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THE CASE OF A VESSEL IN DRY DOCK. EFFECT ON STABILITY DUE

TO VESSEL BEING PARTLY IN MUD.



Effect on passing from Salt to Fresh Water or vice-versa. The

draught being increased when a vessel .passes from salt into fresh
water, there will be a new position of M, because the moment of
inertia, volume of displacement and the vertical centre of buoyancy
will have altered. The position of G, of course, does not alter.
In the diagram shown by Fig. 36 it will be seen that the curve of
rnetacentres descends until a certain draught is reached, and then it
again ascends. The lowest point is /, and is at the draught A. If
the draught in salt wate'r is less than A, say B, then the metacentre

will be lowered when pass-
ing into fresh water, on

account of the vessel sink-



C



I ing deeper. If the draught

in salt water is greater
than A, say C, then the
metacentre becomes higher
when the ship passes into
fresh water, because the
curve is here ascending.
We therefore see that the



stability alters when pass-
ing from sea to fresh water in a manner which is governed according
to the position of the original water-line. The above conditions are
reversed when passing from fresh to salt water. Passing from salt
to fresh water, and increasing the draught, would give less freeboard,
and would slightly reduce the range' of stability. The effect of
freeboard is dealt with in the following chapter in connection with
large angles of inclination.

Loss of Initial Stability due to Grounding. When a vessel first
takes the ground that is, just touches it the stability is not
affected so long as the tide remains as high as the water-line that



82 The Theory and Design of British Shij)building .

she previously floated at, because she will still displace the same
amount of water as before, and thereby still receive the same upward
support ; but suppose the tide to fall, we know that the vessel will
then lose a layer of buoyancy contained between the old and new
water-lines. She will now obtain an amount of support from the
ground, which will act vertically upon the keel if the upright position
is preserved, and this amount of support must be equal to the lost
layer of buoyancy. When the vessel is inclined to a small angle,
so that the bilge does not touch the ground, this new upward force
actually tends to upset the vessel.

In considering the question of this particular loss of stability, a
good way is to take the case of a vessel in dry dock, just after she
has taken the blocks, and without any shores in position, as illustrated
by Figs. 37 and 38. In these sketches

W L The water-line when the vessel is afloat and upright.
W i L i = The water-line when the water has receded and left the

vessel upright upon the blocks, as in Fig. 37.
W 2 L 2 = The water-line when the water has receded, but the

vessel now being inclined upon the blocks, as in Fig. 38.
B = The centre of buoyancy corresponding to W L (afloat).

B i = The centre of buoyancy corresponding to W i L ] .
B 2 = The centre of buoyancy corresponding to W 2 L 2 .

(The centre of buoyancy now having shifted outward

owing to the inclination).
M = The transverse metacentre corresponding to water-line

W 2 L 2 .
W = The total weight of the vessel and everything on board

being equal to the displacement of water when afloat.
G = The centre of gravity of the vessel's weight W.

P = The weight taken by the blocks, which is obviously equal

to the amount of the displacement contained in the layer

between Wi LI and W L, since this amount has been lost

owing to the lowering of the water-line.
W- P = The buoyant support of water acting vertically upwards

through B 2 , being equal to the total displacement minus

the lost layer.
= The position of the " effective metacentre " after the

vessel has taken the blocks.
L = The point of contact of keel and blocks.



The Theory and Design of British Shipbuilding. 83



Fig. 37.




The angle of inclination is supposed to be small, although it is
shown exaggerated in Fig. 38 for clearness. On looking into Fig.
38 it will be seen that we have one downward force W acting through
the vessel's centre of gravity G, and two upward forces, P acting
through L, the point of contact with the blocks ; and W-P acting
through B 2 , the centre of buoyancy of the displacement in the
inclined position. If we can find the point where the resultant
upward force of P and W-P cuts the centre line of the vessel
we will have the position of the " effective metacentre." The result-
ant of these two upward
forces is shown by the
dotted line which passes
through 0, the amount of
this combined force being
exactly equal to the
downward force W act-
ing through G. From the
following it will be seen
how this resultant is
found by the combination
of the two upward forces.
Using the principle of
moments, and taking
" levers " from the line
of force W-P, we have
" moments " about that




line as follows :



Fig. 38.



84 The Theory and Design of British Shipbuilding.

W-P x nil == nil.
P X b = P x b



Totals == W (weight) P x b (moment)



P X 6

therefore - = a, which is the distance

W

from W-P that a resultant W of the two upward forces will act.
We have now only two lines of forces to deal with, W acting down-
wards, and W acting upwards, and if we can find the distance c
between them, we will have the " righting " or " upsetting lever,"
as the case may be (in this case we have a " righting lever/') If
we suppose the vessel to be floating at W 2 L 2 , the righting lever
would be GZj , the distance between the downward force of gravity
and the upward force of buoyancy ; but on account of having to
contend with the upward force P from the blocks, we have seen that
the resultant of the two upward forces is at a distance a inside of
the vertical W-P, and consequently the new effective righting lever
will be shorter than GZ t by the amount a.

Effective righting lever, Gz = G Zj - a.

P X b
In the above we saw that a = -

W
therefore we may write

P X b
Effective righting lever Gz = G Z, -

W

On looking into the figure it will be seen that b = L M sin 9, and
substituting this for 6 in the above equation we have

P x L M sin B
Effective righting lever Gz = G Z 1 -

W
P x L M sin



(i /\ J^ 1YJL 0111 v
being equal to a)

W 7



The Theory and Design of British Shipbuilding. 85

Loss in metacentric height = (G M - G O) = M

G Z, Gz a

sin 9 sin 9 sin G

since the difference between G Z-, and Gz is a.

P x L M sin 9

Above we have seen that a , so therefore the

W

loss in metacentric height = g-^- P x L M~rm&. gin Q cance u_

sin V W x -5TU fl.
ing out we have

P X L M

Loss in metacentric height = - , (which is best for

W

ordinary use in finding the loss in metacentric height), and, when,
deducted from the original amount, readily gives the state of the
vessel's stability for any required condition upon the blocks. This
formula being simple, is easily applicable. P is quickly found from
the displacement scale, being the difference between the displacement
of water when afloat and when upon the blocks. L M is obtained
by reading off the metacentric diagram, the height of M above the
base for the new water-line, and adding to this depth of the keel.
W being the total displacement, has been already found from the
displacement scale when obtaining the displacement of water when
afloat. Of course, throughout the question we have made the
assumption that the inclination is one within the limits for which
the position of M is practically constant, and it should be noted that

B, M = Moment of inertia of the new waterplane.

Volume of displacement up to new waterplane.

Effect on Stability due to a Vessel being partly in mud. In

Fig. 39 we have shown a vessel floating freely in water of density
equal to, say, 1, W Q L Q being water-line and B Q vertical centre of
buoyancy, the vessel's keel just touching the level of a mud line
of specific density of, say, S i . Draught to W Q L Q = D Q .



86



The Theory and Design of British Shipbuilding.



In Fig. 39a the
water-line W Q L Q
has fallen to W, L,
and vessel has
become partly sub-
merged in mud to
water-line W 2 L 2 .
The vessel's
draughts to W 1 L 1
now equal, say, D ,
while the vertical
centre of buoyancy
equivalent to W,
L , is B , . How-
ever, it is quite
obvious that the
greater density of
the mud portion up
, W 2 L 2 acting at its
own vertical centre
of buoyancy B 2
(See Fig. 39b), will
tend to lower B,
to, say, B 3 .

It is desired to
deduce a formulae
to use in obtaining
the depth vessel is
submerged in the
mud, i.e., D 2 , Fig.
39b. We intend to
treat the additional
buoyancy below
W 2 L 2 as an in-
crement of dis-
placement over and
above the displace-
ment up to W, L,.



Do



Bo



'I'linu 111 1 II II ii i

Fig. 39.




Fig. 39a.



W2







// f 1 1 1 nil



Fig. 39b.



That is if W Q be displacement up to W Q L Q and W, be displacement
up to W T L 7 and W 2 (S, 1) is additional displacement up to



The Theory and Design of British Shipbuilding. 87

W 2 L 2 , then W Q = Wj plus W 2 (Sj - - 1), and assuming length
of vessel equal L and breadth = B.

and C = displacement coeff. up to W Q L Q .
C.j = ,, >, ,, ,, W-j L 1 .

^2 = " > " > ** 2 *!

x x D, x C,



Then



x x D x C



f-



X X ^ X D 8 x C, (S, 1)



or 35, L and B cancelling out

D o x C o = D T X C, + D 2 X C 2 (S, - -

D o X C o - D I X C i



D 2 =



C 2 (S T




WATER
'SPECIFIC GRAVITY = I



MUD
SPECIFIC GRAVITY = Si



Fig. 39c.



88



The Theory and Design of British Shipbuilding.



or if vessel of prismatic form



S,- 1 T

It is now worth while considering what effect this condition may
have on the transverse stability. This will involve our obtaining
new metacentric height, new righting lever and new righting moment
supposing the vessel is inclined to a small angle of heel 9, and that
the mud is of even density.

In Fig. 39c we have exaggerated the angle of heel for clearness.
Then in Fig. 39c.

W 3 L 3 = inclined water-line corresponding to W T L T .
W 4 L 4 = mud line W 2 L 2 .

B 4 centre of buoyancy of vessel up to W 3 L 3

(neglecting lowering effect of mud).
B.r, ,, centre of buoyancy up to W 4 L 4 .

G = position of centre of gravity of vessel.

M Q = position of metacentre corresponding to original

water-line W L when vessel floating freely in water

only.
M! = position of metacentre corresponding to W t L T and

neglecting effect of mud.
M 2 = position of metacentre corresponding to W 2 L 2



other letters as in previous figures 39, 39a

W 2 (S
/
Taking moments about K.

K L


and 39b.

, 1) W W

^ A /

6.


I

We have upwards
W 2 (S, - - 1) X a + W( x c =
but a = KM 2 Sin 9


At Ms M

= W x b.



b = KM 3 Sin 9
c = KM, Sin 9



The Theory and Design of British Shipbuilding.



89



KM



9 X W 2 (8, 1) 4- KM, 9 X W, = KM



9 X W .



M 3 being the resultant of these forces and is the " effective meta-
centre " Sin 9 cancels out.



KM 2 x W 2 (S, 1) + KM, x W



and Km Q =



Total loss of metacentric height

= KM - KM 3 ai M o M 3

but new righting moment

= KM 3 Sin 9 X W KG. Sin X W

= Sin 9 x W (KM 3 KG) = Sin X W X GM 3 .

and new righting moment -f- W Q = GZ 2 or new righting lever.



Lo



Do -12'



Bo



L5



Fig. 39d.



90 The Theory and Design of British Shipbuilding.



It should be observed that if KG Sin 9 X W Q is greater than KM 3
Sin 9 X W that GM 3 Sin 9 X W Q would have a negative sign and
an upsetting moment would be present.



Bi

B3



L



TT

D2



Di



Fig. 39e.



Taking a simple example and applying above :

Fig. 39d is a vessel of rectangular section floating freely at water-
line W Q L Q and just touches a mud bank of specific gravity equal
to 2 and say D Q = 12' and B 15' and L = unity. Now supposing
water-line falls 6' to W, L, and vessel is partly submerged in mud
and inclined to small angle



12' (D 2 6)



D 2 -



2 1



2D 2 = 6



D 2 = 3' or D, = 9'



The Theory and Design of British Shipbuilding. 91



I 15 X 15



KM = KB o + B M = KB o + = 6'



V o 12 x 12
= 6' + f -;;- == 7-56'

I, 15 X 15

KM, = KB, + B,M, = KB, H = 4-5 H



,

V, 12 x 9

= 4-5 + tl = 6-58'

I 2 (Si - 1)



KM 2 = KB 2 -f B 2 M 2 = KB 2



V 2 (S, - 1)



15 X 15

= 1-5 -f- = 1-5 + = 7-75'

12 x 3

and KG = say 6'0

7-75 X 45 + 6-58 X 135

then|KM 3 = = 6'88'

180



KM Q KM 3 - 7 56 6-88 = 0-68'



92



The Theory and Design of British Shipbuilding.



CHAPTER VIII.

STATICAL STABILITY AT LARGE ANGLES OF INCLINATION :

DEFINITION. AT WOOD'S FORMULA. ORDINARY CURVE OF

STABILITY ; CROSS CURVES : REMARKS. INFLUENCING THE SHAPE

OF STABILITY CURVES.



Statical Stability at Large Angles. It is the moment of force
(usually expressed in foot-tons] by which a vessel endeavours to right
herself, after having been inclined away from the position of equilibrium.
We will now trace the stability of a vessel through the larger angles
of inclination where the metacentric method no longer holds good.
In Fig. 40 we have a vessel inclined to the angle fr. W L was the
upright water-line, with B its corresponding centre of buoyancy ;
W 7 LT is the new water-line, which has B 1 as its centre of buoyancy.
The displacement is the same up to either water-line. We know
that the length of the righting lever G Z measured in feet and multi-
plied by the vessel's displacement in tons gives the amount of foot-
tons, moment of statical stability. The displacement for any con-
dition is readily obtained from the displacement scale, and the
value of G Z is determined as follows : G Z, we know, is the lever or
couple caused by the action of the buoyancy and the weight of the
vessel, the buoyancy acting upwards through B T and the weight




Fig. 40.



The Theory and Design of British Shipbuilding. 93:

acting downwards through G. The position of G is the same for all
inclinations, but B alters, therefore for each inclination the value
of the shift of B must be found. The righting lever being parallel
to the water-line and at right angles to the vertical forces, we, there-
fore, find the shift of B in a direction parallel to the water-line corre-
ponding to the inclination. It will be obvious that the shift of
B is parallel to a line drawn between the centres of gravity of the
immersed and emerged wedges ; but as we desire it in a direction
parallel to the water-line W^ L^ , we therefore square up on to this
water-line the points g and g, , which gives us h and HI respectively ;.

v X h h^

now - = B R, which is the shift of the centre of buoyancy

V

parallel to W, L 1 or to G Z.

v = volume of either wedge.
V = volume of displacement.

Having found B R, it is easy to find G Z, because on looking into
the figure it will be seen that B R is larger than G Z by the amount
of B r, therefore

BR - Br = G Z.

Now B r = B G sin 9,

v x h h^

therefore G Z = - B G sin 9,

V

this formula being known as Atwood's Formula. The minus sign
is, of course, only in cases where G lies above B ; but if G were below

v X h h^
B, the formula should be written + B G sin 9.



v X h hi is an equation of form only, and if B and G coincided, the
stability would depend solely on the vessel's form and
V v X h h-i

this equation. The values of - representing B R, hav-

V

ing been calculated, they are next corrected for B G sin 9, and then
set off and a curve drawn, as shown by G Z in Fig. 41, where the
horizontal scale is one of degrees of inclination, the vertical scale-



94



The Theory and Design of British Shipbuilding.



representing the lengths of righting levers in feet. In calculating
the stability, it is usual to first of all assume that B and G coincide,

v X h h ,
the formula being therefore reduced to - . This is gener-




ally done on account of the position of G not being known at the
time of the stability calculations being made. In constructing a
curve of stability in this manner, we first of all set off the values of
B R on their respective ordinates. In Fig. 41 this curve is shown



Fig. 41.



by B R. Next, by setting off values of B G sin ft, corresponding to
each angle of inclination used, below the curve B R, B being below
G in this case, we obtain the curve G Z, which is the curve of stability
^corresponding to the given condition of the vessel. This is known
as an Ordinary Curve of Stability. To obtain a guide to the shape
of the curve at its commencement, the amount of met acentric height,
g m, is set off at 57 J deg. of inclination, and the point m joined with
0, as shown in Fig. 41. At its commencement, the curve will lie
upon the line m, as is seen in this Fig. from to z. There are
many methods used to find the value of B R, each involving most
lengthy calculations, which are unnecessary to produce for our
present purpose, a separate calculation being made for a sufficient
number of angles of inclination so as to obtain offsets for construct-
ing the curve. An ordinary curve of stability only refers to one
particular draught and, therefore, can only be used at this draught.



Fig. 42.




TON*



The Theory and Design of British Shipbuilding.



95



Of course, the amount of a vessel's stability may be required at
various draughts, so therefore it is usual to construct Cross Curves,
from which an ordinary curve for any given draught can be readily
obtained. In the case of cross curves, for a base line we have dis-
placements, and the lengths of ordinates being righting levers or
moments, or, perhaps, values of B R. Fig. 42 shows a diagram of
cross curves. On any one ordinate, say, 1,000 tons displacement
is set off the value of B R as found for the various angles used 10,
20, 30, etc., degrees ; this being done on the various ordinates, and
then curves passed through their respective offsets so that each
curve represents the variation of B R for any one degree of inclina-
tion as the displacement alters. Now, suppose that it is required
to draw an ordinary curve of righting levers corresponding to a dis-
placement of 1,000 tons, B below G having been ascertained for the
particular condition of loading. First of all, in Fig. 41 the base
line scale of degrees is drawn, and then the values of B R
corresponding to the curves are lifted from the 1,000 ton



Fig. 43.




IQ ~J|



ordinate of Fig. 53. Transfer these values to their respective
ordinates 10, 20, 30 etc., degrees, in Fig. 41, and draw in the curve
of B R, below which can then be set off the values of B G sin 6 at
the various angles, and the curve of righting levers, G Z, thereby
obtained. Cross curves are therefore seen to be of great use in
enabling an ordinary curve of stability to be quickly constructed
for any particular amount and disposition of loading. The righting
levers having been ascertained for the given condition, they are next
converted into righting moments by multiplying them by the corre-
sponding displacement in tons, which gives the real statical stability
of the vessel. , In Fig. 43 the full lines represent the curves of righting
levers for a vessel in the light and load condition. Converting them
into righting moments the curves become as shown dotted. It will
be seen that while the lever was largest when light, yet the vessel,
in this case, possesses more stability when loaded, the difference in




96 The Theory and Design of British Shipbuilding.

the relation of the curves being caused by the difference in the
displacement.

Two points often mentioned are :
maximum stability and maximum
range. The former occurs at the angle
where the curve reaches the highest
point, as is shown at 62 deg. by x in
Fig. 44 ; and the latter is seen to be
85 deg., because the curve here de-
cends below the base line, and the
vessel no longer possesses a righting tendency, and, therefore,
capsizes. If the vessel is inclined by an external force, such as the
wind, it is obvious that when the force is relaxed she will return to
the upright if the angle of 85 deg. is not exceeded ; but suppose the
vessel to be inclined by weights, which are not relaxed, the vessel
will then settle down to a steady angle of heel, at which the righting
moment of stability equals the upsetting moment caused by the
weights. For instance, say 50 tons are moved 20 ft. across the vessel
causing an upsetting moment of 50 X 20 = 1,000 foot-tons, then
the angle of rest will be 35 deg., since we here have this upsetting
moment balanced by exactly that amount of righting moment.
Now, suppose that a larger upsetting moment causes the vessel to
incline further than 62 deg. (where we have maximum stability), it
is obvious that if the righting moment at x (62 deg.) did not balance
the upsetting moment there is no point after that, where it can be
obtained, so therefore the Vessel will capsize. This instance helps
to emphasise the fact that for all practical purposes maximum
stability should also be considered as the maximum range.

Influencing the Shape of Stability Curves. A vessel's dimensions
control the stability to a large extent, especially in the case
of the breadth, which, as previously mentioned, controls the
moment of inertia. Breadth has practically no effect upon the
height of the centre of gravity, therefore, it only controls the stability
due to form. By increasing the breadth the height of M will be
increased, and, therefore, the metacentric height will be larger,
G remaining unaltered. In Fig. 45 this effect is shown ; G Z
represents a curve of righting levers, the corresponding meta-
centric height g m being set off at 57 J deg. The breadth
being increased results in a larger metacentric height, as shown by
G m 1 . The curve Gj Z T corresponding to the new breadth, lies



The Theory and Design of British Shipbuilding.



97




Fig. 45.

upon the line m 7 at the origin, the increase in value of the righting
levers being in this case more rapid, the curve also showing larger
levers and increased range. Altering the depth of the vessel would
alter the height of the centre of gravity, and thereby the stability ;
but suppose that the depth of the vessel is increased, and arrange-
ments made so that the initial stability and draught remain un-
altered, the additional depth therefore being added to the freeboard.
This additional freeboard is most valuable, the stability being
increased so that the maximum lever is of larger amount and takes
place at 'a greater angle of inclination than previously, the range
being also lengthened (see G 2 Z 2 , Fig. 45), the increased freeboard
causing the deck edge to become immersed at a later angle. The
length of a ship is often said to play no part in the transverse stability,
but there is, however, the possibility of it influencing the amount,
seeing that in a longer ship the probability is to have a longer 'midship
body than in a shorter one, the, longer 'midship body having the
effect to increase the moment of inertia to such an extent that, even



allowing for the increased displacement, the formula - - may pro-

V

duce a higher position of M for the long vessel. Altering the height
of the centre of gravity will obviously influence the stability.



98 The Theory and Design of British Shipbuilding.



CHAPTER IX.

DYNAMICAL STABILITY : DEFINITION. MOSELEY'S FORMULA. CON-
STRUCTION OF CURVE. STABILITY OF SAILING SHIPS ; HEELING
PRODUCED BY WIND PRESSURE ON SAILS, AND EFFECT OF AREA OF
STATICAL STABILITY CURVE. EFFECT OF DROPPING A WEIGHT

OVERBOARD.



Definition of Dynamical Stability. "It is the amount of work
done during the inclining of a vessel to a given angle." We have
already seen that the forces which resist the inclination of the ship
are vertical forces, the weight of the ship acting downwards and the
buoyancy acting upwards with an equal pressure. Throughout an
inclination of the ship we find that the centre of gravity is rising or
the centre of buoyancy lowering, or both taking place. The total
vertical separation of the centres of gravity and buoyancy is the
dynamical lever, and this multiplied by the weight of the ship equals
the work done in foot-tons, or the dynamical stability up to a certain
angle. From Fig 40 it will be seen that the vertical separation of
G and B when inclined to the new water-line W, L, is equal to B,
Z B G.

Now B, Z = B, R + R Z.

B, R being the vertical shift of the centre of buoyancy

v X (gh +g t h,)



is =



V

and R Z is equal to r G or B G cos 0,

v X (gh + 0, h t )

therefore B, Z = - + B G cos .61

V

Dynamical lever or vertical separation :


1 2 3 4 5 7 9 10 11 12 13 14

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