Offshore Wind & Marine Structural Engineering, Wave Loading
Morison Equation Offshore Pile Wave Drag & Inertia Formula
Calculate Peak Wave Drag Force Amplitude F_D (kN/m), Peak Added-Mass Inertia Force Amplitude F_I (kN/m), Naive In-Phase Sum (kN/m), and True Phase-Accurate Peak Morison Force F_max (kN/m).
Wave velocity u(t) and acceleration du/dt are 90° out of phase—NEVER simply add peak drag F_D + peak inertia F_I!
Slender offshore jacket legs, risers, and wind-turbine monopiles of diameter D (m, where D / L_wave < 0.20) experience two hydrodynamic forces per unit submerged length under Morison’s Equation: Drag Force f_D(t) = 0.5 * rho * C_d * D * u(t) * |u(t)| peaking at wave crest (cos(theta) = 1), and Added-Mass Inertia Force f_I(t) = rho * C_m * (PI() * D^2 / 4) * du/dt peaking at zero-crossing (sin(theta) = 1, 90° earlier)! Adding peak amplitudes F_D + F_I naively overpredicts wave load by up to 41%! Differentiating F_D * cos^2(theta) + F_I * sin(theta) with respect to phase theta yields the EXACT Combined Peak Morison Load: if F_I >= 2 * F_D (inertia-dominated large monopiles), F_max = F_I; otherwise (drag-significant slender tubulars), F_max = F_D + (F_I^2) / (4 * F_D)!
Verify on a 1.20 m Offshore Jacket Tubular—Seawater_Density rho = 1,025.0 kg/m³ (B2), Pile_Outer_Diameter D = 1.20 m (B3), Drag_Coeff C_d = 1.05 (marine-growth rough cylinder, B4), Inertia_Coeff C_m = 2.00 (B5), Peak_Orbital_Velocity u_max = 2.40 m/s (B6), and Peak_Orbital_Acceleration a_max = 1.50 m/s² (B7): Peak Drag Amplitude F_D hits 3.72 kN/m (E2), Peak Inertia Amplitude F_I equals 3.48 kN/m (E3), Naive Sum would claim 7.20 kN/m (E4), whereas True Phase-Accurate Peak Morison Force equals 3.72 + 3.48^2 / (4 * 3.72) = 4.53 kN/m (E5)!
| Morison Wave-Force KPI (D=1.2 m, Cd=1.05, Cm=2.0, u=2.4 m/s, a=1.5 m/s²) | Formula | Verified Output |
|---|---|---|
| Peak Drag Force FD (kN/m) | 0.5 * rho * Cd * D * (u_max^2) / 1000 | 3.72 kN/m (15.50 @ D = 5.0 m) |
| Peak Inertia Force FI (kN/m) | rho * Cm * (PI()*D^2/4) * a_max / 1000 | 3.48 kN/m (60.38 @ D = 5.0 m!) |
| Naive Unphased Sum FD+FI (kN/m) | Peak_Drag_FD + Peak_Inertia_FI | 7.20 kN/m (75.88 @ D = 5.0 m) |
| True Phase-Max Morison Force | IF(FI >= 2*FD, FI, FD + FI^2/(4*FD)) | 4.53 kN/m (60.38 @ D = 5.0 m!) |
Exact Phase-Maximized Morison Wave Load per Meter Formula
Validate Seawater_Density > 950, Pile_Diameter D > 0, C_d in [0.4, 2.5], C_m in [1.0, 2.5], u_max > 0, and a_max > 0!
=LET(rh_w, B2, dm_p, B3, cd_f, B4, cm_f, B5, um_x, B6, am_x, B7, IF(OR(rh_w <= 950, dm_p <= 0, cd_f < 0.4, cd_f > 2.5, cm_f < 1, cm_f > 2.5, um_x <= 0, am_x <= 0), "Invalid Morison Input", LET(fd_k, (0.5 * rh_w * cd_f * dm_p * (um_x ^ 2)) / 1000, fi_k, (rh_w * cm_f * (PI() * (dm_p ^ 2) / 4) * am_x) / 1000, ROUND(IF(fi_k >= 2 * fd_k, fi_k, fd_k + (fi_k ^ 2) / (4 * fd_k)), 2))))5.0 m Offshore Wind Monopile Inertia-Dominance Regime Shift Check (Set Diameter D = 5.0 m)
Scale Pile_Diameter B3 from 1.20 m (jacket leg) to 5.00 m (offshore wind monopile): Drag E2 scales linearly with D to 15.50 kN/m, while Inertia E3 scales with D^2 to 60.38 kN/m (>= 2 * 15.50 = 31.00 kN/m)—so True Peak Force E5 equals pure Inertia 60.38 kN/m at the zero-crossing!
=IF(ISTEXT(E5), E5, ROUND(((0.5 * B2 * B4 * B3 * (B6 ^ 2)) + (B2 * B5 * (PI() * (B3 ^ 2) / 4) * B7)) / 1000, 2))Try the example
Use the original values or change them to test your understanding.