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of flotation " sometimes called the ' : tipping centre " which,
on account of being in the after body in this case, has the effect
to increase the draught aft by 2 ft. and reduce the draught forward
.by 3 ft., the difference in these amounts being due to the position
of the centre of flotation. The draughts now become 12 ft. aft
and 7 ft. forward, which gives a mean of 9 ft. 6 in. Taking a
displacement from the scale for this draught of 9 ft. 6 in. would ob-
viously be incorrect, since the inclined displacement is one of the same
amount as when floating at 10 ft. mean draught. In this case the
difference is large, being about 7 per cent., and should be taken
account of in the following manner. Take the mean of the new
draughts :

12 ft. in.
7 ft. in.



2 ) 19 ft. in.



= 9 ft. 6 in.



42 The Theory and Design of British Shipbuilding.

and to this add the distance m between the two level water-lines
as found by :

Total trim

m = x X

Length of water-line

where x is the distance of the centre of flotation aft of 'midships,
the total trim being the difference between the new draughts, 12
ft. aft 7 ft. forward = 5 ft. total trim. In this case

5 ft.

m = = 10 ft x - - = -5 ft. = 6 in.
100 ft,

9 ft. 6 in. -f- 6 in. ==10 ft., the latter being the draught at which
the displacement may be measured from the scale, and in most
cases it will be very nearly correct. If the centre of flotation were
in the fore body and the vessel trimming by the stern, the displace-
ment corresponding to a mean of the draughts will be in excess of
the actual ; therefore the correction for m is deducted in such a
case so as to obtain the level draught at which the displacement
corresponds to that of the inclined condition :

Trim by stern and C F aft of 'midships. add m.

forward . deduct m.

,, head ,, aft ,, . deduct m.

,, forward ,, . add m.

The difference of displacement when using a mean draught is
only worth taking into account when the trim is large and the centre
of flotation far from 'midships ; for instance, in the case of an
ordinary cargo vessel 300 ft. long, with the centre of flotation 5 ft.
aft of 'midships, and having 5 ft. of trim by the stern, the difference
in displacement would be about 25 tons, an amount certainly
worthy of notice.

In the case of an extremely large amount of trim, as for example,
that of a small coasting steamer with engines aft, and in the light
condition, where the draughts may be about 11 ft. aft and 1 ft.
forward, the displacement scale cannot be expected to give the
displacement satisfactorily if the mean draught is taken, and there-
fore a separate calculation should be made with the planimeter,
although by means of the above method a fairly near result may
be obtained. From the above we see that in the case of vessels



The Theory and Design of British Shipbuilding. 43



that are to have much trim, it is very necessary to design their lines
by working parallel to the required trim, and to make the displace-
ment and other calculations from the same.

The curve of block co- efficients is generally drawn as shown in
Fig. 21.

Curve of Longitudinal Centres of Buoyancy. A perpendicular
is erected to represent 'midships, and the positions as calculated
are set off from this on the respective water-lines, and a curve
drawn as shown, from which the position for any intermediate
water-line can be measured.

Curves of Vertical Centres of Buoyancy, etc. The vertical
positions of the C B being found relative to the base line, these
distances are now set off in the diagram by measuring them along
the dotted water-lines for the respective draughts ; for instance,
at the 8 ft. water-line the C B is 5-2 ft. above base ; upon this water-
line the amount is measured off from the perpendicular A B of
the draught scale, as shown by the arrow marks. For the curves
of areas of waterplaiies, tons per inch, areas of 'midship section,
and areas of wetted surface, the results are set off from the per-
pendicular A B. Offsets for area of waterplane and " tons per inch "
are shown upon the 2 ft. water-line ; and for the 'midship section
area and wetted surface, offsets are shown on the 14 ft. water-line.
The curve of centres of flotation is laid off in the same way as
described for the longitudinal centres of buoyancy. It should be
noted that in laying off the curves the results are set off at moulded
draughts, to which the calculations are made. These draughts
are shown by the dotted lines in Fig. 21.

The curves shown upon the displacement scale are extremely
useful in enabling one to obtain particulars of the vessel at any
required draught.



44 The Theory and Design of British Shipbuilding.



CHAPTER IV.

INITIAL STATICAL STABILITY : CONDITIONS. TRANSVERSE META-
CENTRE EXPLAINED. FORMULA FOR CALCULATING THE POSITION
OF THE TRANSVERSE METACENTRE. AVERAGE VALUES OF META-
OENTRIC HEIGHTS. THE METACENTRIC DIAGRAM ; ITS CONSTRUCTION

AND USE.



Conditions of Stable Equilibrium. We have already seen in
Chapter II. that for a vessel to be floating in equilibrium in still water
she must displace an amount of water the weight of which is equal to
the weight of the vessel, and also that the centre of gravity must lie in
the same vertical line as the centre of buoyancy. This is a condition
of equilibrium, and by adding : " And the centre of gravity must
be in such a position (below the metacentre) so that if the vessel were
were inclined, the forces of gravity and buoyancy would tend to bring
the vessel back to her former position of rest," we then have the con-
dition of stable equilibrium. Stable equilibrium means that when
a vessel is inclined from the upright she will return to that position
again as soon as the inclining force is relaxed. Unstable equilibrium
means that when a vessel is inclined from the upright she will not
return to that position, but incline further away from it. Neutral
equilibrium means that when a vessel is inclined from the upright she
will neither return to the upright nor incline further away from it,
remaining in the position she is inclined to. The first portion of
the above conditions is simply a condition of equilibrium, and
holds good in the case of neutral as well as stable equilibrium, since
in both we have the C G and C B in the same vertical line. It can
even be said to apply to a vessel in a condition of unstable equilibrium
while she is in the upright position, since we w^ould again have the
C G and C B lying on the same vertical line, but this would be only
momentary equilibrium, and an impracticable position. In all these
cases the vessel's centre of gravity is assumed to lie on the centre
line.

In Figs. 23, 24 and 25, which are transverse sections of a vessel,
we have the three conditions represented, an upright and inclined
position being shown for each condition. Upon being inclined,
the centre of buoyancy B shifts out to B , , the centre of gravity G
remaining fixed. In Fig. 23, where the condition of stable equilibrium



The Theory and Design of British Shipbuilding.



45



is shown, it will be seen that when the vessel is inclined, the downward
force of the weight acting through G and the upward force of the
buoyancy acting through Bj are acting with a "couple, 5 ' G Z,.




Fig. 23.

which tends to take the vessel back again to the upright. The line^
of the upward force of buoyancy cuts the centre line of the vessel
at a point M, which is known as the Transverse Metacentre. It
should be here noted that M is above G, thereby fulfilling the latter
portion of the condition of stable equilibrium as already given.
Fig. 24 represents the condition of unstable equilibrium and it will
be noticed that the forces are acting with a " couple," Z G, in
directions which tend to incline the vessel further from the upright.
In this case M is below G. The upright position shown in this
figure is the condition of momentary equilibrium, since we here
have G and B in the same vertical line ; but on the slightest inclina-
tion we obtain an upsetting moment, and since it is impossible to-




Fig. 24.



46



Theory and Design of British Shipbuilding.



keep the vessel rigidly upright, the condition is therefore an unstable
one. In Fig. 25 we have an example of neutral or indifferent
equilibrium, since here the vessel possesses neither a " righting
couple " nor a " couple " that will capsize her, because the downward
and upward forces are acting in the same vertical line, this being
caused by G and M coinciding. It will have been noticed that,
in the first two cases, the vessel's stability was dependent upon
the distances between the forces of gravity and buoyancy, which
is known as the " lever " and designated G Z. In the stable con-
dition we have a righting lever, M being above G ; and in the unstable
condition we have an upsetting lever, since M is below G. In the
neutral condition we have neither a righting nor upsetting lever,




Fig. 25.

since the upward force of buoyancy intersects the vessel's centre
line at G, so that we have the position of M coinciding with G. It
will be seen that the positions of G, the centre of gravity, and M,
the metacentre, govern the lever and therefore the stability. When
M is above G, a righting lever is obtained giving a stable condition ;
and when M is below G, an upsetting lever is obtained, resulting
in an unstable condition. When M and G coincide there is no lever,
which therefore produces a neutral condition. When the position
of G is fixed, the vessel's initial stability is then depending solely
upon the position of the Transverse Metacentre, which, we therefore
see, is a most important point.

Definition of the Transverse Metacentre. It is the point vertically
in line with the centre of buoyancy in the upright position, which is
also vertically in line with the new centre of buoyancy, corresponding
to a small transverse inclination to the upright. It is also the point



The Theory and Design of British Shipbuilding.



47



above which the centre of gravity must not be raised for the vessel to
remain in stable equilibrium. The distance between G and M is
the " metacentric height," generally spoken of as G M, being positive
when M is above G, and negative when M is below G. The position
of the Transverse Metacentre for the upright condition is found
by dividing the " transverse moment of inertia of waterplane "
by the volume of displacement up to that waterplane, the result
being the distance of M above the corresponding centre of buoyancy,
the amount being spoken of as B M. We therefore see that a large
moment of inertia and a small displacement give a high position of
M, a large moment of inertia being obtained by having a large width
of waterplane. During inclination, the position of M changes,
because the moment of inertia has been changing, but for all practical
purposes it may be said that in most ships its position remains
constant up to inclinations of 10 and 15 degs., and sometimes even
further ; therefore, within these limits, we can give consideration
to the vessel's stability by basing it solely upon the amount of
" metacentric height."

Formula for calculating the position of the Transverse Metacentre.
Let Fig. 26 represent a vessel transversely inclined over to a small
angle Q (exaggerated for clearness in the sketch) ; W F Wi is the
wedge of emersion and L 1 F L is the wedge of immersion. It will




Fig. 26.



48 The Theory and Design of British Shipbuilding.



be obvious that their areas must be equal so as to allow of the vessel
retaining the same amount of displacement. Areas of the section
are used for the sake of simplicity ; the vessel is supposed to be
prismatic, so that we need take no account of the length ; the areas
therefore, represent volumes. Let

^? = the area of either wedge. (Representing volume.)

g = the centre of gravity of emerged wedge.

gi == the centre of gravity of immersed wedge.

B = the centre of buoyancy in upright position.

B! = the centre of buoyancy in inclined position.

V = the total immersed area of the vessel's section. (Representing

Volume.)
M = the Transverse Metacentre.

On looking into Fig. 26 it will be seen that on account of the in-
clination and the resultant underwater form of the vessel, B has
shifted to B,, and a vertical drawn through this point, at right
angles to the new water-line W, Lj, cuts the former vertical i.e.,
the vessel's centre line at M, the Transverse Metacentre. If we
obtain the value of B BT . we can find B M, because at small angles
of inclination

B B T
B M =

tan 9

or B M tan P = = B B T

We therefore first of all find B B T . From the principle of moments
it will be obvious that B will shift in the same direction as the
wedges i.e., parallel to g g } and that its amount will be equal to

v X g g^

= B B,



V

Let us further simplify this equation. Since the angle of
inclination is supposed to be small, we may take y as being equal
to F W and F L or F W t and F L r , which are each half-ordinates
of the waterplane. We can also assume W T W and L t L to be
straight lines, thereby making triangles of the wedges. The centre
of gravity of a triangle being at f of its height from the apex, g F.
which also equals F g l , will therefore equal f y, or g g , will equal

Sy.



The Theory and Design of British Shipbuilding. 49

The volume of a wedge

y X W i W (emerged y x L i L (immerged
v = - wedge) or - - wedge)

2 2

being simply the area of the triangles.

Since W Wi and L LI = y tan ft, we may write

y x y tan ft

_ = i ?/2 tan w



2

Having now found the values of v and <? </,, we can write
v X # gr, Ji/ 2 tan x -#



BB, =



V V

f y ! tan



= B B



V

The angle being small, so that B M tan = B B 1 ,



tan 9 cancelling out, we have remaining

I y 3

BM = -

V
which is equal to

Moment of Inertia

BM =



Volume of Displacement

2/ 3 is equal to J of cubes' of half-ordinates multiplied by 2 for both
sides, which gives the transverse moment of inertia of the waterplane
about a longitudinal axis passing through its centre line. Knowing
the value of B M, we next find the height of B above the base, and
adding this to B M, we have the height of M above base.

If, from the height of M above base, we take the height of G above
base, we obtain G M, the Metacentric Height.



50 The Theory and Design of British Shipbuilding.

Values of Metacentric Heights. The following are average
figures giving good results in working conditions.

Ft. Ins. Ft. Ins.

Cargo steamers 9 to 1 6

Sailing vessels ... ... ... ... ... 2 6 to 3 6

High speed liners 1 to 2

Fast channel steamers 1 3 to 2

Paddle excursion steamers 1 6 to 2 6

Tugs 1 to 2

Steam yachts 9 to 1 9

Battleships 3 to 3 6

Cruisers 2 to 2 6

Destroyers 1 9 to 2 6

In the designing stages, or at other times when it is impossible
to obtain the calculated value of B M, it can be approximated by
the formula obtained as follows :

Moment of Inertia

BM ==



Volume of Displacement



Approximate Moment of Inertia



D * c b ~ 'Volume of Displacement.



B 2 x i B 2

Cancelling L and B = - - x.

D x c b D

i is a co-efficient which when multiplied by (L x B 3 ) gives the

HI.

c h is the block co-efficient of fineness.
x is the co-efficient obtained by combining i and c b .



Therefore we may write
Breadth 2



X x = Approximate B M.



Draught
An average value for x is -08.



The Theory and Design of British Shipbuilding. 51

The Metacentric Diagram is constructed from the results of
-calculations made for the value of B M at a few draughts. The
following shows a calculation made for the transverse B M. In
the proof of the formula for B M we saw that J of the cubes of half-
ordinates of the waterplane, when multiplied by 2 for both sides
.and divided by the volume of displacement, would give the B M
at the given waterplane. This method will be noticed in the following
where the half-ordinates are first cubed and then put through
Simpson's multipliers, the sum of functions of cubes being multiplied
by \ of the longitudinal interval so as to complete the rule, and then
by J and 2 as required by the B M formula, the result being the
transverse moment of inertia of the waterplane about a longitudinal
axis passing through the centre of the vessel. This is next divided
by the volume of displacement, as taken from the displacement
calculation, which then gives the value of B M i.e., the height of
the metacentre above the centre of buoyancy.



TRANSVERSE METACENTRE.



No. of
Section.


8 ft. Water-line.


.
Half-Ords. Cubes.


1
. S. M.


Functs.
of Cubes.



0-00





1





1


14-60


3,112


4


12,448


2


21-00


9,261


2


18,522


3


14-58


3,099


4


12,396


4


0-00





1






43,366
X J of Longitudinal Interval 25



X of Cubes



For Both Sides X



3)1,084,150

361,383
2



Transverse Moment of Inertia 722,766



52



The Theory and Design of British Shipbuilding.



Tons.
Volume of Displacement = 1,136 x 35 = 39,760 cu. ft.

I 722,766

= 18-18 ft, B M
V 39,760

4- B above Base 5-20 ft.



M above Base = 23-38 ft.



The moulded widths of the waterplane and moulded displacement
are used for this calculation. Having found the position of the
metacentre at a few draughts, as described and shown above, they
are next laid off in diagrammatic form so that the position of M
for any particular draught may be readily ascertained. In Fig.
27, a metacentric diagram is shown. The perpendicular A B is first



2




U-4-5'-H



1



Fig. 27.

drawn, and upon it the scale of draughts is set off. At right angles
to this perpendicular the values, as calculated, are set off upon
lines drawn at the draughts corresponding to the calculations ;
these are shown dotted at the 8 ft,, 14 ft,, and 20 ft, water-lines.



The Theory and Design of British Shipbuilding. 53



As in the case of the displacement calculation and scale, these are
moulded draughts, and, therefore, are situated higher than the
extreme draughts. It is usual to put the curve or vertical centres
of buoyancy upon the diagram. The curve of metacentres is next
laid off by measuring the B M's (i.e., the metacentre above the
centre of buoyancy) from the curve of V C B's, as shown by the
amount 18-18 ft. on the 8 ft. water-line, or 23-38 ft., which is the
distance of M above base, can be measured from the perpendicular
A B. The calculated positions of M having been set off, the curve
can then be drawn. The diagram is completed by drawing across
it level lines at the various draughts, and we then have a means of
quickly obtaining the height of the Transverse Metacentre at any
draught. The value of M above base can be obtained by measuring
the full distance from the draught scale to the curve, or if the B M
is required, the distance between Jbhe two curves is measured. The
metacentric diagram, which gives a graphic representation of the
locus of V C B's and Transverse Metacentres, is extremely useful
to the naval architect or ship's officer, as from it they are able readily
to obtain the position of the Transverse Metacentre at any par-
ticular draught, which position, when used in conjunction with the
position of the Centre of Gravity, then enables them to state the
conditions of the vessel's initial statical stability i.e., by obtaining
the Metacentric Height G M. As an example of its use, take the
case of the vessel dealt with in Chapter III., where a table was
drawn up and the vertical position of the centre of gravity calculated.
When loaded, the vertical height of C G was found to be 17-42 ft.
above base, and when light 16-62 ft. above base. Assume that
the curve of transverse metacentres shown in Fig. 27 is for this
particular vessel, and say that when loaded the draught is 19 ft.,
and when light 11 ft., these assumptions being made for the sake
of this example. Take the load condition first : At 19 ft. draught,
set off the height of the centre of gravity 17-42 ft., giving the point
G as shown in the diagram. Measuring the distance between this
point G and the curve of metacentres, we find the metacentric height
G M to be 2-2 ft. : for the light condition, the height of the centre
of gravity, 16-62 ft., is set off at 11 ft. draught, giving the position
G { , and the resultant metacentric height of 4-5 ft. In both cases G
is below M, and the vessel therefore is in a stable condition ; but
should it have occurred that, say for the load draught, the centre
of gravity had been in the position as shown by P, so that it was
above the metacentre, then the vessel would have been found to



54 The Theory and Design of British Shipbuilding.

be in an unstable condition when loaded, and therefore the dis-
position of the cargo would have to be so amended as to bring the
centre of gravity below the curve. Should this occur in the
designing stages (when the curve of transverse metacentres would
be approximated from a previous similar ship by proportioning
the B M's according to the square of the breadths of the two
vessels

BJ

B M of new vessel = B M of previous similar vessel x -

B 2

where B w = the breadth of the new vessel, and
B = the breadth of the previous vessel),

and should it be impossible to amend the disposition of the specified
cargo, the vessel would then require to be altered in dimensions
so as to obtain stability by means of increasing the breadth and
reducing the depth. The added breadth would cause an increase
of the moment of inertia of waterplane, and, consequently, the
height of the metacentre would be increased. Reducing the depth
would reduce the height of C G, and the combined effect of the
two alterations would tend to give the vessel stability by changing
the metacentric height from a negative to positive quantity. In
designing a vessel where the stability is an important point, the
metacentric height must be carefully looked into before the " lines "
are fixed. The approximate curve, mentioned above, is admissible
at the stage of fixed dimensions, but after the " lines " are drawn
out, the metacentric diagram should be quickly constructed so as
definitely to decide the question before building is actually com-
menced. It may not be necessary to draw out the diagram if the
worst possible condition of the vessel's loading is known, as the
metacentre may be calculated for this condition alone by using the
ordinates of the corresponding waterplane and finding the B M in
the usual way. The positions of the corresponding vertical centres
of buoyancy, however, may not be known at this stage unless the
displacement, etc., calculations are already made ; if not, this
position must be found by an approximate method, such as com-
paring it with another vessel whose " lines " are similar, taking the
position above the keel as a proportion of the draught, the pro-
portion used being obtained from the similar ship. In vessels of
full form the centre of buoyancy above the base is equal to about
55 of the moulded draught, and for finely shaped vessels -6 can be
used when the position is wanted roughly, though, of course, for



The Theory and Design of British Shipbuilding. 55



our present purpose the above-mentioned proportion must be used.
Whichever method is used, the figure obtained for the V C B above
base would not, however, be far from the correct one, and this then
being added to the B M, gives the height of M above base. In the
case of actual vessel, where the effect of a proposed cargo is being
investigated, the only remaining course when a negative metacentric
height is found to be the result, is to alter the position of the centre
of gravity by placing the heaviest weights in the lowest positions,
thus reducing the height of G. If the cargo is a homogeneous one,
the only means left is to fill a bottom ballast tank, the addition of
which will mean the reduction of an equal amount of cargo if the
draught is not to be increased, the combined effect being to provide
the vessel with stability. Another way in which the stability of a
vessel may be altered is by the burning out of the bunkers. Take
the case just mentioned, where a vessel at 19 ft. draught has 2-2 ft.
C M, and suppose an amount of bunkers is burnt out, so that the
draught is reduced to 17 ft. If the bunkers are situated in a low
position, the effect will be to allow the centre of gravity to rise,
since the reduction of weight is taking place in the low position.
Suppose that in this case where G in the load condition is 17-42 ft.
above base, the effect of burning out the bunkers to the above-


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