Hoek-Brown failure criterion

The Hoek-Brown failure criterion is a widely used model in rock mechanics that describes the strength of rock materials in terms of the major and minor principal stresses. It is named after its developers, Edward Hoek and J. A. E. Brown.

The criterion is based on the Mohr-Coulomb failure criterion, but it also takes into account the rock's strength, the rock's deformability, and the rock's cohesion. It is represented by the following equation:

σ_c = (σ_1 - σ_3) * (mi + mb * (σ_1/σ_3)^2)

Where:
σ_c = the critical or failure stress
σ_1 = major principal stress
σ_3 = minor principal stress
mi = the rock's strength parameter
mb = the rock's deformability parameter

The Hoek-Brown criterion can be used to predict the strength of rock under different loading conditions and can be used to estimate the stability of underground excavations.

A solved example :

The rock has a strength parameter mi = 24 and deformability parameter mb = 0.5. The major principal stress is σ_1 = 100 MPa and the minor principal stress is σ_3 = 50 MPa.

σ_c = (100 - 50) * (24 + 0.5 * (100/50)^2)
σ_c = 50 * (24 + 0.5 * 4)
σ_c = 50 * 26
σ_c = 1300 MPa

So, the failure stress for this rock is 1300 MPa.

It's worth noting that Hoek-Brown failure criterion is an empirical model and it's applicable only for certain types of rock and it's not suitable for all kinds of rock

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