Introduction
A ship arriving at a quay carries kinetic energy that the fenders must absorb without damaging the hull or the structure. The energy is small per tonne but the tonnes are many: a laden Suezmax touching at 15 cm/s carries more than a megajoule. PIANC's 2002 guidelines give the standard way to estimate it: the ship's kinetic energy at the design berthing velocity, increased for the water dragged along (added mass) and reduced because an off-centre contact sets the ship rotating (eccentricity), with smaller adjustments for the berth structure and the fender stiffness, then multiplied by a safety factor for abnormal berthings. This calculator applies that method with the Brolsma design velocities and checks a chosen fender's rated energy, reaction and hull pressure.
It is for port and terminal engineers sizing fenders, naval architects checking hull pressures and students of port design; the 2024 PIANC WG211 guidelines supersede the 2002 method for new specifications that cite them.
Formulas
Normal and abnormal berthing energy:
E_N = ½ M_D v_B² C_M C_E C_C C_S E_A = E_N × factor of safety
C_B = M_D / (ρ L_BP B D)Added mass C_M:
PIANC 2002: 1.8 for K_C/D ≤ 0.1; 1.5 for K_C/D ≥ 0.5; 1.875 − 0.75 K_C/D between (then C_C = 1)
Vasco Costa: 1 + 2D/B Ueda: 1 + πD / (2 C_B B)Eccentricity (side berthing, contact at x from the bow, berthing angle α):
K = (0.19 C_B + 0.11) L_BP, R = √((L/2 − x)² + (B/2)²), γ = 90° − α − asin(B / 2R)
C_E = (K² + R² cos²γ) / (K² + R²)Berth configuration C_C (with Vasco Costa or Ueda): solid quay 0.8 (K_C/D ≤ 0.5) or 0.9, partly closed 0.9 or 1.0, open piles 1.0; 1.0 whenever α > 5°. Softness C_S: 0.9 for hard fenders (deflection ≤ 150 mm), else 1.0.
Design velocity: Brolsma curves (BS 6349-4:2014 Fig. 9) by displacement and navigation condition a–e, minimum 0.08 m/s. Abnormal factors (PIANC Table 4.2.5): tankers and bulk carriers 1.25 (large) to 1.75 (small), container ships 1.5 to 2.0, general cargo 1.75, RoRo, ferries, cruise ships and tugs 2.0.
Fender check: E_RPD ≥ E_A (before tolerance, angle, temperature and velocity factors); hull pressure = R_F / panel area against the class allowable (tankers and bulk 150–300 kPa, container 200–400 kPa, general cargo 400–700 kPa).
Assumptions and limitations
- PIANC WG33 (2002) method as used by BS 6349-4 and most existing specifications. PIANC WG211 (2024) removes C_C and C_S and revises velocities; use it where the specification requires.
- Displacement at berthing, not necessarily full load; the velocity is the component perpendicular to the berth.
- The eccentricity formula assumes contact at the parallel body with a small berthing angle; PIANC warns about the special case γ = 90°, and very small γ (contact in line with the velocity) is sensitive.
- Brolsma velocities assume tug assistance and conventional ships; without tugs multiply by 1.3–3 (EAU 2004, ROM 2.0-11). Measured velocities are better.
- Fender rated energy must still be derated for manufacturing tolerance, compression angle, temperature and velocity; multiple-fender contact and panel bending are not treated.
Worked example
A laden 100 000 DWT tanker (M_D 125 000 t, L_BP 236 m, B 43 m, D 15.1 m) berths against dolphins (third-point contact) at 5°, with 3 m under the keel and a measured approach of 0.15 m/s.
- C_B = 125 000 / (1.025 × 236 × 43 × 15.1) = 0.796.
- Added mass (PIANC): K_C/D = 0.199 → C_M = 1.875 − 0.75 × 0.199 = 1.726; C_C = 1.0.
- Eccentricity: K = (0.19 × 0.796 + 0.11) × 236 = 61.7 m; R = √(39.3² + 21.5²) = 44.8 m; γ = 90 − 5 − asin(43/89.6) = 56.3°; C_E = (61.7² + 44.8² cos² 56.3°)/(61.7² + 44.8²) = 0.761.
- Energy: ½ × 125 000 × 0.15² = 1 406 kJ; E_N = 1 406 × 1.726 × 0.761 × 1.0 × 1.0 = 1 847 kJ. Factor 1.25 for a large tanker → E_A = 2 309 kJ.
- A fender rated at 2 500 kJ / 2 200 kN on a 12 m² panel: energy OK with 8 % margin before derating; hull pressure 183 kPa, within the 200 kPa allowable for a VLCC-class hull.
Using the Brolsma curve for condition c (easy berthing, exposed) instead of the measured speed gives 0.12 m/s for 125 000 t and about 36 % less energy.
FAQ
Which velocity should I design for?
Measured data from the berth if available; otherwise the Brolsma curve for the navigation condition, remembering that curves d and e are considered high by PIANC and that berthing without tugs is much faster.
Why does the added mass depend on under-keel clearance?
With little water under the keel the entrained water cannot escape and more of it moves with the ship, raising the effective mass towards 1.8 times the displacement.
What does the eccentricity coefficient do?
An off-centre contact makes the ship yaw; the rotational energy is not absorbed by the fender. For a third-point contact only about three quarters of the kinetic energy goes into the fender.
Normal versus abnormal energy?
The normal energy occurs routinely and must not damage the fender; the abnormal energy covers rare events (exceptional ship, human error, equipment failure) and the fender must still absorb it without significant damage.
How do I choose the hull pressure limit?
From the ship class: tankers and bulk carriers can take 150–300 kPa, container ships 200–400 kPa depending on size, general cargo up to 700 kPa; RoRo and ferries usually have beltings and are not pressure-limited in the same way.
Can I use the 2024 PIANC method here?
Not directly: WG211 drops C_C and C_S (set both to 1.0 here to approximate) and specifies new velocity guidance; the structure of the energy formula is otherwise the same.
Does berthing angle matter much?
Through γ in the eccentricity factor and through the single-fender contact at larger angles; above about 10° the first contact is near the bow and one fender takes the whole energy.
Related calculators
- Hydrostatics basics — displacement and block coefficient from the particulars.
- Anchor cable catenary and scope — the other holding problem at the berth.
- Lifting sling and wire rope load — mooring and lifting hardware ratings.
- Wave properties and encounter period — exposure of the berth to waves.