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should then be found from this edge. Its position will now be
further from the axis of the rudder and the force P will be now
acting through a larger leverage than when the vesesl was moving
in a forward direction. The force P is not, however, so large when
going astern as when ahead, on account of the speed astern being
much less than that ahead. It is usual to take speed astern as being

138

The Theory and Design of British Shipbuilding.

equal to three -quarter speed ahead. The reduced speed, lessening
the rudder pressure, has a greater effect than the increasing of the
lever in the case of ordinary rudders ; therefore, for this type, the
stress when going astern is less than when ahead ; but in the case
of balanced rudders it is generally found that the reduced speed
has a smaller effect than the large increase of lever found in this
type, and, therefore, the maximum stress is obtained when moving
steriiwards, and for balanced rudders the calculations should be
made corresponding to this condition. In the case of figure 62 (A)
the lever y would be used instead of X. To find the centre of pressure
of an actual rudder : Take, for instance, the rudder shown in Fig.
62 (B), and upon it construct a rectangle having the same area and
overall width. Now obtain the centre of gravity G of the actual
shape of the rudder surface and the centre of gravity G T of the
squared-up surface. Next find the centre of pressure p of the squared-
up surface by multiplying the overall width by the co-efficient
corresponding to the angle as given above ; say 10 deg. = W X '24
= d, giving the point p. Since the centre of gravity of the actual

Fig. 62.

surface was found to be forward of the centre of gravity of the
squared-up surface, the actual centre of pressure will also be forward
of p, which corresponds to the squared-up surface. The amount
of correction for this is dependent upon the distance G G^ ,
and if to the distance d we add G G, X -24 (for 10 deg.) the point
P will be obtained, which is very near to the correct position of the
centre of pressure of the rudder surface at 10 deg. inclination to the
line of the motion of the water. It will be noticed that the above
applies to a vessel with balanced rudder and moving sternwards.
The twisting moment in inch-tons will be equal to :

The Theory and Design of British Shipbuilding.

139

(1-12 A v 2 sin ) x I X

12

2240

= T.

And from this the necessary diameter of rudder- head can be found
by means of the following formula :

d = 3 VT x e,

where d = the diameter of rudder head, in inches, and e = a co-
efficient varying according to the material.

For cast steel e = 1-02

For wrought iron e 1-28

For phosphor bronze e = 1*70

Turning Trials. Fig. 63 shows the course taken by a vessel
when steaming aliead with rudder hard over and turning a circle,
the position of the vessel being shown at various points. The outer
line, which is swept out by the vessel's stern, is the boundary within
which the vessel can turn ; the middle line is the one which the
vessel's centre of gravity is traversing, and the inner line is the one
upon which the pivoting point of the vessel lies, and is tangent to
the vessel's centre line (see enlarged sketch Fig.' 64). To make

observations for these trials
a person is situated at the
aft end, and another at the
fore end of the vessel, on
the centre line, as M and
N in Figs. 63 and 64, their
distance apart being known.
represents a buoy which is
moored, and around which
the vessel is making rev-
olutions, the above obser-
vers simultaneously sight-
ing the angles m and n at
times of which they are
acquainted by the blowing of the vessel's whistle. A triangle M N
(see Figs. 63 and 64) can then be constructed, and the [distance of
the vessel from the buoy can then be found by placing the apex upon
0, the position of the buoy. A number of observations are made,
and a third observer having noted the bearing of some fixed object

Fig. 63.

140

The Theory and Design of British Shipbuilding.

on shore, such as A in Fig. 63, or the compass bearing at the times,
the exact positions of the vessel at the various times can be shown
in the diagram, and the path of the vessel can be drawn, as shown
in Fig. 63. In Figs. 63 and 64 it will be seen that the vessel's bow
points inside the tangent to the path of the vessel's centre of gravity
G. The amount of this is known as the drift angle. If a line P
is constructed at right angles to the vessel's centre line, so as to cut

the position of the buoy, the point P is obtained, and this is known
as the pivoting point, being the point where the vessel's centre
line is tangent to a circle (the inner one in Fig. 63), and this point
P is the one about which the vessel is turning, being usually situated
near the extreme fore end of the vessel.

The Theory and Design of British Shipbuilding. 141

CHAPTER XIV.

OF SHIPS, AND STRAINS CAUSED THEREBY. CONSTRUCTUON OF
THE CURVES OF WEIGHT, BUOYANCY, LOADS, SHEARING FORCES
AND BENDING MOMENTS. FORMULA FOR THE DETERMINATION OF
THE AMOUNT OF STRESS. EFFECT OF SUPERSTRUCTURES.

Thereby. Throughout the length of a ship we have a most irregular
distribution of weight, obtaining support from the surrounding
buoyancy which is varying somewhat uniformly all fore and aft.
The effect of the so distributed forces of weight and support is to
up stresses on the vessel's structure. The nature of these excesses
vary between the light and load draughts ; take the vessel shown in
Fig. 65 for instance. If we imagine the vessel to be severed at the
end of each compartment, each portion being water-tight and allowed
to freely take up its independent position, we would find some to

Fig. 65

sink further and others to rise, while they would all, perhaps, alter
their trim (see Figs. 66 and 57). The irregularities of weight and
buoyancy can be best illustrated by means of curves showing their
distribution. The curve of weight is first drawn, the construction
of such a curve for the load condition being shown in Fig. 68, where
it will be seen that each item is taken separately, and built up one
upon another. A vertical scale having been decided, such as J in. = 1
ton, the weight of each item is calculated per 1 ft. of its length,
this amount being set off over the corresponding length covered.
The various items having been set down, the curve produced will
be extremely irregular, as shown in Fig. 68, but the smaller irregular-
ities may be discarded and the curve of weight drawn, as shown in
Fig. 69. The curve of buoyancy is next drawn, the amount of
buoyancy per foot length being found at various positions of the

142 The Theory and Design of British Shipbuilding.

vessel's length, and set off to the same scale as the curve of weights,
as shown in Fig. 69. We now have a graphic representation of
the distribution of the two forces, weight and buoyancy. The
areas of the two curves must be the same, as must also be the

I *

longitudinal position of the centres of gravity of the areas, to con-
form with the conditions of stable equilibrium. From these two
curves we can now find the resultant effect upon the vessel's structure,
as follows : The curve of loads is next constructed, as shown in Fig.
70, this curve representing the differences between the weight and
below the base line A B, and the negative loading, or excess of
buoyancy, being shown above the line. The next curve constructed
is the shearing forces. Commencing at the end A of the curve of
loads, the area, shown shaded, up to x is found, this area being then
converted into tons shearing force in the following way :

,Say the longitudinal scale is : J in. = 1 ft. . . 1 in. = 8 ft.
vertical ,, J in. = 1 ton . . 1 in. = 4 tons.

Then 1 square inch = 32 tons.

If the area up to 9 is three square inches, then 3 X 32 96 tons
of shearing force at this point, which is set off as an ordinate at x
(see Fig. 70). The areas of the loads having been found up to
various points and converted as above, the curve of shearing forces
can be drawn in through the offsets. The curve ascends until the
termination of the positive loading at C, after which, on account of
the load being here above the line A B, and, therefore, minus quan-
tities, the curve then descends. It crosses the base line at D, when
in excess after E is reached, when the ascension ultimately brings
the curve back to the base line at B. From the curve of shearing forces
the curve of bending moments can be obtained in exactly the same

The Theory and Design of British Shipbuilding.

way as the shearing forces were obtained from the loads, again
allowing for the difference in the scales. It will be seen that the
bending moments reach their maximum where the shearing forces
cross the base line. The bending moment is, in this case, one
tending to " hog " the vessel because of the excess of weight at the
ends and the buoyancy excess at 'midships. When without cargo
a " sagging " moment would most probably be found on account
of an excess of weight 'midships and buoyancy excess at the ends.
The above refers to a vessel floating at rest in still water. If we
now consider the vessel placed amongst a series of waves, the bend-
ing moments will be found to be greatly increased. It is usual to

Fig. 67.

suppose the vessel poised upon the crest of a wave whose length
from trough to trough is equal to the length of the ship, and the
wave height from trough to crest to be -jVth of its length when 300
ft. long and below, and ^Vth when above that length. The wave
profile is trochoidal in shape, and its position on the vessel is such
as will produce the same displacement and longitudinal centre of
buoyancy as in the still water condition. The dotted line in Fig.
65 shows the wave profile, while the dotted line in Fig. 69 shows
the resultant buoyancy curve. From the latter figure it will be
seen that a deduction of buoyancy has been made at the ends and
an addition at 'midships, thereby increasing the amounts of positive

Fig. 68.

result being a largely augmented " hogging," bending moment.
When considering the vessel with the wave crest 'midships, it is
usual to assume 'midship bunkers and feed-tanks empty, which,
consequently, tends to increase the " hogging " effect, and for the
trough 'midships to take them as being full, this obviously tending

144

The Theory and Design of British Shipbuilding.

Fig. 69.

to increase the" sagging," bending moment. The bending moments
given by the curve are in foot-tons, and from these amounts the
resultant stress upon the vessel's structure can be ascertained when
used in the following formula :

M x Y

I

= Stress in tons per square inch,

where M = the bending moment.

Y = the distance from the neutral axis to the point at

which the amount of stress is required.
I = the moment of inertia of the cross -section about the

neutral axis.

The moment of inertia of the vessel's cross-section is next to be
found. In this calculation only the longitudinal members of the
structure are taken into account (see Fig. 71). The neutral axis
N A, which passes through the centre of gravity of the cross-section,
axis. The greatest stress will be obtained where Y is greatest ;

Fig. 70.

for instance, if we suppose a " hogging " stress applied to the vessel
represented by Fig. 71, Y is greatest when taken to the top of the
upper deck sheerstrake. This formula is applied to the vessel at
points which appear to be weakest, or where the bending moment
is largest, and the stress upon the points in question being so found,
the strength of the vessel is known. The greatest stress is nearly

The Theory and Design of British Shipbuilding. 145

in all cases found when the vessel is poised upon a wave crest, and,
from experience, it is found that the figure for this stress which will
produce a vessel free from weakness varies with the size of the ship.
In small ships of the coasting types, 2 tons is a safe figure, while
for medium -sized vessels of mild steel construction, about 6 tons
in tension, and 5 tons in compression are suitable. In very large
ships the figures may be further exceeded ; for instance, in the
Cunard Company's Lusitania, while enduring a hogging stress with
a wave crest at 'midships, the corresponding stress is given as 10-6
tons tension on the shelter deck and 7-8 tons compression on the
keel.

In computing the moment of inertia, all longitudinal portions of
the structure, covering, at least, half of the vessel's length, should
be taken into account, allowance being made for all openings that
would cause a weakening. For the parts in tension, a deduction
should be made for rivet holes by taking only T 9 rths of the area of
in finding the M I is as follows : Suppose the shaded portion in
Fig. 72 represents the section of the sheerstrake plate of a vessel,
its area being equal to A. If I = the total amount of inertia of the
plate about the neutral axis N A
and I r = the moment of inertia of the area about its

A x d*
own axis m n, which amount is equal to

12
then I : = (A x h 2 ) -f I,.

It will be seen from the above that to reduce stress it is necessary
to increase the moment of inertia, since this is the denominator of

M x Y
the formula - - which gives the stress, a large moment of

I

inertia, therefore, giving a small stress. To obtain a large M I, the

material farthest away from
the neutral axis should be
.>. made of the heaviest scantling,
because its area is multiplied

- M -A by the square of its distance

C 1

f ^

from that axis, viz. : (A X
** -^ h 2 ). This shows the advan-

Fig. 71. tage of placing material as

11

146 The Theory and Design of British Shipbuilding.

far away from the neutral axis as possible ; but this should only be
done when the added material will stand the large stress that comes
upon it when placed at a large distance from the N A. At first
thought one is apt to think that high superstructures would be the
means of affording further strength to a vessel, because of largely
increasing the moment of inertia ; but when these are only lightly
constructed, such as casings, deckhouses, boat decks, etc., it is
found that the increased length of Y has a greater effect than the

M x Y

increased I, and the stress given by - - is consequently lar-

I

ger than when they were unaccounted for. The light scantlings of
such erections would not stand the increased stress ; therefore, they
must be arranged so that they will take no part in the longitudinal

bending of the vessel, and the vessel's
longitudinal strength be quite inde-
pendent of them. This is done by
fitting an expansion joint, which is
merely an overlap without any riveted
connection, being, therefore, able to
=: work freely during any bending of
the vessel, and thereby preventing
Fig. 72. fracture, which may otherwise occur.

The Theory and Design of British Shipbuilding. 147

CHAPTER XV.

FREEBOARD AND REASONS OF ITS POSITION. RESERVE OF BUOY-
ANCY. EFFECT ON STABILITY AND STRENGTH. BOARD OF TRADE
RULES. AMENDMENTS IN 1906.

Freeboard, and Reasons of its Provision. Roughly speaking,
freeboard may be defined as being the height of the upper deck at
side, amidships, above the load water-line. The purposes for which
it is provided are as follows : 1. To provide a reserve of buoyancy
as a margin against leakage or accident, and to give lifting power
when in a seaway. 2. Sufficient height of deck above water so as
to obtain a safe working platform for those on board, and to give
immunity from danger to deck fittings, etc. 3. Stability. 4. A
upon the vessel's structure. We, therefore, see that buoyancy,
stability, structural strength and freeboard are subjects that must be
closely related to each other, the latter being determined according
to the qualities of the other three..

Reserve of Buoyancy. This is the volume of a ship which is not
immersed and is water-tight. It includes, in addition to the upper
portion of the hull, any erections, such as poop, bridge, forecastle
or raised quarter deck, that have efficient water-tight bulkheads
at their ends. It is possible for a vessel to float with her deck level
with the water, so as to have practically no reserve buoyancy what-
ever, but in such a condition she would have no rising force and,
when in a seaway, every wave would simply wash along or over
the deck, undoubtedly carrying away all deck structures, and pro-
bably eventually causing the foundering of the vessel. In the case
of a vessel which has a reserve of buoyancy and on a wave with the
crest about amidships, we have, for a moment, an excess of displace-
ment, and the vessel immediately tends to rise so that her weight
may no longer be exceeded by the displacement of water. Without
the reserve of buoyancy, the vessel would not possess the power of
heaving and, as in the above case, would be submerged with every
succeeding wave. The second reason is a most obvious one, so as
to allow of the vessel being navigated with a minimum of personal
risk.

148 The Theory and Design of British Shipbuilding.

Effect on Stability. A vessel with a good freeboard is able to
incline to a much larger angle before the deck corner reaches the
water than one whose freeboard is small. This has the important
effect of pulling out the centre of buoyancy and increasing the
righting levers on account of increasing the value of the equation

v X hh l

- which forms part of the formula that gives the righting
V

lever (see Chapter VIII. and Fig. 40). In Chapter VIII. the influence
of freeboard on a vessel's stability was shown in Fig. 45, where the
increased range resulted in the new curve G 2 Z 2 .

Strength. In the previous article we dealt with the causes of
longitudinal bending moments in a vessel ; let us now assume that
we have a bending moment wilich, from experience and such cal-
culations, we know will allow of a safe margin of safety with the
her ; The extra amount of cargo that has to be put on board cannot
very well be placed near amidships on account of the machinery
space, and is, therefore, placed in the holds or upon the deck. The
vessel is now immersed further into the water, the effect being to
increase the amount of supporting buoyancy mostly in the vicinity
of amidships, this being the fullest portion of the vessel. The weights
would increase nearer the ends, and the combined effect of the extra
buoyancy and weight, upon a curve of loads, will obviously have
the resultant effect of increasing the bending moment and, conse-
quently producing an increased stress.

Board of Trade Rules. The rules for the computation of free-
board are arranged for four types of ships, a corresponding table
of freeboards being given for each. Table A is for first-class cargo-
carrying iron or steel steamers not having spar or awning decks
i.e., ships of the full strength. This table corresponds to flush-deck
vessels, but when substantial erections are fitted, allowances are
made for their contribution to the vessel's reserve buoyancy. Table
B is for cargo-carrying spar-deck vessels, which type is of a lighter
construction than the Table A vessel, the freeboards being increased
on that account. Table C is for cargo-carrying awning-deck vessels,
which is a still lighter type of construction, having a greater increase
in freeboard. Table D is for sailing vessels. The tables are drawn
up with columns in which is given the freeboard for vessels of certain

The Theory and Design of British Shipbuilding.

149

depth and proportionate length, a correction being necessary for
a difference in length to that specified in the tables. The freeboards
given for the proportionate dimensions also vary according to the
fineness of the vessel's form, which is represented by a co-efficient,
the fuller vessels having the largest freeboard. In Fig. 73 a diagram
is shown given the variation of freeboard (full lines) on a basis of
length of ship with proportionate table depth. The four tables are

Fig. 73.

represented for typical cargo-carrying vessels, A, B and C being for
steamers of a table co-efficient of -8, while D is for iron sailing vessels
purposes, but is usually about equal to the block co-efficient of
displacement.) The reason for the difference in freeboards in A,
B and C is because of the lighter scantlings of the latter two ; the
addition of freeboard in their cases reducing the weight carried,
which, therefore, tends to make the stress in all cases about equal,
the lighter vessel carrying the smallest amount. Fig. 74 shows
another comparison of the four tables, giving the position of the
respective load-lines for a vessel 300 ft. long, and 25 ft. depth,
moulded, to the uppermost deck, the co-efficient being -74 in all
cases. The relation of displacements is near to the following :

Displacement of B is about 96 per cent, of A.

Coo
55 55

D 97

In the above comparisons it must not be forgotten, however, that
the Table A vessel is one of the most meagre description possible

150

The Theory and Design of British Shipbuilding.

i.e., she is taken as being just equal to the bare table rule. Under
ordinary average circumstances in a vessel of this size and type y
say with a poop, bridge and forecastle covering about half of the
length of the vessel, we would have the following deductions to
make :

For Excess of sheer about 7 ins.

,, Erections 9J ,,

" Iron Deck " If

making a total of 1 ft. 6 in. reduction in the freeboard and giving
the water-line A |} as shown dotted in Fig. 74, the increase of dis-
placement being about 8 per cent. In estimating freeboard the
first thing necessary is to determine the co-efficient of fineness for
use in the tables. This is done by using the registered dimensions
and the under deck tonnage as measured by the Board of Trade
surveyors :

U D Tonnage

- = Co-efficient.
Reg. Dims.

To obtain this accurately many corrections are necessary to the
dimensions and tonnage in almost every case ; but it should be
remembered that the co-efficient sought is one the vessel would have
if framed on the ordinary transverse system i.e., with ordinary

A,

& ^Cvg l?Uw^

Fig. 74.

floors (no double bottom) and frames of Lloyd's 1885 rule scantlings,
also if fitted with hold and spar ceiling ; therefore, when any departure
is made from this, by fitting double-bottom, deep framing, or the
omission of ceiling, corrections should be accordingly made. If the

The Theory and Design of British Shipbuilding. 151

mean sheer is in excess or deficient of the rule standard, the regis-
tered depth to be used for ascertaining the co-efficient is to be in-
creased or reduced respectively by one -third of the difference between
the standard and actual mean sheer of the vessel. The freeboard
can now be determinee for the table length according to the depth,
moulded, and co-efficient. The correction for sheer is next made,
its amount being one-fourth of the difference between the rule and
the actual mean sheer of the vessel, being deducted from the freeboard
for an excess in some cases, and where there is a deficiency it is added.
The correction of length is now accounted for, the amounts being
giving in the tables for every 10 ft. of difference between the table
and actual lengths. It is added when the vessel's length exceeds
that given in the table and deducted when it is less. An allowance
for deck erections is to be made according to the nature of erections,
and based upon the proportion that the combined length of erections
bears to the length of the vessel. It is assumed that an awning-deck
vessel is simply a Table A ship, with an erection extending all fore
and aft, and the difference between the two table freeboards, after

Online LibraryAmos Lowrey AyreThe theory and design of British shipbuilding → online text (page 11 of 14)