Geotechnical Engineering, Landslide Hazard & Hillslope Hydrology
Infinite Slope Stability Factor of Safety & Seepage Formula
Calculate Effective Cohesion Resistance FS_c, Pore-Pressure-Adjusted Frictional Resistance FS_phi, Total Translational Slope Factor of Safety FS, and Landslide Stability Status.
Why storm-saturated parallel seepage cuts frictional slope resistance IN HALF—triggering shallow translational landslides!
Shallow colluvial hillslopes, weathered bedrock mantles, and landfill cover veneers of depth z (m) resting on a planar barrier at slope angle beta (deg) fail via long translational slab slides governed by the Exact Infinite-Slope Effective-Stress Equation! With total soil unit weight gamma (kN/m³), effective cohesion c_prime (kPa), effective friction angle phi_prime (deg), and groundwater phreatic ratio m_w = z_w / z (0.0 for dry slope up to 1.0 when storm seepage reaches ground surface with gamma_w = 9.81 kN/m³), Total Factor of Safety splits cleanly into FS = FS_c + FS_phi = (c_prime / (gamma * z * SIN(beta) * COS(beta))) + ((gamma - m_w * 9.81) / gamma) * (TAN(phi_prime) / TAN(beta))! Notice when gamma ~ 19.62 kN/m³ and rain saturates the mantle to surface seepage (m_w = 1.0), pore pressure cuts (gamma - 9.81) / gamma to 0.50—halving frictional resistance!
Verify on a 24.0° Weathered Hillslope Mantle—Slope_Angle beta = 24.0 deg (B2), Slip_Plane_Depth z = 2.50 m (B3), Soil_Unit_Weight gamma = 19.62 kN/m³ (B4), Effective_Cohesion c_prime = 8.0 kPa (B5), Effective_Friction_Angle phi_prime = 32.0 deg (B6), and Dry Groundwater_Ratio m_w = 0.00 (B7): Cohesion Component FS_c contributes 0.44 (E2), Dry Frictional Component FS_phi contributes TAN(32°)/TAN(24°) = 1.40 (E3), Total Factor of Safety hits 1.84 (E4), and Stability Status confirms "STABLE" (E5, >= 1.30)!
| Infinite Slope KPI (beta=24°, z=2.5 m, gamma=19.62, c'=8 kPa, phi'=32°) | Formula | Verified Output |
|---|---|---|
| Cohesion Safety Term FS_c | c' / (gamma * z * SIN(beta) * COS(beta)) | 0.44 |
| Frictional Seepage Term FS_phi | ((gamma - mw*9.81)/gamma) * (TAN(phi')/TAN(beta)) | 1.40 (0.70 @ mw = 1.0 seepage!) |
| Total Factor of Safety FS | FS_cohesion + FS_friction | 1.84 (1.14 @ mw = 1.0 seepage) |
| Hillslope Stability Status | FS >= 1.30 STABLE; >= 1.00 MARGINAL; < 1 FAILURE | STABLE (MARGINAL @ mw = 1.0) |
Effective-Stress Translational Infinite Slope Factor of Safety Formula
Validate Slope_Angle beta in (1, 75] deg, Slip_Depth z > 0, Soil_Unit_Weight gamma >= 13 kN/m³, c_prime >= 0, phi_prime in [0, 50] deg, and Groundwater_Ratio m_w in [0, 1]!
=LET(bt_d, B2, dp_z, B3, gm_s, B4, cp_k, B5, ph_d, B6, mw_r, B7, IF(OR(bt_d <= 1, bt_d > 75, dp_z <= 0, gm_s < 13, cp_k < 0, ph_d < 0, ph_d > 50, mw_r < 0, mw_r > 1), "Invalid Slope Input", LET(br_d, bt_d * PI() / 180, pr_d, ph_d * PI() / 180, fs_c, cp_k / (gm_s * dp_z * SIN(br_d) * COS(br_d)), fs_f, ((gm_s - mw_r * 9.81) / gm_s) * (TAN(pr_d) / TAN(br_d)), ROUND(fs_c + fs_f, 2))))Extreme Rainstorm Full-Mantle Seepage Halving Check (Set Groundwater Ratio m_w = 1.00)
Simulate an atmospheric-river rainstorm raising Groundwater_Ratio B7 from 0.00 (dry) to 1.00 (surface-parallel seepage): Frictional Resistance E3 drops exactly in half from 1.40 to 0.70, dragging Total FS E4 down from 1.84 to 1.14 ("MARGINAL" in E5—below highway design minimums)!
=IF(ISTEXT(E4), E4, IF(E4 >= 1.3, "STABLE", IF(E4 >= 1, "MARGINAL", "FAILURE")))Try the example
Use the original values or change them to test your understanding.