Rock Mechanics, Tunneling & Underground Mining, Bieniawski RMR89 Classification
Bieniawski Rock Mass Rating (RMR89) & Tunnel Load Formula
Calculate Basic RMR89 (0–100), Strike/Dip Orientation-Adjusted RMR89, Serafim-Pereira Deformation Modulus Em (GPa), and Unal Tunnel Roof Support Load Pressure (kPa).
Ignoring adverse joint dip orientation and fault gouge drops tunnel rock modulus 15x from 33.5 GPa to 2.24 GPa!
**Z. T. Bieniawski’s (`1989`) Geomechanics Rock Mass Rating (`RMR_{89}`)** scores underground tunnel headings and mine drifts across five additive parameters (`0–100` basic points): `1)` Intact Rock `UCS` (`0–15` pts) + `2)` Deere `RQD` (`3–20` pts -> combined `3–35` pts), `3)` Discontinuity Spacing (`5–20` pts) + `4)` Joint Wall Condition (`0–30` pts -> combined `5–50` pts), and `5)` Groundwater Inflow (`0–15` pts), followed by `6)` **Tunnel Strike & Dip Orientation Penalty `Delta_RMR_{orient}`** (`0` very favorable to `-12` pts when slickensided joints dip `20°–45°` against the heading!). From **Adjusted `RMR_{tot}`**, **Serafim & Pereira (`1983`)** predict **In-Situ Rock Mass Deformation Modulus `E_m = 10^((RMR_{tot} - 10) / 40)`** (`GPa`, valid across `RMR <= 50` and widely benchmarked up to `RMR ~ 75`), while **Unal (`1983`)** sizes **Tunnel Crown Support Load Pressure `P_{roof} = ((100 - RMR_{tot}) / 100) * gamma_{rock} * B`** (`kPa`) across Excavation Span `B` (`m`) and Rock Unit Weight `gamma_{rock}` (`kN/m³`)!
Verify on an `8.0` m Span Highway Rock Tunnel in Jointed Granodiorite (`gamma = 26.5` kN/m³)—UCS+RQD_Score = 29 pts (B2), Spacing+Condition_Score = 37 pts (B3), Damp_Groundwater_Score = 10 pts (B4), Fair_Orientation_Penalty = -5 pts (B5), Tunnel_Span `B` = 8.0 m (B6), and Unit_Weight `gamma` = 26.5 kN/m³ (B7): Basic `RMR_{89}` equals `76` (E2), Adjusted `RMR_{89}` equals `71` (`Class II Good Rock`, E3), Rock Mass Modulus `E_m` equals `33.50` GPa (E4), and Unal Crown Support Pressure equals `61.5` kPa (E5)! In a cloud collaboration tunnel-mapping workbook, face geologists (`suggester` role) propose updated `B2:B5` scores per round with whole-cell comments; note that the Arte Sheets ChatGPT `0.1.0` connector is currently `In review` / `Not published` (when used on scoped OAuth cloud workbooks, `read_range` reads raw cell inputs/formulas up to `200` cells without recalculating engine outputs, while `owner`/`editor` reviewers verify live `E2:E5` recalculation in Arte Sheets before accepting suggestions).
| Bieniawski RMR89 Tunnel KPI (Span B=8.0 m, gamma=26.5 kN/m³: Class II vs Fault Class IV) | Formula | Verified Output |
|---|---|---|
| Basic Unadjusted RMR89 (0–100) | UCS_RQD_Pts + Spacing_Cond_Pts + Water_Pts | 76 pts (36 pts sheared fault zone) |
| Orientation-Adjusted RMR89 | Basic_RMR89 + Strike_Dip_Penalty_Pts | 71 pts Class II (24 pts Class IV Poor!) |
| Serafim-Pereira Modulus Em (GPa) | 10 ^ ((Adjusted_RMR89 - 10) / 40) | 33.50 GPa (2.24 GPa 15x softer!) |
| Unal Crown Support Load (kPa) | ((100 - Adjusted_RMR89) / 100) * gamma * Span_m | 61.5 kPa (161.1 kPa steel ribs + shotcrete!) |
Bieniawski RMR89, Serafim-Pereira Deformation Modulus & Unal Support Pressure Formula
Validate `UCS+RQD` in `[3, 35]`, `Spacing+Condition` in `[5, 50]`, `Groundwater` in `[0, 15]`, `Orientation_Adjustment` in `[-12, 0]`, positive `Adjusted RMR` in `[1, 100]`, `Span B` in `[1.5, 30]` m, and `Unit_Weight` in `[18, 33]` kN/m³!
=LET(p1_s, B2, p2_s, B3, p3_s, B4, pa_d, B5, bs_m, B6, gr_k, B7, rt_s, p1_s + p2_s + p3_s + pa_d, IF(OR(p1_s < 3, p1_s > 35, p2_s < 5, p2_s > 50, p3_s < 0, p3_s > 15, pa_d < -12, pa_d > 0, rt_s < 1, bs_m < 1.5, bs_m > 30, gr_k < 18, gr_k > 33), "Invalid RMR89 Input", ROUND(((100 - rt_s) / 100) * gr_k * bs_m, 1)))Water-Bearing Clay-Gouge Fault Zone (`RMR = 24` Class IV Poor Rock) Squeeze Check
Advance the heading into a dripping, adversely dipping fault shear zone (`B2 = 14`, `B3 = 18`, `B4 = 4`, `B5 = -12`): Adjusted `RMR_{89}` plunges to **`24`** (`Class IV Poor Rock`, E3), Deformation Modulus E4 collapses `15x` to `2.24` GPa, and Unal Roof Support Load E5 jumps `2.6x` to **`161.1` kPa**—demanding forepoling umbrella arch pipes, lattice girders, and immediate fiber-reinforced shotcrete!
=IF(ISTEXT(E5), E5, ROUND(10 ^ (((B2 + B3 + B4 + B5) - 10) / 40), 2))Try the example
Use the original values or change them to test your understanding.