| name | ship-stability |
| description | Ship stability — metacentric height GM (initial stability), righting lever GZ curves (large angle stability), Archimedes' principle and displacement, free surface effect (FSE), wall-sided formula, cross curves of stability, IMO intact stability criteria (IS Code 2008), damage stability (SOLAS floodable length), inclining experiment, and deadweight/trim/draft calculations. |
| metadata | {"priority":7,"promptSignals":{"phrases":["ship stability","metacentric height","GZ curve","righting moment","vessel stability","naval architecture stability"],"minScore":3}} |
Ship Stability — Complete Skill
Archimedes' Principle and Displacement
Buoyancy Fundamentals
Displacement:
Δ = ρ_sw × ∇ [Δ = displacement mass [tonnes]; ρ_sw = 1.025 t/m³; ∇ = displaced volume [m³]]
Δ_weight = m_ship = Δ [ship mass equals displacement in equilibrium]
Draft and waterplane:
TPC (Tonnes Per Centimetre immersion) = A_wp × ρ_sw / 100 [A_wp = waterplane area [m²]; TPC in t/cm]
MCTC (Moment to Change Trim 1 cm) = Δ × GML / (100 × L_bp) [MCTC in t·m/cm; GML = longitudinal metacentric height; L_bp = length between perpendiculars]
Deadweight (DWT):
DWT = Δ_loaded − Δ_lightship [cargo + fuel + stores + crew + ballast water capacity]
Block coefficient:
C_B = ∇ / (L_pp × B × T) [L_pp = waterline length; B = beam; T = draft]
Typical: bulk carrier C_B = 0.80–0.85; container ship = 0.60–0.70; warship = 0.45–0.55
Initial Stability — Metacentric Height GM
Metacenter and Buoyancy Shift
Metacentric radius BM:
BM = I_T / ∇ [I_T = second moment of waterplane area about centerline; ∇ = displaced volume]
For rectangular waterplane: I_T = L × B³ / 12 → BM = (L × B³/12) / (L × B × T) = B²/(12T)
Metacentric height GM:
GM = KB + BM − KG [K = keel; B = center of buoyancy; M = metacenter; G = center of gravity]
KB ≈ T/2 (approx for box barge); exact from Bonjean curves for ship form
KM = KB + BM [metacenter above keel]
KG = height of ship CG above keel (from lightship + loading condition)
Initial stability (small angles θ < 10°):
Righting moment = Δ × GZ ≈ Δ × GM × sin(θ) ≈ Δ × GM × θ [θ in radians for linear approximation]
GM > 0: stable (upright equilibrium restoring)
GM < 0: loll — ship heels to angle where GZ = 0 beyond upright
Typical GM values:
Fully loaded tanker: GM = 0.5–2.0 m
Container ship (in port, light): GM may be negative if stacking too high (dangerous)
Ferry/passenger ship: GM ≥ 0.15 m minimum (IS Code)
Free Surface Effect (FSE)
Reduction in GM
Free surface correction:
FSE = Σ (i_s × ρ_liquid) / Δ [i_s = second moment of free surface area of each slack tank; ρ_liquid = density of liquid in tank]
For rectangular tank: i_s = l × b³ / 12 [l = tank length; b = tank breadth]
Corrected metacentric height:
GM_corrected = GM_solid − FSE [always reduces stability; reduce by filling tanks completely or dividing tanks with longitudinal bulkheads]
Example:
Ship Δ = 5000 t; slack tank 10 m × 8 m of seawater (ρ = 1.025 t/m³)
i_s = 10 × 8³ / 12 = 426.7 m⁴
FSE = 426.7 × 1.025 / 5000 = 0.0875 m
If GM_solid = 0.6 m → GM_corrected = 0.6 − 0.0875 = 0.513 m
Longitudinal division: dividing tank into 2 equal parts reduces FSE by 4× (b → b/2, i_s → 2×(l×(b/2)³/12) = l×b³/48 = i_s/4)
Large Angle Stability — GZ Curves
Righting Lever GZ
GZ at large angles (wall-sided formula — valid for ±30°):
GZ = sin(θ) × (GM + BM × tan²θ / 2) [wall-sided: ship sides vertical; exact for box forms]
or: GZ = sin(θ) × (GM + (1/2)BM × tan²(θ)) — equivalent
Cross curves of stability (KN curves):
KN(θ, ∇) = KS(θ, ∇) + S(θ, ∇) × sin(θ) [tabulated for given volume/displacement at each angle]
GZ = KN − KG × sin(θ) [for given KG; allows GZ curve construction for any loading condition]
Cross curves computed at constant displacement ∇ for θ = 10°, 20°, 30°, ..., 90°
GZ Curve Shape
Key GZ curve parameters:
Maximum GZ (GZ_max): righting lever peak [m]; should occur at 25°+ and GZ_max ≥ 0.20 m (IS Code)
Range of positive stability: angle from 0° to angle of vanishing stability (AVS); IS Code: AVS ≥ 30° for large ships
Area under GZ: energy of righting; IS Code areas:
- 0° to 30°: ≥ 0.055 m·rad
- 0° to 40° (or flooding angle if less): ≥ 0.090 m·rad
- 30° to 40°: ≥ 0.030 m·rad
IMO Intact Stability Criteria (IS Code 2008)
Intact Stability Requirements (Chapter 2)
Basic IS Code criteria (all ships ≥ 24 m):
| Criterion | Minimum Value |
|---|
| Area 0°–30° | ≥ 0.055 m·rad |
| Area 0°–40° | ≥ 0.090 m·rad |
| Area 30°–40° | ≥ 0.030 m·rad |
| GZ_max at ≥ 25° | ≥ 0.200 m |
| Initial GM₀ | ≥ 0.150 m |
| Angle of GZ_max | ≥ 25° |
Severe wind and rolling criterion (weather criterion):
Wind heeling moment: M_w = P_w × A_lateral × h_lateral [P_w = 504 Pa steady; 1.5× gust for dynamic]
Stability criterion: b ≥ a [area under GZ from leeward roll angle to capsize; must exceed area of wind moment arm]
Passenger Ship Additional Criteria
Passenger crowding (all to one side):
P = 75 kg × N_passengers at side; heeling angle ≤ 10°
Passenger to boat deck: heeling angle ≤ 15°
High-speed craft criteria: MSC/Circ.1072
Damage Stability (SOLAS Probabilistic Method)
Attained Subdivision Index A
Probabilistic damage stability (SOLAS 2009, Reg. II-1):
Attained subdivision index: A = Σ pᵢ × sᵢ [sum over all damage cases; pᵢ = probability of flooding zone i; sᵢ = probability of survival given flooding]
Required: A ≥ R [R = required index from ship type and length tables]
Flooding angle and GZ requirement post-damage:
After flooding: final waterline below lowest opening; residual GZ ≥ 0.05 m over range ≥ 7°; GM ≥ 0.05 m
Floodable length:
L_f(x) = maximum damage extent centered at x such that ship survives flooding of that zone
Permissible length = L_f × Factor F [F = 1.0 for one-compartment; 0.5 for two-compartment standard]
Trim and Drafts
Trim Calculation
Trim change from loading:
δTrim = w × d / MCTC [w = mass added [t]; d = distance from midship [m]; MCTC [t·m/cm]]
Draft change at bow: δT_F = δTrim × (L/2 − LCF) / L_pp [LCF = longitudinal center of flotation]
Draft change at aft: δT_A = δTrim − δT_F
Draft change from added mass:
δT_mean = w / TPC [in cm; w in tonnes; TPC = tonnes per centimetre]
Trim by stern: positive trim = aft deeper than forward (normal ship condition underway)
Inclining Experiment
Determination of KG and GM
Purpose: determine ship's KG and GM experimentally after construction or major refit
Procedure:
- Ship in calm water; minimum free surfaces; all liquid tanks measured
- Known mass m_i shifted transversely by distance d_i
- Pendulum or inclinometer records heel angle θ
- GM calculated: GM = (m_i × d_i) / (Δ × tan θ)
- Subtract BM_calculated → KB_calculated → get KG = KM − GM
Stability booking:
KG = KG_experiment ± corrections for free surfaces, removed masses, added masses
Standards and References
| Standard | Scope |
|---|
| IMO IS Code 2008 | International Code on Intact Stability |
| SOLAS 2020 | Safety of Life at Sea — damage stability |
| IMO MSC.267(85) | IS Code adoption resolution |
| DNV Rules for Ships | Ship stability calculations and verification |
| ISO 18072-2 | Ships and marine technology — ship stability calculations |
| IACS UR S22 | Longitudinal strength of hull girder |
Output
Provide: ship particulars (Lpp [m]; B [m]; T [m]; Δ [t]; C_B; ρ_sw [t/m³]), hydrostatics (KB [m]; BM = I_T/∇ [m]; KM = KB+BM [m]; TPC [t/cm]; MCTC [t·m/cm]; LCF [m from midship]), loading condition (mass items: cargo, fuel, ballast, stores; KG per item; total Δ [t]; KG_solid [m]; FSE = Σ(i_s×ρ_liq)/Δ [m]; GM_corrected = KM − KG_solid − FSE [m]), GZ curve (cross-curves or wall-sided formula at θ = 0°–90° at 10° intervals; GZ_max [m] at angle [°]; AVS [°]; areas: 0-30°, 0-40°, 30-40° [m·rad]), IS Code check (all 6 criteria; pass/fail; minimum GM = 0.15 m check), trim (LCG − LCB = trim lever [m]; trim = Δ × trim_lever / MCTC×100 [cm]; fwd and aft drafts [m]), damage stability (if applicable: SOLAS A ≥ R; one/two-compartment standard; residual GM and GZ range post-flood), and applicable standard (IMO IS Code 2008; SOLAS 2020; DNV Classification Rules).
