NATM Shotcrete Rebound Delivery & Rockbolt Safety Formula

Multiplying 6.5 m³ of crown shotcrete by 1.18 instead of dividing by (1 - 0.18) leaves the nozzle short—while widening rockbolts from 1.5 to 2.2 m drops wedge safety to 0.78!

During **New Austrian Tunneling Method (`NATM`)** heading excavation, field engineers routinely commit two dangerous support errors: `1)` **Shotcrete Rebound Accounting**—because coarse aggregate bounces off hard granite overhead at **Rebound Fraction `R` (`8%–18%` robotic wet-mix with silica fume vs `25%–40%` dry-mix!)**, achieving **In-Place Design Volume `V_{in} = A_{crown} * (t_{mm} / 1,000)`** (`m³`) requires **Batch-Plant Truck Delivery `V_{del} = V_{in} / (1 - R / 100)`** (`NOT V_{in} * (1 + R / 100)`, which under-orders concrete when rebound spikes!); and `2)` **Tributary Rockbolt Dead-Weight Suspension**—across square **Bolt Pattern `s x s`** (`m`), a detached roof block of equivalent apex/thickness `h_w` (`m`) and Rock Unit Weight `gamma_r` (`kN/m³`) hangs **`W_{bolt} = s^2 * h_w * gamma_r`** (`kN`) on each grouted rebar/Swellex tendon (`Yield Capacity T_y` kN), demanding **`FS = T_y / W_{bolt} >= 1.50` (`>= 2.0` permanent crown)**!

Verify on a `4`-m NATM Advance Round—Sprayed_Crown/Wall_Area `A_{crown}` = 65.0 m² (B2), Design_Thickness `t` = 100 mm (B3), Overhead_Wet-Mix_Rebound `R` = 18.0% (B4), Square_Bolt_Spacing `s` = 1.50 m (B5), Potential_Fallout_Depth `h_w` = 1.80 m (B6), Rock_Unit_Weight `gamma_r` = 26.5 kN/m³ (B7), and `25`-mm_Rebar_Yield `T_y` = 180 kN (B8): In-Place Lining Volume equals `6.50` m³ (E2), Required Batch Delivery equals `7.93` m³ (E3, vs `7.67` m³ naive `+18%` error!), Tributary Load per Bolt equals `107.3` kN (E4), and Rockbolt Safety Factor equals **`1.68`** (`>= 1.50` safe! E5)!

ACI 506R Shotcrete Rebound Mass Balance & Tributary Rockbolt Suspension Formula

Validate `Crown_Area` in `[5, 1,000]` m², `Lining_Thickness` in `[30, 400]` mm, `Rebound_Loss` in `[2, 55]`%, `Bolt_Grid s` in `[0.6, 3.5]` m, `Wedge_Depth h_w` in `[0.3, 6.0]` m, `Rock_Unit_Weight` in `[18, 34]` kN/m³, and `Bolt_Capacity` in `[50, 600]` kN!

=LET(ac_m, B2, th_m, B3, rb_p, B4, sb_m, B5, hw_m, B6, gr_k, B7, ty_k, B8, wb_k, (sb_m ^ 2) * hw_m * gr_k, IF(OR(ac_m < 5, ac_m > 1000, th_m < 30, th_m > 400, rb_p < 2, rb_p > 55, sb_m < 0.6, sb_m > 3.5, hw_m < 0.3, hw_m > 6, gr_k < 18, gr_k > 34, ty_k < 50, ty_k > 600), "Invalid Support Input", ROUND(ty_k / wb_k, 2)))

Dry-Mix Overhead Rebound (`36%`) + Stretched Bolt Grid (`s = 2.2` m) Crown Collapse Check

Switch to high-pressure dry-mix gunite (`B4 = 36%` overhead rebound) while crews stretch bolt spacing from `1.5` to `B5 = 2.20` m: Concrete Truck Delivery E3 jumps to **`10.16` m³** (`3.66` m³ of rebound muck on the invert!), while Tributary Load E4 **more than doubles (`2.2^2 / 1.5^2 = 2.15x`!) to `230.9` kN**, crashing Bolt Safety Factor E5 to **`0.78`** (immediate tendon snap)!

=IF(ISTEXT(E5), E5, ROUND((B2 * (B3 / 1000)) / (1 - B4 / 100), 2))

Further reading

• Arte Sheets Lesson — NGI Barton Tunneling Q-System & Rockbolt/Shotcrete Support
• Arte Sheets Lesson — Soil-Nail Wall Pullout, Tendon Yield & Global Spacing Formula