Bernard Courtney Laws.

Stability and equilibrium of floating bodies online

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B.Sc. (ENG.), A.E.C.Sc. (LoND.), A.M.I.C.E., M.I.N.A.







THE present treatise is an attempt to set forth briefly the
principles underlying the stability and equilibrium of bodies
floating partially or wholly submerged in water, and in air.

Hitherto published matter bearing on stability has for the
most part been confined to ship forms. Submarines and
aerial machines claim a more recent development, and problems
relating to the stability and equilibrium of these bodies may
be said to be still under investigation. It is necessary now to
approach the subject in a more liberal manner, treating the
bodies as subject to active as well as passive forces, and to call
into requisition the principles of fluid pressure whether
liquid or gaseous in their action upon bodies at rest and in
motion. A knowledge of the salient features of rigid dynamics
and hydromechanics is required in order to enable the reader
to take a comprehensive view of the subject under discussion.

In the introductory chapter the author has endeavoured to
set out the essential points bearing upon this phase of the
subject, and in Chapter I. discusses generally definitions, the
nature and conditions of equilibrium, and the important
formulae, in order to enable the reader to pursue without
interruption the chapters dealing with specific types of body,
which follow.

Chapter II. treats of the stability of ships, and an endeavour
has been made to render the matter comprehensive without
touching upon the historical side of the subject ; those readers
desirous of pursuing to the end the history and development
of stability as applied to ships are referred to Sir E. J. Reed's
classic work.

The consideration of floating docks is included in Chapter IV.,
and, so far as the author is aware, has not been dealt with
hitherto in any published work.



Chapters III. and V., treating of submarines and air craft,
indicate the manner in which the problem may be attacked
without entering into what, to a large extent, could be only
approximate data with reference to the forces acting on these
bodies. It is hoped that the treatment of the subject may be
found instructive to those interested in the study of submarine
navigation and aerial flight.

Chapter VI. deals concisely with caissons.

The subject-matter for the most part has been derived from
the author's notes culled over a period of intimate association
with the scientific side of shipbuilding ; the data and experi-
mental results where given are reliable, and no effort has been
spared to make the book trustworthy. Where necessary
reference has been made to works and papers dealing with
the subject.

The author is under obligation to Mr. Lyonel Clark, M.I.C.E.,
and to Mr. P. Hillhouse, B.Sc., for their patience and kindness
in reading over the MS., and to Mr. A. E. Berriman, late
Technical Editor of Flight, for reading the MS. dealing with
air craft.





Types of floating body Definition of a fluid Pressure at any
point in a fluid Kesultant pressure of a fluid on a body is
equal to weight of body Equilibrium of a body under con-
straint Fluid pressure on a body in motion Experiments of
Lord Kaleigh, Beaufoy, Froude, Mallock, Finzi, Soldati,
Zahm, Dines, and M. Eiffel . . . . pp. 1 11


Plane, and surface of flotation Surface of buoyancy Curves of
buoyancy and flotation Eelation between surface of flotation
and form of body Locus of centre of buoyancy a closed sur-
face Surface of flotation a closed surface Positions of a body
for stable and unstable equilibrium occur alternately Con-
sideration of prismatic bodies (Def .) Metacentre Expression
for " height of metacentre above centre of buoyancy "
Attwood's formula for "statical stability" (Def.) " Meta-
centric height " or " initial stability " Expression for amount
of work done in inclining a body Effect of contained fluid,
free to move, on stability of body Prometacentre Locus of
prometacentre the evolute of curve of buoyancy Conditions
for stability of body under constraint Angular displacement
of a body floating in a perfect fluid Angular displacement of
a body floating in an imperfect fluid Oscillation of a floating
body pp. 12 45



Factors governing design No absolute standard of stability
Shifting metacentre Consideration of various types of sea-
going vessels Stability determined by joint effect of " form "
of vessel and general distribution of weight Vessels with top
hamper not necessarily unstable Winged weights Meta-
centric diagram Consideration of rectangular prismatic


vessel Stability curves Eange of stability Effect of change
of draught and beam on stability of rectangular shaped vessel
Freeboard, beam, depth and draught considered collectively
Calculations of stability Heck's mechanical methods
Amsler's integrator Crank, tender and steady ships Stiff
ships Negative GM Inclining experiment Stability curves
of typical sea-going vessels Cross curves of stability
Coefficients of displacement, and moment of inertia of water-
plane area Morrish's rule for vertical position of centre of
buoyancy Statical stability curves for specific vessels
Analysis of effect of change in dimensions, freeboard, position
of centre of gravity, etc., on curves of stability in specific
cases Longitudinal stability Trim, and change of trim
Moving weights on board Expression for " moment to change
trim one inch " Determination of change of draught with
change of trim Placing weights of small and large value on
board Effect on trim Bonjean curves Long's trim curves
Change of trim necessary to secure a specific draught forward
or aft Mobile cargoes, effect on safety of vessel Effect of
bilging a compartment Case of box-shaped vessel Vessels
with deep tanks Sub -division of vessels into watertight com-
partments Bulkhead Committee's (1891) recommendations
regarding spacing of watertight bulkheads Stability of vessels
partially waterborne Stability of vessels at launching
Sir E. J. Eeed's report on Daphne disaster Effect of action of
rudder and propeller on stability Stability of vessels running
at high speeds Dynamical stability Moseley's formula for
dynamical stability Relation between dynamical stability
and statical stability curve Stability of vessels subject to
wind pressure Stability of vessels amongst waves . pp. 46 160


Amsler's integrator Action of instrument Proof of principles
Tchebycheff's rules Application of rules Proof of rules

pp. 161167


Stability in longitudinal direction Form of submarine Methods
used to obtain submersion Action of rudders Diving con-
dition Action of hydroplanes or fins " Dynamic " type of
vessel Action of forces to produce horizontal movement of
vessel Conditions for motion in a straight line Mean straight
path Transverse stability Effect of vertical rudder on
transverse stability pp. 168 181



Types of dock Generally only necessary to consider small angles
of heel Metacentric method of stability generally sufficient
Critical period Reason of broad lower walls Air box Paths
of centre of buoyancy, and centre of gravity, in docking and
undocking Free water in dock Action of offshore and out-
rigger docks Action of booms Static effect of immersing or
emerging the dock Action of pontoon in outrigger dock
Stability curves relating to outrigger dock Loci of centre of
buoyancy Stability calculation Strength of booms Action
of cams Angular movement of docks, geometric considera-
tions . . . pp. 182 213


Means of sustaining bodies in the air Types of air machines
Tendency of air craft to oscillation Angle of incidence
Mechanics of motion of the aeroplane Immovable and
movable planes Force diagram Tail plane Elevator
Direction of propelling force Effect of change of position of
centre of gravity on longitudinal stability Longitudinal
stability varies directly as the speed, and inversely as angle of
incidence Automatic longitudinal stability Vertical keel
plane Principle of vertical keel plane considered mathe-
matically Warping Mechanical stability devices Pre-
ference of monoplane over biplane Waterplanes or hydro-
aeroplanes Form of floats Number of floats Mechanics of
the waterplane Dirigibles Buoyancy of airships Stability
of airships Stabilising planes . . . .pp. 214 235



Purpose of caissons Watertight sub-division Consideration of
high and low-water levels Method of working caissons
Tidal deck Calculations and diagram relating to stability
Condition that caissons may float Capability of flotation at
high and low-water levels Form of caissons Bute Dock
caisson Barry Dock caisson Keyham caisson Sliding and
rolling caissons ....... pp. 236 247

INDEX pp. 249251





As a matter of ordinary experience we know that fluids exert,
and are capable of resisting, pressure.

Without going analytically into the question of classification,
we may state broadly that fluids are of two kinds, viz. :

Compressible and Incompressible,

preferring that the reader who may be desirous of obtaining a
complete knowledge of the general properties of fluids should
consult those published works which are devoted to the study
of hydromechanics.

What affects us more in the present case is the consideration
of the properties of water and air so far as concerns their
ability to support and exert pressure on bodies, whether at
rest or in motion, floating at the surface of the former or
entirely within either. Consequently in this work the term
" floating body " will generally embrace not only those bodies
usually met with in practice, the normal position of which is
at the surface of the water, viz., bodies of ship-shape form,
floating docks, and caissons, but also those of more recent
development specially designed for service in the air or beneath
the water surface ; these latter embody respectively aeroplanes
and airships, and submarines.

In order to understand the application of the principles
governing the stability of these several types of body, it will be
well to first establish, or record without proof, certain funda-
mental points, a knowledge of which will be found to be
essential to the investigation.

S. B


Air, of course, is compressible ; water, although not really
incompressible, is generally regarded as such, because so great
a force is required to compress it into a volume even a little less
than that occupied under normal conditions.

Def. A fluid is a substance which yields continually to the
slightest stress in its interior, i.e., it may be divided easily along
any plane, provided the fluid be at rest.

If the fluid have motion or be of a viscous nature it still may
be so divided if sufficient time be allowed. Hence when a
fluid is at rest the tangential stress in any plane within it
vanishes, and the resultant stress at any point in that plane
will be entirely normal to it.

We may therefore conclude at once that in any fluid :

(a) The pressure on a surface at any point is normal to that

(6) The pressure on any elemental area is independent of the
inclination of the plane containing that area, and is
therefore the same in any direction.

The pressure at a point situated at a distance z below the
surface of a fluid is given by

P = Po + w - *,

where : #o = pressure at the fluid surface due to the atmo-
sphere or otherwise

w = weight per unit volume of the fluid.
From this it follows that the pressure at all points in the same
horizontal plane is the same.

The pressure at any point of a surface being perpendicular
to the surface, it follows that the resultant pressure on a plane
surface is equal to the sum of all the normal pressures, acts in a
direction at right angles to the surface, and is given by :

If (Po + w.z) da = A (p Q + w . z),
where : A = area of the surface

z distance of the centre of area below the fluid


This resultant may be resolved into a vertical and horizontal
component the directions of which lie in the same plane with
the resultant.

If through any point in the boundary of the given surface a
vertical line be drawn to the surface of the fluid, and this line
be caused to traverse, parallel to itself, the whole boundary so


as to trace out the surface of a vertical cylinder enclosing a
mass of fluid, then the reaction of the plane surface resolved
vertically is equal to the weight of fluid in the cylinder and acts
in a vertical line through the centre of gravity ; the point at
which this line cuts the plane is the " centre of pressure."

Consider a body of any form of surface floating freely at rest,
and wholly or partially submerged. The resultant pressure of
the fluid on the body may be resolved into a vertical pressure,
and two horizontal pressures in directions at right angles to
each other. Obviously the latter must be each zero, since there
is no horizontal movement of
the body, and the resultant
vertical pressure must equal
the weight of the body.

Suppose a vertical line
touching the surface of the
body to trace out a cylinder
enclosing the immersed portion.
The surface will be divided into
two parts, on one of which
abc, Fig. 1 the resultant ver-
tical pressure acts upwards,
and on the other aedc down-
wards. The difference between
these pressures is the resultant vertical pressure on the body,
and is evidently equal to the difference between the weight of
the fluid included in the cylinders wabcl and waedlc, i.e., is equal
to the weight of fluid displaced by the body, and must act
upwards through the centre of gravity of the fluid displaced.

Therefore : The weight W of the body must equal the
weight V . w of the fluid displaced ; and the centre of gravity G
of the body and of the fluid displaced H must be situated in the
same vertical line, where V is the volume of the fluid displaced.

These are the necessary and sufficient conditions of equili-
brium of a body floating freely and at rest.

Def. The centre of gravity of the displaced fluid is desig-
nated the centre of buoyancy of the body.

With constraint the conditions of equilibrium are altered.
Suppose a body abed, Fig. 2 floating at the water line bd
to be supported at a by a force or reaction R.

B 2


Let W and P denote the weight of the body and of the fluid
displaced respectively. The resultant horizontal pressure of
the fluid on the body is nil, and therefore the direction of R
must be vertical, and we have :

K = w- P

Hence, in the case of a body floating in equilibrium under
constraint, the constraining force is vertical and is equal to the
difference between the weight of the body and that of the fluid

The foregoing refers to the equilibrium of a body floating at
rest. When we come to consider the conditions of equilibrium
of a submarine a body the function of which is to move at
and below the surface of the water we shall require to approach

the questions of equilibrium and stability under new con-
ditions, for upon the relative speed of the body and fluid and
the angle at which the planes or fins superposed on the body
and used for the purpose of stabilisation are presented to the
direction of motion will depend the resistance experienced by
the body, and the pressure on the planes ; as a result of which
the body is enabled to assume certain required and definite
positions in the fluid. It will therefore be necessary to con-
sider concisely certain leading principles relating to the pres-
sure of water on planes moving through the fluid or against
which the fluid flows.

Similarly it will be necessary to know what is the nature of
the pressure of a gaseous fluid on a body moving through it in
order to be able to apply our knowledge to aeroplanes and air-


ships when we come to consider the equilibrium and stability
of these bodies.

Suppose a thin lamina to be moving through water, the
direction of motion being parallel to the moving surface. The
resistance experienced will be purely frictional.

If the direction of motion be perpendicular to the moving
surface, the resistance
will be entirely head or
direct resistance.

Suppose now the
lamina to be moving in
such a manner that the
direction of motion is FIG. 3.

inclined to the moving

surface, i.e., that the surface is moving obliquely, the resist-
ance will partake of both frictional and head.

In the case of motion which is either oblique or perpen-
dicular to the moving surface, an eddying wake will form behind
the lamina somewhat as indicated in Fig. 3 and 4 respectively,
the arrows indicating the direction of motion of the body
relative to the fluid. The dotted lines indicate the stream lines

or paths pursued by the
water, and the broken
_2 - _-__-_-iL water produced immedi-
^V-"^ ~ ately behind the lamina
i^z^rJIz constitutes an element of
-^Jlz^jr resistance additional to
EL-L-IT. that measured by the speed
p^- = " IL -~ at which the lamina is


FlG 4 Beaufoy, in his experi-

mental work in this depart-
ment, ascertained that so long as the lamina was actually
immersed, so that no surface disturbance of the fluid resulted,
the resistance to motion was independent of the depth below
the fluid surface, i.e., was independent of hydrostatic pressure ;
and that the resistance per square foot in the case of a surface
moving in a direction perpendicular to itself is 112 Ibs.,
corresponding to a uniform speed of 10 ft. per second ; for any
other speed the resistance varies as the square of the speed.


Lord Raleigh, as the result of his experiments on surfaces
relatively narrow, determined a formula connecting the oblique
with direct resistance as follows :

_ P (4 + 77) sin a
4 + 77 sin a

where : P x = normal pressure on the face of the plane.

P = pressure of a head due to the relative speed of

the water and plane,
a = inclination of the direction of motion to the

plane (see Fig. 3).
If a = 90 then P t = P.

Beaufoy stated that, from his experiments, when a = 90
the resistance was greater than P by some 16 per cent., from
which it would appear that Lord Raleigh's formula takes no
account of the negative pressure on the back surface of the

The position of the point of application of the pressure on
a plane moving through water has been worked out by Lord
Raleigh, and has been determined experimentally by Mr.
Froude and others.

For a rectangular plate of breadth b and length I, of
which I is perpendicular to the direction of motion, the
distance x of the centre of pressure from the forward
edge is expressed in the following formula given by Lord
Raleigh :

6 f _ 3 cos

x =- \ 1


sin a) J *

2 ( 2 (4 + 77 sin

where a is the angle made by the plane with the direction of

There is obviously great difficulty in determining by
experiment the value of x for small angles of inclination, but
for angles greater than 10 the results may be considered as
more trustworthy.

Froude's results for plates of certain proportionate dimen-
sions are shown graphically in Fig. 5, and are taken from his
Reports and Memoranda.

With reference to frictional resistance, the exhaustive
experiments conducted by Mr. Froude and recorded in the


Reports of his researches are probably the most reliable.
The following leading factors should be observed :

(1) Frictional resistance is independent of the pressure
between the fluid and solid.





to* IQ" 20' 30' .40' 50 60
Fia. 5. Diagram illustrating Froude's Experiments on Planes moving in Water.

Centre of Pressure Curves.

(a) r- = 2 plane inclined about long edge.

(b) I = 6 plane square.

(c) y = 2 plane inclined about short edge.

(2) Frictional resistance is proportional to the area of the

surface presented to the fluid.

(3) Frictional resistance oc v 2 (approximately) where v is

the velocity of the surface relative to the fluid.

(4) Frictional resistance depends upon the density of the




(5) Frictional resistance is sensibly affected by the length

of surface in the direction of motion.
We may take R = / . A . v 2
where : R = resistance

A = area of surface

/ is a coefficient dependent upon the nature and
degree of roughness of the surface.

Direction of pla,ne moving
through a/r

FIG. 6.

Direction of curt/ed /a/n/na,
moving through

FIG. 7.

When a plane is moving through air the nature of the
stream lines and the eddies behind the plane are of a very

similar character to the
phenomenon in the case of
a lamina moving through
water. Figs. 6 and 7 illus-
trate examples of a plane
and curved lamina respec-

Our knowledge relating
to air pressure and the
position of the centre of
^ pressure on surfaces mov-
ing through the air is not
definite, although experi-
ments have been and are still being carried out.

Mr. Mallock states * that he has made several series of
experiments in air with reference to the forces acting on planes
and curved laminae. Taking r x and r y to represent the
resistances parallel and perpendicular to the direction of

0" 4 'or 5 "(about)

FlG. 8.

* Report of Advisory Committee on Aeronautics.


motion or flow, and 9 to denote the angle of incidence,*
his results are represented approximately by the diagram
in Fig. 8, where the abscissae denote angles of incidence and
the ordinates the value of the ratio f f /r.t

The maximum value of the ordinate of the curve appears
to correspond with about four or five degrees of incidence,
and may be taken at an average of 7 the values obtained
during the experiments varying between 5 and 10.

With curved surfaces the angle of incidence 6 has been
reckoned relative to the chord of the arc, thus :


Mr. Mallock evidently considered his results as being very

Experiments carried out by Finzi and Soldate { gave the
following results :

Angle of incidence 0, 5, 10, 15.

, ) Flat surface 11 '3 5'6 3'7

Ratio fJr m for \ ~ * . A * /* * o A a

j Curved surface 1*4 10*5 8'9 5'2

The most reliable work with reference to the surface friction
of thin plates in air, according to Dr. Stanton,{ appears to
have been done by Zahm (see Philosophical Magazine, Vol. 8,
1904), who experimented on boards of various lengths placed
in an air channel with an air current varying from five to
twenty-five miles per hour. The results generally show that
the resistance per square foot of area oc v 1 ' 85 .

Compared with Froude's experiments on plates in water
(British Association Report, 1872), this leads to the conclusion
that under similar conditions the resistance in the two media
due to surface friction ex densities.

* In this work the angle of incidence is taken to be the angle at which the air
meets the plane.

t Ratio of lifting to propulsive force in flying machines.

{ Engineering, March 17th, 1905.

6 must be negative if r v is to be zero with the curved surface.


If v = velocity of the wind in feet per second
and P = direct pressure in Ibs. per square foot,
then P = 0023 v 2 approximately, and the normal pressure of
the plane for an angle of incidence 9, = P sin 6.

Generally the formula for P may be taken as P = csv 2 , and
the normal pressure for small angles of incidence R = c-^sv^B,
where c and c l9 are coefficients dependent upon the " aspect
ratio "* of the surface, and s is the area of the surface.

The mean value of c may be taken as '08, and c x as about
four times this value for angles up to 7J degrees, and aspect

iF pressure
in terms

1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

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