Hydrostatics

A floating body pushes aside water until the weight of the water pushed aside equals its own weight. That single sentence is Archimedes' principle, and everything in this chapter is a consequence of it.

Two quantities come out of it, and it matters that you keep them separate. The volume of displacement, written ∇, is the volume of hull below the waterline — pure geometry. The displacement, written Δ, is the weight of the water that volume holds, which depends on how dense the water is.

So a ship moving from the sea into a river does not get heavier or lighter. Δ stays at whatever the ship weighs. What changes is ∇: fresh water is less dense, so a larger volume is needed to hold the same weight, and the ship settles deeper until it has one.

There is a second condition hiding behind flotation. The weight acts down through the centre of gravity G, and the buoyancy acts up through the centre of buoyancy B — the centroid of the underwater volume. For the ship to float without trimming, those two must be in the same vertical line. Load a weight forward and G moves forward; the ship trims until B has moved forward to sit under it again. That is the whole mechanism behind the trim calculation later in this chapter.

Form coefficients compare the actual hull with a box of the same length, breadth and draught. Each one answers a different question, and knowing which is which is worth easy marks.

• Block coefficient C_B — how much of the whole box the hull fills. A tanker is around 0.80, a container ship around 0.60, a fast naval hull below 0.50.
• Midship coefficient C_M — how full the midship section is. Nearly 1.0 on a full-bodied ship with a flat bottom and hard bilge.
• Prismatic coefficient C_P — how the volume is spread along the length. This is the one that matters for resistance, because longitudinal fullness drives wave making.
• Waterplane coefficient C_W — how full the waterplane is. It controls TPC and, through the second moment of area, BM.

The relation C_P = C_B/C_M falls straight out of the definitions and is quoted often enough to be worth memorising rather than deriving under time pressure.

A useful sense check: C_W is always larger than C_B for a normal ship, because the waterplane is the fullest horizontal slice of the hull while the block coefficient is dragged down by the fined ends below the water.

The waterplane is only a flat shape, but almost every hydrostatic quantity you will be asked for comes from it.

Its area gives TPC. Sink the ship by one centimetre and you add a slab of underwater volume one centimetre thick with the area of the waterplane. The mass of water in that slab is the tonnes needed to sink the ship by that centimetre.

Its second moment of area gives BM. Heel the ship a little and the wedge of buoyancy that emerges on one side and immerses on the other shifts B sideways. How far it shifts depends on how much area the waterplane has far from the centreline — which is exactly what the second moment of area measures.

Because I_T depends on breadth cubed, a beamier ship is dramatically stiffer — this is why beam, not depth, is the lever a designer pulls for stability.

Its centroid gives F, the centre of flotation. This is the point about which the ship trims. It is normally aft of amidships on a merchant hull, because the after waterplane is fuller than the fine forward end. Getting this wrong is the single commonest error in trim problems, and it is dealt with in section 5.

You are given a table of offsets and asked for an area. Simpson's rules fit a smooth curve through the points and integrate it exactly — the first rule fits a parabola through every three points, the second fits a cubic through every four.

Count intervals, not ordinates. Seven ordinates is six intervals — even, so the first rule works. Six ordinates is five intervals, and neither rule applies without splitting the range or adding a half-ordinate near the end.

The same machinery gives moments, and this is where most of the marks are. Multiply each ordinate by its lever from the reference axis before summing, and you get the first moment; multiply by the lever squared and you get the second moment. Divide the first moment by the area and you have the centroid — that is how LCF is found.

One habit prevents most errors: lay the work out as a table with a column each for ordinate, multiplier, product, lever and moment. Examiners award method marks for a correct table even when the arithmetic goes astray at the end.

Loading a weight anywhere other than at F does two things at once, and the way to keep it straight is to do them one at a time.

Step one — put the weight at F. The ship sinks bodily, parallel to its original waterline, with no change of trim at all. How far it sinks is the weight divided by TPC.

Step two — slide the weight from F to where it really is. Shifting a weight that is already on board changes no displacement, so there is no further sinkage. What it produces is a trimming moment, w × d, where d is measured from F.

Then the trim has to be shared between the two ends. The ship pivots about F, so the end further from F moves more. If F is 3 m aft of amidships on a 120 m ship, it is 63 m from the forward perpendicular and 57 m from the after perpendicular, and the trim divides in that ratio.

Finally, combine: add the bodily sinkage to both draughts, then apply the trim change — one end up, the other down. A quick check at the end catches most slips. Did the mean draught increase for a weight added? Does the change of trim you calculated match the difference between the new draughts?

Everything here follows from one fact established in section 1: the ship's weight does not change when the water does.

Move from sea water to fresh, and the same 9000 t of ship must now be supported by water that is 2.5 % less dense. The volume of displacement therefore rises by about 2.5 %, and the ship sinks until it has found that extra volume.

The general result is the fresh water allowance — the sinkage in going from standard sea water to fresh water.

For dock water somewhere between the two, scale the allowance by how far the density has actually dropped:

This is the reason load line marks include a fresh water allowance above the summer line. A ship loading in a river to her fresh water mark will rise to exactly her summer marks by the time she reaches the open sea, having burned nothing and discharged nothing.

Three problems in the style GATE actually sets. Work each one on paper before reading the steps — recognition is not the same as being able to execute under time pressure.