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Calculating KB, BM, and metacentric diagrams 109
The following is a description of the method used in constructing this diagram. The scale on the left-hand side represents a scale of metres, and it is from this scale that all measurements are to be taken.
First the curve of the Centres of Buoyancy is plotted.
For each draft plot the corresponding KB. For example, plot 6.65 m @ 13m, 6.13m @ 12m draft and so on to 1.55m @ 3m draft.
Join these points together to form the KB curve. In practice it will be very close to being a straight line because the curvature will be so small. See Figure 12.5.
Next the KM curve or Locus of Metacentres. For each draft plot the corresponding KM value given in the table.
At 13 m plot 11.60 m. At 12 m plot 11.30 m and so on down to plotting 20.54 m KM @ 3 m draft.
These points are then joined by a smooth curve as shown in Figure 12.5.
Note how it is possible for two different drafts to have the same value of KM in the range of drafts from 7 m to 13 m approximately.
For any draft being considered, the vertical distance between the KB line and the KM curve gives the BM value.
To ®nd the KB's and KM's the vertical distances are measured from the base line to the curves.
Example 1
Construct the metacentric diagram for a box-shaped vessel 64m long, 10m beam,and6mdeep,forevenkeeldraftsat0.5mintervalsbetweenthelight draft 1 metre and the load draft 5 m. Also, from the diagram ®nd:
(a) The minimum KM and the draft at which it occurs, and (b) The BM at 3.5 m.
Draft KB 12 draft BM B2/12d KM KB BM
1m 0.5m 8.33m 8.83m 1.5m 0.75m 5.56m 6.31m 2m 1.0m 4.17m 5.17m 2.5m 1.25m 3.33m 4.58m 3.0m 1.5m 2.78m 4.28m 3.5m 1.75m 2.38m 4.13m 4.0m 2.00m 2.08m 4.08m 4.5m 2.25m 1.85m 4.10m 5.0m 2.5m 1.67m 4.17m
See Figure 12.6(a) for KB and KM plotted against draft.
Explanation. To ®nd the minimum KM, draw a horizontal tangent to the lowest
point of the curve of metacentres, i.e. through A. The point where the tangent
cuts the scale will give the minimum KM and the draft at which it occurs.
Note. It is shown below that for a box-shaped vessel the minimum KM and the p
draft at which it occurs are both given by B/ 6, where B is the beam.