Probabilistic Damage Stability: The SOLAS Approach

Chapter 5 opened a compartment somebody had chosen and asked whether the ship survived it. This chapter stops choosing. It asks instead what fraction of all the damages that could happen to her she would come through, and answers with a single number.

6.1 A different question

The weakness of a deterministic rule is not that it is wrong. It is that somebody has to pick the damage. Pick No.3 hold and MV Ninja passes comfortably. Pick No.1 hold and she is within two thirds of a metre of the margin line. The rule cannot tell you which of those two answers describes the ship, because the ship is not going to be rammed where the rule says.

The probabilistic method takes the choice away. It starts from what is actually known about collisions: where along the ship damage tends to fall, and how long it tends to be. From those it builds a probability for every possible damage, works each one out by the methods of Chapter 5, and gives it a mark between nought and one for survival. The two are multiplied and added up.

A = Σ p_i s_i   and A must be not less than R

p depends on the watertight arrangement and on nothing else: not on the loading, not on the stability, not on the freeboard. s depends on everything else.

6.2 Subdivision length and the required index

The length used here is the subdivision length Ls, the greatest projected moulded length of that part of the ship at or below the deck limiting the vertical extent of flooding at the deepest subdivision draught. For MV Ninja it comes to her length between perpendiculars, 148.00 m.

R = 1 − 128 / (Ls + 152) = 1 − 128 / 300 = 0.57333

A longer ship must be better subdivided, which is the factor of subdivision idea from Chapter 5 expressed as a number instead of a length.

6.3 The zones

The ship is divided at her watertight bulkheads. A collision floods one zone, or a run of adjacent zones, and nothing else. MV Ninja has eight, and they allow thirty six damages.

The five holds are the rectangular boxes of Chapter 5: bulkheads at 20.80, 44.90, 68.70, 92.90, 117.20 and 139.60 m from the after perpendicular, the breadths listed, and an effective flat floor 2.20 m above the keel standing in for the inner bottom and the hopper tanks. The after peak bulkhead at 7.00 m, the machinery space (same floor) and the two peaks are this chapter's additions, prismatic, with breadths a fifth (peaks) or a twentieth (machinery) less than the neighbouring hold's; the eight boxes cover 3322 m2, 97 per cent of her tabulated waterplane of 3442 m2 at the deepest subdivision draught. A real ship is also divided transversely and horizontally, and each zone splits again by how far the damage penetrates and how high it reaches. Those are the r and v factors; this chapter takes both as one.

6.4 How long is a collision damage?

Everything in p comes from one distribution, fitted to several hundred collision casualties and written as two straight lines meeting at a knuckle. J is the damage length divided by Ls.

The four coefficients come out as exact round numbers: b11 = -65.34, b12 = 11.00, b21 = -7.26, b22 = 2.20. That is not a coincidence. Four conditions fix them, that the density integrates to one, is continuous at the knuckle, accumulates eleven twelfths of its probability before it, and falls to nothing at the longest damage, and the constants were chosen to make the arithmetic tidy.

6.5 Three draughts, and how they are weighted

The index is worked at three draughts and averaged. The deepest subdivision draught ds is the summer waterline, the light service draught dl is her ballast condition, and the partial draught dp sits six tenths of the way between. The conditions are built on the consumables of Chapter 1 (heavy fuel oil 509 t at 12.65 m, diesel oil 35 t at 11.45 m, fresh water 165 t at 11.86 m, seven slack tanks, free surface moments 815.2 t m): the summer departure condition, fluid KG 8.113 m; a part loaded condition at the partial draught with the same cargo centre of gravity, 8.217 m; and the normal ballast condition with the double bottom tanks and both peaks pressed, 6.020 m. The KG values are fluid KG throughout.

dp = dl + 0.6 (ds − dl) = 4.019 + 0.6 × 5.581 = 7.368 m
A = 0.4 As + 0.4 Ap + 0.2 Al

Notice the permeability column. The probabilistic method varies it with draught, 0.70 at the deepest, 0.80 at the partial, 0.95 in the light condition, on the reasoning that a lightly loaded ship has emptier holds. Chapter 5 used 0.60 for a loaded hold, and the difference is not academic.

6.6 The probability of surviving it

For a cargo ship the s factor reduces to one expression taken off the residual righting lever curve. The factor for the intermediate stages of flooding applies to passenger ships and to cargo ships fitted with cross flooding devices, and the allowance for heeling moments to passenger ships only; for a cargo ship without cross flooding both are taken as one.

s = K [ (GZmax / 0.12) × (Range / 16) ]^0.25

GZmax is counted no higher than 0.12 m, Range no further than 16 degrees, and K penalises a ship that comes to rest heeled: one up to 25 degrees for a cargo ship, nothing at 30. Because both caps are low, MV Ninja, with a residual GM above two metres and a residual lever at 16 degrees five to six times the cap, scores either one or nothing. Her index is settled by buoyancy and freeboard, not by stability.

6.7 The result

Seven of the eight zones survive. No.1 hold does not. Flooded at the summer marks with a permeability of 0.70 she sinks to 10.428 m and trims 567 cm by the head, until the forward draught reaches 13.600 m, above the freeboard deck at 13.50, so the hatchways are in the sea and s is nought, whatever the residual GM of 2.174 m may say. Chapter 5 worked the same damage at a permeability of 0.60 and found her forward draught 12.804 m, clearing the margin line by 62 centimetres. A change of a tenth in the permeability is the difference between passing and failing.

A = 0.69941 against R = 0.57333. She complies, with about 22 per cent in hand. At the deepest draught seven single zones and one pair (No.4 with No.3 holds) survive; at the partial draught the extra freeboard lets No.1 hold and four pairs through; at the light draught three pairs and the three midship holds together. Each partial index also clears the regulation's floor of 0.5 R = 0.2867.

6.8 What the index is actually for

An index on its own tells a master very little. What it does, and what no deterministic rule can do, is put a price on a design decision. Move a bulkhead and the index moves with it, by an amount you can read off.

The index is flat, then jumps. Bring the collision bulkhead aft from 139.60 to 139.28 metres, a third of a metre off the length of No.1 hold, and that hold stops immersing the deck when it floods at the summer draught: its 0.086 of probability changes sides and A rises from 0.6994 to 0.7344. Keep going aft and the index stays between 0.732 and 0.744, with a step at 135.86 m where a two hold case at the light draught crosses the 7.40 m trim limit of the hand method, until at 133.80 m the fore peak, now 14.2 m long, immerses the deck when it floods in its turn and the index falls back to 0.716. There is a window, from 133.8 to 139.3 m, and the calculation finds it. A deterministic rule could not have shown that the window existed, or where its edges were.

Run the second slider instead. A metre may be added to her centre of gravity at every draught without moving the index at all, because her survival is decided by whether the deck goes under, not by whether she has stability left. At a metre and a half the index has slipped to 0.670, at 1.88 m it meets R, and by two metres it has fallen to 0.482 as the flooded conditions at the partial draught lose their residual GM.

6.9 What the index does not tell you

It measures subdivision, not seamanship: it assumes the watertight boundaries are watertight, which on the day depends on whether the doors were shut. It is built on collision statistics alone, so it says nothing about grounding, which is why bottom damage sits outside it as a separate deterministic rule. And a single number conceals its own composition: two ships with the same A can fail in quite different ways, which is the method’s founding assumption and its most questioned feature.

For the officer in the chair the residue is smaller and firmer. The index was calculated at three draughts and three centres of gravity, and those calculations are what the limiting KG curve in the stability book is drawn from. Stay inside that curve and the ship is in the condition her index assumed.

Test yourself

Questions

1. Explain the difference between the deterministic and the probabilistic approach to damage stability, and state the weakness in the deterministic method that the probabilistic one is intended to remove.
2. Define the attained subdivision index A and the required subdivision index R, and state what each depends on.
3. Calculate R for a cargo ship of subdivision length 148.00 m, and again for one of 220 m. Comment on the direction of the change.
4. Define the factors p and s, and state which properties of the ship each of them depends upon.
5. Explain why the probabilities of all the damage cases considered must add to one, and what it would mean if they did not.
6. State the three subdivision draughts, explain how the partial draught is derived, and state the weighting applied to each partial index.
7. The permeability of a dry cargo space is taken as 0.70 at the deepest subdivision draught and 0.95 at the light service draught. Explain the reasoning, and explain why a higher permeability is not always the more severe assumption.
8. MV Ninja fails on No.1 hold at the deepest subdivision draught but passed the same damage in Chapter 5. Identify the single assumption that accounts for the difference and explain its effect.
9. Explain why the s factor for MV Ninja takes the values one and nought and almost nothing in between, and what kind of ship would behave differently.
10. A shipyard proposes to move a bulkhead two metres. Explain how the probabilistic method allows the proposal to be priced, and why no deterministic rule could do so.

Looking ahead

Having taken the trouble to stop choosing damages, Chapter 7 goes straight back to choosing them. The International Bulk Chemical Code and the International Gas Carrier Code are deterministic, and a great deal stricter than anything in Chapter 5. A chemical tanker is assigned to ship type 1, 2 or 3 according to how dangerous her cargo is, and the damage she must be able to take follows from that assignment. The cargo decides the standard, and the standard decides the ship. It is the opposite way round from everything so far, and for good reason.