Engineering
Coulomb Active Retaining-Wall Earth Pressure (Ka) & Horizontal Thrust
Calculate Charles-Augustin de Coulomb's Active Earth Pressure Coefficient K_a (accounting for soil friction phi, wall-soil friction delta, and sloping backfill beta) and horizontal overturning thrust P_ah.
Account for rough concrete wall friction and sloped backfill behind retaining walls!
Simpler Rankine earth pressure theory (K_a = (1 - SIN(phi)) / (1 + SIN(phi))) assumes a frictionless smooth wall (delta = 0) and level backfill—overpredicting horizontal thrust on rough concrete stems while failing when backfill slope beta approaches phi! Charles-Augustin de Coulomb's limit-equilibrium wedge analysis incorporates BOTH soil-wall interface friction delta and backfill slope angle beta (< phi)!
For a vertical back face, Coulomb's Active Coefficient is K_a = COS(phi)^2 / (COS(delta) * (1 + SQRT((SIN(phi + delta) * SIN(phi - beta)) / (COS(delta) * COS(beta))))^2)! Setting delta = 0 and beta = 0 recovers Rankine K_a = 0.3333 at phi = 30 deg; introducing realistic concrete wall friction delta = 20 deg drops K_a to 0.2973 and inclines resultant thrust P_a = 0.5 * gamma * H^2 * K_a by delta, reducing horizontal overturning force P_ah = P_a * COS(delta) to 90.52 kN/m on a 6 m wall!
| Retaining Wall Case (Granular Backfill phi = 30 deg, gamma = 18 kN/m^3, H = 6 m) | Wall Friction delta & Slope beta | Coulomb K_a | Horizontal Thrust P_ah (kN/m) |
|---|---|---|---|
| Smooth Wall, Level Backfill (Rankine) | delta = 0 deg, beta = 0 deg | 0.3333 | 108.00 kN/m |
| Rough Concrete Wall, Level Backfill | delta = 20 deg, beta = 0 deg | 0.2973 | 90.52 kN/m (16.2% lower!) |
| Rough Wall, 10 deg Sloped Backfill | delta = 20 deg, beta = 10 deg | 0.3400 | 103.52 kN/m |
Coulomb Active Earth Pressure Coefficient K_a (Vertical Wall Face)
Enforces backfill slope beta <= soil friction phi (otherwise dry granular slope is unstable!) and evaluates Coulomb K_a.
=IF(OR(B2 <= 0, B4 > B2), "Unstable Backfill (Slope Beta > Friction Phi)", ROUND((COS(RADIANS(B2)) ^ 2) / (COS(RADIANS(B3)) * ((1 + SQRT((SIN(RADIANS(B2 + B3)) * SIN(RADIANS(B2 - B4))) / (COS(RADIANS(B3)) * COS(RADIANS(B4))))) ^ 2)), 4))Horizontal Overturning Thrust P_ah = 0.5 * gamma * H^2 * K_a * COS(delta) (kN/m)
Computes horizontal component of active earth thrust per meter of wall length from unit weight B5 (kN/m^3) and stem height B6 (m).
=ROUND(0.5 * B5 * (B6 ^ 2) * ((COS(RADIANS(B2)) ^ 2) / (COS(RADIANS(B3)) * ((1 + SQRT((SIN(RADIANS(B2 + B3)) * SIN(RADIANS(B2 - B4))) / (COS(RADIANS(B3)) * COS(RADIANS(B4))))) ^ 2))) * COS(RADIANS(B3)), 2)Try the example
Use the original values or change them to test your understanding.