Adesso, Gemini. Ho fatto 2 tentativi, nel primo ho allegato il manuale di roclab, con risposta non proprio congruente, poi ho chiesto di verificare la sua tesi con fonti affidabili presenti in rete. La risposta, che adesso presenta precisi riferimenti di letteratura, è del tutto coerente con il programma ed è la seguente. Ometto la prima risposta, che però sarebbe ibnteressante nell'ottica della compresione del funzionamento delle AI.
Hoek-Brown Conversion Parameter for Shallow Foundations The Determination of $\sigma'_{3\max}$ for Shallow Foundations on Rock
The determination of the upper limit of the minor principal stress ($\sigma'_{3\max}$) is a critical, yet empirically driven, step in converting the non-linear Generalized Hoek-Brown (H-B) failure criterion into an equivalent linear Mohr-Coulomb (M-C) envelope ($c'$ and $\phi'$) for use in classical geotechnical design methods like shallow foundation bearing capacity.
Based on authoritative scientific literature, the consensus recommendation for determining $\sigma'_{3\max}$ for low-stress applications, which includes shallow foundations with typical excavation depths ($\approx 1 \text{ m}$), is provided by E. Hoek (2002).
1. Official Consensus and Recommended Formula
The most widely cited and generally suggested value for the upper stress limit ($\sigma'_{3\max}$) is directly related to the uniaxial compressive strength of the intact rock ($\sigma_{ci}$).
Recommended $\sigma'_{3\max}$ Value (Hoek, 2002)
The general recommendation for $\sigma'_{3\max}$ in low-confinement scenarios, used for fitting the equivalent Mohr-Coulomb envelope, is:
$\sigma'_{3\max} = 0.25 \sigma_{ci}$
Where:
- $\sigma'_{3\max}$ is the maximum minor effective principal stress used to define the stress range for curve-fitting (in MPa).
- $\sigma_{ci}$ is the Uniaxial Compressive Strength of the intact rock material (in MPa).
Degree of Consensus
- Official Consensus: High. This formula, or variations closely related to it, is the industry standard recommendation provided by the primary developer of the criterion, Evert Hoek, particularly in the widely accepted "Hoek-Brown failure criterion – 2002 Edition" paper. It is commonly implemented in commercial geotechnical software (e.g., Rocscience's RocLab).
- Diverging Opinions/Nuance: The choice of $\sigma'_{3\max}$ is explicitly stated by Hoek as a matter of experience and trial-and-error, as no theoretically established method exists. The $0.25 \sigma_{ci}$ value represents a general practical upper limit for failure processes in low-stress fields, such as those governing shallow foundation failure (bearing capacity) or slope stability. Some diverging practices may use a value based on the estimated in-situ stress state or the calculated ultimate bearing capacity, $q_{ult}$, to constrain the fitting range more narrowly around the expected failure stress.
2. Rationale and Application to Shallow Foundations
Stress Regime
Shallow foundations placed on rock typically induce a failure zone characterized by low confining stress ($\sigma'_{3}$) because the foundation depth ($D$) and width ($B$) are small, leading to a shallow failure mechanism.
The H-B criterion is non-linear in the $\sigma'_{1}$ vs. $\sigma'_{3}$ space, meaning its equivalent strength parameters ($c'$ and $\phi'$) change depending on the stress level.
Purpose of $\sigma'_{3\max}$
The value of $\sigma'_{3\max}$ defines the upper bound of the stress range ($\sigma'_{t} \le \sigma'_{3} \le \sigma'_{3\max}$) over which a linear Mohr-Coulomb envelope is mathematically fitted to the H-B curve. The objective of the fitting process is to balance the area above and below the M-C plot to obtain representative average strength parameters, $c'$ and $\phi'$, that are most relevant to the failure zone.
Using $0.25 \sigma_{ci}$ ensures that the equivalent M-C parameters are derived from the lower, flatter portion of the H-B envelope, which is representative of the low confinement stresses associated with near-surface failure mechanisms, thus providing actionable parameters for bearing capacity calculations using the conventional Terzaghi/Meyerhof framework.
Reference
- Hoek, E., Carranza-Torres, C. & Corkum, B. (2002). Hoek-Brown failure criterion—2002 edition. Proceedings of the 5th North American Rock Mechanics Symposium.
- Gharsallaoui, M., et al. (2019). Bearing capacity of surface foundations resting on Hoek – Brown materials using equivalent Mohr – Coulomb parameters. ECSMGE-2019 Proceedings. (Citing the Hoek 2002 recommendation for $\sigma'_{3\max}=0.25\sigma_{ci}$ in the context of surface foundations).
