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Overview of the Hoek-Brown Failure CriterionThe Hoek-Brown failure criterion is an empirical, non-linear model used to estimate the strength of rock masses, particularly in geotechnical engineering applications such as tunnels, slopes, and foundations. It relates the major principal stress (
\(\sigma_1\)) to the minor principal stress (
\(\sigma_3\)) at failure, accounting for intact rock strength (
\(\sigma_{ci}\)), geological strength index (GSI), material constant (
\(m_i\)), and disturbance factor (D). The generalized form is:
\sigma_1' = \sigma_3' + \sigma_{ci} \left( m_b \frac{\sigma_3'}{\sigma_{ci}} + s \right)^awhere
\(m_b\),
\(s\), and
\(a\) are rock mass parameters derived from GSI and D. For practical analysis, especially in numerical modeling or limit equilibrium methods that require linear Mohr-Coulomb (MC) parameters (cohesion
\(c'\) and friction angle
\(\phi'\)), the Hoek-Brown envelope is approximated by fitting a linear MC tangent over a specific stress range:
\(\sigma_t < \sigma_3' < \sigma_3'_{\max}\), where
\(\sigma_t\) is tensile strength (often negative or zero) and
\(\sigma_3'_{\max}\) is the maximum effective minor principal stress. The choice of
\(\sigma_3'_{\max}\) significantly affects the equivalent
\(c'\) and
\(\phi'\), with higher values potentially overestimating cohesion and underestimating friction.
Context of \(\sigma_3'_{\max}\) in Shallow FoundationsIn shallow foundations on rock masses, bearing capacity calculations often involve approximating the non-linear Hoek-Brown criterion with equivalent MC parameters to use classical theories (e.g., Prandtl's mechanism) or numerical methods like finite element analysis (FEA). Shallow foundations typically experience low confinement (low
\(\sigma_3'\)) compared to deep excavations, but the bearing pressure can induce higher stresses beneath the footing. The Hoek-Brown criterion is applicable for intact rock or heavily jointed masses behaving isotropically, but less so for anisotropic or sparsely jointed rocks. For bearing capacity, methods include:
- Kinematic upper-bound limit analysis: Using modified Hoek-Brown with support functions for failure mechanisms.
- Carter & Kulhawy method: Assumes weightless rock, sets \(\sigma_3' = 0\) in one zone, and derives bearing capacity as \(q_u = \sigma_{ci} \cdot (m_b \cdot \sigma_{ci}/4 + s \cdot \sigma_{ci}^2)^{1/2}\).
- Serrano et al. method: Uses slip line theory with \(q_u = \sigma_{ci} \cdot (N_\phi + d)\), where \(N_\phi\) and \(d\) depend on Hoek-Brown parameters.
- FEA or probabilistic methods: For spatially variable rock masses.
Analysis of \(\sigma_3'_{\max}\) Values from SourcesSources consistently reference
\(\sigma_3'_{\max}\) for fitting equivalent MC parameters, but specific values vary by application. For shallow foundations (bearing capacity problems), no unique formula like those for slopes or tunnels is provided; instead, a fixed fraction of
\(\sigma_{ci}\) is used due to the absence of depth/height parameters. Key findings:
- Consensus Value for Shallow Foundations: Multiple sources cite \(\sigma_3'_{\max} = 0.25 \sigma_{ci}\) as the standard for bearing capacity, derived from trial-and-error fitting in Hoek & Brown (1997) to ensure consistent MC parameters. This range (0 < \(\sigma_3'\) < 0.25 \(\sigma_{ci}\), using 8 points) is recommended for low-confinement scenarios like foundations, avoiding overestimation at higher stresses. It is explicitly suggested for bearing capacity verification, with warnings to validate against problem-specific stresses.
- Comparisons and Variations:
- Some analyses test broader ranges (e.g., 0 < \(\sigma_3'\) < 0.75 \(\sigma_{ci}\)) to assess sensitivity, but 0.25 \(\sigma_{ci}\) yields more conservative and consistent results for shallow foundations.
- For intact rock triaxial testing, the original Hoek-Brown (1980) used 0 < \(\sigma_3'\) < 0.5 \(\sigma_{ci}\), but for rock masses in foundations, 0.25 \(\sigma_{ci}\) is preferred.
- No sources use rock mass strength (\(\sigma_{cm}\)) directly for foundations; \(\sigma_{ci}\) is the basis.
- In contrast, for slopes: \(\sigma_3'_{\max} = 0.72 (\sigma_{cm} / \gamma H)^{0.91} \sigma_{cm}\); for tunnels: \(\sigma_3'_{\max} = 0.47 (\sigma_{cm} / \gamma H)^{-0.94} \sigma_{cm}\). These are not applied to foundations.
- Critiques and Limitations: The choice is empirical; higher \(\sigma_3'_{\max}\) may overestimate strength in low-overburden cases like shallow foundations. Direct Hoek-Brown implementation (e.g., via support functions) avoids fitting but is complex. Ductile transition at \(\sigma_1' = 3.4 \sigma_3'\) limits high-confinement applicability.
Conclusion: Correct Value for \(\sigma_3'_{\max}\) in Shallow FoundationsBased on analyzed sources (academic papers, Rocscience documents, conference proceedings), the correct and most commonly accepted value is
\(\sigma_3'_{\max} = 0.25 \sigma_{ci}\). This is empirically derived for consistent MC fitting in low-confinement bearing capacity problems and should be verified case-by-case. No conflicting "correct" value emerged; variations are sensitivity tests.
