**The stability of
floating bodies.**

We cannot discuss the design of floating bodies, such as ships, floating cranes and oil rigs, until we understand the physics of a simple body floating at rest in a liquid, typically water, that is also is at rest.

I want to explain the principles governing stability and not get tangled up with hulls of many shapes and for my purpose the cuboid will be most suitable. Let me start with a cuboid having a uniform density less than that of water. If it has the correct proportions, not long and thin, it will float with two opposite faces parallel to the free surface as shown in figure 2-18

We could do things to this cuboid. We could push it further into the water and release it, or we could tilt it to one side and release it, and, if we did either or both of these things it would ultimately recover its original attitude. It follows that the cuboid must to be in vertical equilibrium and that equilibrium must be stable both vertically and laterally. We can examine the requirements for these two states of equilibrium separately. Let us start with the vertical equilibrium.

If the body has mass *m* the gravitational force exerted on it
is *m.g* and this acts at the centre of
mass G. For the body to be in vertical equilibrium a force that is equal and
opposite to the weight of the cuboid and passes through G must be exerted on
the body. This force must be exerted on the body by the water and must be the
consequence of the pressure exerted on the part of the surface of the body that
is submerged (wetted). This is shown in figure 2-19. This force can usefully be
called the upthrust U and figure 2-20 shows the equal forces and U. We know the position of G because the
body is of uniform density but the point of action of U is not immediately
obvious. However in connection with the element shown in figure 2-4 we said
that the external forces exerted on the element must equal the weight of the
element and be equal and opposite and act at the centre of mass of the element.
The fact that we now have similar external forces acting on a solid surface
which cuts the free surface makes no difference, the water behaves as if the
space occupied by the ship is full of water and the pressure distribution
results in a force equal to the weight of the water which could occupy the
space acting vertically where the centre of mass of this water would be.

There is a concept of a centre of volume and this concept is useful here, indeed, it is hard to see any other application for it. The centre of volume is the same point as the centre of mass of a solid of uniform density and, as the centre of mass is the point through which an accelerating force can be exerted on the body without causing rotation when the body is free to move, it can only be found by taking moments of volume. If we decide to use the idea of a centre of volume then we can locate the point of action of U as being on the axis of symmetry of the cuboid and at half the depth of immersion and this is shown in figure 2-20. This point of action is called the centre of buoyancy and denoted B. (A body which floats is said to be buoyant.) Clearly for this cuboid G is above B.

This gives the condition for vertical equilibrium but does not tell us whether it is stable. The normal test for stability is to disturb the equilibrium and see whether a system of restoring forces come into existence. So if we push the cuboid down the upthrust exceeds the weight of the cuboid and a restoring force does comes into existence. If the cuboid is raised, is greater than U, and a downward force tends to restore the body to its original position. The body is in stable vertical equilibrium.

However we have G above B and this arrangement is not what one might associate immediately with stable lateral equilibrium. We can now apply the ordinary test for stability. Figure 2-21 shows the cuboid tilted to one side in some way that still leaves us with U equal to . (There is no way to do this but that does not affect the argument.) The position of G relative to the body is unchanged but, as U still equals , the volume displaced by the cuboid is not altered. But the shape of the displaced volume has altered and the position of the centre of volume, that is the centre of buoyancy, has moved to B'. Clearly B' and G are now no longer in the same vertical line and there is a couple exerted on the body tending to restore it to its original position. So we can see that the equilibrium will be stable if the relative positions of G and B after tilting combine to produce a restoring or righting couple.

Figure 2-22 shows the relative
positions of G, B and B' and the centre line of the cuboid when the cuboid is
in the tilted position. The forces and U are equal. If P is the point of
intersection of the line of action of U and the centre line and q is the angle of tilt, the
restoring couple equals *.* If PG were to be large we would think
of the cuboid as being very stable and if it were to be small not very stable.
It is evident that we have a concept of the "degree" of stability
even if we have no word for it. The degree of stability is in some way linked
to the magnitude of PG. We need to find a useful expression for PG.

Now P and G are two independent points. The position of G for this cuboid is fixed because we let it be of uniform density but it could have been anywhere on the axis of symmetry.[1] P is a point that will depend on the value of q and the consequent position of B¢. P has a position even when q is zero and that position turns out to be very important. It is so important that it has special name, the metacentre[2], and it is given the symbol M. We can now draw a diagram showing the relative positions of B, M and G.

However we cannot expect to find MG directly because P is a function of the shape of the cuboid and G depends on its content, we must find the distance PM that is a function of shape, instead. For this we must find the horizontal shift of B to B' and, because the practical application requires a position for P when is zero, we must find BP for small values of .

In
figures 2-23 a and 2-23 b the cuboid is shown when it is floating
freely and when it is tilted through a small angle *q*. As
the volume of water displaced is unchanged the original and final positions of
the waterline intersect on the centreline of the cuboid. The change in shape of
the displaced volume is equivalent to cutting off a wedge of volume from one
side and adding it to the other. In figure 2-23c the displaced volume is shown
in the upright position. The shape can be considered to be made up of two
parts, a part that is symmetrical about the centre line and a part that is
equivalent to the sum of the volume of the cuboid that was raised above the
water by the tilt and the volume that was submerged. For our purpose all we
need is the horizontal shift of the centre of volume. For this we need only to
find the moment of the two wedges of volume about the line represented in
figures 2-23 b and 2-23 c as OO and divide by the displaced volume.
There is an advantage in dealing with this by dividing one of the wedges into
elemental volumes as shown in Figure 2-24 because the outcome will be
general and not limited to this cuboid.

Provided that the angle *q** * is small, the volume of the element is given by :-

and its moment about OO is:-

or we can say that the moment of the element is:-

Then the horizontal shift of
the centre of volume, which equals the moment of two wedges of volume about OO
divided by the displaced volume *V* is
given by:-

where *b/2* is half the
width of the cuboid.

Now is really the summation of the elemental
quantities made up of an elemental area multiplied by *x *twice. This occurs
sufficiently often in mathematics and engineering to have a name. It is called
the second moment of area and is given the symbol *I*. So the horizontal shift of the centre of volume of the displaced
volume is given by :-

, where I is the second moment of area of the water plane section about its longitudinal axis of symmetry (line OO)

However, as ,
*,* which, when *q* is
small sin*q* =tan*q** *can be
written :-

** **

**
BP = I/V. **

This is now a special case of *P*
being the position of *P* when the cuboid is floating without heel. It is
called the metacentre and denoted *M. *

From figure 2-21 we found that
the righting couple was given by and now, because *P* is the metacentre *M*,
we can write this as and go on to give the righting couple as:-

righting couple = *,* from which

**righting
couple = m.g.(I/V—BG).sin**

This result is interesting. It looks to be of little value because it applies to this cuboid floating in still water and to small angles of heel. But its value does not lie in calculating righting couples. We have derived it for a cuboid for which it is clearly correct. But boats are not shaped like cuboids at all. They have sides that might well be vertical over the middle part of the length but they have shaping at the bow and stern and generally appear to be so far removed from the cuboid to require a whole new analysis. However boats, whatever their shape still have a centre of buoyancy and a displaced volume and a section at the water plane. It follows that the value of BM can be calculated for a boat or any other floating vessel at the design stage. It is a dimension of the hull just like its beam and its length. We must see how it is used.

We can start by considering the stability of ships or, more correctly, the ability of ships to survive at sea, that is, their sea keeping qualities.

Ships are designed and built to make long voyages across any of the oceans. They must have an acceptably high probability of arriving undamaged at their destination although, as we hear of ships foundering almost daily, it is evident that the totally safe operation of ships is unlikely to be possible.

Osborne Reynolds put the problem in simple terms in a lecture on the safety of life boats in 1886. He said “.......the peculiar construction of boats of all sizes is the result of a long process of trial and failure, and that, although certain general principles, connecting the (sea-keeping) qualities of ships to their shapes, have been discovered and recognised in the last thirty years, still the recognition of these principles has not resulted in any considerable improvement to be effected in what were before high class vessels, such as yachts and fast sailing vessels, but rather have confirmed the form previously arrived as the best, and led to their being copied in larger vessels.”

He went on “(Whilst) the discovery and recognition of certain general principles have undoubtedly been of immense service in improving large modern vessels the improvement is not as great as might have been expected. But this is mainly because with large ships there is not the same opportunity for trial and failure as with the small, the number being so much smaller, and experiments are so much slower and more costly; but the main reason is that the circumstances which call out the highest qualities of the large vessels become so extremely rare. There is no doubt that many large vessels pass through their whole lives without meeting weather which tests their sea-going qualities in the way in which those of fishing boats are tested many times every winter.”

He went on to argue that there was a case for building scale models of new designs for large ships to a size that permitted them to be operated as vessels in their own right so that they would meet the “circumstances which call out their highest qualities” more quickly. This argument has not been accepted even now as the proponents of the wide beam frigate as a replacement for the long thin frigates lost during the Falklands war discovered.

Reynolds is saying that ship design is difficult and that it proceeds by evolution. It is difficult because the water in which the ship must operate is not at rest. Surface waves may frequently be 20 metres in height with wavelengths up to 600m and very occasionally as much as 35 metres in a single rogue wave. These conditions cannot be quantified and so we cannot construct mathematical models of a ship and the sea conditions it might meet. A strategy is needed to facilitate the evolutionary process. The physics above has helped in this.

The starting point is that we
must accept that a ship has an underwater shape which has been chosen as a
compromise between various requirements and that it will only work properly when
the ship is loaded to the intended depth. In this condition the shape of the
displaced volume is known and this means that the displaced volume *V* and
the position of the centre of buoyancy B can be calculated. In addition the
value of I, the second moment of area of the water-plane section of the ship
about the longitudinal axis, can be calculated and this means that the position
of M can be found. The positions of B and M are as much a part of the geometry
of the ship as the length and breadth. Now, in some way, the stability of a
ship is linked to the distance between G and M and this distance is called the
metacentric height (and even in legal proceedings called MG for short). The
strategy involves the keeping of records, by interested bodies such as the
insurers, of values of MG for types of ship and records of their service. This
leads to a knowledge of the values of MG which have proved to be successful.
Typical values are :- for ocean
going ships.........0.3 to 1.5 metres[3] for river craft ..............4 metres.[4]

The value for ocean going ships seems to be ludicrously small but there are good reasons for this low value. It is well known that ships pitch and roll simultaneously in response the fluctuating forces caused by wave motion. Further the water forces may combine to subject the ship to both twisting and bending. All ships have concentrated masses in them such as the engines, the fuel bunkers, and the cargo. As the ship moves these masses have the same accelerations impressed on them as the motion of the water is impressing on the ship. Nowhere else will masses of this magnitude be tossed about in this way. The forces involved are very large indeed and they are exerted by the water acting through the structure of the ship. The structure of the ship has to be strong enough to withstand these forces and Reynolds was observing that eventually every ship will meet conditions which are so severe as to lead to failure.

However we can see that the magnitude of the forces on a given ship are in some way connected with the value of the metacentric height and that they increase as MG is increased and so an upper limit must be placed on MG.

There must also be a lower limit for MG because with a low value of MG the roll rate is so slow that the ship cannot right itself sufficiently quickly to shed one lot of seawater from the deck before the next wave breaks over it.

The best design of ship is the one that makes the best compromise between strength and therefore cost to build, cost of operation, and safety. Perhaps it is not just accidental that the value of MG that suits the structure of the ship also gives the best ride, in the circumstances, for passengers. Now we must consider the factors that determine the value of MG for a given ship when it leaves port. Ships have marks painted on the hull at bow and stern to show the depth to which it should be loaded. If they are loaded to these marks the position of M is known. But the matacentric height is the distance between the metacentre and the centre of mass and the position of the centre of mass can be altered. In a passenger liner the scope for moving the centre of mass is small but for a ship carrying mixed cargo the position of the centre of mass will depend on the disposition of the cargo. As the best value of MG is small, a small shift of G can materially affect the value of MG. It is not surprising to hear claims for the provision of computers to help load-masters on vehicle ferries.[5]

In order to operate these computer programmes for a container ship it is necessary to have a position for the centre of mass of every container. This is not possible but nor is opting out possible. Container ships have to be loaded and be safe to operate. I think that the way that this has been managed is interesting. Initially each container is assumed to have a centre of mass at one of two or three positions eg mid way up, 1/3 up and 1/4 up from the bottom. Then each container is assigned a position for its centre of mass according to the content of the container from these pre-selected positions. Just knowing that a container is loaded with motor-cycles might be enough to make the choice as might a knowledge that it holds overstuffed furniture. Then the ship is loaded using the computer programme to oversee the loading so that its calculated centre of mass is known. The fact that the ship carries many containers almost certainly means that errors tend to average out. From a record over a significant time of the behaviour of the ship and of other ships using the same loading procedure the assigning parameters are refined as part of an on-going process to give better loading. It is what I would call a strategy but I find others pooh-pooh this idea just because the word is not recognised in engineering.

It will be noted that the metacentric height of river craft is much greater than that for ocean going ships. This merely reflects the fact that river craft do not meet excessively rough seas. However there is a group of vessels which are used at sea but only when conditions are calm. These are the floating cranes, pontoons, dry docks and floating drilling rigs. For these it is necessary to note that in addition to the requirement for stability there is the need to use the condition for vertical equilibrium. When, say, a floating crane lifts a load its centre of mass moves laterally and in order to restore vertical equilibrium the vessel must tilt so that the under-water shape changes to bring the centre of buoyancy below the centre of mass. As an extension of this problem the sequence of events required to tilt a drilling rig, which has been built on its side and floated to position, into the upright position is quite complicated.

[1] If the centre of gravity is on the axis of symmetry there is the possibility that the cuboid will float “on an even keel”. If it is off the axis the cuboid could never float level.

[2] The word metacentre clearly has its roots in imaginary centre, presumably of rotation, but it is best to forget this idea.

[3] I do not know values of MG for the extraordinarily large cruise liners that are now in operation but it is doubtful that they will be much larger.

[4]This is by no means the only parameter used in ship design and operation but it is probably the most used.

[5] This problem was understood in the days of sail when dense cargoes would be supported on "dunnage" which was brushwood gathered for the purpose. Ballast was needed to load a ship to its marks and mahogany first arrived in Britain as ballast as did tortoises.