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similar effect. While depth may be termed the cheapest dimension,
yet it is one in which great care is necessary in its decision. Depth
very largely decides scantlings, particularly in the case of framing,
while its relation to the length of ship is an important factor in
determining the scantlings of longitudinal material in the top sides.
Again, in the freeboard rules and tables we find spots w r here,
by careful juggling, large advantages can be obtained. There-
fore, by careful consideration with regard to the various stipulations
contained in the laws for tonnage and freeboard and classification
societies' rules, we are able by judicious handling to obtain, from
an owner's point of view, a vessel which will be economical in initial
cost, maintenance and working. In the determination of such
dimensions, however, we must not lose sight of other important
questions, such as speed and stability. In addition to the necessary
consideration of fineness of the vessel's form, as previously men-
tioned, the proportions of the dimensions to each other should be
suitable for the speed, and also from the important point of view
of stability. Having now obtained suitable dimensions, the next
proceeding is to estimate the lightweight. From the approximate
estimate, the power necessary to drive the vessel at the given speed
can be calculated, and again from this the weight of the machinery.
Calculations are then made for the weight of the hull iron and
steel, timber and outfit the sum of which added to the weight
of machinery gives the lightweight. Next, the load draught is
found by means of first estimating the freeboard according to the
Board of Trade rules and tables. (The subject of Freeboard is
dealt with in a later chapter.) This is a most intricate calculation,
and there are many points to be watched in dealing with the various
types of vessels if an accurate result is required. It is very im-
portant that such a result should be obtained, because in economical
designing it is most desirable to know the exact draught that the
vessel will be able to load down to, instead of having to allow a
margin by means of a few inches in the vessel's depth. The free-
board being ascertained, the load draught is found as in the following
example :



The Theory and Design of British Shipbuilding. 29



ft. in.

Depth, Moulded ... 23

Depth of Keel H

Thickness of Deck Stringer Plate OJ

Statutory Deck Line above Stringer Plate... 2

Extreme Side 23 4

Certificate Freeboard 3 2|

Load Draught = 20 1J to bottom of

Keel.
The above particulars are shown in Fig. 18.




We now have the dimensions, lightweight, given deadweight
and the load draught. Lightweight plus deadweight gives the
load displacement. With this load displacement, length, breadth
and draught we now obtain the co-efficient resulting from this
preliminary estimate. It may be here necessary to again slightly
modify the dimensions if the co-efficient is not near enough to that
suitable for the vessel ; however, at this stage we are able to finally
decide the dimensions of the proposed vessel. Having^ arrived at
the required dimensions, displacement and load draught, the drawing
of the lines, commonly known as the sheer draught, can be pro-
ceeded with. If the trim of the vessel is specified, care must be



30



The Theory and Design of British Shipbuilding.



taken in fixing the position of the centre of buoyancy as explained
in Chapter II., where we saw that, for the vessel to be floating
freely and at rest, it is necessary to have the centre of gravity of the
vessel's weight, and the centre of buoyancy in the same vertical
line, therefore at the desired trim we must have this occurring.
We must first of all estimate the position of the centre of gravity,
and then the position of the centre of buoyancy can be fixed to give
the required trim. By means of an ordinary calculation of moments
the position of the centre of gravity can be found, as shown in the
following :











1








^Vertical




tal Lever




Item.


Weight


Levei-


* Vertical


i from Aft


Horizontal




in Tons.




Moment


Perpen-


Moment






Feet.




dicular.
,' in Feet




Hull, Iron and Steel


1,320


15-5


20,460


158


208,560


Wood and Outfit


350


26-0


9,100


170


59,500


Machinery


320


11-0


3,520


138


44,160


Lightweight


1,990


16-62


33,080


156-88


312,220


Stores and Fresh Water


25


23-0


575


130


3,250


Bunkers


315


21-0


6,615


157


49,455


Cargo


4,300


17-5


75,250


168


722,400


Load Displacement . . .


6,630




115,520




1,087,325



* Vertical centre of gravity in load condition :

115,520
- = 17-42 ft. above keel.



6,630

Longitudinal centre of gravity in load condition :
1,087,325



= 164 ft. forward of aft perpendicular.



6,630



* The vertical position is required for stability purposes, as afterwards
mentioned.

Suppose a design is being got out for a vessel to carry 4,640 tons
deadweight on a draught corresponding to Lloyd's summer free-



The Theory and Design of British Shipbuilding. 31



board, and of 9 knots speed, the particulars shown in the foregoing
table representing the vessel. Let the proposed dimensions be :

Length, B.P 320ft.

Breadth, Moulded 45 ft.

Depth, Moulded 23 ft,

The calculated freeboard being 3 ft. 2J in., the draught is obtained
as shown in the recent example, where the figures for this vessel
were used. The extreme load draught i.e., to the bottom of the
keel is 20 ft. 1| in. Above, the lightweight is given as 1,990 tons.
This, added to the required deadweight of 4,640 tons, gives a load
displacement of 6,630 tons, as is also shown. We have now to
design the form of the vessel to give a displacement of 6,630 tons
at 20 ft. 1J in. extreme draught. In designing the form the lines
are drawn to the moulded dimensions i.e., to the inside of the
plating. The plating itself contributes an amount of displacement,
and since we are to design to the moulded form it is necessary to
deduct from the total displacement, the displacement of the plating,
as well as that of any other appendages, such as keel, bilge-keel,
rudder, &c., so as to find the amount of moulded displacement
required. For the shell and appendages in a vessel such as the above
a deduction should be made of about -7 per cent, of the total dis-
placement :

6,630 tons - -7 per cent.
- 6,633 46-4 = 6,583'6 tons,

which is the moulded displacement required. The draught to
which this displacement is to be obtained is also to be moulded
i.e., the extreme draught reduced by the thickness of the keel.
In this case, where we have an extreme draught of 20 ft. 1| in. and
a keel 1J in. deep, the moulded draught is 20 ft. We have now to
design the moulded form of a vessel of the above dimensions to
give 6,584 tons displacement at 20 ft. moulded draught. The block
co-efficient for this would be :

6,584 x 35

: = -g



320 x 45 x 20

which, for the dimensions and required speed, is fairly suitable.
Should the vessel be required to float at even keel when loaded,
it will be obvious that the longitudinal position of the Centre of
Buoyancy must be placed at 164 ft. forward of the aft perpendicular,



32 The Theory and Design of British Shipbuilding,



so as to be immediately under the position of the Centre of Gravity,
which was calculated in the above. In other words, the centre of
support is to be placed directly under the centre of the weight.
The drawings of the vessel's form, giving the required displacement
and Centre of Buoyancy, may now be proceeded with.

The Sheer Draught is the name given to the plans upon which
the shape of the vessel's form is illustrated. The form is obtained
and faired up by ordinary geometrical methods, using elevation
plan and sections. The designing and fairing of a ship's lines in
the Sheer Draught can be claimed to be the most beautiful and
interesting problem in solid geometry. In ship work the elevation
is termed Sheer Plan or Profile. In the present articles it will be
called Sheer Plan. Fig. 19 shows the Sheer Draught for the above
vessel. The dimensions are first of all laid off in block form as
follows : A base line, A B, for the Sheer Plan is drawn, upon which
the length of the vessel is measured and perpendiculars erected
at each end. The length used is the length B P, one perpendicular,
therefore, being the after side of the stern post, and the other the
fore side of the stem, as explained in Chapter I. Measuring above
this base-line, the depth moulded is set off, and a fine D M is drawn
parallel to the base-line. This is termed the depth moulded line.
We have now completed a rectangle which represents, in block form
the length and depth of the vessel. At a convenient distance below
another block is constructed, in which the plan view is drawn, this
being known as the half-breadth plan. The vessel's centre line
C L is drawn, and then the half-breadth moulded is set off from it
(both sides of the vessel being alike, only one side need be drawn),
and the half -breadth line H H is drawn. The plan in which the
shape of the sections is shown is called the body plan, a block being
next constructed for this. This plan usually has the same base-line
as the sheer plan and is placed either at one end, clear of this plan,
or at the middle of its length, as shown in Fig. 19. The centre line
of the body plan being drawn, the breadth moulded of the vessel is
set off and verticals erected. The depth moulded is also drawn in
this plan, which line is already drawn if the body plan is placed at
mid-length of the sheer plan. In the body plan is shown the shape
of the vessel at various points, which are equally spaced throughout
the vessel's length, these points being spaced off and perpendiculars
erected at them in the sheer and half -breadth plans. It must here
be decided where the displacement is to be measured from i.e.,



The Theory and Design of British Shipbuilding. 33



the aft perpendicular or the fore side of the propeller aperture.
In this case the length for displacement is taken from the aft per-
pendicular, and the sections spaced accordingly. The number of
sections required is dependent upon whichever rule is to be em-
ployed in calculating the displacement, &c. Simpson's First Rule
is used in most cases. The form of the stem and stern and the
amount of sheer having been decided, the shapes of the displacement
sections can now be sketched in the body plan. It is usual to
sketch these sections by using the body plan of a previous similar
ship for a guide, which enables one to obtain " lines " which will
be nearly fair, as well as giving a displacement somewhat slightly
more or less than that required. It is not, however, absolutely
necessary to have another body plan as a guide ; the sections can
be sketched in by carefully using the eye to obtain fairness, as far
as possible, in the shape of each section, as well as symmetry re-
garding their longitudinal spacing. These preliminary sections
are now to be faired up. This is commenced by laying off water-
lines in the half-breadth plan, lifting the widths for any particular
water-line at each section from the corresponding water-lines in
the body plan and setting them off on the respective sections in
the half -breadth, and through the spots so obtained to draw the
water line. It may be found necessary to depart from some of the
spots so as to obtain a fair line, but by this means the form of the
vessel is gradually faired up. After a few water-Hnes are laid off,
one or two buttocks may be drawn in the sheer plan in the following
method : Take the 12 ft. buttock in the after body of Fig. 19, for
instance. In the body plan, lift the heights above the base-line
at which this buttock is cut by the sections ; these spots are shown
by the black dots in that plan. Transfer these heights to their
corresponding section in the sheer plan, again shown by black dots
in Fig. 19. There are other points through which the buttock
should pass. At the points in the half -breadth plan where this
buttock is intersected by the water-lines we have spots, shown by
the black dots, by which, when squared up to their corresponding
water-lines in the sheer plan, further spots are obtained. This is
shown by the dotted vertical lines in Fig. 19. These are all the
points through which a buttock can be drawn. By working in
this way and making modifications here and there, the three plans
are gradually brought to agreement with each other, resulting in
the faired up form of the vessel. The above brief description of
the construction and fairing of the sheer draught, while not intended



34 The Theory and. Design of British Shipbuilding.



Fig. 19.




The Theory and Design of British Shipbuilding. 35



as a treatise on laying off, shows the adopted means whereby the
" lines " of a vessel are determined in that plan. This part of the
subject can only be learnt and mastered by practical experience ;
therefore, to all students of this particular item of ship-design the
actual construction of a sheer draught is recommended. The
faired up form having been eventually determined, the next step
is to calculate the displacement and the position of the centre of
buoyancy longitudinally. This is done by finding the area of
each section, and then by putting these areas through Simpson's
rule to find the volume as was explained and done in connection
with Fig. 6 in Chapter I. The area of each separate section can
be first found by using Simpson's rule, but since this calculation
is only of a preliminary nature, so as to find how the results obtained
from the " lines " compare with the required displacement and
centre of buoyancy, a quicker method is employed by means of the
use of the planimeter, an instrument of great value in rapidly
ascertaining areas.

Calculating the Displacement, etc. The planimeter being fixed
in position, the reading of each half-section (bounded by the centre-
line, frame-line and the water-line to which the displacement is
required 20 ft. in this case) is obtained by tracing the pointer of
the instrument around each required area. The planimeter readings
having been obtained, they are then put through Simpson's rule
for the purpose of finding the volume of displacement, also being
multiplied by " levers " to find the " longitudinal centre of buoy-
ancy." By use of the planimeter, the displacement and longitudinal
.centre of buoyancy, as represented by the " lines," are therefore
quickly obtained. Of course, one cannot always expect to obtain
the correct displacement and L C B at the first attempt, and from
this preliminary calculation the required amount of alteration is
ascertained, and the " lines " can be modified in accordance thereto
when another planimeter calculation is made. In this way we
eventually obtain the form that will fulfil the required conditions
of displacement and longitudinal centre of buoyancy. After the
" lines " are fixed according to the planimeter calculation, the final
displacement sheet calculations may be commenced. For these
calculations the widths of the water-lines are measured at the various
sections, and then by use of Simpson's rules the areas of water-lines
can be found, or by using the widths of the various water-lines at
.any particular section the area of that section can be found. If



36 The Theory and Design of British Shipbuilding.

the areas of water-lines or of sections as found in this way are then
put through Simpson's rule, the volume of displacement can be
found. It is usual to use both methods, working vertically with
the water-line areas and longitudinally with sectional areas, and
by use of " levers " and " moments " to find the position of the
centre of buoyancy both vertically and longitudinally. The areas
of the water-lines having been found, their centres of notation
(or centre of gravity of area) is then found by the use of " levers J?
and "moments." The tons per inch, being dependent upon the
area of water-line, is also found at this stage by dividing the area
by 420, as was seen in Chapter II. The area of 'midship section,
being often required, is also calculated by using the water-line
widths upon the displacement section at 'midships and putting
them through Simpson's rule.

When dealing with questions of resistance, the area of the im-
mersed surface is often required, and, therefore, this generally
forms another branch of the present calculations. The displacement
of the shell can also be found from the area of the wetted surface
when multiplied by its mean thickness.

The following shows a sample calculation for wetted surface
area and the shell displacement, the minor appendages being added.
The method adopted is to take the half-girths of the sections and
put them through the rule multipliers, and find the mean immersed
half-girth by dividing the sum of functions so obtained by the



1?

tie*J>tauiu*4f.

Fig. 20.

sum of the multipliers used. The half -girths are obtained from
the body plan of the sheer draught or the model, by measuring
round the outside of the section from the centre line at base up to
the required water-line. The mean immersed half -girth so found
being multiplied by the mean length of water-line and then by 2
for both sides, gives the total area of wetted surface. The mean
length of water-line can be found in a similar way to that used in
finding the mean half-girth of the sections, although it is quite near
enough to take the length of a water-line at half the required
draught. For instance, suppose that the given draught is 14 ft.
Let the water-line shown in Fig. 20 be one at half of this draught



The Theory and Design of British Shipbuilding.



37



viz., 7 ft. Measure round the outside of this water-line, as shown
by 1, and use this length to obtain the wetted surface.

WETTED SURFACE UP TO 14 FT. W L.



No. of
Section


Half-
girths


Simpson's
Multipliers


Functions





14-00


1


14-00


1


22-50


4


90-00


2


30-80


2


61-60


3


22-60


4


90-40


4


14-00


1


14-00



Sum of multipliers = 12 ) 270-00



22-5 ft. = mean immersed half-girth.

Mean length of water-line

= 304 ft.



6,840

2 sides.



13,680 sq. ft. wetted surface.
Mean thickness of

Shell x 1 J = (i-Jin. X 1| = fin.) = -06 of a foot.
(I r{j = thickness of plates

820-8 cb. ft. displacement of shell.
4- Rudder, propeller and

bilge keels = 19-2



35 ) 840-0

24 tons displacement of shell and
minor appendages.

NOTE. This calculation does not refer to the vessel represented
in Fig. 19.

It is only when the wetted surface is being calculated for the
various draughts that the shell displacement is found in the manner
shown above. When no wetted surface calculation is being made
the amount of shell displacement is taken as being a percentage of



38 The Theory and Design of British Shipbuilding.

the moulded displacement, the following being good figures to use
for such : Load Light

Draught Draught

Fine vessels (about -5 block co-efficient) ... 1-00% 2-0%

Full vessels (about -8 block co-efficient) ... -65% 1-5%

However, when the areas of wetted surface are being calculated,

advantage may be taken to calculate from this the shell displacement,

as shown in the foregoing example.

The calculations for the different items, as mentioned above,
are made for a number of different draughts, and the results so
obtained are set off in diagrammatic form in the displacement scale,
as described in the following paragraph.

The "Displacement Scale " and the various Curves shown thereon.

In Fig. 21 we have shown the displacement scale and the other
curves as constructed from the results obtained by means of the
before -mentioned calculations. Vertically, we have the draught
scales from which the results are set off in a horizontal direction:
at their respective draughts. Two scales are shown one being
the moulded draughts measured above the base-line, which is the
top of keel ; the other being the extreme draughts, taken from the
bottom of the keel. The calculations being made to the moulded
draughts, as used in the sheer draught, the results must therefore
be set off at these draughts in the displacement scale. ' The water-
lines used in the calculations, 2 ft., 8 ft., 14 ft. and 20 ft., up to
which the various results were obtained, are drawn across the diagram
at right angles to the draught scale, as shown dotted in Fig. 21.
The curve of displacement is first laid off by taking the displacement,
as found in the calculation, at each of the water-lines and measuring
it to a suitable scale, on the horizontal lines drawn at the respective
draughts. Through these spots a curve is drawn, the scale
used being shown at the top of the diagram. This curve is
extremely useful in obtaining the displacement at any particular
draught, or the draught corresponding to any given displacement.
For instance, suppose this vessel to be floating at a draught of
15 ft. 6J in. forward and 16 ft. 9 J in. aft, and it is required to find the
displacement corresponding to this condition, the mean draught
is first found : F 15 ft 6 i in

A 16ft. 9J in.

2)32 ft. 4 in.

Mean draught = 16 ft. 2 in.



The Theory and Design of British Shipbuilding,



39




Fig. 21.



40 The Theory and Design of British Shipbuilding.

At this draught on the extreme scale, a line x y is squared across
until it cuts the displacement curve, and from the point of inter-
section y, a perpendicular is erected which cuts the scale for dis-
placement at 3,170 tons, this being the displacement corresponding
to the mean draught of 16 ft. 2 in.

Again, suppose that it is required to find the mean draught at
the time of the vessel having 2,300 tons of cargo on board, her
light displacement being 1,500 tons.

1,500 light displacement.
2,300 cargo.



3,800 total displacement at the time.

From 3,800 tons displacement in the scale, draw the perpendicular
cutting the curve at b, from which point the horizontal line ba is
next drawn, and which intersects the draught scale at the draught
corresponding to the above amount displacement. In Fig. 21
it is seen to be 18 ft. 6 in.

The figures obtained from the displacement scale for a mean
draught are fairly accurate except when the trim is excessive.
Taking the mean of the forward and aft draughts, we make the
assumption that the actual inclined water-line W 1 L 1 and a level
water-line W L each giving the same displacement) are intersecting
at 'midships. (See Fig. 22.) This may not be the case, as is shown
in this sketch, where these two water-lines are intersecting at F, which
is 10 ft. aft of 'midships. For the displacement to be equal in the
inclined and level conditions, the amount contained in the immersed
wedge W 1 F W must be exactly equal to the amount contained
in the emerged wedge L 1 F L. If this is the case, and the water-
lines do intersect at 'midships, the assumption is correct ; but should
we assume the intersection to be at 'midships, and the contents
of the corresponding wedges as produced by the dotted level
water-line be not equal, then the result is obviously incorrect.
In the case of a vessel trimming by the stern at the load draught,
the displacement, corresponding to the mean draught, as taken
from the scale, is generally less than the actual amount, due to
the reasons explained in the following : In Fig. 22 we have shown
a vessel with a large amount of trim by the stern, the water-line
being W 1 L 1 . With an equal amount of displacement, but floating
at level draught, the water-line is W L. The volumes of the wedges



The Theory and Design of British Shipbuilding.



41



W 1 F W and L 1 F L are equal since the displacement has not
changed. The end draughts being 12 ft. and 7 ft., we have a mean
at 'midships of 9 ft. 6 in. Setting this off at 'midships, as shown
in the sketch, it will be seen that we obtain a different water-line
(shown dotted) to the actual level water-line W L, which is equal
to 10 ft. draught, and cuts the inclined water-line W 1 L 1 at F.




Fig. 22.

The " mean draught displacement " would therefore be less than
the actual by the amount contained between the w^ater- lines at
9 ft. 6 in. and 10 ft. level draughts. To look at this in another way,
suppose that the vessel originally floats at 10 ft. mean draught,
and then, on account of a weight being shifted the vessel changes
trim, going down by the stern and up by the head. The point at
which the water-lines intersect is approximately F, the " centre


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