Geotechnical Excavation Support, USACE Cantilever Steel Sheet-Pile Wall Mechanics
Cantilever Sheet-Pile Zero-Pressure Depth & Moment Formula
Calculate Rankine Active Ka & Passive Kp Earth Coefficients, Zero-Net-Pressure Depth z0 Below Dredge Line (m), and Active Wedge Overturning Moment at z0 (kN·m/m).
Below the excavation dredge line, active earth pressure does NOT vanish—passive resistance has to out-climb gamma*Ka*(H + z) first!
An unanchored **Cantilever Steel Sheet-Pile Wall** retaining granular cut height `H` (m, Unit Weight `gamma` kN/m³, Friction Angle `phi` deg) derives its entire stability from passive soil resistance mobilized in front of the driven toe! Above the excavation dredge line, active pressure climbs linearly to `sigma_{a,dredge} = gamma * K_a * H`, where **`K_a = (1 - SIN(phi)) / (1 + SIN(phi))`**. Below depth `z` under the dredge line, front-side passive resistance `gamma * K_p * z` (**`K_p = (1 + SIN(phi)) / (1 - SIN(phi)) = 1 / K_a`**) opposes continuing active pressure `gamma * K_a * (H + z)`, crossing **Zero Net Lateral Pressure at Pivot Depth `z_0 = (K_a * H) / (K_p - K_a)`** (`m`)! Integrating the dual-triangle active diagram from `0` down to `z_0` yields Exact Active Thrust `R_a = 0.5 * gamma * K_a * H * (H + z_0)` (`kN/m`) acting at centroid `(H + 2 * z_0) / 3` above `z_0`—driving **Pivot Overturning Moment `M_{z0} = R_a * ((H + 2 * z_0) / 3)`** (`kN·m/m`, the governing bending demand used to select Z-profile section modulus `S_x`)!
Verify on a `4.20` m Temporary Granular Excavation Cutoff—Retained_Cut_Height `H` = 4.20 m (B2), Compact_Sand_Unit_Weight `gamma` = 18.5 kN/m³ (B3), and Drained_Friction_Angle `phi` = 32.0° (B4): Rankine `K_a` equals `0.307` (E2), `K_p` equals `3.255` (E3), Zero-Net-Pressure Depth `z_0` sits `0.438` m below dredge level (E4), and Active Overturning Moment at `z_0` reaches `93.7` kN·m/m (E5)!
| USACE Cantilever Sheet-Pile KPI (H=4.20 m, gamma=18.5 kN/m³, phi=32° vs 25°) | Formula | Verified Output |
|---|---|---|
| Rankine Active Coefficient Ka | (1 - SIN(phi_rad)) / (1 + SIN(phi_rad)) | 0.307 (0.406 @ loose sand 25°) |
| Rankine Passive Coefficient Kp | (1 + SIN(phi_rad)) / (1 - SIN(phi_rad)) | 3.255 (2.464 @ loose sand 25°) |
| Zero-Net-Pressure Depth z0 (m) | (Ka * Cut_Height_H) / (Kp - Ka) | 0.438 m (0.828 m deeper @ 25°!) |
| Overturning Moment at z0 (kN·m/m) | (0.5*gamma*Ka*H*(H+z0)) * ((H+2*z0)/3) | 93.7 kN·m/m (154.8 kN·m/m = +65% jump!) |
Rankine Active-Passive Pressure Crossover Depth z0 & Bending Moment Formula
Validate `Cut_Height H` in `[0.5, 12.0]` m, `Unit_Weight gamma` in `[13.0, 24.0]` kN/m³, and granular `Friction_Angle phi` in `[18°, 45°]`!
=LET(hc_m, B2, gm_k, B3, ph_r, B4 * (PI() / 180), ka_c, (1 - SIN(ph_r)) / (1 + SIN(ph_r)), kp_c, (1 + SIN(ph_r)) / (1 - SIN(ph_r)), zz_m, (ka_c * hc_m) / (kp_c - ka_c), ra_k, 0.5 * gm_k * ka_c * hc_m * (hc_m + zz_m), IF(OR(hc_m < 0.5, hc_m > 12, gm_k < 13, gm_k > 24, B4 < 18, B4 > 45), "Invalid Sheet-Pile Input", ROUND(ra_k * ((hc_m + 2 * zz_m) / 3), 1)))Loose Silty Sand (phi = 25°) +65% Bending-Moment Escalation Check (Set Friction Angle B4 = 25°)
Encounter looser `phi` B4 = 25° sand at the toe: `K_a` E2 rises to `0.406` while `K_p` E3 drops to `2.464`, nearly DOUBLING Zero-Pressure Depth `z_0` E4 to `0.828` m and driving Bending Moment E5 up `+65%` to `154.8` kN·m/m—requiring a heavier AZ-series sheet-pile section or tieback waler!
=IF(ISTEXT(E5), E5, LET(ph_r, B4 * (PI() / 180), ka_c, (1 - SIN(ph_r)) / (1 + SIN(ph_r)), kp_c, (1 + SIN(ph_r)) / (1 - SIN(ph_r)), ROUND((ka_c * B2) / (kp_c - ka_c), 3)))Try the example
Use the original values or change them to test your understanding.