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Retaining Wall Design, Lateral Earth Pressure & Limit Equilibrium

Coulomb Active Earth Pressure Coefficient (Ka) & Thrust Formula

Calculate Coulomb Active Coefficient K_a (with Wall Friction delta, Battered Stem beta & Backfill Slope alpha), Inclined Active Thrust P_a (kN/m), Horizontal Sliding Thrust P_ah (kN/m), and Stabilizing Vertical Shear P_av (kN/m).

Wall-soil friction delta tilts active thrust downward—cutting horizontal overturning while adding stabilizing heel weight!

Simple Rankine earth pressure assumes a frictionless vertical wall—ignoring real concrete-to-sand interface friction delta (typically 0.5 * phi to 0.67 * phi), battered backface stem angles beta, and sloping highway backfills alpha (< phi)! Coulomb’s Limit-Equilibrium Wedge Theory solves the exact Active Coefficient: K_a = (SIN(beta + phi)^2) / ((SIN(beta)^2) * SIN(beta - delta) * ((1 + SQRT((SIN(phi + delta) * SIN(phi - alpha)) / (SIN(beta - delta) * SIN(beta + alpha))))^2))! Crucially, Total Active Force P_a = 0.5 * gamma * H^2 * K_a acts at angle delta below the normal to the wall backface—resolving into Horizontal Sliding Component P_ah = P_a * COS(delta + 90° - beta) and Stabilizing Vertical Downward Component P_av = P_a * SIN(delta + 90° - beta) that helps resist wall overturning!

Verify on a 6.0 m Concrete Cantilever Retaining Wall—Soil_Friction phi = 32.0 deg (B2), Wall_Interface_Friction delta = 16.0 deg (B3), Stem_Backface_Angle beta = 90.0 deg (vertical, B4), Backfill_Slope alpha = 10.0 deg (B5), Retained_Height H = 6.0 m (B6), and Granular_Unit_Weight gamma = 18.5 kN/m³ (B7): Coulomb K_a hits 0.3145 (E2), Total Inclined Thrust P_a equals 0.5 * 18.5 * 36 * 0.3145 = 104.74 kN/m (E3), Horizontal Sliding Thrust P_ah equals 100.68 kN/m (E4), and Stabilizing Downward Shear P_av equals 28.87 kN/m (E5)!

Coulomb Retaining Wall KPI (phi=32°, delta=16°, beta=90°, alpha=10°, H=6 m)FormulaVerified Output
Coulomb Active Coefficient KaExact Coulomb Wedge Limit Formula0.3145 (0.3073 @ delta=0, alpha=0)
Total Inclined Thrust Pa (kN/m)0.5 * gamma * (H^2) * Ka104.74 kN/m (102.32 @ Rankine)
Horizontal Thrust Pah (kN/m)Pa * COS(delta + 90° - beta)100.68 kN/m (102.32 @ Rankine)
Stabilizing Vertical Shear PavPa * SIN(delta + 90° - beta)28.87 kN/m (0.00 @ delta=0!)

Exact Coulomb Active Earth Pressure Coefficient Ka Formula

Validate phi in [15, 50] deg, delta in [0, phi], beta in [60, 110] deg, alpha strictly < phi (a cohesionless slope >= phi slides before touching the wall!), H > 0, and gamma > 10!

=LET(ph_d, B2, dl_d, B3, bt_d, B4, al_d, B5, ht_m, B6, gm_s, B7, IF(OR(ph_d < 15, ph_d > 50, dl_d < 0, dl_d > ph_d, bt_d < 60, bt_d > 110, al_d < 0, al_d >= ph_d, ht_m <= 0, gm_s <= 10), "Invalid Coulomb Wall Input", LET(pr_r, ph_d * PI() / 180, dr_r, dl_d * PI() / 180, br_r, bt_d * PI() / 180, ar_r, al_d * PI() / 180, sq_t, SQRT((SIN(pr_r + dr_r) * SIN(pr_r - ar_r)) / (SIN(br_r - dr_r) * SIN(br_r + ar_r))), ROUND((SIN(br_r + pr_r) ^ 2) / ((SIN(br_r) ^ 2) * SIN(br_r - dr_r) * ((1 + sq_t) ^ 2)), 4))))

Rankine Frictionless Level-Backfill Benchmark Check (Set delta = 0°, alpha = 0°)

Set Wall_Friction B3 = 0 deg and Backfill_Slope B5 = 0 deg on a 90° vertical wall: Coulomb K_a E2 collapses identically to Rankine TAN(45° - 32°/2)^2 = 0.3073, Horizontal Thrust E4 equals Total Thrust E3 = 102.32 kN/m, and Stabilizing Vertical Shear E5 drops to 0.00 kN/m!

=IF(ISTEXT(E2), E2, LET(pr_r, B2 * PI() / 180, dr_r, B3 * PI() / 180, br_r, B4 * PI() / 180, ar_r, B5 * PI() / 180, sq_t, SQRT((SIN(pr_r + dr_r) * SIN(pr_r - ar_r)) / (SIN(br_r - dr_r) * SIN(br_r + ar_r))), ka_c, (SIN(br_r + pr_r) ^ 2) / ((SIN(br_r) ^ 2) * SIN(br_r - dr_r) * ((1 + sq_t) ^ 2)), pa_t, 0.5 * B7 * (B6 ^ 2) * ka_c, ROUND(pa_t * SIN(dr_r + (PI() / 2 - br_r)), 2)))

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