Civil, Structural & Geotechnical Engineering
Caquot-Kerisel Log-Spiral Passive Earth Pressure (K_p) Formula vs Planar Coulomb Error
Calculate safe horizontal passive earth pressure coefficient K_p,h under wall-soil interface friction delta using Albert Caquot & Jean Kerisel’s (1948) / Eurocode 7 curved logarithmic-spiral slip surface, avoiding Coulomb’s dangerous overprediction.
Why planar Coulomb K_p overpredicts passive resistance by > 60% when wall friction delta > phi / 2!
When a sheet-pile bulkhead, bridge abutment, or anchor block pushes horizontally into backfill sand (friction angle phi), upward wall-soil shear mobilizes interface friction angle delta (typically delta = (1/2) * phi to (2/3) * phi)! Assuming a straight planar failure wedge, Coulomb’s 1776 equation predicts K_p,Coulomb = (COS(phi)^2) / (COS(delta) * ((1 - SQRT((SIN(phi + delta) * SIN(phi)) / COS(delta)))^2)) — which blows up toward infinity as delta approaches phi because real soil shears along a CURVED downward-dipping Logarithmic Spiral r = r_0 * EXP(theta * TAN(phi))!
Solved rigorously by French geotechnical pioneers Albert Caquot and Jean Kerisel (1948, Tables for the Calculation of Passive Pressure, Active Pressure and Bearing Capacity) and codified in Eurocode 7 (EN 1997-1 Annex C, Kerisel & Absi / Lancellotta closed-form analytical log-spiral formulation), Horizontal Log-Spiral Passive Earth Pressure Coefficient for vertical wall & horizontal backfill is K_p,h = ((1 + SIN(phi)) / (1 - SIN(phi))) * EXP((2 * theta_0) * TAN(phi)) * COS(delta) — or in Soubra / Kerisel-table power-reduction form K_p,CK(delta) = K_p,max * ((1 - 0.02 * (90 - phi) * (1 - delta / phi)) -> exact Eurocode 7 analytical Sokolovskii-Lancellotta closed form): let 2*psi_w = ASIN(SIN(delta) / SIN(phi)) + delta, giving K_p,h = (COS(delta) / (1 - SIN(phi))) * (COS(delta) + SQRT((SIN(phi)^2) - (SIN(delta)^2))) * EXP((ASIN(SIN(delta) / SIN(phi)) + delta) * TAN(phi))!
| Sand Friction phi = 35° & Wall Friction Ratio delta / phi | Smooth Rankine K_p (delta = 0) | Curved Log-Spiral K_p,h (Eurocode 7 / Caquot-Kerisel) | Unsafe Planar Coulomb K_p,h |
|---|---|---|---|
| Smooth Wall (delta = 0°) | 3.690 | 3.690 (Identical) | 3.690 |
| Moderate Roughness (delta = 17.5° = 0.5*phi) | 3.690 | 6.120 | 6.619 (+8.2% High) |
| Full Roughness (delta = 35.0° = 1.0*phi) | 3.690 | 8.324 (Safe Curved Wedge) | 26.66 (+220% UNSAFE!) |
Curved Log-Spiral Horizontal Passive Coefficient K_p,h (Caquot-Kerisel / Lancellotta EC7)
Evaluates exact Sokolovskii-Lancellotta curved log-spiral horizontal K_p,h across soil friction angle phi (B2 deg in 15..45) and wall-soil friction angle delta (B3 deg in 0..B2).
=IF(OR(B2 < 15, B2 > 45, B3 < 0, B3 > B2), "Requires 15 <= phi <= 45 deg and 0 <= delta <= phi", ROUND((COS(RADIANS(B3)) / (1 - SIN(RADIANS(B2)))) * (COS(RADIANS(B3)) + SQRT(MAX(0, (SIN(RADIANS(B2)) ^ 2) - (SIN(RADIANS(B3)) ^ 2)))) * EXP((ASIN(MIN(1, SIN(RADIANS(B3)) / SIN(RADIANS(B2)))) + RADIANS(B3)) * TAN(RADIANS(B2))), 3))Planar Coulomb Horizontal Passive Coefficient K_p,h (Shown to Quantify Overprediction)
Computes COS(delta) * K_p,Coulomb so engineers can immediately audit how much unconservative excess planar wedges introduce when delta > phi / 2.
=IF(OR(B2 < 15, B2 > 45, B3 < 0, B3 > B2), "Requires 15 <= phi <= 45 deg and 0 <= delta <= phi", ROUND((COS(RADIANS(B2)) ^ 2) / ((1 - SQRT((SIN(RADIANS(B2 + B3)) * SIN(RADIANS(B2))) / COS(RADIANS(B3)))) ^ 2), 3))Try the example
Use the original values or change them to test your understanding.