math></EquationSource> </InlineEquation>, and the normalized radius of the elasto-plastic boundary (rep/ri) as state variables. Goodness-of-fit (GoF) tests are performed for the state variables, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3272_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma_{\theta ,\max } /p_{0}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mrow> <mi>θ</mi> <mo>,</mo> <mo movablelimits="true">max</mo> </mrow> </msub> <mo stretchy="false">/</mo> <msub> <mi>p</mi> <mn>0</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> and rep/ri for four candidate probability distributions, viz., normal, log-normal, Weibull, and gamma. Statistical analysis shows that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3272_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma_{\theta ,\max } /p_{0}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mrow> <mi>θ</mi> <mo>,</mo> <mo movablelimits="true">max</mo> </mrow> </msub> <mo stretchy="false">/</mo> <msub> <mi>p</mi> <mn>0</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> follows a normal distribution, while rep/ri honours a log-normal distribution. Additionally, a First-Order Reliability Method (FORM) analysis is conducted to evaluate the probability of failure (Pf) corresponding to limiting values of rep/ri and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3272_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma_{\theta ,\max } /p_{0}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mrow> <mi>θ</mi> <mo>,</mo> <mo movablelimits="true">max</mo> </mrow> </msub> <mo stretchy="false">/</mo> <msub> <mi>p</mi> <mn>0</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. The results from FORM and MCS demonstrate good agreement, with FORM consistently yielding lower Pf estimates. At a value of rep/ri as 1.5 in the limit state function, FORM predicts Pf values of 10.5%, 28.3%, 35.5%, and 38% for COVs of 5%, 10%, 15%, and 20%, respectively, compared to 11.4%, 32.5%, 43.5%, and 48.1% from MCS. Similarly, for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3272_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma_{\theta ,\max } /p_{0}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mrow> <mi>θ</mi> <mo>,</mo> <mo movablelimits="true">max</mo> </mrow> </msub> <mo stretchy="false">/</mo> <msub> <mi>p</mi> <mn>0</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> = 1.8, FORM predicts Pf values of 85.9%, 66%, 56.7% and 52.8%, while MCS gives 86%, 66%, 56.7%, and 52.8% for the same COVs.

Probabilistic Analysis of Circular Tunnel Stability Using an Elasto-Plastic Model Based on Ladanyi’s Criterion

期刊论文
发表日期：

2025年8月21日

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Sagar Jaiswal,
Priti Maheshwari

摘要

This study presents a probabilistic elasto-plastic analysis of a circular tunnel in a rock mass using Monte Carlo Simulations (MCS). Ladanyi’s closed-form solutions as modified by Hoek and Brown, is used as the deterministic model. Uniaxial compressive strength of intact rock (σ_ci), Hoek–Brown material constants for original rock mass (m_b, s_b), and broken rock mass (m_r, s_r) are treated as random variables. σ_ci, m_b, and m_r are assumed to follow the normal distribution, while s_b and s_r are treated as log-normal distributed random variables. The realizations of these random variables are generated using MATLAB’s inbuilt functions and then fed to the deterministic model for analysis. The number of simulations is determined to achieve a 95% confidence level with a 0.5% error margin in the mean of random variables. The study evaluates the radial stress (σ_r), tangential stress (σ_θ), critical stress concentration factor $\left( {\sigma_{\theta ,\max } /p_{0} } \right)$ $(σ_{θ, max} / p_{0})$ , and the normalized radius of the elasto-plastic boundary (r_ep/r_i) as state variables. Goodness-of-fit (GoF) tests are performed for the state variables, $\sigma_{\theta ,\max } /p_{0}$ $σ_{θ, max} / p_{0}$ and r_ep/r_i for four candidate probability distributions, viz., normal, log-normal, Weibull, and gamma. Statistical analysis shows that $\sigma_{\theta ,\max } /p_{0}$ $σ_{θ, max} / p_{0}$ follows a normal distribution, while r_ep/r_i honours a log-normal distribution. Additionally, a First-Order Reliability Method (FORM) analysis is conducted to evaluate the probability of failure (P_f) corresponding to limiting values of r_ep/r_i and $\sigma_{\theta ,\max } /p_{0}$ $σ_{θ, max} / p_{0}$ . The results from FORM and MCS demonstrate good agreement, with FORM consistently yielding lower P_f estimates. At a value of r_ep/r_i as 1.5 in the limit state function, FORM predicts P_f values of 10.5%, 28.3%, 35.5%, and 38% for COVs of 5%, 10%, 15%, and 20%, respectively, compared to 11.4%, 32.5%, 43.5%, and 48.1% from MCS. Similarly, for $\sigma_{\theta ,\max } /p_{0}$ $σ_{θ, max} / p_{0}$ = 1.8, FORM predicts P_f values of 85.9%, 66%, 56.7% and 52.8%, while MCS gives 86%, 66%, 56.7%, and 52.8% for the same COVs.