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Mariner's Formulae Quick reference guide
Transverse Stability
 
STABILITY NOMENCLATURE
  1. K = Keel
  2. G = Center of gravity
  3. B = Center of Buoyancy
  4. M = Metacentre
  5. ø = Angle of Heel
  6. BM = Metacentric Radius
  7. GM = Metacentric Height
  8. GZ = Righting Lever measured from G
  9. KB = Height of Center of Buoyancy from keel
  10. KG = Height of Center of Gravity from keel
  11. 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

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