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The conditions for the stability of floating bodies and ships with solid loads. Includis an introduction to Metacentric heights and centre of buoyancy

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Introduction

In 1628 the Swedish warship Vasa was launched in Stockholm harbour.

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At some time during her construction it had been decided to increase the size and weight of the cannons on the upper gun deck. At the time of her launch she was ballasted but was not fully loaded. A full load would have increased her stability but whether it would have been enough to prevent a catastrophe, is not known.What is known is that she sailed a few yards, heeled over and sank. Sadly the ship builders of the time did not understand the requirements for a stable ship. This page, the first of three, examines these requirements and includes worked examples where the load or ballast is fixed. The two further pages will consider floating bodies with liquid loads or ballast and the period of roll.

Centre of Buoyancy and Stability.

The Buoyancy Force act through the Centre of Gravity of the Displaced Fluid and is called " The Centre of Buoyancy"

There are three Types of Equilibrium.

Stable. The body returns to it's original position if given a small angular displacement.
Neutral. The body remains in a new position if given a small angular displacement.
Unstable. The body heals further over if given a small angular displacement.

The Stability of Fully Submerged Bodies.

Let

V = Volume of Body.
w = Specific weight of the fluid.
W = Weight of the Body.
G is the Centre of Gravity.
B is the Centre of Buoyancy and is the centre of gravity of the displaced liquid.

If B and G are coincident the the Body will be in Neutral Equilibrium.

B is below G then the Body is in Unstable equilibrium.

B is above G then the body is in Stable equilibrium.

This last case has a righting couple of $W\;B\;G\;\sin \theta$ Where $\theta$ is the angle of tilt. Note that the Position of B and G relative to the Body does not change when the body is rotated.

The Stability of Partially Submerged Bodies.

G is the C of G. B is the C. of B and The line o o is the original water surface.

After tilting O' O' is the new water line and the angle of Tilt is $\theta$

G remains in the same position relative to the ship but the Centre of Buoyancy moves to $B_1$

M is the "META CENTRE" and is defined as the point where the vertical through the new Centre of Buoyancy meets the original vertical through the Centre of Gravity after a very small angle of rotation.

MG is called the METACENTRIC HEIGHT

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Therefore for stable equilibrium for a floating Partially Submerged Body the Meta centre must be above the Centre of Gravity G. If the Metacentric height is zero the Body will be in Neutral equilibrium.

In ship design the choice of the Metacentric height is a compromise between stability and the amount that the ship rolls. In British Dreadnaught Battle ships, for instanace, the metacentric height was so great that they had a tendency to roll badly, even with large bilge keels.

The Righting couple $=\:W\;GM\;\sin \theta$

The Determination of Metacentric Height.

1) Experimentally

Let W be the weight of the Boat plus it's Load

A small load w is moved a distance x and causes a tilt of angle $\theta$ . The Boat is now in a new position of equilibrium with B' and G' lying along the Vertical through M.

The Moment due to the movement of the load is given by:-

$wx\;=\;W\times GM\;\theta \;=\;moment\;due\;to\;movement\;of\;C\;of\;G$

(1)

$\therefore\;\;\;\;\;\;GM\;=\;\frac{w\;x}{W\;\theta }$

(2)

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2) Theory

The Ship tilts from it's old waterline O O to a new waterline O' O' as it moves through an angle $\theta$ . Due to the movement of the wedge of water from A' A C to D' D C, the Centre of Buoyancy moves from B to B'

The Change in the moment of the buoyancy Force = wV X B B'

$=\;w\;V\times BM\;\theta \;\;\;\;where\;\theta \;is\;small$

(3)

The Volume of the Wedge A C A'

$=\;\frac{1}{2}\times \frac{b}{2}\times \frac{b}{2}\theta \times L\;=\;\frac{b^2\,L}{8}\;\theta$

(4)

Therefore the Moment of the Couple due to the movement of the wedge $w\times \frac{w b^2L\,\theta }{8}\times \frac{2}{3}b$

$wV\times BM\;\theta \;=\;\frac{wb^3\;L}{12}\;\theta$

(5)

$\therefore\;\;\;\;\;BM\;=\;\frac{b^3\,L}{12\,V}\;=\;\frac{I}{V}$

(6)

Where I is the Second Moment of Area of the Water Plane Section and V is the volume of water Displaced.

Thus if the positions of B and G are known or can be calculated , then the distance GM can be determined since:-

GM = BM + BG = Bm + BO - OG

There are in fact two Metacentric heights of a ship. One for Rolling and the other for Pitching. The former will always be less than the latter and unless otherwise stated, the Metacentric given will be for Rolling.

Worked Examples

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Example 1

A Pontoon measuring 20ft.by 12ft. and 4ft deep, weighs 12 tons. It carries a load of 8 tons. The Pontoon sits in sea water with a density of 64 lb/cu. ft. Find it's Metacentric height and establish the angular tilt which will result if the load is moved by one ft. sideways.

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Example 2

State the conditions for stable equilibrium of a body floating partially submerged in a liquid.

A cylinder of circular section, diameter d, made of material who's specific gravity is S, floats in a liquid of specific gravity $S_O$ . Find the maximum lehgth of cylinder if the axis is to remain vertical. (B.Sc. Part 1)

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Example 3

State the conditions to be met to ensure the stable equilibrium of a body partly immersed in water.

A right solid cone with apex angle equal to $60^0$ is of density k relative the that of the liquid in which it floats with the apex downwards. Determine what range of k is compatible with stable equilibrium. (B.Sc. Part 1)

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Example 4

A hollow cylinder with closed ends is 12 in in diameter, weighs 60 lb. and has a small hole in the bottom.

Calculate

The depth to which it will sink .

The height to which the water will rise.

The final air pressure in the cylinder.

Disregard the wall thickness of the cylinder but assume that the compression of the air is isothermal. ( Atmospheric pressure is $14.7\;lb./in^2$ )

Determine also whether the cylinder will be stable in the vertical position when in equilibrium. (B.Sc. Part 1)

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Example 5

A rectangular pontoon of breadth 12 ft., when on an even keel has a draught of 3 ft. The C. of G. is 5ft. 8in. above the bottom. Show that the pontoon is unstable on an even keel and calculate the angle through which it will heel to reach stable equilibrium. Work from first principles. (B.Sc. part 2)

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Example 6

A ship displaces $4\times 10^4\;ft.^3$ in fresh water when the centre of buoyancy is 6.4 ft. below the water plane. The second moment of area of the water plane about the fore and aft axis is $3.6\times 10^5\;ft.^4.$ Calculate the change in the transverse metacentric heoght that will ensue if the vessel passes from fresh water to salt water, the density of which is $64\;lb./ft^3$