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

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)

3

1. δLat = difference in Latitude between the 2 points, N or S

2. δLong = difference in Longitude between the 2 points, E or W

3. θm = mid-Latitude

4. θc = Course

5. D = Distance

6. p = Departure

COURSE

tan θc = cos θm x δLong

δLat

DISTANCE

Distance = 1

cos θc

ARRIVAL POSITION

Difference of Latitude = D x cos θc

Difference of Longitude = D x sin θc

cos θm

4.

PARALLEL OPERATION OF 2 PUMPS

- with single common Intake and Discharge

Total combined Discharge Head or Pressure

= ½ (Discharge Head at 1st Pump + Discharge Head at 2nd Pump)

Total combined Discharge Capacity

= (Capacity at 1st Pump + Capacity at 2nd Pump)

SERIES OPERATION OF 2 PUMPS

- with single common Intake and Discharge

Total combined Discharge Head or Pressure

= (Discharge Head at 1st Pump + Discharge Head at 2nd Pump)

Total combined Discharge Capacity

= ½(Capacity at 1st Pump + Capacity at 2nd Pump)

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5. HYDROSTATIC PRESSURE

Pressure at any point in a fluid = ρ x g x h [KN/m2]

where ρ = density of liquid [tonnes/metre3]

g = 9.81 [metre/second2]

h = distance of the point from liquid surface [metre]

HYDROSTATIC LOAD

(1) Load acting on an immersed plate = Pressure x Area

= (ρ x g x h) x A [KN]

where ρ = density of liquid [tonnes/metre3]

g = 9.81 [metre/second2]

h = centroid of the immersed area from liquid surface* [metre]

A = immersed area of the plate [metre2]

(2) Load taken by a stiffener of an immersed plate = ρ x g x h x A [KN]

where ρ = density of liquid [tonnes/metre3]

g = 9.81 [metre/second2]

h = centroid of the immersed panel area from liquid surface

= depth of immersion of the plate divided by 2 [metre]

A = immersed area of the rectangular panel plate supported by

the stiffener

= width of panel x depth of immersion [metre2]

CENTRE OF PRESSURE

- the point of an immersed plate at which the resultant hydrostatic load acts.

Centre of pressure from liquid surface

= 2nd moment of area of immersed area about surface

1st moment of immersed area about surface

= I + A(h)2 [metre]

A(h)

where I = I NA [metre4]

= 2nd moment of area of the immersed area about

the neutral axis which is parallel to the liquid surface

A = immersed area [metre2]

h = position of the neutral axis from the surface [metre]

= centroid of the immersed area from the liquid surface

CALCULATION OF CENTROID

Centroid of immersed area from liquid surface = Σ (A x y)

ΣA

where Σ(A x y) = Moment of area about liquid surface

= (A1.y1 + A2.y2 + A3.y3 + ....)

yn = distance of centroid of each immersed area, An

from the liquid surface

ΣA = total immersed area

6. SLIP

Slip = 100% - Efficiency

Efficiency = observed speed or distance

engine speed or distance

Mean Apparent Slip = distance run by propeller - distance run by ship

per Day distance run by propeller

Dist. run by propeller in n.m. = pitch [m] x total engine revolution per day

1852

6.SWL, MSL and Breaking Strain

BREAKING STRAIN

For Manilla, Breaking Strain = Circumference2 [tonnes]

2.5

For Wire, Breaking Strain = Circumference2 x 2.5 [tonnes]

SAFE WORKING LOAD

SWL = Breaking Strength

Safety Factor

MAXIMUM SECURING LOAD (MSL)

Material MSL

Shackles, rings, deck eyes, turnbuckles made of mild steel 50% of Breaking Strength

Fibre rope 33% of Breaking Strength

Wire rope (single use) 80% of Breaking Strength

Wire rope (re-useable) 30% of Breaking Strength

Steel band (single use) 70% of Breaking Strength

Chains made of mild steel 50% of Breaking Strength

Chains made of high tensile steel 33% of Breaking Strength

In a combination of securing gear with different MSL for lashing, the overall lashing strength would equal the weakest link used, i.e. the gear with the smallest MSL.

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7. PROPELLER PITCH

- the distance advanced by one turn of the propeller

Pitch = 2 x π x r x y [metre]

x