\sigma_1' = \sigma_3' + \sigma_{ci} \left( m_b \frac{\sigma_3'}{\sigma_{ci}} + s \right)^a

• 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.

• 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.

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).

Synthesis: Simple Envelope-Fitting Bounds for a Flat Rock Surface

When fitting a straight Mohr–Coulomb line to the curved Hoek–Brown envelope over
0 ≤ σ′₃ ≤ σ′₃₋ₘₐₓ, you need a practical choice for
σ′₃₋ₘₐₓ that doesn’t rely on knowing the ultimate capacity or running
finite-element models. Three widely used, flat-surface bounds are:

1. 
   Constant-Fraction of Intact Strength
   
   
   
   • 
     Formula:
     σ′₃₋ₘₐₓ = 0.25 · σci
     
   
   
   • 
     Rationale:
     Hoek & Brown (1980) observed that, for shallow loads, the minor principal
     stress range in jointed rock lies between 0.1·σci and 0.4·σci. A midpoint of
     0.25·σci gives a representative upper bound without further data.
     
   
   
   
   



2. 
   At-Rest Horizontal Stress (K₀) Approach
   
   
   
   • 
     Formula:
     σ′₃₋ₘₐₓ = K0 · γ · Df, K0 ≈ 1 − sin φ′
     
   
   
   • 
     Rationale:
     Prior to loading, a rock mass at depth Df supports a vertical
     stress γ·Df. Taking its natural horizontal stress
     σh′ = K₀·γ·Df as σ′₃₋ₘₐₓ ties the
     envelope limit to the actual in-situ confinement.
     
   
   
   
   



3. 
   Mid-Envelope Point via Rock Mass Parameters
   
   
   
   • 
     Formula:
     
σ′₃₋ₘₐₓ = (s / mb) · σci

mb = mi · exp((GSI − 100) / (28 − 14D))

s = exp((GSI − 100) / (9 − 3D))

     
   
   
   • 
     Rationale:
     The term (mb·σ₃′/σci + s)a shifts the
     Hoek–Brown curve from intact rock (s = 1) toward jointed mass (s < 1). Setting
     mb·σ₃′/σci = s locates the stress where intact-rock
     and joint-mass contributions are equal—an intuitive “mid-curve” bound.
     
   
   
   
   

Conservatism Comparison

*Example assumes σci = 50 MPa, φ′≈30° ⇒ K₀≈0.5,
γ=25 kN/m³, Df=2 m, GSI=50,
D=0.5.