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be swung and a batten placed in position. Now commence to fill
the tank. When full, take note of the deviation of the plumb line..
The draughts of the vessel having been taken, the displacement can
be read off the scale and the height of the metacentre from the
diagram. He would then be in possession of everything necessary
to give him the amount of metacentric height at the time, or the
height of the vessel's centre of gravity above the keel. Say dis-
placement, including the weight of the added fresh water, was
3,600 tons, and M 18 ft. above keel, the deviation obtained with a
20 ft. plumb line being *5 ft. The inclination obtained by filling
the tank at a distance of 20 ft. off the centre line is obviously equal
to moving the same amount of weight that distance across the deck,
since had it been placed upon the centre line no inclination would
have resulted, therefore, the added weight of 18 tons is multiplied
by 20 ft. to obtain the inclining moment.

w x d
18 x 20

G M = deviation - = 4 ft.

W X - 3,600 X ^

length of plumb line

M = 18 ft. above keel
G M = 4 ft. (as found by experiment)

therefore G = 14 ft. above keel.

The Theory and Design of British Shipbuilding. 69



Co-efficients for Heights of Centre of Gravity. In the designing
stages the height of the centre of gravity is generally required, and
is found either by the lengthy detailed calculation comprising every
item of the hull and machinery, or by the quicker method of com-
parison with a previous similar ship whose height of G, as obtained
by inclining experiment, is known. The method adopted for the
latter is to obtain the height of G for the previous ship, as a pro-
portion or co-efficient of the depth, moulded. Say, a vessel of 20
ft. depth, moulded, has her centre of gravity at 12 ft. above the
base line (i.e., the top of keel), the height of G would be :


- = -6 of the depth, moulded.

Say, the new vessel has a depth, moulded, of 22 ft., then 22 x -6
= 13-2 ft., which would be very near to the height of G for this
vessel, assuming her to be exactly similar in arrangement and
distribution of weight. Should there be differences, they should be
corrected for ; for instance, suppose the above new vessel has an
addition in the shape of a longer bridgehouse, which is situated at
26 ft. above the keel, the extra weight being 10 tons and the total
weight of the new vessel 1,520 tons, then

Weight. Lever. Moment.

Vessel exactly similar to the j

previous ship, i.e., without the I 1,510 tons x 13-2 19,932
amended bridge... ... ... j

Addition in arrangement over the ) ^ ^ 26 ^

basis ship J

Total weight of new ship ... 1,520 tons. 20,192

20,192 - 1,520 = 13-28 ft., which is the corrected height of G for
the new ship.

70 The Theory and Design of British Shipbuilding.

Of course, this is a very simple case, with only the one amendment,.
but it will suffice as an example. It is very probable that when
comparing a proposed ship with a previous vessel a fairly large
amount of corrections will be necessary, but, if carefully made.
the result obtained by use of this method will be very nearly correct.
and in some cases may be even more accurate than a figure obtained
by means of the independent calculation. The following are average
figures for the Co-efficient for Height of Centre of Gravity of hull

and machinery complete :

Co- Depth, Moulded,
Type. efficient. taken


to shelter deck ~

Cargo and Passenger, about 16 knots with shelter ) _ , , , ,

deck and erections above ...... ... (

Cargo Vessel with shelter deck ... ... ... '54 to shelter deck.

Cargo vessel with poop, bridge and forecastle i

covering about half length ......... / '60 to upper deck.

Fast Channel Steamer with awning deck ... -60 to awning deck.

Coasting Steamer with long raised quarter deck, \ [to a mean be-

bridge and forecastle, also double-bottom '72 -I tween main and

j (r.q. decks.

Steam Trawler ............... -75 to main deck.

Sailing Vessel ... ... ... ... ... '70 to upper deck.

Effect of Free Water in the Tanks, or Liquid Cargo in the Holds of
Vessels. A great loss of stability is caused by a free surface of water,
etc., in tanks or holds of vessels. The free surface of a vessel's
cargo, such as oil or grain, has a most dangerous effect if not guarded
against, since if an external force inclines the ship, the cargo will
move in the inclined direction and, of course, further increase the
amount of inclination due to the applied force. When such cargoes
are carried, every precaution should be taken to prevent a free
surface by keeping the holds quite full and to erect longitudinal
divisional bulkheads, so that if a free surface does occur, the move-
ment will not be large. A loose way of speaking is to say that a
free surface raises the centre of gravity ; what is meant is that it
" virtually " raises the centre of gravity. Take the case of a vessel
with a free surface of water in a double-bottom tank. Fig. 31
represents a vessel with a double-bottom compartment partly filled
and heeled over to a small angle, though exaggerated in the sketch
for the sake of clearness.

w I = Free water surface in the tank, and is parallel to WL, the
water-line of the vessel before inclination.

The, Theory and Design of British Shipbuilding.


?/?, l } = The new surface of water in the tank, and is parallel to
W, LI, the water-line of the vessel after inclination.

Let v f = the volume of water contained in either of the wedges
in the tank, w A w\ or l } A I,

g and g\ = their centres of gravity,

b - the centre of gravity of the water before inclination.

i>\ ,, after

Vf == the total volume of water in tank,

x g


= b b\


m - the point of intersection of a perpendicular through b\ and
the centre line of the vessel upon which lies the position of 6.

bb, v f X g g^

b m is obviously equal to - - and since b b\ -


we have b m =

v f x g g

V f X tan 9

i where i = the moment of inertia of free water

or b m = - surface about the fore and aft axis passing through

V f A, and V f the volume of water in tank, the proof

Fig. 31

72 The Theory and Design of British Shipbuilding.

of this latter equation is the same as that for B M previously dealt
with. Although b is the " actual " centre of gravity of the water
in the tank, its effect upon the ship is as though it were at m, which
is termed the " virtual " centre of gravity of water in tank. It
is like the case of a pendulum where the centre of gravity is at the
point of suspension, which in this case is m. In the sketch G re-
presents the position of the centre of gravity of the vessel, including
the water, while in the upright position : but on the slightest in-
clination the centre of gravity shifts to G,, because of the centre of
gravity of the water being virtually shifted to m. The effect is
similar to shifting the amount of water in tank a vertical distance
of b m. The value of the shift GG ; , which is the loss of metacentric
height, is therefore equal to

W f X b m where W f = the weight of the water in tank (tons)
and W = the total displacement of vessel,

W including water in tank (tons).

Converting these weights into volumes we have

V f X b m

- GG f (W being multiplied by 35 for volume),

W x 35
V, X b m

= GG f , or it may be written


V, i

- x b m = GG f . But we have already seen that b m = - ,

therefore x - = G G/ .


V f cancelling out, we have

- = GG f . We therefore see that the alteration GG f

depends entirely upon the amount of " moment of inertia of the
free water surface " and not the amount of water in the tank. A
small quantity of water may have more effect than a large quantity,
on account of the small quantity having a larger surface. It being
difficult to obtain the correct moment of inertia of the free water

The Theory and Design of British Shipbuilding. 73

surface, and since such is necessary so as to obtain the true position
of ra, this method is not quite satisfactory in the case of making a
correction to the result of an inclining experiment. A better and a
more reliable method is to take the two wedges w A w i and I A 1 1
and treat them as is done with the wedges of immersion and emersion
in an ordinary case for finding the shift of a vessel's centre of buoy-
ancy that is, to find the moment of transference of the wedges
and adding it on to the inclining moment caused by the moving of
the weights across the deck, thereby obtaining the total moment
which produced the angle of inclination. The addition to the
moment will, therefore, be the weight of a wedge multiplied by g g \ ,

v f
the weight of a wedge being equal to - Therefore,

35 for salt water
v f

- x g g\ = the addition to be made to the moment caused by

shifting the weights. G M at the time of experiment will, therefore,

wxd + ( - xgg*)

W x tan 9

If the double-bottom were divided by a longitudinal centre water-
tight girder, as shown in Fig. 32, the inclined water surface would
take up the positions as shown by w 2 l z and w 3 / 3 . In this case
we have a wedge w a 2 w 2 being shifted through a distance of g 2 g%
to 1 2 a 2 A, and another similar wedge, upon the opposite side of
the girder, being shifted a similar distance. It will be seen that
the volume of either of the transferred wedges is only one -quarter
of the volume of a transferred wedge when there was no division
in the tank, and that the transverse shift g 2 g 2 or </ 3 g 3 is only one-
half as great as g gr t , therefore the shift of a single wedge causes
a moment which is only (J x J) = Jth of the amount caused by
shifting a wedge when the tank had no centre division. But we
now have a shift of two wedges occurring, one on each side of the
centre line, therefore the shift of the two smaller wedges will cause
a moment equal to J x 2 = J of that caused when the tank was

74 The Theory and Design of British Shipbuilding.

not divided, and when only one large wedge was transferred. In
the above it is supposed that the vessel is prismatic (being of the
same section throughout), but, roughly speaking, we could say
that in the case of ordinary vessels the effect of free water in a vessel
with a water-tight centre girder is only one-quarter of that in a
vessel without same. In the case of a vessel where the centre
girder is not water-tight, but the holes through same are only small
such as drainage holes, it is practically impossible to estimate the
effect, as the levels of the water on each side may be at some inter-
mediate position to those shown in Figs. 31 and 32 at the time of
her beginning to roll back to the upright, since it will not have had
time to flow through the small holes and obtain the same level ;

Fig. 32.

but in the case of an inclining experiment, when the vessel is held
over to an angle for some time, the water has time to run through
and find its level on the opposite side. From this it will be seen
that great advantage is obtained by cutting nothing more than
small drainage holes through the centre girder, and a still greater
advantage by making it water-tight. While " corrections " for
free water may be made as above described, yet the description
goes to prove the great necessity for having no loose water in a vessel

The Theory and Design of British Shipbuilding.


at the time of an inclining experiment, it being difficult to estimate
the required amount of " correction " to the degree of accuracy
necessary when making corrections to the result obtained from that
operation. The above also applies to cargoes with a free surface
in the holds of vessels ; the effect on the vessel's stability due to
the erection of longitudinal bulkheads or shifting boards being
exactly similar to the case of fitting the water-tight girder in the
above ballast tank.

It has been shown that moment of inertia of the surface of free
water, divided by the volume of the vessel's displacement, equals
the loss in metacentric height caused by the free water surface.

= loss if the liquid is water ; but if it is other than water, and

the specific gravity different to that of the water in which the vessel
is floating, then GG f , or the loss of

i X weight per cubic foot of the liquid, in tons.
GM =

displacement in tons.

Effect on Initial Stability due to adding Water Ballast. The

present-day means of ballasting ships is to build tanks into the
structure, into which water is allowed to enter when extra immersion
is required or for the purpose of providing stability. The most
usual positions for water ballast are shown in Fig. 33. The tank
marked 1 is the " double-bottom " tank, being shown all fore and
aft in the hold space, and is also seen in sections a, b, and c ; 2 is






Fig. 33.

76 The Theory and Design of British Shipbuilding.

the after peak tank, and 3 the fore peak tank ; 4 is a ' ; deep " tank
divided longitudinally by a centre line water-tight bulkhead, also
shown in section a ; 5 are 'tween deck side tanks, being shown in
section by b. Another means of ballasting which, in some cases,
has been fitted, is to build small tanks upon deck between the
hatchways, etc. ; these are shown in the elevation, and also in section
c by 6. The filling of the various tanks has important effects on
the vessel's stability due to altering the position of the centre of
gravity, and also the immersion ; the alteration in the immersion
produces a new position of the transverse metacentre, since the
moment of inertia of waterplane and the volume of displacement
are changed, while the centre of gravity changes its position by
moving in the direction of the added ballast. Both of these altera-
tions must be taken into account when determining the change in
stability, and for this a metacentric diagram is of much use. Taking
the above-mentioned tanks, we may note their resultant effects
upon a vessel whose curve of transverse metacentres is show r n in
Fig. 34. At the light draught, 7 ft., the position of G is shown to
be 20 ft. above the base, the GM, therefore, being 5 ft., since the
curve of metacentres gives the height of M as 25 ft. for this draught.
Now, suppose the double-bottom tanks, the capacity of which is
400 tons, to be filled, the effect being to reduce the position of G to
15 ft. above the base, and to increase the draught to 10 ft. At
this new draught M is now found to be 22 ft. above the base, the
curve at the lower draughts descending and thereby causing this
alteration. G M is now 7 ft., which shows that the vessel possesses
a larger amount of G M in the new condition. The metacentric
height alone is not, however, the true means of comparing the
stability of the two conditions, the amount of righting moment
being the truest comparison. When dealing with metacentric
stability, we saw that G M sin == G Z, the righting lever, and
that the righting lever G Z multiplied by the displacement W gave
the righting moment, therefore W X G M sin H = righting moment.
We have seen that the above vessel in the light condition has a G M
of 5 ft., therefore at 10 deg. of inclination the righting lever will
be 5 X sin 10 deg. = 5 X -1736, and if the corresponding dis-
placement is 1,500 tons, then 1,500 X 5 x -1736 = 1,302 foot-tons
righting moment at 10 deg. of inclination.

When the double-bottom tanks are filled, the displacement is
increased to 1,900 tons, and we have already seen that the G M is
then 7 ft. The righting moment for the ballasted condition at 10

The Theory and Design of British Shipbuilding. 77

deg. of inclination is therefore 1900 X 7 x -1736 = 2,309 foot-tons.
The righting moment for the ballasted condition is seen to be about
77 per cent, more than when light, the difference being caused by
an increase of displacement and an increased metacentric height.
Comparing the metacentric heights only, the increase is 40 per
cent. The vessel will be much stiffer in the ballasted condition,
and will most probably roll heavily. This difference in stability
between the light and ballasted conditions is very representative
of the modern cargo tramp of full form, the ballasting in this manner
of such vessels in many cases having a similar effect, as shown in
this example, and is responsible for the uncomfortable rolling and
the heavy strains obtained in these vessels when ballasted and in
a seaway. By filling the aft and fore peak tanks, Nos. 2 and 3 in
Fig. 33, the change in the vertical position of the centre of gravity
is usually of small amount, although they may effect the immersion
to a fair extent. They are each extremely useful for trimming
purposes, on account of the large " moment " produced by their
being placed at the extremities of the vessel. When a vessel is-
without cargo it is very necessary that, apart from the necessity of
the provision of stability when such is required, ballast should be
provided so as to obtain greater immersion for the propeller and
thereby obtain greater efficiency, and save time when in a seaway
as well as to minimise the risks of the breaking of the tail shaft due
to racing when the vessel is pitching and heaving ; the increase of
draught also making the vessel much easier to navigate under such
circumstances. As in the above example, the latter necessities
are often provided at the expense of certain disadvantages. Of
course, it would be possible to obtain the required ^amount of im-
mersion for the propeller by placing a sufficient quantity of ballast
in the after peak. In such a manner the increase of immersion aft
could be obtained without making the vessel excessively " stiff "
as is often the case in the above-mentioned type, when the ballast
is fitted to a low position all fore and aft, but the trimming of the-
vessel would lift her higher at the fore end, and possibly be dis-
advantageous from the point of view of navigation, since with a
beam wind the tendency is for the vessel's head to fall off.

The position of M being governed by the underwater form and
shape of the vessel, it will be seen that the only means of controlling
the stability in such a case is by the position of G. In some modern
vessels of large beam the excess of stiffness prevails to a large extent.

78 . The Theory and Design of British Shipbuilding.

but in many cases of this type the disposition of ballast is now
arranged so that the position of G produced by filling the tanks is
such as will give a metacentric height which will result in easy rolling
when in a seaway as well as a reasonable amount of stability. In
such cases additional ballast tanks are fitted, which not only have
the effect of further increasing the immersion, but to raise the centre
of gravity to a higher position than would be obtained in the usual
way by filling the double-bottom and peak tanks only. The addi-
tional ballast is placed in " deep " ' 'tween deck side," or " deck "
tanks, as previously mentioned and shown in Fig. 33. Deep tanks
are usually situated near 'midships and are used as holds when the
vessel is in a " load " condition. Side tanks in the position as
shown by Fig. 33 may be used as bunkers when required. While
the effect of " deep " tanks is to reduce the excess of " stiffness "
when ballasted, yet in many cases where they are fitted this is still
found to exist. Such tanks also possess the great disadvantage of

setting up serious strains,
owing to a large weight
being confined to one par-
ticular point. In the case
of the patent ceiitilever-
framed type of ship built
by Sir Raylton Dixon and
X^ Co., Limited, both the

^^^^ above disadvantages are
done away with. The ad-
ditional ballast being
placed in compartments

-extending all fore and aft below the deck and next to the vessel's
side, has the effect of raising the centre of gravity, and thereby
preventing the excessive " stiffness " and heavy rolling, while the
distribution is such as will cause no strain to the vessel's structure.
A large number of vessels of this type have been built, giving excellent
results. When the " double -bottom " and the additional tanks are
filled, the immersion is such as will allow of easy and safer navigation
when without cargo in bad weather, the position of the additional
ballast producing ideal stability conditions.

Suppose that by filling the aft peak, fore peak and deep tanks,
the above vessel's draught is increased to 12 ft., the displacement
there being 2.200 tons, and the centre of gravity now being 17*5 ft.

The Theory and Design of British Shipbuilding. 79

above the base. The curve of metacentres here shows the metacentre
to be 21 ft. above the base, giving a G M of 3-5 ft. For this con-
dition at 10 deg. of inclination the righting moment = 2,200 x 3-5
X -1736 = 1,336-7 foot-tons, which shows the vessel to have a
righting moment which is very little in excess of the amount when
in the absolutely " light " condition.

The following is an interesting point in connection with the
ballasting of a certain vessel known to the author. It is generally
the case that when a double-bottom compartment is completely
filled, the value of G M has been increased ; the fact of the addition
of ballast at such a low position giving one, at first sight, the im-
pression that G has been lowered and the G M thereby increased.
This case shows how this might not always happen, the G M being
less when in the ballasted condition than when light. Fig. 35 is
the metacentric diagram for the vessel in question. Before filling
the tanks the draught is a, and after filling it is b. It will be noticed
that by the addition of the ballast we have lowered the centre of
gravity from G to G T , which fact, at first sight, makes one think
that G M is now larger. It will be seen that in this instance such
is not the case, because by adding the extra weight we have im-
mersed the vessel to a deeper water-line, where M is much lower
than it was when at the light draught, the reduction in the height of
M being greater than in the case of G. In ordinarily proportioned
cargo vessels of full form, M is usually decreasing rapidly about the
light draught, in this case the decrease being exceptionally rapid,
and the amount being more than the decrease in height of G, there-
fore, at b draught the metacentric height G , M T is smaller than G M
at a draught. However, in such a case, notwithstanding the
reduction in G M, it may happen that owing to the increase of
displacement, the righting moment at b draught may be greater
than when at a draught.

In the curves of metacentres for ordinary vessels, and especially
when the lines are of full form, it will be observed, as in Figs. 34 and
35, that M is highest at the light draughts at which the curve is
rapidly descending. Near to the load draught it is seen to be again
ascending. The reason for this variation is as follows : At the light
draughts of such vessels the waterplane is fairly large and wide,
producing a proportionate transverse moment of inertia, and the
volume of displacement being small, the value of B M is large since

80 The Theory and Design of British Shipbuilding.

= B M.


As the draughts increase, the height of M is lowering owing to the
volume of displacement increasing at a greater rate than the moment

of inertia of waterplane,
since the fulness of the
waterplanes is increasing
slowly. although the
height of B is increasing
and consequently tending
to lift M, yet the effect

due to the moment of


^-^ "" inertia and volume of dis-

" placement is predominant.
When the deeper draughts
Fig- 35. are reac hed, the displace-

ment does not increase so rapidly, because the vessel is of a wall-
sided shape over a large portion of her length at these draughts,
and, therefore, the rate of increase of moment of inertia and volume
of displacement becoming nearly equal, the value of B M is now
decreasing so slowly that the increasing height of B has the effect
again to raise M, since B is increasing in height at a quicker rate
than B M is lessening at these draughts.

The Theory and Design of British Shipbuilding. 81




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Online LibraryAmos Lowrey AyreThe theory and design of British shipbuilding → online text (page 6 of 14)