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Underground Caverns, Drill-and-Blast Tunneling & NATM, NGI Barton Q-System

NGI Barton Tunneling Q-System & Roof Support Formula

Calculate Logarithmic NGI Rock Mass Quality Q (0.001–1,000), Equivalent Excavation Dimension De (m), Maximum Permanent Unsupported Span (m), and Crown Support Pressure (kPa).

Clay-coated slickensided joints drop Barton’s Q-index across six orders of magnitude—multiplying permanent cavern roof support pressure 10x!

Developed at the **Norwegian Geotechnical Institute (`NGI`) by Nick Barton, Lien & Lunde (`1974`)**, the **Tunneling `Q`-System** spans six logarithmic decades (`0.001` squeezing clay fault to `1,000` massive unjointed granite) as the product of **Three Physical Quotients**: **`Q = (RQD / J_n) * (J_r / J_a) * (J_w / SRF)`**—where `RQD / J_n` (`Joint Set Number 0.5–20`) sizes **Relative Block Volume**, `J_r / J_a` (`Roughness 0.5–4` over `Clay Alteration 0.75–20`) measures **Inter-Block Shear Friction `~tan(phi_r)`**, and `J_w / SRF` (`Joint Water / Stress Reduction Factor`) captures **Active Effective Stress**! Scaling Excavation Span `B` (`m`) by **Excavation Support Ratio `ESR`** (`1.0` road/rail tunnel & powerhouse, `1.6` mine haulage, `0.8` nuclear/public station) defines **Equivalent Dimension `D_e = B / ESR`**, **Maximum Permanent Unsupported Span `S_{max} = 2 * ESR * Q^0.4`** (`m`), and **Barton Permanent Roof Support Pressure `P_{roof} = (200 / J_r) * Q^(-1/3)`** (`kPa`)!

Verify on a `9.60` m Span Underground Hydro Powerhouse Access Tunnel (`ESR = 1.0`)—Deere `RQD` = 72% (B2), Two_Joint_Sets+Random `J_n` = 6.0 (B3), Smooth_Undulating_Roughness `J_r` = 2.0 (B4), Unaltered_Joint_Walls `J_a` = 1.0 (B5), Minor_Inflow_Medium_Stress `J_w / SRF` = 0.264 (`0.66 / 2.5`, B6), Excavation_Span `B` = 9.60 m (B7), and `ESR` = 1.0 (B8): `Q` equals `6.336` (`Fair/Good`, E2), `D_e` equals `9.60` m (E3), Max Unsupported Span equals `4.19` m (`< 9.60` m -> systematic rockbolting + `50` mm fiber shotcrete required! E4), and Permanent Crown Support Pressure equals `54.0` kPa (E5)!

NGI Barton Q-System KPI (Span B=9.60 m, ESR=1.0: Good Gneiss vs Swelling Clay Fault)FormulaVerified Output
NGI Rock Mass Quality Index Q(RQD / Jn) * (Jr / Ja) * Jw_Over_SRF6.336 Fair/Good (0.046 Extremely Poor!)
Equivalent Excavation Dim De (m)Tunnel_Span_B_m / Support_Ratio_ESR9.60 m
Max Unsupported Span Smax (m)2 * ESR * (Q ^ 0.4)4.19 m (0.58 m immediate face collapse!)
Permanent Roof Support Load (kPa)(200 / Roughness_Jr) * (Q ^ (-1/3))54.0 kPa (558.9 kPa cast concrete arch!)

Barton, Lien & Lunde (1974) Three-Quotient Q-Index & Crown Support Formula

Validate `RQD` in `[10, 100]` (`NGI` rule: assign `RQD = 10` whenever nominal `RQD < 10`!), `J_n` in `[0.5, 20]`, `J_r` in `[0.5, 4.0]`, `J_a` in `[0.75, 20]`, `J_w / SRF` in `[0.005, 2.0]`, `Span B` in `[1.5, 60]` m, and `ESR` in `[0.5, 5.0]`!

=LET(rq_d, B2, jn_s, B3, jr_n, B4, ja_n, B5, jw_s, B6, bs_m, B7, es_r, B8, qv_l, (rq_d / jn_s) * (jr_n / ja_n) * jw_s, IF(OR(rq_d < 10, rq_d > 100, jn_s < 0.5, jn_s > 20, jr_n < 0.5, jr_n > 4, ja_n < 0.75, ja_n > 20, jw_s < 0.005, jw_s > 2, bs_m < 1.5, bs_m > 60, es_r < 0.5, es_r > 5), "Invalid Barton Q Input", ROUND((200 / jr_n) * (qv_l ^ (-1 / 3)), 1)))

Crushed Clay-Gouge Fault Zone (`Q = 0.046` Extremely Poor Rock) 559 kPa Load Check

Intersect a sheared smectite-clay fault (`RQD = 22`, `J_n = 12`, `J_r = 1.0`, `J_a = 4.0`, `J_w / SRF = 0.10`): `Q` E2 plunges `138x` to **`0.046`**, Unsupported Span E4 shrinks to `0.58` m, and Permanent Crown Support Pressure E5 surges `10.3x` to **`558.9` kPa** (`5.6` bar)—mandating spiling, steel ribs, and reinforced concrete lining!

=IF(ISTEXT(E5), E5, LET(qv_l, (B2 / B3) * (B4 / B5) * B6, ROUND(2 * B8 * (qv_l ^ 0.4), 2)))

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