The shift of the centre of gravity
A weight w moved a transverse distance d moves the ship’s centre of gravity G across by GG1 = w x d / displacement. For 12 t moved 8 m on 18 000 t displacement: GG1 = 12 x 8 / 18 000 = 0.00533 m.
Reference tool — the list from a weight shift, nothing more
List from a weight shift.
Shift of G · list · crane lift
Weight and ship.
Type a line (12 t moved 8 m displacement 18000 t GM 1.2 m) or fill in the fields: the line and the fields stay in step.
Examples:
All 130 engineering tools
Enter the weight, the transverse distance it is moved, the displacement and the metacentric height (GM), and the calculator gives the list of the ship. For a lift with the ship’s crane, add the height from the load to the jib head: the calculator counts the rise of the centre of gravity and the reduced GM. With an allowed list it works the other way and gives the distance or the weight. It shows the working with your numbers, draws the heeled midship section, and gives a line you can copy or add to a request.
The formula tan(angle) = w x d / (displacement x GM) holds for small angles only, and the calculator notes a list above 10°. GM comes from the stability booklet or the loading computer; at zero or below the calculator says plainly that the formula does not apply. No allowed list and no rule figures here: the limits come from the ship’s stability booklet and the lifting plan.
How it is worked out
The triangle of G, G1 and M.
The list
The ship lists until the shifted G is under the metacentre M: tan(angle) = GG1 / GM = w x d / (displacement x GM). With GM 1.2 m: tan(angle) = 0.00533 / 1.2 = 0.00444, a list of about 0.25°. This holds for small angles only, while the metacentre can be taken as fixed.
A crane lift
From the moment it is lifted, the weight acts at the jib head: G rises by GG1 = w x h / displacement, and the list is worked with the reduced GM. For 5 t with the jib head 20 m above the load, an outreach of 15 m, 9000 t displacement and GM 0.9 m: GG1 = 0.0111 m, the reduced GM is 0.889 m, and the list is about 0.54°.
| Quantity | Relation | Note |
|---|---|---|
| Shift of G across | GG1 = w x d / displacement | m; w and the displacement in t, d in m |
| List | tan(angle) = GG1 / GM = w x d / (displacement x GM) | small angles only |
| Distance for a list | d = displacement x GM x tan(angle) / w | the reverse |
| Weight for a list | w = displacement x GM x tan(angle) / d | the reverse |
| Crane lift: rise of G | GG1 = w x h / displacement | h the height from the load up to the jib head |
| Crane lift: reduced GM | G1M = GM - GG1 | used for the list |
| Crane lift: weight for a list | w = displacement x GM x t / (d + h x t) | t = tan(angle) |
| Units | 1 ft = 0.3048 m; 1 kg = 0.001 t | 1 rad = 180 / pi degrees |
The weight is taken as already on board and inside the displacement. A lift from the quay or over the side also changes the displacement and the height of the centre of gravity, which the loading computer works out.
Sources
The shift of the centre of gravity GG1 = w x d / displacement, and the list tan(angle) = GG1 / GM from the triangle of G, G1 and M, are given in the standard ship stability textbooks for deck officers and in independent training material on small-angle stability. That the formula holds only at small angles, while the metacentre can be taken as fixed, is stated both in the textbooks and in reference articles on the metacentric height (which put that at up to about 10 to 15°). That a lifted weight acts at its point of suspension, the jib head of the crane, so that the ship’s centre of gravity rises and GM is reduced, and that the list is then worked with the reduced GM, is described in the stability textbooks, in shipboard training material on heavy lifts and in worked training problems.
Before you work it out
Questions about the list from a weight shift
How is the list from a weight shift worked out?
Moving a weight w a transverse distance d moves the ship’s centre of gravity G across by GG1 = w x d / displacement. The ship lists until the shifted G is under the metacentre M, and in the triangle of G, G1 and M: tan(angle) = GG1 / GM = w x d / (displacement x GM). For 12 t moved 8 m on a ship of 18 000 t displacement with GM 1.2 m: GG1 = 12 x 8 / 18 000 = 0.00533 m, tan(angle) = 0.00533 / 1.2 = 0.00444, a list of about 0.25°.
Why does a crane lift list the ship more than the same weight on deck?
From the moment it is lifted, the weight acts at the jib head, not where it lay. G rises by GG1 = w x h / displacement, h the height from the load up to the jib head, and the GM is reduced by as much; the list is then worked with the reduced GM. For 5 t with the jib head 20 m above the load, an outreach of 15 m, 9000 t displacement and GM 0.9 m: GG1 = 5 x 20 / 9000 = 0.0111 m, the reduced GM is 0.889 m, G moves 5 x 15 / 9000 = 0.00833 m across, and tan(angle) = 0.00833 / 0.889 = 0.00938, a list of about 0.54°; with the GM left at 0.9 m it would be 0.53°.
How far can a weight go for an allowed list, and how heavy can it be?
The reverse of the same relation: d = displacement x GM x tan(angle) / w. For an allowed list of 2°, 8 t, 12 000 t displacement and GM 1.0 m: d = 12 000 x 1.0 x 0.03492 / 8 = 52.4 m. The weight for a list: w = displacement x GM x tan(angle) / d; 1.5° at 10 m with 15 000 t and GM 1.1 m gives 43.2 t. In a crane lift the reduced GM depends on the weight itself, and the weight comes out as w = displacement x GM x t / (d + h x t), with t = tan(angle).
Why does the formula hold for small angles only?
The triangle takes the metacentre M as a fixed point on the centreline, which it is only at small angles of heel; the stability texts put that at up to about 10 to 15°. Above that M moves, and the list has to come from the GZ curve in the stability booklet or from the loading computer. The calculator notes every list above 10°, the lower of those figures.
Where does GM come from, and what if it is zero or below?
GM is the metacentric height of the ship in the present loading condition, from the stability booklet or the loading computer; the calculator does not work it out. At zero or below the formula does not apply: the ship has no positive initial stability and does not settle at a small list, and the calculator says so instead of giving a figure. In a crane lift the same holds for the reduced GM, GM - w x h / displacement.
What list is allowed during a lift?
The calculator gives no allowed list and no rule figures: the limits come from the ship’s stability booklet and the lifting plan. The largest list of a cargo operation is worked out in advance, and the GM checked before the lift. The calculator gives the list for the figures entered, and the largest distance or weight for an allowed list taken from those documents. Units: t or kg, m or ft, degrees or radians.
Crane or deck equipment parts?
Send the line and the equipment.
The calculator gives the list from a weight moved or lifted. When parts for a ship’s crane, deck equipment or lifting gear are needed, send the line from the calculator with the equipment details and a photograph of the nameplate, and Vessel Core checks the details before an offer goes back.
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