Necessary formulas for ship stability exam
Vertical Center of Gravity
KG = VCG = ( ΣMZi + ΣMZj ) / D
Vertical Center of Buoyancy obtained by Displacement from Hydrostatic Properties Table
VCB
Transversal Metacentric Radius obtained by Displacement from Hydrostatic Properties Table
TBM
Height of Metacentre
KM = VCB + TBM
Metacentric Height
GM = KM – VCG = (VCB + TBM) – KG = 9,951 – 5,19 = 4,76 m
The Formula of Righting Lever
GZi = KNi – KG * sinΘi
Righting Lever Curve
Θi 0º 10º 20º 30º 40º 50º 60º 70º 80º
sineΘi 0
KG * sin Θ 0
KNi 0
GZi 0
KNi taking from STABILITY CROSS CURVES (PANTACARENAS)
A0º-30º = The Area of Triangle = 0,5 · ac · cb
A0º-40º = The Area of Triangle = 0,5 · ae· ed
A30º-40º = (A0º-40)º - (A0º-30º)
(According to the International Code on Intact Stability,
2008, the following Criteria are mandatory for Passenger
and Cargo Ships )
(According to the International Code on Intact Stability, 2008, the following Criteria are mandatory for Passenger
and Cargo Ships )
(The area under the righting lever curve (GZ curve) should not be less than A 0º - 30º > 0,055
1 0,055 metre-radians up to 30° angle of heel.) metre-radians
X
(The area under the righting lever curve (GZ curve) should not be less than A 0º - 40º > 0,090
2 0,09 metre-radians up to 40° angle of heel or the angle of downflooding if metre-radians
this is less than 40°.) X
(The area under the righting curve between the angles of heel of 30° and 40° A 30º - 40º >
3 or between 30° and the angle of downflooding if this angle is less than 40°, 0,030
should not be less than 0,03 metre-radians.)
metre-radians
X
(The righting lever GZ should be at least 0,20 m at an angle of heel equal to GZ > 0,2 m.
4 or greater than 30°.) Θ > 30º
X
5 (The maximum righting arm GZmax should occur at an angle of heel Θmax > 25º
preferably exceeding 30° but not less than 25°.) X
6 (The initial metacentric height GM should not be less than 0,15 m.) GM > 0,15 m.
X
7 (Severe wind and rolling criterion (weather criteria) K not less than 1) K>1
X
make a dynamic stability diagram.
Θi 0º 10º 20º 30º 40º 50º 60º 70º 80º
sinΘi 0
KG * sin Θ 0
KNi 0
GZi 0
∑∑lct(integralines 0
kreives)
Ld=0,0872*∑∑lct 0
Find Minimum dynamic upsetting moment and dynamic upsetting angel
Calculate weather criteria
1. lw1 – Heeling Moment Lever caused by Steady Wind (in Drawing from Port Side)
given by the following Formula
lw1 = P * A * (Z / W)
P = 0,0514 (t/m2)
S – Underwater Projected Lateral Area of Hull
S=d·L
A – Projected Lateral Area of Hull, Superstructure and cargoes on Deck above
Waterline WL
A = At – S
At taking from ship particulars
Tl – The Breadth of Lateral Area of Hull, Cargoes and Superstructure above
Waterline WL Projection
Tl = A / L
Z – Vertical Distance between the Center of Area A and the Center of Underwater
Projected Lateral Area of Hull
Z = d/2 + Tl/2
lw2 = 1.5 x lw1
Θo – Angle of Heel under Action of Steady Wind.
Θl – Angle of Roll to Windward due to Wave Action
Θl = 109 · K · X1 · X2 · √r · s
Koeficiento X1 reikšmės
B/d < 2,5 2,6 2,7 2,8 2,9 3,0 3,1 3,2 3,3 3,4 >
2,4 3,5
X1 1,0 0,98 0,9 0,95 0,93 0,91 0,9 0,88 0,86 0,84 0,82 0,8
6
X2 we are finding from Cb = W / (1,025 * L * B * d )
Koeficiento X2 reikšmės
Cb < 0,45 0,5 0,55 0,6 0,65 > 0,7
X2 0,75 0,82 0,89 0,95 0,97 1,0
To find : r = 0,73 + 0,6 * ( OG / d )
OG = KG – d = KG – Tm.
Koeficiento s reikšmės
T <6 7 8 12 14 16 18 > 20
s 0,100 0,098 0,093 0,065 0,053 0,044 0,038 0,035
T = 2*B*C / √2 GM
C = 0,373 + 0,023 * B/d – 0,043 * L/100
Θr = Θo – Θl
Md.max. = GZd.max. · D
Msq. = lw2 · D
K = Md.max. / Msq
