Chapter 15 tested a loading condition against six criteria, and the test took a KN table, a page of Simpson sums and a drawn curve. No officer runs that machinery for every proposed lift. Instead the stability information book carries a Maximum KG table: for each displacement, the highest KG at which every criterion still passes. The day’s check then collapses to one comparison: the actual KG from the moment table against the maximum from this table. This closing chapter builds that table for MV Ninja from her own KN data, and then uses it on the two questions it exists to answer: may we load this, and will we still be legal when we arrive.
The limiting KG at a given displacement is found by running the Chapter 15 compliance test repeatedly: raise the trial KG one centimetre at a time and stop the moment any criterion breaks. The last KG at which all six lines pass is the limit. The quicker way is to solve each criterion directly for the KG at which it is just met, because the righting lever GZ = KN − KG sinθ is a straight line in KG: the GM criterion gives KM − 0.15; the lever criterion gives (KN30 − 0.20)/sin 30°; each area criterion gives (Simpson sum of KN − required area) divided by the same Simpson sum of sinθ (0.13398 to 30°, 0.23396 to 40°, 0.09998 for the band); and the peak criterion, which on the booklet’s ten angles means that the 30° lever must not be below the 20° lever, gives (KN30 − KN20)/0.15798. The smallest of the six ceilings is the limit. One important point sits in the phrase all six: the table does not merely guarantee a GM of 0.15 m; it guards every criterion, and at deep draughts it is not the GM that sets the number.
Find the limiting KG for MV Ninja at the summer draught of 9.60 m (displacement 30456 t, KM 10.330 m, angle of flooding 52.8°), and compare it with the Appendix A table.
The KN rows at 29000 t and 30500 t bracket the displacement, fraction (30456 − 29000)/1500 = 0.971: KN10 = 1.807, KN20 = 3.633, KN30 = 5.149, KN40 = 6.334 m. The flooding angle exceeds 40°, so the areas run to 40°.
The six ceilings on KG. GM: 10.330 − 0.150 = 10.180 m. Lever at 30°: (5.149 − 0.200)/0.500 = 9.898 m. Area to 30°: second rule sum 3 × 1.807 + 3 × 3.633 + 5.149 = 21.469, area under KN 0.065450 × 21.469 = 1.4051 m rad, ceiling (1.4051 − 0.055)/0.13398 = 10.077 m. Area to 40°: first rule sum 4 × 1.807 + 2 × 3.633 + 4 × 5.149 + 6.334 = 41.424, area 0.058178 × 41.424 = 2.4100 m rad, ceiling (2.4100 − 0.090)/0.23396 = 9.916 m. Band 30–40°: (2.4100 − 1.4051 − 0.030)/0.09998 = 9.751 m. Peak at 25° or more (30° lever not below the 20° lever): (5.149 − 3.633)/0.15798 = 9.596 m.
The smallest is the last, so the limiting KG at the summer displacement is 9.596 m, 9.60 m to two decimals, set by the angle of the maximum GZ. Check by trial: at KG 9.596 m the levers at 20° and 30° are equal, 0.351 m, the areas are 0.120, 0.165 and 0.045 m rad and the GM is 0.73 m, so every line passes; at 9.61 m the levers are 0.346 and 0.344 m, the peak has moved back to 20°, and the condition fails.
Notice what set it. The GM at the limit is 10.330 − 9.60 = 0.73 m, nearly five times the legal minimum. Had the GM line been the only rule, the limit would have been 10.180 m. The shape criteria stopped the KG 0.58 m below that: at summer draught, the curve governs, not the GM. Appendix A prints 9.753 m at 30000 t and 9.631 m at 30500 t, which interpolate to 9.642 m: 4 cm above the hand figure, because the designer’s table was computed on the faired curve through the KN points, where the peak can sit anywhere between 20° and 30°. Where the GM or an area governs, the hand method and the table agree to a few millimetres; where the peak governs, expect a few centimetres. The ship’s table is the authority.
Repeating Worked example 16.1 at each tabulated displacement gives the table. From 15000 t to 28000 t the limit simply tracks the falling KM at a constant 0.15 m below it: the GM criterion governs, and the hand method reproduces Appendix A to within 4 mm at every row. Above 28000 t the curve criteria take over, first the area to 40 degrees, then the band between 30 and 40 degrees, then the position of the peak, and the limit drops away from the GM line, 0.54 m below it at the summer displacement (Appendix A 9.642 m against the GM ceiling 10.180 m). Below 15000 t the lightly loaded ship has a very large KM but the peak criterion holds the limit down again. The widening gap at the deep end is the reason the table exists: it is the region where a GM check alone would pass a condition that fails the criteria.
The everyday use of the table is a two line moment sum followed by one comparison. The routine, exactly as the examiner wants it shown: state the current condition and its maximum from the table; run the moment table for the proposed change; enter the table again at the new displacement, interpolating between rows; compare.
MV Ninja is at 26941 t with KG 10.08 m. Is the condition satisfactory, and may 900 t of deck cargo be loaded at Kg 15.0 m?
Current: Appendix A gives 10.235 m at 26500 t and 10.220 m at 27000 t; at 26941 t, fraction 441/500 = 0.882, the maximum KG is 10.235 + 0.882 × (10.220 − 10.235) = 10.222 m. Actual 10.08 m: satisfactory, with 0.142 m in hand.
The proposed lift, moments about the keel: 26941 × 10.08 = 271565.3; plus 900 × 15.0 = 13500.0; total 285065.3 t m over 27841 t, so the new KG = 10.239 m.
Maximum KG at 27841 t, between the 27500 t (10.206 m) and 28000 t (10.195 m) rows, fraction 341/500 = 0.682: 10.206 + 0.682 × (10.195 − 10.206) = 10.198 m.
Actual 10.239 m against maximum 10.198 m: over by 0.041 m, so it would not be safe to load the deck cargo. The current condition was legal; the lift itself breaks the limit, and the table said so in four lines of arithmetic. The greatest height at which this 900 t could go is found by setting the new KG equal to the new maximum: 900 × Kg = 10.198 × 27841 − 271565.3 = 12357.2, so Kg = 13.73 m, barely above the weather deck at 13.50 m.
A condition that is legal at the berth can become illegal at sea, because the ordinary changes of a winter passage tend to push the same way. In a ship whose bunkers are carried in double bottom tanks, burning them removes low weight, and removing low weight raises G; the tanks drawn on acquire free surfaces, which raise the fluid KG directly; fresh water from low tanks acts the same way; and ice forms high on hatch tops and rigging. The displacement falls at the same time, which on the deep part of the curve raises the maximum a little, but seldom as fast as the KG rises. The arrival condition must therefore be tested before departure. Worked example 16.3 takes double bottom bunkers as a stated assumption because it is the worst case; MV Ninja’s own fuel and fresh water tanks in Appendix A are wing and topside tanks with centres between 10.27 m and 12.65 m, so consumption from them lowers G slightly, while the free surface and the ice act in the same way on any ship.
MV Ninja sails at a draught of 9.40 m, displacement 29751 t, with a fluid KG of 9.85 m and all tanks pressed up or empty. Take her bunkers and fresh water to be carried low, as in a ship with double bottom fuel tanks (a stated assumption). On passage she burns 340 t of bunkers (Kg 0.85 m), consumes 55 t of fresh water (Kg 5.60 m), collects 45 t of ice on deck (Kg 15.2 m), and the slack tanks at arrival have free surface moments totalling 2600 t m. Test both ends of the voyage.
Departure: Appendix A gives 10.007 m at 29500 t and 9.753 m at 30000 t; at 29751 t, fraction 251/500 = 0.502, the maximum is 10.007 + 0.502 × (9.753 − 10.007) = 9.879 m. Actual 9.85 m: satisfactory, with 0.029 m in hand.
Arrival moments about the keel: 29751 × 9.85 = 293047.4; minus 340 × 0.85 = 289.0; minus 55 × 5.60 = 308.0; plus 45 × 15.2 = 684.0; plus the free surface moments, 2600.0. Total 295734.4 t m over 29401 t: arrival KG = 10.059 m fluid.
Maximum KG at 29401 t, between the 29000 t (10.082 m) and 29500 t (10.007 m) rows, fraction 401/500 = 0.802: 10.082 + 0.802 × (10.007 − 10.082) = 10.022 m.
Arrival 10.059 m against maximum 10.022 m: over by 0.037 m, so the ship will not arrive in a satisfactory condition. She sailed three centimetres inside the limit; on passage the fluid KG rose by 0.209 m while the maximum rose by only 0.143 m, and the voyage spent 0.066 m of a 0.029 m margin. The free surface alone accounts for 2600/29401 = 0.088 m of the rise: had the tanks been managed so as to arrive pressed up or empty, the arrival KG would have been 293134.4/29401 = 9.970 m, inside the limit by 0.052 m. The cure is planned before departure: press up or empty the slack tanks, keep double bottom ballast pressed up, budget for the ice, or stow less high.
The Maximum KG table is a good place to end, because everything in this volume stands behind it. The hydrostatic booklet of Chapters 6 to 10 supplies the displacements and the KM; the KN data read in Chapters 13 and 15 supply the levers; Chapters 11, 12 and 14 showed what docking, damage and motion do to the margins the table protects; and Chapter 15 built the compliance engine that each row of the table runs. The officer’s daily check, one KG against one maximum, is short precisely because all of that work was done in advance. That is what a stability information book is: the whole subject, folded down until the decision takes one line.
The limiting KG is the highest fluid KG at which all six criteria pass, found by inverting each criterion for KG (or by running the Chapter 15 test at rising trial values) and taking the smallest ceiling.
The Maximum KG table guards every criterion, not just the GM: from 15000 t to 28000 t the GM governs, above that the curve does, 0.54 m stricter at the summer displacement; where the peak governs, the hand figure and the designer’s table can differ by a few centimetres (9.596 against 9.642 m at 30456 t).
To use it: moment table for the proposed condition, enter the table at the new displacement (interpolating), compare actual against maximum, and always show the maximum in the working.
Check the arrival condition as well as departure: burning low fuel, free surfaces and ice all raise the fluid KG together.
One comparison, the whole volume behind it: that is what the stability book is for.