Cruise liners and tankers

Cruise liners and tankers can be about the same length and some are very long at 300 m or more. Cruise liners are built like a floating block of flats yet tankers are almost wholly submerged. Tankers do not put to sea with empty tanks so there is some aspect of sea keeping that is not just obvious. We can explore the mechanics of the cuboid to learn more.

I have drawn a cuboid as it might float. I have given it a length , a beam of  and a draught of . I propose to give the cuboid an aspect ratio that I shall call  so that . Then the draught will be  where  is some number less than 1. With these ratios I can alter the proportions of the submerged volume at will.

 

All that I want to do is explore the expression  for different aspect ratios and for different values of . We know that the second moment of area for a rectangle about its long axis is   and that  so Mathcad can be used to draw some graphs of  against length for values of  and .

 

I have to choose some practical values for the aspect ratio. Generally speaking, aspect ratios of 4, that is having a length equal to four times the beam, is more usual for smaller boats. Cruise liners and super-tankers will be up in the region of eight. Queen Mary 2 has an aspect ratio of about 8.5. I have drawn three graphs of the variation of MB with length of my cuboid for aspect ratios of 4,6, and 8. They are graphs 2.1, 2 and 3.

 

 

 

I have said that the range of values for the metacentric height of ships is from about 0.3 to 1.5 metres. Now I have values of BM that for large ships are much greater. As B is about half the draught above the bottom of the ship the metacentre is a long way above the waterline, in some cases several times the desirable metacentric height.

 

This raises the question of the validity of my cuboid when thinking about cruise liners and super-tankers In truth the shape of the displaced volume is not so much different from my cuboid. For a cruise liner the value of k would be about 0.25 that is the draught is about ¼ of the beam Then the value of BM for the cruise liner of 300 metres in length is about 25 metres. That means that the centre of gravity is about 24 metres above the centre of buoyancy. No wonder the cruise liner has so many decks and two swimming pools on the top one.

 

By comparison the super-tanker will have a draught equal to about half the beam and an aspect ratio of about 8. Then the value of BM is much smaller. I suppose that having a double bottom by law helps in getting the centre of gravity up to the required level.

 

One could go on exploring the stability of this cuboid to take into account the position of the centre of buoyancy etc. but I think that this is sufficient for this text.