Ship Stability and Hydrostatics

Weight change effects on draft and centre of gravity

When a weight w is added to a floating ship at a point with height z above the keel and longitudinal position x from midship, the displacement becomes \Delta' = \Delta + w. The new centre of gravity is the mass-weighted average of the old and new positions, so KG' = (\Delta\,KG + w\,z)/\Delta' and LCG' = (\Delta\,LCG + w\,x)/\Delta', with aft positions taken as negative. The extra weight is carried by extra buoyancy, so the ship sinks until the additional displaced volume equals w/\rho. For a small parallel sinkage the change in mean draft is \delta d = w/\text{TPC}, where \text{TPC} = \rho A_{wp}/100 gives tonnes per centimetre when \rho is in t/m^3 and A_{wp} is in m^2. A body that is partly immersed when attached to the hull contributes only its net load. Its net addition to the ship is its mass minus the buoyancy of the volume it displaces, w_{net} = m - \rho V. For a box-shaped pontoon, the mass of the displaced water equals the total floating mass, so comparing two loading states of the same hull eliminates the unknown hull mass and allows the draft or side length to be found.

Ship subdivision and floodable length

Subdivision divides the ship's interior into watertight compartments by transverse and longitudinal bulkheads, so that flooding after damage is confined to the damaged spaces. This keeps the ship afloat and preserves its residual stability in damaged conditions. Watertight bulkheads also act as stiffeners, which increases hull girder strength and resists local buckling. Separate compartments allow different cargo types to be carried in segregated spaces. Intact stability, however, depends on the hull form and the vertical position of the centre of gravity, so subdivision does not reduce it. The margin line is a line drawn 76 mm below the upper surface of the bulkhead deck at the side of the ship, parallel to the deck. The floodable length at a point is the greatest length of a compartment that can be flooded while the margin line stays above the waterline. For a barge with fully loaded end compartments, there is no free surface, so the transverse metacentric height is GM = KB + BM - KG with KG found from the symmetric arrangement of the cargo.

Centroid and buoyancy of submerged body

The buoyant force on a submerged or floating body equals \rho g \nabla, where \nabla is the displaced volume. Its line of action passes through the centroid of the displaced volume, called the centre of buoyancy B. For a floating body the stiffness in heave is k = \rho g A_{wp}, and the equation of motion uses the effective mass M + a_{33}, where a_{33} is the added mass. With no damping, the steady-state amplitude under a harmonic force F_0\sin\omega t is X = (F_0/k)/|1 - r^2|, with frequency ratio r = \omega/\omega_n and \omega_n = \sqrt{k/(M + a_{33})}. A body of revolution about the x axis, such as a streamlined underwater vehicle, is symmetric about its axis, so many added-mass coupling terms vanish. Terms such as surge–pitch A_{15} and roll–yaw A_{46} are zero for such a body. Fore–aft asymmetry, for example a semi-ellipsoidal bow with a hemispherical stern, couples heave with pitch and sway with yaw, so A_{35} and A_{26} can be nonzero. The block coefficient is C_B = \nabla/(LBT), and the prismatic coefficient satisfies C_P = C_B/C_M, where C_M is the midship section coefficient.

Metacentric radius and waterplane inertia

The initial metacentric radius is BM = I/\nabla, where I is the second moment of the waterplane area about the longitudinal centreline for transverse stability and \nabla is the displaced volume. The metacentre height is KM = KB + BM. For a rectangular waterplane of length L and breadth B, I = LB^3/12, and for a rectangular prism the centre of buoyancy lies at mid-draft, so KB = T/2. Separating the two demi-hulls of a catamaran increases I through the parallel axis theorem, I = \sum (i + a x^2), where i and a are the second moment and area of each demi-waterplane and x is its offset from the centreline. Because the displacement is unchanged, the metacentric radius is larger than that of a mono-hull with the same waterplane area. For a hull with a triangular cross-section, the waterline breadth and immersed area both vary with draft, so BM must be expressed as a function of draft. The minimum draft for stability is then the draft at which KM = KG.

Free surface effect on stability

A liquid in a partly filled tank has a free surface that tilts as the ship heels, which shifts the centre of gravity of the liquid and produces a virtual rise of the centre of gravity. The apparent rise is GG' = \rho_l \sum i / \Delta, where i is the second moment of each free surface about its own longitudinal centroidal axis, \rho_l is the density of the liquid and \Delta is the displacement in mass units. The corresponding virtual reduction of metacentric height is \delta GM = \rho_l \sum i/\Delta. For transverse stability in a rectangular tank, i = l\,b^3/12, where l is the longitudinal length and b is the transverse breadth. Because b is cubed, a tank with a small athwartship breadth has a small free surface effect. A rectangular tank has the same free surface area whatever the fill depth, so partial filling of one tank gives the same free surface effect as a full tank of the same plan area. Splitting a given liquid volume between tanks brings more free surfaces into service, and their moments add. When the liquid density equals the water density, the density terms cancel and \delta GM = i/\nabla.

Ship hydrostatic particulars

The hydrostatic particulars of a ship are tabulated against draft. The tonnes per centimetre immersion (TPC) is the mass that changes the mean draft by 1 cm, and the moment to change trim by one centimetre (MCT) is the moment that alters the trim by 1 cm. The longitudinal centre of buoyancy (LCB) and the longitudinal centre of flotation (LCF) locate the trim-producing buoyancy and the axis about which the waterplane pivots. When a ship passes from fresh water to sea water with unchanged displacement, the submerged volume decreases by \Delta(1/\rho). The mean draft falls by this volume divided by the waterplane area. If the ship is on an even keel and the LCB coincides with the LCF, no trimming moment arises, so both ends rise by the same amount. The block coefficient is C_B = \nabla/(LBT), so for a given displaced volume the relations between L, B and T follow directly from the definition when the proportions are specified.

Angle of loll on GZ curve

A ship with negative initial metacentric height has a negative righting lever near the upright position, so the GZ curve starts below the axis. The angle of loll is the heel angle at which the GZ curve crosses zero from negative to positive, marking a stable equilibrium at a finite heel angle. A ship at this angle is stable, but it is not stable when upright. The angle of vanishing stability is a different point, located on the right-hand side of the curve where GZ returns to zero after its positive range. Deck edge immersion and trim are not identified by a zero crossing of the GZ curve.