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