# Floatation & Stability

Urban and Civil

Nov 16, 2013 (4 years and 7 months ago)

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Chapter IV

(Ship
Hydro
-
Statics

& Dynamics)

Floatation & Stability

4.1 Important Hydro
-
Static Curves or
Relations

(see Fig. 4.11 at p44 & handout)

Displacement Curves
(displacement [molded, total]
vs. draft, weight [SW, FW] vs. draft (
T
))

Coefficients Curves
(
C
B

, C
M

, C
P

, C
WL
, vs.
T
)

VCB (KB,
Z
B
):
Vertical distance of Center of
Buoyancy (C.B) to the baseline vs.
T

LCB (LCF,
X
B
):
Longitudinal Distance of C.B or
floatation center (C.F) to the midship vs.
T

4.1 Important Hydro
-
Static Curves or
Relations (Continue)

TPI:
Tons per inch vs.
T

(increase in buoyancy due
to per inch increase in draft)

Bonbjean Curves
(p63
-
66)

a) Outline profile of a hull

b) Curves of areas of transverse sections (stations)

c) Drafts scales

d) Purpose: compute disp. & C.B., when the vessel
has 1) a large trim, or 2)is poised on a big wave crest or
trough.

How to use Bonjean Curves

Draw the given W.L.

Find the intersection of the W.L. & each station

Find the immersed area of each station

Use numerical integration to find the disp. and C.B.

4.2 How to Compute these curves

Formulas for Area, Moments & Moments of Inertia

0
0
2 2
2
2
0.0
0
) Area ,
) Moments ,
Center of Floatation /
) Moments of Inertia
,
L
A
L
M
I A
L
C F
a d ydx A ydx
b d xydx M xydx
x M A
c d x d x ydx
I x ydx I I x A
 
 

 
  

Examples of Hand Computation of
Displacement
Sheet

(Foundation for Numerical Programming)

Area, floatation, etc of 24’ WL (Waterplane)

Displacement (molded) up to 8’ WL

Displacement (molded) up to 24’ and 40 ‘WL (vertical
summation of waterplanes)

Displacement (molded) up to 24’ and 40 ‘WL (Longitudinal
summation of stations)

Wetted surface

Summary of results of Calculations

4’wl

area

8’wl

area

16’wl

area

24’wl

area

32’wl

area

40’wl

area

Up to
4’wl

Up to

8’wl

Up to

24’ & 40

wl

Disp. Up
to 24’wl

Disp. Up
to 40’wl

Disp. Up
to 16’wl

Disp. Up
to 32’wl

MTI

MTI

Wetted
surface

Summary

Red sheet will be studied in
detail

1
-
6 Areas & properties (F.C.,
Ic, etc) of W.L

7
-
11 Displacement,
Z
B

, and
X
B

up W.L., vertical
integration.

12
-
15 Transverse station area,
longitudinal integration for
displacement,
Z
B

, and
X
B

16
-
18 Specific Feature (wetted
surface, MTI, etc.

19 Summary

1
0 1
2 3
0 1
2
1 2
2
2
3
The distance
between the
two stations
2
Simpson's 1st
3
2 1
2
3 2
2
2 1
3 4
3
2..
4
2 Symmetric
m S
S
y y
y y
y y
y y
  

  

 

  

 

2
2
3
3
3
4
5 6
Formulas for the remaining coefficents
2
2
3
the distance between the two stations
2
Simpson's 1st rule coeff.; 2 - Symmetr
ic
3
2
2
3
1 2 1
2 ( from )
3 3 3 3
2
2,
3 3
i
m
m S
S
m S
y
m S
h
m S m
  

  
   
   
2
2
2
3 3
h
S
   
Illustration of Table 4
:

C1 Station FP
-
0

AP
-
10 (half station)

C2 Half Ordinate copy from line drawing table ( 24’ WL).

(notice at FP. Modification of half ordinate)

C3 Simpson coefficient (Simpson rule 1) (1/2 because of half station)

C4 = C3 x C2

(area function) displacement

C5 = Arm (The distance between a station and station of 5 (Midship)

C6 = C5 x Function of
Longitudinal Moment

with respect to Midship (or station 5)

C7 = Arm (same as C5)

C8 = C6 x C7 Function of
Longitudinal moment of inertia

with respect to Midship.

C9.= [C2]
3

C10. Same as C3. (Simpson Coeff.)

C11. = C9 x C10. Transverse moment of inertia of WL about its centerline

Table 5 is similar to Table 4, except the additional computation of appendage.

Illustration of

Table 8

For low WLs, their change is large. Therefore, it is first to use planimeter or other
means to compute the half
-
areas of each stations up to No. 1 WL (8’ WL).

C1.

Station

C2.

Half area (ft
2
) of the given station

C3.

C3/(h/3) ( divided by h/3 is not meaningful, because it later multiplying by

h/3) (h = 8’ the distance between the two neighboring WLs)

C4.

½ Simpson’s Coeff.

C5.

C4 x C3

C6.

Arm

distance between this station and station 5 (midship)

C7

C5 x C6

f(M)

Illustration of

Table 9

C1.

WL No.

C2.

f(V) Notice first row up to 8’. f(v)

C3.

Simpson’s coeff.

C4. C2 x C3

C5. Vertical Arm above the base

C6.

C4 x C5.

f(m) vertical moment w.r.t. the Baseline.

*
Notice up the data in the first row is related to displacement up to 8’
WL. The Table just adding V)

Illustration of

Table 12

C1. Station No.

C2. under 8’ WL.

(From Table 8)

C3.

8’ WL x 1

C4.

16’ WL x ¼ (SM 1 + 4 + 1)

C5.

24’ WL x 1

C6.

(C2 + C3 + C4 + C5) Function of Area of Stations

C7.

Arm

(Distance between this station to midship)

C8.

C7 x C6

(Simpson rule)

C9.

C6*h/3

4.3 Stability

A floating body reaches to an
equilibrium

state, if

1) its weight = the buoyancy

2) the line of action of these two forces become
collinear
.

The equilibrium:
stable, or unstable or neutrally stable
.

Stable equilibrium
: if it is slightly displaced from its

Unstable equilibrium
: if it is slightly displaced form its
equilibrium position and tends to move farther away from
this position.

Neutral equilibrium
: if it is displaced slightly from this
position and will remain in the new position.

Motion of a Ship:

6 degrees of freedom

-

Surge

-

Sway

-

Heave

-

Roll

-

Pitch

-

Yaw

Axis

Translation

Rotation

x Longitudinal

Surge Neutral S.

Roll S. NS. US

y Transverse

Sway Neutral S.

Pitch S.

z Vertical

Heave S. (for sub, N.S.)

Yaw NS

Righting & Heeling Moments

A ship or a submarine is designed to float in the
upright
position
.

Righting Moment
: exists at any angle of inclination where
the forces of weight and buoyancy act to move the ship
toward the upright position.

Heeling Moment
: exists at any angle of inclination where
the forces of weight and buoyancy act to move the ship away
from the upright position.

G
---
Center of Gravity, B
---
Center of Buoyancy

M
---

Transverse Metacenter
, to be
defined later
.

If M is above G, we will have a
righting moment
, and

if M is below G, then we have a
heeling moment
.

W.L

For a displacement ship,

For submarines (immersed in water)

G

B

G

If B is above G, we have
righting moment

If B is below G, we have
heeling moment

Upsetting Forces (overturning moments)

Beam
wind
,
wave

&
current

pressure

the harbor.)

Offside weight (C.G is no longer at the center line)

The loss of part of buoyancy due to damage (partially
flooded, C.B. is no longer at the center line)

Turning

Grounding

Longitudinal Equilibrium

For an undamaged (intact) ship, we are usually only
interested in determining the ship’s draft and trim regarding
the longitudinal equilibrium because the ship capsizing in the
longitudinal direction is
almost impossible
. We only study
the
initial stability for the longitudinal equilibrium
.

Static Stability & Dynamical Stability

Static Stability:

Studying the magnitude of the
righting moment given the inclination (angle) of the
ship*.

Dynamic Stability:

Calculating the amount of work
done by the righting moment given the inclination of
the ship.

The study of dynamic Stability is based on the study of
static stability.

Static Stability

1)

The
initial stability

(aka stability at small inclination) and,

2)

the
stability at large inclinations
.

The initial (or small angle) stability:

studies the
right
moments or right arm

at small inclination angles.

The stability at large inclination (angle):

computes the right
moments (or right arms) as function of the inclination angle, up
to a limit angle at which the ship
may lose

its stability
(capsizes).

Hence, the initial stability can be viewed as a special case of the
latter.

Initial stability

Righting Arm:
A symmetric ship is inclined at a small angle
dΦ. C.B has moved off the ship’s centerline as the result of the
inclination. The distance between the action of buoyancy and
weight,
GZ
, is called
righting arm
.

Transverse Metacenter:
A vertical line through the C.B
intersects the original vertical centerline at point,
M
.

sin
if 1
GZ GM d
GMd d
 
  
Location of the Transverse Metacenter

Transverse metacentric height

:
the distance between the
C.G.

and
M

(
GM
). It is important as an index of
transverse stability at small angles of inclination.
GZ
is
positive, if the moment is righting moment.
M

should be
above C.G, if
GZ

>0.

If we know the location of
M
, we may find
GM
, and thus the
righting arm
GZ

or righting moment can be determined
given a small angle dΦ.

How to determine the location of
M
?

When a ship is inclined
at small angle dΦ

WoLo

Waterline (W.L) at upright position

W
1
L
1

Inclined W.L

Bo

C.B. at upright position, B
1

C.B. at inclined position

-

The displacement (volume) of the ship

v
1
, v
2

The volume of the emerged and immersed

g
1
, g
2

C.G. of the

emerged

and
immersed

wedge, respectively

Equivolume Inclination

(
v
1

=v
2

)

If the ship is
wall
-
sided
with the range of inclinations of a small
angle dΦ, then the volume
v
1

and
v
2

, of the two wedges between
the two waterlines will be
same
. Thus, the displacements under
the waterlines WoLo and W
1
L

1

will be same. This inclination is
called
equivolume inclination
.

Thus, the intersection of WoLo,
and W
1
L
1

is at the
longitudinal midsection
.

For most ships, while they may be
wall
-
sided

in the vicinity of
WL near their
midship section
, they are not
wall
-
sided near
their sterns and bows
. However, at a small angle of
inclination, we may still
approximately

treat them as
equivolume inclination
.

When a ship is at
equivolume inclination
,

According to a theorem from mechanics, if one of the bodies
constituting a system moves in a direction, the C.G. of the
whole system moves in the same direction parallel to the shift of
the C.G. of that body. The shift of the C.G. of the system and
the shift of the C.G of the shifted body are in the inverse ratio of
their weights.

1 2
0 1 1 2
,
vg g
B B v v v
  

3
0
0 1
1 2
0
3 3
1 2
0 0 0
0 0
2
3
,
tan( ) tan( )
1 2 2 2
( tan ) (2 ) tan,
2 3 3 3
the moment of inertia of WL w.r.t. the l
ongitudinal axis
L
x
L L L
x
x
y dx
B B I
vg g
B M
d d
vg g y y d y dx d y dx I y dx
I
 
 
   
  
     

  
For a ship inclined at a small angle , t
he location of
its transverse metacenter is approximate
ly above its
C.B. by , which is independent of .
.. (Metacenter measured from keel ), o
r
is
x
M
d
I
d
K M
H

.
the height of metacenter above the basel
ine.
,
where is the vertical coordinates of th
e C.B.
The vertical distance between the metace
nter & C.G,
x
M B
B
x
M G B G
I
K.M. = H = +Z
Z
I
GM H Z +Z Z

   

If we know the vertical position of the
C.G., and the
C.B., the righting arm at small angles
of inclination, ,
and the righting moment is
.
G
B
x
B G
x
w B g
Z
Z d
I
GZ GM d Z Z d
I
M Z Z d

 
      
 

 
 
    
 

 
Examples of

computing
KM

d

B

3
2
2
3
2
2
) Rectangular cross section
1
, ,
2 12
12
12 2
) Triangular cross section
2 1 1
, ,
3 12 2
6
2
6 3
B x
x
B
B x
x
B
a
d
Z I LB LBd
I
B
BM
d
B d
KM BM Z
d
b
d
Z I LB LBd
I
B
BM
d
B d
KM BM Z
d
   
 

   
   
 

   
d

B

Natural frequency of Rolling of A Ship

2
2
2
Free vibration
0
where is the inertia moment of the shi
p w.r.t. C.G.
A large leads to a higher natural freq.
X w
w
X
X
M GM
t
GM
M
M
GM

 
    

 

4.4Effects of free surfaces of
liquids on the righting arm

pp81
-
83

When a liquid tank in a ship is not full,

there is a free surface in this tank.

The effect of the free surface of liquids

on the initial stability of the ship is to

decrease the righting arm.

For a small parallel angle inclination,

the movement of C.G of liquid is

0 1
tan
OL
k
I
G G d

The increase in the
heeling moment

due to the movement
of C.G. of liquid

tan 0 1
heeling F k F OL
M G G I d
  
   
If there is no influence of free
-
surface liquids, the
righting
moment

of the ship at a small angle

is:

ox
w B g w
I
M GM d Z Z d
   
 
        
 

 
In the presence of a free
-
surface liquid, the righting moment
is decreased due to a heeling moment of free
-
surface liquid.
The reduced
righting moment

M’

is

ox ol
F
heeling w B g
w
I I
M M M d Z Z

 

 

      
 
 
 
The reduced metacentric height
GM’:

OX F OL
B g
w
I I
GM Z Z

   
 
Comparing with the original
GM
, it is decreased by an
amount
,

.
OL
F
w
I

The decrease can also be viewed as an increase in height
of C.G. w.r.t. the baseline.

OL
F
g g
w
I
Z Z

 

How to decrease I
OL
:

Longitudinal subdivision: reduce the width
b,
and

thus reduces

Anti rolling tank

3
OL
I b l

4.5 Effects of a suspended
weight on the righting
arm

When a ship inclines at a
small

angle

, the
suspended object

moves transversely

Transverse movement of the weight
=
h dΦ

, where
h

is the distance
between the suspended weight and
the hanging point

The increase in the heeling moment
due to the transverse movement

heeling
M w h d

  
In the presence of a suspended object, the righting moment &
righting arm are decreased due to a heeling moment of the
suspended object. The reduced
righting moment

M’

&
metacentric height

GM’

are:

ox
heeling w B g
w
ox
B g
w w
I
w
M M M d Z Z h
I
w w
GM GM h Z Z h
 

 
 

      
 
 
 

     
  
In other words, the C.G of a suspended object is actually at its
suspended point

Because the suspension weights & liquid with free
surface tend to
decrease the righting arm
, or
decrease the
initial stability
, we should
avoid

them.

1.

Filling the liquid tank (in full)

to get rid of the
free surface. (creating a expandable volume)

2.

Make the
inertial moment of the free surface as
small as possible by adding the

separation

3.
Fasten the weights

to prevent them from moving
transversely.

4.6 The Inclining Experiment (Test)

Purpose

1.
To obtain the
vertical position of C.G

(Center of Gravity) of the ship.

2.
It is required by “International convention
on Safety of Life at Sea.” (Every
passenger or cargo vessel
newly built or
rebuilt
)

4.6 The Inclining Experiment (Continue)

Basic Principle

M:
Transverse
Metacenter (
A vertical
line through the C.B
intersects the original
vertical centerline at
point,
M
)

Due to the movement of
weights, the heeling
moment is

heeling
M wh

where
w

is the
total weight

of the moving objects and
h

is the
moving distance.

4.6 The Inclining Experiment (Continue)

The shift of the center of gravity is

where
W
is the total weight of the ship.

The righting moment = The heeling moment

1
wh
GG
W

1
tan cot( )
tan( )
wh
GM W wh GM GG
W
 

     

1.

w

and
h

are recorded and hence known.

2.

is measured by a pendulum known as stabilograph.

3.

The total weight
W

can be determined given the draft T. (at
FP, AP & midship, usually only a
very small trim

is allowed.)

4.

Thus
GM

can be calculated,

4.6 The Inclining Experiment (Continue)

,
x
M g M B
I
GM H Z H Z
   

The metacenter height and vertical coordinate of C.B have
been calculated. Thus, C.G. can be obtained.

g M
Z H GM
 
Obtaining the longitudinal position of the gravity center of
a ship will be explained in section 4.8.

4.6 The Inclining Experiment (Continue)

.

1.
The experiment should be carried out in
calm water & nice weather
. No
wind, no heavy rain, no tides.

2.
It is essential that the ship
be free to incline

(mooring ropes should be as
slack as possible, but be careful.)

3.
All weights capable of moving transversely should be
locked in position

and there should be
no loose fluids in tanks
.

4.
The ship in inclining test should be as
near completion

as possible.

5.
Keep as few people on board as possible.

6.
The angle of inclination should be
small enough

with the range of validity
of the theory.

7.
The ship in experiment
should not have a large trim.

4.7 Effect of Ship’s Geometry on Stability

Transverse metacenter height
GM = BM

(Z
G

Z
B
)

3 2 2
1 1
1
( )
where dpends on waterplane.
2
x
g B
x
B B
x
x
I
GM Z Z
I C LB C B B
C C
C LBT C T T
I
d
dB dT
I
B T
  

  

 

 

 

4.7 (Continue)

a) Increase only: 2 ( increases)
b) Decrease T only: ( decreases)
c) Change B & T but keep fixed:

x
B
x
x
x
x
x
I
d
dB
B C LBT
I
B
I
d
dT
I
T
dB dT
B T
I
d
I
 
 

 
  
 
 

 
 
 

 
  
 
 

 
  
 
 

 
 
 

 
3
dB
B

Conclusion
:
to increase
GM

(
Transverse metacenter
height
)

1.
increasing the beam,
B

2.
decreasing the draft,
T

3.
lowering
C.G (Z
G
)

4.
increasing the
freeboard

will increase the
Z
G
, but will
improve the
stability at

large inclination angle
.

5.
Tumble home or flare will have effects on the stability at
large inclination angle.

6.
Bilge keels
,
fin stabilizers
, gyroscopic stabilizers,
anti
-
rolling tank

also improve the stability (at pp248
-
252).

4.7 (Continue)

Suitable metacenter height

It should be large enough to satisfy the requirement of
rules.

GM
~0.04B.

However, too large
GM

will result in a very small rolling
period. Higher rolling frequency will cause the crew or
passenger uncomfortable. This also should be avoided.

(see page 37 of this notes)

4.8
Longitudinal Inclination

Longitudinal Metacenter:
Similar to the definition of the
transverse meta center, when a ship is inclined longitudinally
at a small angle, A vertical line through the center of
buoyancy intersects the vertical line through (before the
ship is inclined) at .

1
B
0
B
L
M

The Location of the Longitudinal Metacenter

For a small angle inclination
,

volumes of forward wedge immersed
in water and backward wedge emerged out of water are:

1
0
2 1 2
0
0 0
(2 ) ( tan ) where is the half breadth.
(2 ) ( tan ), .
Thus, 2 2, which indicates:
moment of area forward of = moment of a
rea after .
is the cente
l
L l
l L l
v y x dx y
v y x dx v v v
yxdx yxdx
F F
F

 
   

 
0 0
0 0
r of (mass) gravity of waterline , &
is called of . Therefore, for equal
volume longitudinal inclination the new
waterline always
passes through the (C.F
W L
W L
center of flotation
center of flotation
).

Location of the Longitudinal Metacenter

Using the same argument used in obtaining transverse metacenter
.

0 1 1 2
0
1 2
0
0
2 2
0
0 1 0
/,
(2 ) ( tan ) 2 tan
tan 2 2 tan
is the moment of inertia with respect t
o the
transverse axis passing the center of fl
otation.
t
l
L l
l
FC
L l
FC
B B vg g
vg g y x xdx yx xdx
yx dx yx dx I
I
B B B M
 
 

 
   
 
  
 
 

 
 
0
0
an, .
.
FC
FC FC
ML B B L B g
I
B M
I I
H B M Z Z GM Z Z

      
 

Location of the Longitudinal Metacenter

2
0,
0,
Usually Floatation Center (C.F) of a wat
erplane is not at the
midship,
,
where is the moment of inertia w.r.t. t
he transverse axis
at midship (or station 5) and is the d
istance from F
FC T
T
I I Ax
I
x
 
.C. to
the midship.

Moment to Alter Trim One Inch (MTI)

MTI: (moment to alter (change) the ship’s trim per inch) at
each waterline (or draft) is an important quantity. We may
use the longitudinal metacenter to predict MTI

MTI ( a function of draft)

Due to the movement of a weight, assume that the ship as 1” trim,
and floats at waterline W.L.,

0 1
0 1
0 1 0
1"1
tan, where is in feet.
12
Due to the movement of the weight, moves
to ,
,
tan tan
tan tan
1
12
w
L L G
FC
w L G w B G
FC
w B G
L
L L
G G
M w h G G
G G G M HM Z
I
M HM Z Z Z
I
Z Z
L

 
   

 

     
  
 
      
 

 
 
   
 
 
 

MTI ( a function of draft)

If the longitudinal inclination is small,
MTI

can be used to find
out the
longitudinal

position of gravity center ( ).

3
1
,
12 12
64 lb/ft, Long Ton = 2240 lb, MTI (ton
-ft)
420
FC FC w FC
G B w
FC
w
I I I
Z Z M
L L
I
L

    
   
 
1
G
X
0 1
0 0 0
1 0 1
Trim
tan
tan,
Since is in the same vertical line as .
under ,
F A
FC FC FC
B G G B
FC
G B B
T T
T
L L L
I I I
T
G G Z Z Z Z
L
G C B W L
I
T
X X G G X
L

  

 
    
 
  
 

   