Rock Mechanics, Open-Pit Slope Stability & Deep Caverns, Hoek-Brown 2002 Criterion
Generalized Hoek-Brown Rock Mass GSI & Strength Formula
Calculate Generalized Hoek-Brown Reduced Constant mb, Rock Mass Cohesion Parameter s, Non-Linear Exponent a, and Triaxial Major Principal Failure Stress sigma_1 (MPa).
Heavy production bench blasting (Disturbance D = 0.8) slashes jointed open-pit rock mass triaxial strength by 36% without changing intact drill-core UCS!
Intact diamond-drill rock cores tested in a lab press (`Intact Uniaxial Compressive Strength sigma_{ci}` MPa, `Petrographic Constant m_i` from `7` limestone to `32` granite) wildly overestimate jointed field rock masses cut by interlocking fractures! **Evert Hoek, Carranza-Torres & Corkum’s (`2002`) Generalized Hoek-Brown Failure Criterion** scales intact rock properties down using **Geological Strength Index `GSI`** (`10` sugar-cube sheared mylonite to `85` massive blocky rock) and **Blast/Stress-Relief Disturbance Factor `D`** (`0.0` controlled TBM / presplit smooth wall to `0.8–1.0` heavy production pit blasting): **`m_b = m_i * EXP((GSI - 100) / (28 - 14 * D))`**, **`s = EXP((GSI - 100) / (9 - 3 * D))`**, and **`a = 0.5 + (1 / 6) * (EXP(-GSI / 15) - EXP(-20 / 3))`**, predicting **Curved Major Principal Failure Stress `sigma_1 = sigma_3 + sigma_{ci} * (m_b * (sigma_3 / sigma_{ci}) + s)^a`** (`MPa`) at Minor Confining Stress `sigma_3` (`MPa`)!
Verify on a `GSI = 58` Blocky Quartz-Diorite Cavern Pillar under Smooth-Blasted `D = 0.0`—Intact_Core_UCS `sigma_{ci}` = 95.0 MPa (B2), Intact_Hoek_Constant `m_i` = 17.0 (B3), Geological_Strength_Index `GSI` = 58 (B4), Blast_Disturbance `D` = 0.0 (B5), and Confining_Stress `sigma_3` = 4.50 MPa (B6): Reduced `m_b` equals `3.793` (E2), Cohesion Constant `s` equals `0.00940` (E3), Exponent `a` equals `0.5033` (E4), and Triaxial Rock Mass Strength `sigma_1` reaches **`45.58` MPa** (E5)!
| Hoek-Brown (2002) Rock Mass KPI (UCS=95 MPa, mi=17, GSI=58, sig3=4.5 MPa) | Formula | Verified Output |
|---|---|---|
| Reduced Frictional Constant mb | mi * EXP((GSI - 100) / (28 - 14 * D)) | 3.793 (1.395 @ heavy blast D=0.8!) |
| Rock Mass Cohesion Parameter s | EXP((GSI - 100) / (9 - 3 * D)) | 0.00940 (0.00172 @ D=0.8) |
| Envelope Curvature Exponent a | 0.5 + (1/6) * (EXP(-GSI/15) - EXP(-20/3)) | 0.5033 |
| Triaxial Strength sigma_1 (MPa) | sig3 + sigci * ((mb*(sig3/sigci) + s) ^ a) | 45.58 MPa (29.02 MPa = -36.3% @ D=0.8!) |
Hoek, Carranza-Torres & Corkum (2002) GSI-to-Triaxial Failure Envelope Formula
Validate `Intact_UCS sigma_{ci}` in `[1, 400]` MPa, `Hoek m_i` in `[4, 35]`, `GSI` in `[10, 95]`, `Disturbance D` in `[0, 1]`, and `Confining_Stress sigma_3 >= 0` MPa!
=LET(sc_m, B2, mi_c, B3, gs_i, B4, df_b, B5, s3_m, B6, mb_r, mi_c * EXP((gs_i - 100) / (28 - 14 * df_b)), sm_r, EXP((gs_i - 100) / (9 - 3 * df_b)), ae_x, 0.5 + (1 / 6) * (EXP(-gs_i / 15) - EXP(-20 / 3)), IF(OR(sc_m < 1, sc_m > 400, mi_c < 4, mi_c > 35, gs_i < 10, gs_i > 95, df_b < 0, df_b > 1, s3_m < 0), "Invalid Hoek-Brown Input", ROUND(s3_m + sc_m * ((mb_r * (s3_m / sc_m) + sm_r) ^ ae_x), 2)))Heavy Open-Pit Production Blasting (`D = 0.8`) 36% Bench-Wall Strength Loss Check (Set D B5 = 0.8)
Omit presplit buffer rows on an open-pit catch bench so explosive gas heave dilates joints (`D` B5 = 0.8): `m_b` E2 plunges `63%` (`3.793 -> 1.395`), `s` E3 drops `5.5x` (`0.00940 -> 0.00172`), and Triaxial Rock Mass Strength `sigma_1` E5 falls **`-36.3%`** (`45.58 -> 29.02` MPa)—explaining why un-presplit pit batters ravel and wedge-fail!
=IF(ISTEXT(E5), E5, ROUND(B3 * EXP((B4 - 100) / (28 - 14 * B5)), 3))Try the example
Use the original values or change them to test your understanding.