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mentioned extent is to increase the height of G by 1 ft. 3 in., which
now gives G at 18-67 ft. above base, as shown by B in Fig. 27. The
metacentric height is now only 1 ft. against 2-2 ft. in the fully load
condition. This being a large reduction, shows that but for the
vessel having a fairly large amount of G M in the original condition,
she might have been left with a dangerously small amount, or even
in an unstable condition. The metacentric diagram is therefore
seen to be a valuable asset to the ship's officer, who is "nowadays,
in some cases, supplied with it. The metacentre being solely
dependent upon the vessel's form, it is therefore fixed for any
draught, the stability being then dependent upon the disposition
of the weights. Having the diagram, it then remains to estimate
the position of G for the particular condition of loading, and knowing
the position of G for the light condition, the cargo, bunkers, etc.,
are then added, and by the principle of moments the combined
centre of gravity for the total weight is found, the positions of the
cargo, etc., being obtained from the capacity plan supplied to the
vessel. The condition of the vessel can therefore be obtained before
loading is commenced, or by these means the positions of cargo
can be obtained so as to produce any required condition, or again,



56



The Theory and Design of British Shipbuilding.



it can be ascertained if ballasting will be necessary for any particular
disposition of cargo. A large metacentric height produces a " stiff "
vessel and one that will roll heavily ; this could be avoided by making
the above investigations and fixing an amount so as to obtain a
safe quantity of stability without making the vessel excessively
stiff. The following is an example of such an estimate for stability
as made previous to the loading of a vessel, the " levers " being
the distances of centres of gravity of the various quantities above
the base :

Items.

Vessel in Light Condition

Bunkers in Hold

Bunkers in Bridge

Cargo in Fore Hold

Cargo in After Hold

Fresh Water in Tanks in Bridge

Stores in Poop



Weight
in Tons.


Levers
in Feet


Moments
Foot-tons.


1,000


... 15


... 15,000


100


... 10


... 1,000


100


... 22


... 2,200


860


... 10


... 8,600


800


... 11


... 8,800


10


22


220


5


24


120



2,875



35,940



35,940 -=r 2,875 = 12*5 ft, = = Height of G above base.
Say 12-0 ft. = = Height of M above base.

G would therefore be -5 ft. above M, and the vessel in an unstable
condition. Should the above quantities and positions of cargo be
imperatively fixed, the only remaining means of providing stability
is by ballasting. Say 400 tons of water is added in the double-
ballast bottom tanks, the effect is found as follows :

Weight Levers

in Tons. in Feet.

2,875 ... 12-5
400 1-5



Vessel in the above condition
Ballast in double-bottom tanks



Moments
Foot-tons.
35,940
600



3,275



36,540



36,540 -^ 3,275 = 11-14 ft. = Height of G above base. The addition
of 400 tons will sink the vessel down to a deeper draught, at which
the position of M will have changed ; therefore this new position,
for the new draught, must be taken from the metacentric diagram.

Say at the new draught the height

of M is 11-75 ft. above the base.
G as altered by the ballast = 11-14 ft. ,,

which gives a positive G M of -61 ft.,

the vessel being therefore in a stable condition.



The Theory and Design of British Shipbuilding. 57



CHAPTER V.

METACENTRIC STABILITY. EFFECT OF INCLINATION AND SUCCESSIVE
METACENTBES. To DETEBMINE PBACTICALLY, BY INCLINING Ex-
PEBIMENT, THE HEIGHT OF A VESSEL'S CENTRE OF GRAVITY ; THE
PBOCEDUBE AND NECESSARY PRECAUTIONS. EXAMPLE OF IN-
CLINING EXPERIMENT.



Metacentric Stability. In Fig. 26 it will be seen that the righting
lever G Z is equal to G M sin 9. It is usual to speak of statical
stability as a " moment " of foot- tons, it being equal to the weight
of the vessel, in tons, multiplied by the '"' righting lever " measured
in feet. The weight being equal to the displacement and the righting
lever equal to G Z, we therefore have the

Moment of statical stability = W X G Z,

or W X G M sin 9,

W being the displacement in tons and 9 the angle of inclination.
As previously mentioned, the position of M can be assumed to remain
constant, in most cases, up to angles of 10 to 15 degrees ; therefore,
within these limits it will be seen that the amount of statical stability
can be determined from the metacentric height, and is consequently
spoken of as metacentric stability.

Effect of Inclination. As a vessel inclines the width of the
waterplane increases, and, consequently, the moment of inertia
becomes larger ; and since the volume of displacement is constant,
the value of B M is increased. From the formula it will be seen
that the moment of inertia varies as the cube of the breadth, there-
fore a small increase in breadth can be responsible for largely
increasing the moment of inertia, which again causes M to rise.
A position can, therefore, in some cases be reached where the in-
creased moment of inertia of waterplane will produce a height of
M so that the vertical through the new centre of buoyancy will
intersect the vessel's centre line at G, and then a position of rest
will be attained, since both the forces of weight and buoyance will
be acting on the same vertical line, although in the upright position
the vessel was unstable on account of M being below G.

Fig. 28 illustrates this point. In the upright position the centre
of buoyancy, metacentre, and centre of gravity are represented by



58



The Theory and Design of British Shipbuilding.



B, M and G respectively. M being below G, the vessel is, therefore,
in an unstable condition. On inclination, the centre of buoyancy
travels in the direction shown by B, B 2 B 3 , etc., and corresponding
to these positions we have lines drawn perpendicular to the
corresponding water-lines. The moment of inertia being calculated
for each water-line, and divided by the volume of displacement,
gives the B M, which is set off on these perpendiculars giving M 2 M 3 ,
etc. For the first inclination, which is a small angle, the B M
coincides with the upright value, the perpendicular cutting the centre
line at M ; but on increasing the inclination the altered waterplane




J n



Fig. 28.



The Theory and Design of British Shipbuilding. 59

width causes a rapid increase in moment of inertia, which now gives
larger values for B M, as shown for the succeeding angles. It
will be seen that the upward forces of buoyancy no longer intersect
the vessel's centre line at M, but at points situated higher. The
perpendiculars B 2 M 2 and B 3 M 3 intersect below G, and therefore
the vessel is still unstable, and will continue inclining ; but on
reaching B 4 M 4 this perpendicular is seen to intersect at G, where
we now have the forces of weight and buoyancy lying upon the same
vertical line, therefore producing equilibrium, this then becoming
the position of rest. Now, suppose the vessel to be still further
inclined by an external force, such as the wind, so that the centre
of buoyancy reaches B.-,, the B M here being B 5 M% the perpendicular
now intersecting the centre line at a point above G, therefore causing
a righting lever G Z (the distance between the two forces), which
will bring the vessel back to the position corresponding to B 4 G M 4 .
Of course, it is only in some cases where this occurs ; it may happen
that successive perpendiculars through the new centres of buoyancy
would intersect at points lower than M, instead of increasing in height
as described in this case. The above changes are caused by the
varying moment of inertia, which, while increasing, gives a large
B M and a large shift of the centre of buoyancy, the latter being the
means of changing the point of intersection of the buoyancy per-
pendicular and the vessel's centre line, upon which the centre of
gravity lies. B,-, corresponds to an inclination of 45 degrees where
the deck-edge enters the water, and at which position the moment
of inertia is at its maximum, since the waterplane width is greatest
here. The moment of inertia now reduces as the waterplane width
becomes less, and the shift of the centre of buoyancy does not increase
so rapidly as previously. At 90 degrees of inclination, B 1 is the
corresponding centre of buoyancy, and BI O # the line of the upward
force, which is here seen to be intersecting below G. If perpendiculars
are produced from B,, , B 7 , B s and B 9 , it will be found that the
points of intersection with the centre line gradually reduce in height
from the position obtained by the perpendicular B ; -, M r , .

In the case of a vessel carrying deck loads, such as timber, where
the centre of gravity is raised to a high position on account of the
cargo on deck, a negative Metacentric Height is sometimes obtained
when in the upright position. The result is that the vessel inclines
until she reaches an angle where the moment of inertia of waterplane
will produce a value of B M so that the vertical through the new



(50



The Theory and Design of British Shipbuilding.



centre of buoyancy will intersect the vessel's centre line at G, and
thereby obtain the position of rest as mentioned in the a,bove.
The following interesting case of a timber-laden ship was brought
to the notice of the author. A certain steamer loading in the Baltic
gradually listed as the deck cargo was being put on board. By
ballasting she was brought as near to the upright as the capacity
of ballast would produce, eventually sailing with a few degrees list
to starboard, which during the crossing of the North Sea, increased
until the alarming angle of 35 degrees was reached, when an amount
of deck cargo on the lower side (starboard) broke adrift, causing the
vessel to give a few rolls and to finally settle to a much smaller angle
of inclination, but to the port side, which, after trimming some
cargo from port to starboard, was further reduced. The causes of
the above were as follows : As the bunkers were being burnt out
from a low position, the centre of gravity arose from GI to G 2 (Fig.
29), and therefore the vessel further inclined so as to obtain stability




in the way mentioned above. When the cargo broke adrift the
vessel rolled owing to the sudden withdrawal of weight from the
starboard side, and the reason of her taking up the smaller list was
because the position of the centre of gravity was now
reduced in height, and therefore required a less amount of
moment of inertia to obtain the coincidence of the buoyancy per-
pendicular with that point. An amount of cargo being deducted



The Theory and Design of British Shipbuilding. 61



from the starboard side, the list was necessarily to port, the trimming
of the excess cargo from port to starboard then obviously reducing
the list,

Throughout the above it will have been observed that, while a
vessel is in her initial upright position, the amount of stability is
depending upon the situation of the points M and G, the latter
being fixed and the former varying according to the vessel's under-
water form, it being determined by the line of the upward force of
buoyancy. M being found by means of the calculations described
in the above, it therefore remains to obtain the position of G, the
centre of gravity, It must first be found for the light condition,
accounting for every item of the hull and machinery by means of
the principle of moments. This is an enormous calculation, in-
volving many figures, introducing the possibility of error ; besides,
for a large amount of the weight, it is difficult to obtain an accurate
position of the centre of gravity. It is, therefore, usual to obtain
the position of G by means of an Inclining Experiment performed
at the finishing stages of the vessel's construction. For a load
condition, the amounts of deadweight are added, and G found as
shown in a previous example.

To determine practically, by Inclining Experiment, the height of
a vessel's Centre of Gravity : We know that if a weight is moved
across the deck of a vessel an alteration in the position of the centre
of gravity occurs, the C G moving from G to GI (see Fig. 30). The
distance moved is G G l . We also know that, at all positions of rest,
the vessel's centre of gravity and centre of buoyancy lie in the same
vertical line, this vertical being perpendicular to the water-line at
which the vessel is floating. This perpendicular cuts the centre
line of the vessel's section at a point M which, for small transverse
inclinations, is the Metacentre, Now, if we know G G! we can find
G M (which is the Metacentric Height], it being obvious that

G G,

G M = - 9 being the angle of inclination.

tan 9

w X d

G G ] is readily found by - , where w - = the weight moved

W

across the ship a distance of d, and W being the total weight of
the ship at the time. To find the angle of inclination that is caused



The Theory and Design of British Shipbuilding.



by the shift of the weight, an Inclining Experiment must be per-
formed as is afterwards described. Having found the value of



ri r\

l (

G M by means of - I or



w x d



which is the same



tan 9



W x tan



thing), we can find the position of G relative to the keel by first of
all finding the position of M, the Transverse Metacentre, as was
described in Chapter IV., where we saw that M was determined by



B M =



Transverse Moment of Inertia



Volume of Displacement



Calculating the position of B above keel, or scaling it off the curve
of vertical centres of buoyancy, usually shown on the displacement
scale, and adding this height to B M, we have the position of M
above keel. If from this we subtract G M, as found above, we will
have the height of the centre of gravity of the ship above keel, G, of
course, lying below M, so as to allow of the vessel possessing stable
equilibrium.



v4Ctti|Ht A




Fig. 30.



The Theory and Design of British Shipbuilding. 63

Procedure and Precautions. It will be seen that the following
information is required for the Inclining Experiment : The weights
that are moved and the distance, the total weight of the ship, the
angle of inclination, the height of the transverse metacentre at the
time (this either being obtained from the metacentric diagram or by
independent calculation for moment of inertia of waterplane, volume
of displacement and vertical position of the centre of buoyancy), the
weights and positions of all things that have either to go aboard,
come ashore or be shifted after the experiment has been performed.

In Fig. 30

G = Centre of gravity in upright position before weight is moved.

G, = Centre of gravity in inclined position after weight is moved.

M = The transverse metacentre.

B = The centre of buoyancy in upright position.

B[ == The centre of buoyancy in inclined position.

P = Point of suspension of the plumb line.

y = Length of plumb line to batten in upright position.

x = Deviation of plumb line.

The batten, upon which the inclinations of the plumb line are
marked, must be rigidly fixed. To prevent the plumb line trembling
and to help it in becoming steady, before noting the inclination, a
bucket of water may be used, as shown in the sketch. Before
commencing the experiment a rough estimate of the position of
G should be made to enable a preliminary calculation to be made
to ascertain the probable weight required to give the vessel the
angle of heel aimed at. The vessel having been brought to the
upright, or as near as can be obtained, a batten is erected upon two
trestles at right angles to the plumb line, as shown. The weights
that are used should be sufficient to produce a heel of 4 to 5 degrees.
The following precautions should be taken : Make a personal
inspection of the vessel and ascertain that she is properly afloat ;
boilers full to working level ; ballast tanks quite full or empty (it
is even better to have them full than to have a small quantity of
water rolling from side to side during the experiment) ; fresh water
and service tanks full ; that all movable weights are secured ; any
boats, floating stages, etc., that are moored to the vessel are removed
or slacked off, and all gangways removed. The mooring ropes
should be slacked off, and anchors hauled up. No men should be
on board other than those actually engaged in the experiment.



64 The Theory and Design of British Shipbuilding.



The experiment should be performed when there is little or no wind ;
if any, the vessel's head should be put to it, if possible.

The weights that are used in the experiment are divided, half
being placed on the starboard side and the remainder on the port
side, the distance from the centre line being the same in each case.
The men employed in the work of moving the weights from side to
side must take up a position on the centre line of the vessel when not
engaged in the operation.

Everything now being ready, the draughts forward and aft should
be taken, so that the actual displacement, transverse moment of
inertia, and the position of the centre of buoyancy might be ascer-
tained for the vessel's conditions during the experiment. Before
moving the weights the position of the plumb line should be marked
on the batten, and the distance from the point of suspension P to
the batten noted. This is shown in Fig. 30 as y. Now move the
weight A on to the top of B, or in a position so that its distance
from the centre line is the same as in the case of B. The distance
moved is d feet. Note the inclination of the pendulum for future
reference by marking it on the batten. The weight A should now
be moved back to its former position, when the plumb line should
correspond with the mark which was placed on the batten before
the weight was formerly moved. Should the line and mark not
coincide, it proves that there must have been some loose w r ater or
something which at the first inclination rolled out of its place and
has not been able to regain it. Next move the weight B over to a
position corresponding with A, and note the deviation as before,
and which should be the same. Care should be exercised in taking
the deviations after the ship has become steady. If there happens
to be a slight difference in the deviations for these two shifts (one
port and one starboard), the mean should be taken. Generally,
two or three plumb lines are used, being placed one in the forward
holds, being allowed to swing in the hatchways ; the jsecond at the
fore end of the stokehold or some other suitable position in the
engine and boiler space, the trestles being, in both cases, placed
upon the tank top ; the third would be placed in the after holds,
the trestles being placed upon a lower deck hatch top, so as to be
clear of the tunnels. The mean deviation is taken at each plumb
line, and the tangent of the angle found by dividing the deviation
x by the length of the plumb line y. This being done for the three



The Theory and Design of British Shipbuilding.



plumb lines, a slight difference may be found in the " tangents,"
but the mean of the three being found, we may use it in the equation

w X d

= G M



W x tan

Should the vessel be floating with a large amount of trim, it is
preferable to make an independent calculation for the displacement,
centre of buoyancy and transverse moment of inertia for the draughts
at which the vessel floated during the experiment. If the trim is
small, these particulars may be taken off the curves, as shown upon
the displacement scale, which for ordinary vessels is drawn up for
even -keel draughts. From the above it will be seen that the height
of the centre of gravity above the top of keel =

height of transverse metacentre above keel - - G M

= M - - G M

= (B M + h) - G M

Gr\
\x\



tan
w x d

x

W x-

y

Where I = Transverse moment of inertia of waterplane.

V = Volume of displacement.

h = Centre of buoyancy above keel.

W = Total weight of ship plus the weights used in experiment.

The weights and positions of all things that have either to go aboard,
come ashore, or be shifted after the performance of the experiment,
having been rioted, the corrections for them are made as follows :
If a weight has to be taken ashore (such as those used in the ex-
periment), multiply the weight by the vertical distance of its centre
of gravity from the centre of gravity of the total, as found by the
experiment, and divide by the total weight of displacement at the
time of the experiment, reduced by the weight of the article that
is being taken off, and the result will be the shift of the vessel's

6



66 The Theory and Design of British Shipbuilding.



centre of gravity. This shift will be upwards if the weight was
removed from a position below G, and downwards had the weight
been taken from a position above G.

w x d r
Shift of C G vertically = -

W - - w

d r = distance between centre of gravity of weight and vessel's C G,
including the weight.

If a weight were to be added, multiply the amount by the distance,
measured vertically, from its centre of gravity to the vessel's centre
of gravity, and divide by the weight of the vessel, at time of ex-
periment, plus the added weight, and the result will be the shift of
the centre of gravity of the ship. If the added weight is placed
above the former position of G, the shift will be upwards, and if
below, it will be in a downward direction.

w x d v
Shift of C G vertically =

W + w

d" = distance between centre of gravity of weight and vessel's C G.
If the weight has only to be shifted to another position in the ship,
the correction would be :

Weight multiplied by the distance moved vertically, divided by

w x d v

the displacement = - , G altering in the same direction as

W

the moving of the weight.

Example of inclining experiment performed on steamship 254
ft. x 38 ft. x 18 ft. 8 in. moulded dimensions.

Displacement at time, 1,590 tons.

Weight moved, 25 tons a distance of 30 ft.

Two plumb lines used.

Deviations of forward plumb line, 16 ft. long.

1st shift to jtort ... -93 ft. ) )

1st starboard -96 ft. / * ( -95 ft. mean

2nd port ... -94 ft. 1 [ deviation.

, \ '955 ft. mean [
2nd starboard -97 ft.



The Theory and Design of British Shipbuilding.



Mean deviation,



95 ft.



= -0594 tangent.



Length of plumb line, 16-0 ft.
Deviations of after plumb line, 12 ft. long.



1st shift to port ... -71 ft. |
1st ,, starboard -70 ft. I

2nd



ft. mean ,

i -71 ft. mean



2nd port ... -72 ft. I
2nd starboard -725ft. J

Mean deviation,



mean



I deviation.



71 ft.



Length of plumb line, 12-0 ft.



= -0591 tangent.



( -05Q4 1
Mean of tangents \ * -05925 ft.

I *v/Ot7-L I



w



25 x 30



G M =



= 7-96 ft.



W x tan 61 1,590 X -05925
B M ... = 15-05 ft.

B, above keel = 4-56 ft.



M, above keel = 19-61 ft.
Less G M ... 7-96 ft.

= G above Keel = 11-65 ft. at time of experiment.
Corrections as under :



67







CG above






Weight


keel


Moments.




Tons.


Ft.




Vessel at time of experiment


1,590


11-65


18,523-4


Add fittings not yet aboard


*15


12-1


181-5


Tons


1,605





18,705-0


Deduct Inclining weights ... 25 "\
Water ballast ... 265 !


305


19-0
1-5


475-0 ^
397-5 1 l 022 . 5


Staging, tools, rubbish, j
etc 15)




10-0


150-0 J



1,300 tons.



17,682-5



17,682-5 -r 1,300 = 13*6 ft. The corrected height of the centre of gravity

above the keel.

("This total is obtained from a separate calculation taking into account all the
various items).



68 The Theory and Design of British Shipbuilding.



It will be seen that the inclining experiment, while an important
thing, is a very simple undertaking. All that is required is to obtain
a deviation on the plumb line by listing the ship by means of a known
transverse moment obtained, as in the above example, by shifting
a weight, or it could be done by the adding of a single weight on one
side of the vessel. The whole thing is therefore, within the reach
of every ship's officer, provided he is supplied by the builders with
the displacement scale and metacentric diagram. For instance,
suppose a vessel is about to have a fresh water tank filled, the
position of which is at one side of the vessel say, 20 ft. off the
centre line and the quantity 18 tons. The weather conditions
being favourable, no loose water in tanks, mooring ropes slacked
off, and everything else in satisfactory condition, a plumb line can


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Online LibraryAmos Lowrey AyreThe theory and design of British shipbuilding → online text (page 5 of 14)