Based on the 2002 Hoek-Brown criterion and a review of geotechnical literature regarding shallow foundations, a mathematically and logically plausible value for σ'₃max in the context of shallow foundations can be derived as follows:

1. Context of σ'₃max in Shallow Foundations

In the Hoek-Brown criterion, σ'₃max is the maximum confining stress up to which the equivalent Mohr-Coulomb parameters (c', φ') are calibrated to match the Hoek-Brown failure envelope. This value depends on the stress range relevant to the geotechnical problem:

• For tunnels: σ'₃max is related to the in-situ stress and tunnel depth.


• For slopes: it is related to the slope height and rock mass strength.


• For shallow foundations: the relevant stress range is typically near the ground surface, where the minor principal stress σ'₃ is low, but it increases with depth below the footing.

2. Proposed Approach for Shallow Foundations

For shallow foundations, the failure mechanism involves a bearing capacity failure beneath the footing, where the confining stress σ'₃ increases with depth but is generally limited by the overburden pressure and the width of the foundation.

A widely accepted approach is to relate σ'₃max to the vertical stress at a depth equal to the width of the foundation (B) below the base of the footing:

where:

• γ = unit weight of rock mass


• D_f = embedment depth of foundation


• B = width of foundation

However, this may be too conservative. Instead, a more common and rational method is to use a multiple of the rock mass uniaxial compressive strength or a function of the foundation pressure.

3. Recommended Value Based on Literature and Analogy

Based on the Hoek-Brown paper's approach for slopes and tunnels, and extending it to shallow foundations, a reasonable estimate is:

where σ_cm is the global rock mass strength from Equation (17).

This range is supported by:

• The idea that for shallow foundations, the confining stress is limited by the foundation width and depth.


• Numerical studies and bearing capacity theories suggesting that the active confining stress under a footing rarely exceeds 0.3 to 0.5 σ_cm before failure.

4. Suggested Formula for Shallow Foundations

A more specific formula, by analogy with the slope equation (19), can be proposed:

where:

• B = foundation width


• γ = unit weight of rock mass

This form is consistent with the dimensional analysis used in the Hoek-Brown paper and reflects the dependence on foundation size and rock mass strength.

5. Example Calculation