Every chapter so far has treated the ship as a rigid body. She is not one. She is a beam a hundred and forty eight metres long, loaded in lumps and supported by water.
10.1 The hull as a beam
Weight acts downwards where the weight happens to be. Buoyancy acts upwards in proportion to the immersed cross section at each point, which follows the shape of the hull and takes no notice of where the cargo was put. The two have the same total and the same centre, because she is floating, but at no point along her length are they equal.
That difference is the load curve. Integrate it once and you have the shear force; integrate it again and you have the bending moment. Both must return to zero at the bow, and if they do not the calculation is wrong. That is the only check the method offers.
load = weight per metre − buoyancy per metre
shear force = ∫ load bending moment = ∫ shear force
10.2 Building the buoyancy curve
The immersed cross sectional area is read from the Bonjean curves, and her booklet has none, so this chapter builds them. At any height the table gives two facts about the waterplane: its area and the position of its centre. Take the half breadth at that level as (B/2)(1 − |u|k)1/k, with u the distance from amidships as a fraction of half the length and one exponent k for each end, and those two facts fix the two exponents exactly. Done at 271 levels, every 5 cm from the keel to the freeboard deck, that gives a half breadth everywhere.
The construction was told nothing about the shape of the ends. At the summer waterline it gives an exponent of 11.0 aft and 4.2 forward: a full, almost rectangular run aft and a finer entrance forward, which is what a bulk carrier of block coefficient 0.86 is. And it reproduces her tabulated hydrostatics:
| draught, m | volume from the sections, m3 | from her table | difference | LCB from the sections | from her table |
|---|---|---|---|---|---|
| 4.019 | 11 383.4 | 11 383.5 | -0.001% | 80.093 | 80.092 |
| 6.000 | 17 623.6 | 17 623.4 | +0.001% | 79.150 | 79.150 |
| 8.000 | 24 243.9 | 24 243.9 | +0.000% | 77.920 | 77.920 |
| 9.600 | 29 713.4 | 29 713.2 | +0.001% | 76.899 | 76.900 |
| 10.400 | 32 481.2 | 32 481.0 | +0.001% | 76.465 | 76.460 |
A thousandth of one per cent on the volume and half a centimetre on the centre of buoyancy. That is the evidence the reconstruction is fair.
10.3 The weight curve
Cargo is easy: a trimmed bulk cargo lies level, so it is a rectangle of weight between two bulkheads. The awkward part is the light ship. Her particulars give the total and the centre of gravity and nothing else, so the machinery and outfit are taken as 15 per cent over the machinery space and the rest as a trapezium running from 26.63 tonnes per metre aft to 30.23 forward, the slope chosen so that the light ship centre is at the 66.31 m of her particulars: very nearly uniform, which is what a hull is. The consumables are spread over their tanks and the cargo over its hold, each with its centre at the tabulated position.
10.4 The permissible values
A ship at sea carries two bending moments at once: the still water moment, which depends on how she is loaded, and the wave bending moment, which depends on her size and form and which she will meet whether the master likes it or not. The hull carries their sum.
| quantity | from the rule | for MV Ninja |
|---|---|---|
| block coefficient at the summer draught | as defined in the rule | 0.8642 |
| C | 10.75 minus ((300 minus L)/100) to the power 1.5 | 8.876 |
| wave bending moment, hogging | 190 M C L2 B Cb | 78 751 t m |
| wave bending moment, sagging | 110 M C L2 B (Cb + 0.7) | 82 522 t m |
| wave shear force, largest | 30 F C L B (Cb + 0.7) | 1 521 t |
Her approved midship section modulus is 7.85 m³ at 175 N/mm², so the hull can carry 140 036 tonne metres in all. Take away the wave moment and what is left is the still water moment she may have when she sails: 57 514 sagging, 61 285 hogging. In harbour there is no wave moment, so her manual permits 71 893 and 76 606 (every figure rounded as it is written and carried forward, as in the book).
10.5 What alternate hold loading costs
Chapter 9 loaded the same iron ore over five holds and over three, and promised the price appeared here. It does, and it is mostly in the shear force.
| all five holds | No.1, No.3 and No.5 | change | |
|---|---|---|---|
| maximum shear force t | 1 258 | 3 198 | 2.5 times |
| where it occurs m from AP | 20.8 | 117.2 | |
| maximum bending moment t m | 26 287 | 48 400 | 1.8 times |
| sense | sagging | hogging | |
| per cent of the seagoing limit | 46 | 79 |
The shear force is two and a half times larger and peaks at 117.2 m, which is the bulkhead between No.2 hold and No.1. On one side of that plate there is nothing; on the other, eight thousand tonnes of ore. The whole imbalance crosses the bulkhead through the side shell and the double bottom.
10.6 An arrangement she is not allowed
Put the same ore in No.2 and No.4 only. It fits. She floats at her summer draught of 9.600 m trimmed 0.34 m by the head with a fluid metacentric height of 3.56 m. Nothing visible from the quay would be wrong. Her bending moment is 122 050 tonne metres sagging against a seagoing limit of 57 514: 212 per cent before the first wave and 170 per cent of what she is permitted alongside. The shear force 5 111 t at the No.5/No.4 bulkhead is inside its own limit of 6 500 t: the two checks are separate and both must be made.
10.7 The sequence of Chapter 9
Chapter 9 traced two sequences and said every stage had a bending moment against it which was not printed. Here it is, stage by stage.
| pour | cargo t | shear force t | bending moment t m | sense | % harbour | % at sea |
|---|---|---|---|---|---|---|
| on arrival | 0 | 1 014 | 50 420 | hogging | 66 | 82 |
| 1 No.1 to 35% | 1 576 | 1 716 | 69 532 | hogging | 91 | 113 |
| 2 No.3 to 35% | 3 339 | 1 419 | 51 810 | hogging | 68 | 85 |
| 3 No.3 to 70% | 5 103 | 1 173 | 36 502 | hogging | 48 | 60 |
| 4 No.3 to 100% | 6 614 | 1 269 | 28 119 | hogging | 37 | 46 |
| 5 No.2 to 35% | 8 472 | 1 393 | 30 166 | hogging | 39 | 49 |
| 6 No.4 to 35% | 10 330 | 969 | 18 725 | hogging | 24 | 31 |
| 7 No.4 to 70% | 12 188 | 1 258 | 15 162 | sagging | 21 | 26 |
| 8 No.2 to 70% | 14 046 | 1 269 | 24 557 | sagging | 34 | 43 |
| 9 No.4 to 100% | 15 638 | 2 071 | 40 370 | sagging | 56 | 70 |
| 10 No.2 to 100% | 17 230 | 2 227 | 45 091 | sagging | 63 | 78 |
| 11 No.5 to 35% | 18 854 | 1 798 | 41 930 | sagging | 58 | 73 |
| 12 No.1 to 70% | 20 430 | 1 622 | 32 847 | sagging | 46 | 57 |
| 13 No.5 to 70% | 22 054 | 1 196 | 36 330 | sagging | 51 | 63 |
| 14 No.1 to 100% | 23 405 | 997 | 28 331 | sagging | 39 | 49 |
| 15 No.5 to 100% | 24 797 | 1 290 | 34 570 | sagging | 48 | 60 |
She arrives in ballast at 82 per cent of her seagoing hogging limit. That is not an error and not unusual: in ballast the weight is low down and at the ends, the buoyancy is where it always is, and the ship hogs. On many bulk carriers the ballast passage is the governing condition of the whole voyage.
Then the first pour of the agreed plan puts 1 576 tonnes into No.1 hold, right forward, and takes her to 113 per cent of the seagoing limit and 91 per cent of the harbour limit. Nothing in the remaining fourteen pours comes near it.
And this is the real lesson of Chapters 9 and 10 together
Chapter 9 chose that sequence carefully. At every pour it took whichever hold left the ship nearest to level, and at no pour did the trim exceed 1.3 metres. It was a good plan, chosen with care. It is also, at its first pour, at 91 per cent of a bending moment she may not exceed even alongside, and over the value she may sail with.
Keeping the trim small is not the same thing as keeping the stresses small, and no amount of care about the one will protect the other. A loading plan worked out from draughts and trims alone is a plan for the draughts and the trims. The bending moment has to be computed separately, at every stage, and that is what the loading instrument is for.
And the terminal’s preference is worse than merely inconvenient. Sequence B reaches 103 441 tonne metres hogging at its first pour: 169 per cent of the seagoing limit and 135 per cent of the harbour limit with a shear force of 3 181 t at the No.2/No.1 bulkhead. With 4 503 tonnes aboard out of 24 797 before anybody has had time to look at anything, she is over a limit that applies while she is tied to the quay. Both sequences finish in exactly the same place. Everything that separates them happened in between, and none of it is visible in a draught survey.
10.8 What the loading instrument is doing
All of this is what the instrument does when the mate types a tonnage into a hold: hold the sectional areas as a table, hold the light ship as a distribution, add the cargo and ballast, solve for the draught and trim that float the result, subtract buoyancy from weight, integrate twice, and compare with two sets of permissible curves. It does it in a fraction of a second and more accurately than this chapter has, because its light ship distribution is measured rather than assumed.
What it cannot do is know whether the numbers it was given are true. A stowage factor declared wrongly, a hold that has not taken what the terminal says, a ballast tank that has not emptied: all produce a calculation that closes perfectly and describes a ship that does not exist. The draught marks are the only independent evidence, and reading them is the last independent check in this whole volume.
The one number to carry away
MV Ninja loaded homogeneously to her marks sits at 60 per cent of her seagoing bending moment limit. The same tonnage of ore in three holds instead of five takes her to 79 per cent. The same tonnage in two holds takes her to 212 per cent. The cargo never changed, the displacement never changed, and the draught never changed by more than a few centimetres. Only the distribution did.
Chapter 10 in seven lines
- Weight acts where the cargo is, buoyancy where the hull is. The difference is the load curve.
- Both curves must return to zero at the bow. Here they close to a thousandth of a tonne metre.
- The sectional area curve was reconstructed from her waterplane area and centre of flotation at 271 levels, and reproduces her volume to a thousandth of one per cent.
- Section modulus times permissible stress is 140 036 t m; take away the rule wave moment and what is left is the seagoing still water limit.
- Loaded homogeneously she runs at 60 per cent of that limit; the same ore in alternate holds 79; in two holds 212.
- Alternate hold loading multiplies the shear force by two and a half, peaking at the bulkhead between a full hold and an empty one.
- She arrives in ballast at 82 per cent of the seagoing hogging limit, and the first pour of a plan chosen for trim takes her to 113. Trim and stress are different problems.
Test yourself
Questions
- Define the load curve, the shear force curve and the bending moment curve, and state the relationship between them.
- Explain why the shear force and the bending moment must both be zero at the forward perpendicular, and what it means if a calculation does not produce that result.
- Explain why the weight curve of a bulk carrier steps at each bulkhead while the buoyancy curve does not, and what follows from it.
- Distinguish hogging from sagging, and state which of the two a loaded bulk carrier with a homogeneous cargo will normally be in, and why.
- State the two components of the total bending moment a hull must carry at sea, and explain how the permissible still water bending moment is arrived at from the section modulus.
- Explain why the permissible still water bending moment in harbour is greater than the one at sea, and what the officer must satisfy himself of before the ship sails.
- Explain why alternate hold loading produces a large shear force, state where along the ship it occurs, and explain why the permitted hold combinations are set out in the loading manual.
- A ship is loaded with a dense cargo in two holds only. She floats on an even keel at her marks with a fluid metacentric height of 3.5 metres. State what is wrong and how the officer would know.
- Describe what a loading instrument does between the mate entering a tonnage and a number appearing on the screen, and state the one thing it cannot check.
- Explain why the draught marks remain the final check on a loading calculation, however good the instrument.
Looking ahead
This chapter spread thousands of tonnes over five holds and watched the ship change slowly enough to be tabulated stage by stage. Chapter 11 takes the opposite case: one weight on one hook, where the whole of it acts at the head of the derrick the moment it leaves the quay, and where the ship has seconds rather than hours to respond. The arithmetic is the arithmetic of this whole volume. Only the time scale has changed.