Transverse Stability
STABILITY NOMENCLATURE K = Keel G = Center of gravity B = Center of Buoyancy M = Metacentre ø = Angle of Heel BM = Metacentric Radius GM = Metacentric Height GZ = Righting Lever measured from G KB = Height of Center of Buoyancy from keel KG = Height of Center of Gravity from keel KM = Height of Metacenter from keel KM = KG + GM KM = KB + BM GZ = KN - KG x sin ø where KN can be found from KN curves Righting Moment = Δ x GZ where Δ = displacement CALCULATION OF KG KG = Total Vertical Moment of Weights about keel [metre.tonnes]                                        Δ [tonnes] GG1 = Moment of Weight,W shifted over Distance, D [metre.tonnes]                                        Δ [tonnes]        : vertical shift of G CALCULATION OF KM KM = KB + BM KB = 0.53 x Draft   [metre] BM =    2nd moment of waterplane area       =   I   [metre]                 volume of displacement                    V               where I = L x B3  [metre4]  for a rectangular barge                                12 CALCULATION OF GM GM = KM - KG VIRTUAL LOSS OF GM DUE TO FREE SURFACE GGv =          s.g. of Liquid in the Tank             x    I     x    1             s.g. of Water in which vessel floats           V          n2 where GGv = virtual rise in G or deduction in G               I =  2nd moment of the free surface about the centre line                 =  L x B3  [metre4]  for a rectangular compartment                       12               L = Length of the Tank [metre]               B = Breadth of the Tank [metre]               V = Volume of the Tank [metre3]               n = number of longitudinal compartments into which the tank is                     subdivided LARGE ANGLE STABILITY - where the force of buoyancy can no longer be considered to act vertically upwards through the initial metacentre, M GZ = (GM + ½BM.tan2 ø) sin ø       : for ship's side that are "WALL-SIDED" in the vicinity of the water line Note:-   for small angle stability, the term ( ½BM.tan2 ø) becomes negligible             ∴     GZ = GM.sin ø
CHANGE OF TRIM - the difference between initial trim and final trim i.e. change of draught forward + change   of draught aft Trimming Moment = Weight x Distance shifted = W x d [tonnes.metre] Change of Trim, t = Trimming Moment [metre]                                100 x MCT.1cm MCT.1cm = Moment To Change Trim by 1 cm               = Δ x GM L [tonnes.metre]                   100 x L               ≅ Δ x BM L [as GML is small when compared with BML]                   100 x L                where Δ = displacement [tonnes]                      GM L = Longitudinal Metacentric Height                      BM L = height of the longitudinal metacentre, ML                                above centre of buoyancy, B GML = KB + BML - KG [metre] where BML = long. 2nd mmt of waterplane about centre of flotation, F                                               volume of displacement                  =  I L [metre]                      V Change of draught aft, ta =  l a x change of trim [metre]                                      L Change of draught forward, tf =  l f x change of trim [metre]                                             L Change in mean draught = Weight loaded or discharged [metre]                                                     TPC TPC = Tonnes per Centimetre Immersion        = Aw x ρ             100          where Aw = area of waterplane [metre2]                         = L x B x Cw (waterplane area coefficient)                      ρ = density of sea water [tonnes per metre3] LARGE CHANGE IN DISPLACEMENT Trimming Moment = Δ x (longitudinal separation LCG and LCB)            where LCG = Longitudinal centre of gravity [metre]                      LCB = Longitudinal centre of buoyancy [metre] CHANGE IN DENSITY Change in Mean draught due to change in density =   Δ   x (ρ1 - ρ2)                                                                             Aw     (ρ1.ρ2) Trimming Moment = Δ x (horizontal shift of LCB)      or (mass of layer of water added or removed due to change in density) x           (horizontal distance between initial LCB & final LCF of waterplane)