The Severe Wind and Rolling Criterion

Every criterion in Volume Two was worked in still water with the ship upright. This one is not. It puts a gale on the beam, sets her rolling into it, and then asks whether there is enough righting energy left when the gust arrives.

2.1 Why the intact criteria are not the whole story

The six criteria of Chapter 15 of Volume Two all measure the same thing: the shape of the GZ curve. They ask whether the curve is tall enough, whether it peaks late enough, and whether the area beneath it is large enough. What none of them asks is what is pushing the ship over. They assume the only heeling influence is whatever the officer has already accounted for in the loading condition.

That assumption fails as soon as the ship leaves the berth. A beam wind is a heeling moment that no moment table contains, and it grows with the area of the ship that the wind can see. Two ships with identical GZ curves, one a laden tanker and the other the same hull carrying a deck stow eighteen metres high, are not equally safe in a gale, and no amount of examining the GZ curve alone will show the difference.

The severe wind and rolling criterion, usually called the weather criterion, closes that gap. It appears in the 2008 Intact Stability Code alongside the general criteria and applies to the same ships. Of the Part A criteria that apply to a cargo ship, it is the only one that brings a heeling moment from outside the ship into the calculation, which is one reason it is easily overlooked.

2.2 The two levers

The wind is turned into a heeling lever in metres, so that it can be drawn on the same axes as GZ. The steady wind heeling lever is

lw1 = P × A × Z / (1000 × g × displacement)

where P is the wind pressure, taken as 504 pascals for a ship in unrestricted service, A is the lateral projected area of the ship above the waterline in square metres, Z is the vertical distance from the centroid of that area down to about half the draught, and g is 9.81. Section 7 of the MV Ninja booklet tabulates A and the height of its centroid against draught, and the lever Z is already worked out in the fourth column.

The gust lever is simply lw2 = 1.5 lw1. The gust is taken as half as strong again as the steady wind, and both levers are treated as constant with angle of heel.

One point catches people out. When there is cargo on deck, the profile area is no longer the tabulated one. The deck stow must be added to A, and the centroid of the combined area must be recomputed as a weighted mean. That is where most of the interest in this chapter lies.

2.3 The three angles

theta 0, the steady angle of heel

The ship settles where the righting lever equals the steady wind lever, that is where the GZ curve crosses the lw1 line. The Code sets a ceiling on this angle: it must not exceed 16 degrees, or 80 per cent of the angle at which the deck edge immerses, whichever is the smaller. On a deeply laden ship the deck edge angle is small and that second test governs.

theta 1, the angle of roll to windward

theta 1 = 109 × k × X1 × X2 × √(r × s) degrees

X1 depends on the ratio of breadth to draught, X2 on the block coefficient, and k on the area of bilge keels relative to the product of length and breadth. The factor r is 0.73 + 0.6 OG / d, where OG is KG less the draught, so it is negative when G lies below the waterline and the ship rolls less. The factor s depends on the rolling period T, itself found from T = 2 C B divided by the square root of GM, the same formula used in Chapter 14 of Volume Two.

theta 2, where the story stops

The area to leeward is measured up to theta 2, the smallest of three angles: 50 degrees, the angle of flooding, and the angle at which the GZ curve falls back through the gust lever on the far side of its peak.

2.4 Worked example 2.1: the loaded ship

Take the homogeneous full load departure of Chapter 1: 30456 t, fluid KG 8.113 m (solid KG 8.086 m plus 0.027 m for the free surface of the seven slack tanks, 815.2 t m), draught 9.60 m, GM 2.217 m. At that draught the booklet gives a profile area of 1044.2 m² with its centroid 14.08 m above the base line, so the lever Z is 9.28 m.

lw1 = 504 × 1044.2 × 9.28 / (1000 × 9.81 × 30456) = 0.01635 m, and lw2 = 0.02452 m. Those are levers of about sixteen and twenty five millimetres. They heel her to theta 0 = 0.42 degrees. She is deeply laden, with a summer freeboard of 3.92 m to the deck line, so her deck edge immerses at only 17.95 degrees and the governing limit is 80 per cent of that, 14.4 degrees, rather than 16. She is nowhere near it. She rolls 16.22 degrees to windward, theta 2 is the 50 degree cap, and by Simpson's rule area a is 0.094 and area b 0.678 metre radians: area b is 7.2 times area a. She complies by a wide margin.

2.5 Worked example 2.2: the same ship with cargo on deck

Now load her with a light bulk cargo stowing at 2.60 cubic metres per tonne, filling all five holds, and put a deck stow 5.00 m high across 148 metres of deck over the hatch covers. She comes to 24318.6 t at a fluid KG of 10.299 m, against a booklet maximum of 10.347 m, so she passes the intact criteria.

The windage tells a different story. The deck stow adds 740 m² to the 1308.0 m² of bare ship, the combined centroid rises to 14.08 m and the lever Z to 10.16 m. lw1 becomes 0.04396 m, nearly three times the laden figure, and because GM has fallen to 0.199 m she now heels to theta 0 = 9.01 degrees rather than under half of one degree. She still passes, with area b 11.5 times area a, but the margin that was very large in worked example 2.1 is now one that a loading officer can use up.

2.6 Worked example 2.3: spending the margin

Replace the deck stow with something light and tall: empty containers, taken here at 17.5 cubic metres per tonne in the stow envelope, each tier 2.59 m high (481.9 t a tier). Add them a tier at a time and watch what happens.

At six tiers she is at theta 0 = 9.34 degrees and complies. At seven tiers, 4117 m² of profile and a fluid KG of 10.564 m, theta 0 reaches 17.25 degrees. The limit is 16. She fails, by a degree and a quarter.

The important part is the column beside it. At seven tiers the maximum KG table allows 10.620 m and her fluid KG is 10.564 m. She is 0.056 m inside the limit the booklet prints. Every one of the six intact criteria passes. Her areas under the weather criterion pass too, with area b 2.5 times area a. She fails the one test that the maximum KG table does not include: the angle to which a steady beam wind heels her. At eight tiers the fluid KG of 11.055 m does finally exceed the maximum, so the booklet would have caught that one. The seven tier case is the dangerous one, because every check the officer normally makes is satisfied.

2.7 The cure

The first thought is usually to take cargo off. That is usually the wrong remedy, because the problem is not the weight but the height of G relative to the windage. Adding weight low down fixes it, and adding weight low down is what double bottom ballast is for.

Press up the No.1 double bottom water ballast tanks, port and starboard: 2 × 386.5 m³ of salt water, 792.3 t at Kg 1.43 m (booklet section 5). The displacement rises from 21430.3 to 22222.6 t and the fluid KG falls from 10.564 to 10.238 m, so GM rises from 0.206 to 0.440 m. Theta 0 falls from 17.25 to 13.43 degrees and area b rises to 4.6 times area a. The deck stow has not been touched. Of the two effects of the ballast, the fall in the wind lever is under five per cent (0.1536 to 0.1467 m); the rise in GM is a factor of 2.1, and nearly all of the recovery comes from it.

Note that the ballast brought her back with two and a half degrees in hand rather than on to the limit. Sailing on the exact limit of a criterion means sailing with no allowance for the ballast that will be consumed, the fuel that will be burnt from low tanks, or the water the deck stow will absorb. The No.1 tanks are right forward, so the trim must be checked alongside this calculation.

2.8 What this does to the maximum KG table

Chapter 16 of Volume Two derived MV Ninja’s maximum KG table by running the six intact criteria at rising trial values of KG until one of them failed. The weather criterion can be run the same way, and the correct procedure is to run both and print the lower of the two.

With a bare deck, and even with three tiers of boxes, the bulk carrier’s wide, shallow form gives her so much righting lever at moderate angles that the weather criterion is still satisfied when GM has all but vanished; the Code’s floor of GM at least 0.15 m, which sets the booklet maximum at these displacements, is the binding limit. At five tiers the two limits are within 20 millimetres of each other. At seven tiers the weather criterion governs, and by 134 millimetres. The gap opens as the windage grows, so the printed table is least reliable in the conditions in which it is most needed. This is why an approved book covering deck cargo carries limiting curves for each deck cargo arrangement rather than one curve for the ship. A limiting KG figure is only valid for the windage it was calculated with.

Test yourself

Questions

1. Using section 7 of the booklet, calculate lw1 and lw2 for MV Ninja at a draught of 6.60 m, displacement 20065 t, with no deck cargo.
2. Explain, with reference to worked example 2.1, why the 16 degree ceiling on theta 0 was not the governing limit for the loaded condition, and state what was.
3. A ship has GM 0.90 m, B 24.20 m, draught 7.00 m and length 148 m. Calculate her rolling period and hence the factor s.
4. State the three angles from which theta 2 is chosen, and explain why the smallest is taken.
5. MV Ninja carries a deck stow 148 m long and 9.00 m high with its centroid at 18.00 m above the base line. At a draught of 7.00 m, section 7 of the booklet gives a bare profile area of 1434.6 square metres with a centroid at 12.51 m by interpolation. Calculate the combined profile area, its centroid, and the lever Z.
6. Explain why adding weight low down reduces theta 0 in two separate ways, and identify which of the two is the larger effect.
7. In worked example 2.3 the ship passed all six intact criteria and passed the area comparison of the weather criterion, yet did not comply. State which test she failed and explain, in your own words, why the maximum KG table could not have warned her master.
8. A limiting KG curve in an approved book is annotated as valid for one deck cargo arrangement only. Explain the reason for that annotation.
9. The factor r is 0.73 + 0.6 OG / d. State the sign of OG for the Chapter 1 full load departure and for the seven tier condition, and explain what the difference in r means for the roll to windward.
10. A ship fails the weather criterion on the area comparison rather than on theta 0. State three ways in which the loaded condition could be altered to correct this, and identify which of them also improves the intact criteria.

Looking ahead

The wind was the first heeling influence from outside the loading condition. The next one comes from inside it. Chapter 3 takes up the International Grain Code, in which a bulk cargo that has settled during the voyage shifts across the hold and produces a heeling moment that does not go away when the weather improves. The shape of the calculation is the one just learned: a heeling lever drawn across the GZ curve, an angle of equilibrium where the two meet, and a residual area measured to leeward. Only the source of the lever changes, and the numbers come from section 9 of the booklet rather than section 7.