**Ship Stability – Part 1**

*-Raghu Varier*

The most important and the critical consideration made when ships are to be designed is the stability of ship. Stability of ship encompasses the study of ship stability at various stages namely the initial stability, the dynamic conditions and the damage cases (popularly called as the damage stability). The stability criterias evolved when some of the mishaps that have happened in the recent past. Let us take a look at some of the mishaps in the pastbefore we jump right into the technical stuff.

The sinking of the naval ship Captain in Biscay Bay in1870 in a heavy sea state while the naval ship Monarch of similar dimensions surviving it surprised the naval architects then. And in 1939, in his doctoral thesis Rahola proposed certain stability criteria after carrying out extensive statistics and analysis on different ships, both stable and unstable. These criterias were further developed and adopted by the classification societies. The criteria so developed are the basic minimum requirements, however naval architects try to out-perform these margins by certain factor from a safety consideration.

It was Archimedes who laid down the foundations for a quantitative assessment of ship’s stability about 2000 years ago, (the EUREKA moment). Prior to the development of stability rules, seafarers and shipbuilders relied more on intuitions to deal with the ship stability. This can be seen even in the netflix series Vikings, if you are interested, you may watch that, to get an idea.

It was in the 18th century that Bouguer from France and Euler from Russia independently arrived at ship stability theories. And the criteria that they developed, is the basis of intact stability wherein the concept of metacentre (M) and metacentric height (GM) was introduced. Having talked about that let us now look at the initial stability of ships.

The Archimedes Principle states that ‘ A body floating in water experiences a buoyant force whose magnitude is given by the mass of the water it displaces.’ Now for a floating body, the weight and the buoyancy forces balance. This can be demonstrated as below:

But just try to comprehend, what would happen when a ship tilts due to some reason, as shown below. In order to maintain the same force balance, the submerged volume magnitude will have to be retained as per Archimedes principle. While doing this, the center of buoyancy changes due to the shape of submerged portion of ship. Now even though the force balances, there is a moment due to this buoyant force acting and this moment tries to restore the ship back to its initial state. This moment is therefore called as the restoring moment.

Brace for some nerdy science ahead.

Every force is associated with a line of action. The intersection of this line of action of the buoyant force with the initial line of action of weight (simply said the symmetric axis as shown) is called the Metacentre (M) and the height is called metacentric height when measured from G to M, hence GM.

Having talked about this, let us now try to investigate if metacenter M has an influence on ship stability. Refer the image above. The force of buoyancy acting through B1 can be given as:

Now the moment due to this force is given as

Where GZ is the moment arm.

One thing to be emphasized here is that, the centre of buoyancy (let us denote by B) could move not just horizontally but also vertically. So it is very important to consider both these shifts in B. But for small angles, the vertical shift can be neglected, and so the simple trigonometry would apply to give us:

Now based on the position of M with respect to G the value GM can take the following values. A positive value of GM means positive value of GZ and so the moment is restoring the ship back to its initial state as shown above. If G and M coincide, i.e GM = 0, then the ship is in a neutral state of equilibrium since there is no moment acting as the line of action of weight and buoyancy coincide.

However if M is below G, then GM is negative, and as can be seen from below image that the moment due to buoyancy instead of restoring the ship back, causes it to capsize.

So it can be concluded that the value of GM is to be kept positive.

Now if you look at the image below, what we can see is that at the section we are considering, a small triangle portion has risen out of water on the port side (left) whereas the same equal amount has submerged on the starboard side(right). Now let’s go back to school geometry and we know that centroid of a right angled triangle lies 2/3^{rd} of the distance from the vertices of hypotenuse (refer image below). So to make things simple, we can say area of such a triangle on the port side has shifted by to the starboard side.

Since we are dealing with small angles, the approximation below is valid.

And since such right angled triangles of varying base values can be found over the length, the easiest way is to integrate them. (for those who do not know integration, I shall recommend you may refer some sources on the internet , preferably on youtube)

Now it is intuitive that the value of integrating such small areas of triangle over the length will give volume:

* The total shift of the whole underwater volume of ship from initial B to final B1 position is due to this shift of small volume by 4y/3*.

So we can write:

For the waterplane area of ship assumed to be almost a rectangle the following substitution can be deemed valid, where ‘I’ is the transverse second moment of area.

The below substitution can also be made from the diagram below and hence we can arrive at an expression for the term BM.

From this we can arrive at a value for GM using the below expression. From this we can evaluate GZ (Also called the righting lever arm) and hence the value of the restoring moment (also called the righting moment).

**GM = KB + BM – KG** where K is the keel point

It can thus be concluded that a higher value of GM means a more stable ship.

But what if the angle is not small and is considerable enough that we cannot neglect the effect of vertical shift in B. Well we shall look into it in the next post.

*Next post* : *Large Angle Stability*