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Chapter IV

(Ship Hydro-Statics & Dynamics)Floatation & Stability

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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 (CB , CM , CP , CWL, vs. T)

VCB (KB,ZB): Vertical distance of Center of

Buoyancy (C.B) to the baseline vs. T

LCB (LCF,XB): Longitudinal Distance of C.B or

floatation center (C.F) to the midship vs. T

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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 scalesd) Purpose: compute disp. & C.B., when the vessel

has 1) a large trim, or 2)is poised on a big wave crest or

trough.

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How to use Bonjean CurvesDraw 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.

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4.2 How to Compute these curves

Formulas for Area, Moments & Moments of Inertia

0

0

2 2

22

0 . 00

) 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 Ac d x d x ydx

I x ydx I I x A

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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

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4wl

area

8wl

area16wl

area

24wl

area

32wl

area40wl

area

Up to

4wl

Up to

8wl

Up to

24 & 40

wl

Disp. Up

to 24wl

Disp. Up

to 40wl

Disp. Up

to 16wl

Disp. Up

to 32wl

MTI

MTI

Wettedsurface

Summary

Red sheet will be studied in

detail

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

7-11 Displacement,ZB , and

XB up W.L., vertical

integration.

12-15 Transverse station area,

longitudinal integration for

displacement,ZB , andXB

16-18 Specific Feature (wetted

surface, MTI, etc.

19 Summary

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1

0 1

2 3

0 12

1 2

22

3

The distance

between the

two stations

2Simpson's 1st

3

2 12

3 2

2

2 1

3 4

32 ..

4

2 Symmetric

m S

S

y y

y y

y y

y y

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2

2

3

3

3

4

5 6

Formulas for the remaining coefficents

2

23

the distance between the two stations

2Simpson's 1st rule coeff.; 2 - Symmetric

32

23

1 2 12 ( from )3 3 3 3

22,

3 3

im

m S

S

m S

ym S

hm S m

22 2

3 3

hS

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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 ofLongitudinal Moment with respect to Midship (or station 5)

C7 = Arm (same as C5)

C8 = C6 x C7 Function ofLongitudinal 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.

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Illustration ofTable 9

C1. WL No.

C2. f(V) Notice first row up to 8. f(v)

C3. Simpsons 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)

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Illustration ofTable 12

C1. Station No.

C2. under 8 WL. (From Table 8)

C3. 8 WL x 1

C4. 16 WL x (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

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4.3 Stability

A floating body reaches to an equilibrium state, if1) 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

equilibrium position and will return to that position.

Unstable equilibrium: if it is slightly displaced form its

equilibrium position and tends to move farther away fromthis position.

Neutral equilibrium: if it is displaced slightly from this

position and will remain in the new position.

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Motion of a Ship:

6 degrees of freedom

- Surge

- Sway

- Heave

- Roll

- Pitch

- Yaw

Axis Translation Rotationx Longitudinal Surge Neutral S. Roll S. NS. USy Transverse Sway Neutral S. Pitch S.z Vertical Heave S. (for sub, N.S.) Yaw NS

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Righting & Heeling Moments

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

position.

Righting Moment: exists at any angle of inclination wherethe 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 awayfrom the upright position.

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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,

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For submarines (immersed in water)

G

B

G

If B is above G, we have righting momentIf B is below G, we have heeling moment

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Upsetting Forces (overturning moments)

Beam wind, wave & current pressure

Lifting a weight (when the ship is loading or unloading in

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

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Longitudinal Equilibrium

For an undamaged (intact) ship, we are usually onlyinterested in determining the ships 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.

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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.

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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.

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Initial stability Righting Arm: A symmetric ship is inclined at a small angle

d. C.B has moved off the ships centerline as the result of the

inclination. The distance between the action of buoyancy andweight, 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

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Location of the Transverse Metacenter

Transverse metacentric height : the distance between the

C.G. andM(GM). It is important as an index of

transverse stability at small angles of inclination. GZ is

positive, if the moment is righting moment. Mshould be

above C.G, ifGZ >0.

If we know the location ofM, we may find GM, and thus the

righting arm GZ or righting moment can be determined

given a small angle d.

How to determine the location ofM?

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When a ship is inclined

at small angle d

WoLoWaterline (W.L) at upright position

W1L1Inclined W.L

BoC.B. at upright position, B1C.B. at inclined position

- The displacement (volume) of the ship

v1, v2The volume of the emerged and immersed

g1, g2C.G. of the emerged and immersed wedge, respectively

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Equivolume Inclination (v1 =v2 )

If the ship is wall-sided with the range of inclinations of a smallangle d, then the volume v1 and v2 , of the two wedges between

the two waterlines will be same. Thus, the displacements under

the waterlines WoLo and W1L 1 will be same. This inclination is

called equivolume inclination. Thus, the intersection of WoLo,

and W1 L1 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 ofinclination, we may still approximately treat them as

equivolume inclination.

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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 andthe shift of the C.G of the shifted body are in the inverse ratio of

their weights.

1 20 1 1 2,

vg gB B v v v

3

00 1 1 20

3 3

1 20 0 0

0 0

2

3 ,tan( ) tan( )

1 2 2 2( tan ) (2 ) tan ,

2 3 3 3

the moment of inertia of W L w.r.t. the longitudinal axis

L

x

L L L

x

x

y dxB B Ivg g

B M d d

vg g y y d y dx d y dx I y dx

I

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For a ship inclined at a small angle , the location of

its transverse metacenter is approximately above its

C.B. by , which is independent of .

. . (Metacenter measured from keel ), or

is

x

M

d

I

d

K M

H

.

the height of metacenter above the baseline.

,

where is the vertical coordinates of the C.B.

The vertical distance between the metacenter & C.G,

xM B

B

xM G B G

IK.M. = H = + Z

Z

IGM H Z + Z Z

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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

xB G

x

w B g

Z

Z d

IGZ GM d Z Z d

I

M Z Z d

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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

dZ I LB LBd

I BBM

d

B dKM BM Z

d

b

dZ I LB LBd

I BBM

d

B dKM BM Z

d

d

B

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Natural frequency of Rolling of A Ship

2

2

2

Free vibration

0

where is the inertia moment of the ship w.r.t. C.G.

A large leads to a higher natural freq.

X w

w

X

X

M GMt

GMM

M

GM

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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 liquidson 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

IG G d

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The increase in the heeling moment due to the movement

of C.G. of liquid

tan 0 1heeling F k F OLM G G I d If there is no influence of free-surface liquids, the righting

moment of the ship at a small angle d is:

oxw B g w

IM 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 momentMis

ox olFheeling w B g

w

I IM M M d Z Z

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The reduced metacentric height GM:

OX F OL

B gw

I IGM Z Z

Comparing with the original GM, it is decreased by an

amount,.OLF

w

I

The decrease can also be viewed as an increase in height

of C.G. w.r.t. the baseline.OLF

g g

w

IZ Z

How to decrease IOL:Longitudinal subdivision: reduce the width b, andthus reduces

Anti rolling tank

3

OLI b l

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4.5 Effects of a suspended

weight on the righting

arm

When a ship inclines at a small

angle d, the suspended object

moves transversely

Transverse movement of the weight

= h d , where h is the distance

between the suspended weight andthe hanging point

The increase in the heeling moment

due to the transverse movement heelingM w h d

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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 momentM &

metacentric heightGMare:

oxheeling w B g

w

oxB g

w w

I wM M M d Z Z h

Iw wGM GM h Z Z h

In other words, the C.G of a suspended object is actually at itssuspended point

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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 theseparation

longitudinal plates (bulkhead).

3. Fasten the weights to prevent them from moving

transversely.

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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 orrebuilt)

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4.6 The Inclining Experiment (Continue)

Basic Principle

M: Transverse

Metacenter (A verticalline through the C.B

intersects the original

vertical centerline at

point, M)

Due to the movement of

weights, the heeling

moment is

heelingM wh

where w is the total weight of the moving objects and h is the

moving distance.

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The shift of the center of gravity is

where Wis the total weight of the ship.

The righting moment = The heeling moment

1

whGG

W

1tan cot( )tan( )

whGM W wh GM GGW

1. w and h are recorded and hence known.

2. is measured by a pendulum known as stabilograph.3. The total weight Wcan be determined given the draft T. (at

FP, AP & midship, usually only a very small trim is allowed.)

4. Thus GMcan be calculated,

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, xM g M BIGM H Z H Z

The metacenter height and vertical coordinate of C.B have

been calculated. Thus, C.G. can be obtained.

g MZ H GM

Obtaining the longitudinal position of the gravity center of

a ship will be explained in section 4.8.

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.

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 positionand 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.

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4.7 Effect of Ships Geometry on Stability

Transverse metacenter height GM = BM

(ZGZB)

3 2 2

1 11

( )

where dpends on waterplane.

2

xg B

x

B B

x

x

IGM Z Z

I C LB C B BC C

C LBT C T T

IddB dT

I B T

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4.7 (Continue)

a) Increase only: 2 ( increases)

b) Decrease T only: ( decreases)

c) Change B & T but keep fixed:

x

Bx

x

x

x

x

Id

dBB C LBT

I B

Id

dT

I T

dB dT

B TI

d

I

3dB

B

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Conclusion: to increase GM( Transverse metacenter

height)

1. increasing the beam,B

2. decreasing the draft, T

3. lowering C.G (ZG)

4. increasing the freeboard will increase theZG, but will

improve the stability atlarge 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).

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4.7 (Continue)

Suitable metacenter height

It should be large enough to satisfy the requirement of

rules.

Usually under full load condition, GM~0.04B.

However, too large GMwill 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)

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4.8 Longitudinal InclinationLongitudinal Metacenter: Similar to the definition of the

transverse meta center, when a ship is inclined longitudinallyat a small angle, A vertical line through the center of

buoyancy intersects the vertical line through (before the

ship is inclined) at .

1B

0B

LM

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The Location of the Longitudinal MetacenterFor a small angle inclination, volumes of forward wedge immersed

in water and backward wedge emerged out of water are:

10

2 1 20

0 0

(2 ) ( tan ) where is the half breadth.

(2 ) ( tan ) , .

Thus, 2 2 , which indicates:

moment of area forward of = moment of area 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 Lcenter of flotation

center of flotation ).

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Location of the Longitudinal MetacenterUsing the same argument used in obtaining transverse metacenter.

0 1 1 2

0

1 20

0

2 20

0 1 0

/ ,

(2 ) ( tan ) 2 tan

tan 2 2 tan

is the moment of inertia with respect to the

transverse axis passing the center of flotation.

t

l

L l

l

FCL 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

IB M

I IH B M Z Z GM Z Z

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Location of the Longitudinal Metacenter

2

0,

0,

Usually Floatation Center (C.F) of a waterplane is not at themidship,

,

where is the moment of inertia w.r.t. the transverse axisat midship (or station 5) and is the distance from F

FC T

T

I I Ax

I

x

.C. to

the midship.

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Moment to Alter Trim One Inch (MTI)

MTI: (moment to alter (change) the ships trim per inch) ateach waterline (or draft) is an important quantity. We may

use the longitudinal metacenter to predict MTI

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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" 1tan , where is in feet.

12

Due to the movement of the weight, moves to ,

,

tan tan

tan tan

1

12

w

L L G

FCw L G w B G

FCw B G

LL L

G G

M w h G G

G G G M HM Z

IM HM Z Z Z

IZ Z

L

MTI ( a function of draft)

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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 1264 lb/ft , Long Ton = 2240 lb, MTI (ton-ft)

420

FC FC w FC G B w

FCw

I I IZ Z M

L LI

L

1GX

0 1

0 0 0

1 0 1

Trimtan

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 ITG G Z Z Z Z L

G C B W L

I TX X G G X

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