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Text-Books of Science —continued.
VIii. THE STRENGTH OF MATERIALS AND STRUCTURES:
The Strength of Materials as depending on their quality and as ascertained
by Testing Apparatus; the Strength of Structures as depending on their
form and arrangement, and on the Materials of which they are composed.
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TEXT-BOOKS OF SCIENCE
ADAPTED FOR THE USE OF
ARTISANS AND STUDENTS IN PUBLIC AND OTHER SCHOOLS.
THE STRENGTH OF MA TERIALS
AND  STRUCTURES.








THE
STRENGTH OF MATERIALS
AND STRUCTURES.
PART I.
THE STRENGTH OF MATERIALS, AS DEPENDING ON
THEIR  QUALITY, AND  AS ASCERTAINED  BY
TESTING - APPARA TUS.
PART II.
THE STRENGTH OF STRUCTURES, AS DEPENDING ON
THE FORM AND ARRANGEMENT OF THEIR PARTS,
AND ON THE MATERIALS OF WHICH THEY ARE
CONS TR UC TED.
BY
JOHN ANDERSON, C.E. LL.D. F.R.S.E.
Sulerintendent of Machinery to the War Defartment; formerly Lecturer
on Afllied Mechanics at the Royal Military Academy, Woolwicl;
at the Royal Engineer Establisrhment, Chatham; and at the
Royal School of Naval Architecture, South Kensington.
D.  APPLETON   AND   CO.
549 & 5 5  BROADWAY,
NEW  YORK.
I872.








PREFACE.
THIS ELEMENTARY TREATISE is divided into two distinct
parts.
The First Part treats of the natural properties of various
materials employed in construction, more especially in
regard to their strength and elasticity, and their adaptation for particular practical purposes in the arts.  The
object of this portion of the work is to describe the qualities
and characteristics of materials, so far as they are of importance to the engineer, or are exhibited in the results of
experiments made with the testing-machine.
An acquaintance.with the natural properties of materials,
as forming part of the great field of applied mechanics,
is indispensably necessary for the young mechanic or engineer, who desires to be something more than an artisan.
That real knowledge, which consists in understanding the
materials which he handles, and in a familiarity with their
points of agreement or difference, both in regard to elasticity and strength, cannot fail to give a charm to his daily
duty.
As it cannot be expected that all will have the opportunity of making experiments for themselves, the first three
chapters are devoted to the testing of materials, and to the
practical manipulation of a testing-machine.   In these
chapters reference is made to the physical properties of




vi                     Preface.
some common materials, by which the student will be able
to comprehend the nature of such experimental investigations, and the labour and care needed in order to arrive
at true results; and it is hoped that he will find the subject
treated in such a clear and simple manner, that he may
understand it without much difficulty.
The fourth, fifth, sixth, seventh, and eighth chapters
refer more especially to cast iron, wrought iron, steel,
copper, alloys, and timber, and are intended to describe
their qualities and leading peculiarities.
The ninth, tenth, and eleventh chapters treat, more generally, of the resistance of materials to torsion, shearing, and
punching, and to transverse strains, conjoined with impact
and vibration.
The first eleven chapters, therefore, have regard to the
nature of materials, and the remaining six chapters-which
constitute the second part of the volume-are devoted
to the strength of structures, when made of the materials
previously treated of.
In the Second Part, the student will learn the correct
forms which must be given to the various structures in
order to obtain the requisite strength, and likewise the best
arrangement of materials, as depending on their respective
properties, so that by the practical application of correct
principles, the maximum of strength may be attained with
the minimum of weight and cost.
The experiments most frequently referred to, and which
are quoted as' Woolwich experiments,' have all been made
in the Royal Arsenal for various purposes during the past
eighteen years, and chiefly with the American testingmachine, which is described in the third chapter, the only
exception being certain experiments, to ascertain the strength




Preface.                   vii
of ropes under various conditions, which were carried out
with a hydraulic testing-machine, recently transferred from
Her Majesty's Dockyard to the Royal Arsenal.
The author has taken pains to ensure accuracy; still, when
so many figures have been necessarily transcribed several
times, more particularly in the Tables, some errors may
probably exist. In this part of the work Mr. Charles Topple
and Mr. George Cuthbert have both rendered valuable
assistance; the former, more especially, in regard to the
chapters on structures, where calculations were concerned.
Most of the examples given are taken from actual works,
which have passed through the author's hands during the
past few years.
The sources from whence the results of other experiments
have been drawn, or from which extracts have been made,
are generally quoted. Reference has very frequently been
made to the Blue Book, containing the Report of the
Commissioners appointed to enquire into the Application
of Iron to Railway Structures.
WOOLWICH: August 1872.








CONTENTS.
PART I.
ON THE STRENGTH OF MATERZALS EMPLOYED ZN
CONS TRUCTIOV.
CHAPTER I.
ON SOME OF THE  PHYSICAL PROPERTIES OF MATERIALS.
Fitness of Materials for Special Purposes. Rigidity only apparent.
Elasticity. Hooke's Law. Solids, Plastic Substances, Liquids and
Gases. Permanent Set of Solids. Cases in which the Effects of
Elasticity need to be counteracted.... PAGE I-9
CHAPTER II.
ON THE EXPERIMENTAL TESTING OF MATERIALS.
Need of Accuracy. Several Specimens should be tested. Stretching
of Long and Short Specimens. Force to be applied in Axis of
Specimen. Effect of Time. Limit of Elasticity. Form of Specimens. Testing Machines.                              9 —15
CHAPTER III.
ON A MACHINE FOR TESTING THE STRENGTH AND ELASTICITY
OF MATERIALS.
Woolwich Machine. Balance of Machine. Application of Load.
Construction of Specimen Holders. Behaviour of Specimens. Flow
of Solids. Compressive Resistance. Transverse Resistance. Torsion. Deformation.                                  I5-29




Contents.
CHAPTER IV.
CAST IRON.
Production of Cast Iron. Influence of Carbon on Cast Iron. Mixing
Cast Iron. Limit of Elasticity. Tenacity. Selection and Testing
by Founder. Testing Girders. Woolwich Experiments. Experiments on Io-feet Bars. Compressive Strength. Experiments on
50-feet Bars.   Woolwich Compressive Experiments.  Renlelted
Iron.  Strength at various Temperatures.. PAGE 30-45
CHAPTER V.
WROUGHT IRON.
Production of Wrought Iron. Characteristics. Conditions determining
Quality. Tenacity of Bars and Plates.  Strength at high and low
Temperatures.  Appearance of Fracture.  Effect of Vibration.
Annealing. Safe Load. Red short and Cold short iron. Limit of
Elasticity. Behaviour under Test of hard and ductile Specimens.
Strength of Welds. Experiments on 50o-feet Bars.  Modulus of
Elasticity.  Compressive Resistance.... 46-67
CHAPTER VI.
STEEL.
Production of Steel. Bessemer Steel. Effect of working. Whitworth's Treatment. Mild Steel. Tempering in Oil. Strength of
Steel. Tempering Barrels for Guns. Experiments on Steel. Compressive Resistance.                                68-76
CHAPTER VII.
ON COPPER AND  OTHER METALS, AND THEIR ALLOYS.
COPPER. Its Tenacity.  Forging.  Effect of Exposure.  Specific
Gravity. Melting Point. Wiredrawing.  Casting.  Use of Phosphorus.  Experiments on Tenacity. BRONZE AND GUN-METAL.
Specific Gravity. Fusibility. Manufacture. Experiments on Gunmetal. Resistance to Tension and Compression. BRASS. Manufacture.  Specific Gravity.  Melting Point.  Influence of Lead.




Contents.                         xi
MUNTZ METAL. Tensile Strength. STERRO-METAL. ALUMINIUM BRONZE. BELL METAL. BABBITT'S METAL. Melting Point
of Alloys.  Additional Experiments... PAGE 77 —I00
CHAPTER VIII.
TIMBER.
Variability in Quality. Density. Conditions influencing Quality.
Seasoning. Laws of Shrinkage. Ash. Beech. Elm. Pine and
Fir. Hornbeam. Mahogany. Oak. Teak. Tensile and Compressive Resistance of Timber. Resistance to Shearing. Transverse Strength..                       0-121
CHAPTER IX.
TRANSVERSE STRENGTH  OF IRON  AND RESISTANCE TO
IMPACT.
Breaking Weight and Deflection of Bars. Effect of Impact. Transverse Flexure. Effect of repeated Deflections.. 122-130
CHAPTER X.
RESISTANCE TO TORSION AND SHEARING.
Shafting. Law of Resistance to Twisting. Wrought and Cast Iron
Shafting. Cast Steel and Copper under Torsion. Torsional Stiffness. Sudden Variations of Section. Experiments with Bars. Rupture by Torsion. Torsive Resistance of different Materials. Strength
and Stiffness of Shafting. Resistance to Shearing and Punching.
Gradual Shearing by inclined Instruments. Shearing Action on
Links. Punching and Drilling..            I3o-I-49
CHAPTER XI.
ON THE IMPORTANCE OF UNIFORMITY OF SECTIONAL AREA.
Surplus Material in parts a source of weakness. Screw Bolts. Chains.
Proof Strains for Chains. Rules for Strength of Chains. Strength
of Ropes. Rope Slings.                             I49 —I56




xii                       Contents.
PART II.
ONV THE STRENGTH OF STRUCTURES.
CHAPTER XII.
BEAMS AND GIRDERS.
Mathematics useful. Dependence on Formulae.  Beams and Girders:
Relative Strength as depending on Mode of Application of Load and
Support. Strength as affected by Arrangement of Material. Neutral
Surface. Tension and Compression in Beams. Flanged Girders.
Rectangular Beams. Tubular Beams. Beams of uniform strength.
Constants for Strength and Deflection  of Beams.   Cast and
Wrought Iron Girders.  Deflection of Beams.  Resilience of
Beams.......            PAGE I57 —87
CHAPTER XIII.
ON THE STRENGTH OF GEARING.
Teeth of Wheels.  Strength of Teeth.  Flanging Wheel Teeth.
Wear of Teeth. Wrought Iron Wheels. Malleable Cast Iron
Wheels.  Cast Steel Wheels.  Power transmitted by Wheels.
Strength of Screws. Cutting Instruments... 187-I95
CHAPTER XIV.
ON THE STRENGTH OF LONG COLUMNS.
Modes in which Columns yield. Short Columns. Euler's Theory.
Hodgkinson's Experiments. Practical Deductions. Wrought Iron
Columns.  Cast Iron Columns.  Strength of Rectangular and
Cylindrical Tubes.  Steel Shear Poles.              i95 —2o6




Contents.                       xiii
CHAPTER XV.
ON  THE  STRENGTH  OF  CRANEG  AND  ROOF TRUSSES AS
EXAMPLES OF COMPLEX STRUCTURES.
Mode of dealing with Complex Structures.  Bracing.  Triangular
arrangement of parts. Common Roof Truss. Trussed or Braced
Girder. Stresses on Hydraulic Crane. Strength of Sheer Legs.
Construction and Calculation of Stress on Members of a Steam
Crane. Crane Shafts. Crane Chain. Crane Foundations. Concrete Foundations.  Cylinder Foundation.   Pier-head Foundation.                                        PAGE 206-240
CHAPTER XVI.
STRENGTH OF RIVETED STRUCTURES-STEAM BOILERS, ETC.
Strength of Boiler Plates. Of Joints in Plates. Rivets. Proportions
of Joints. Oval Rivets. Double-cover Plates. Single and Double
Riveting. Plates with thick Edges. Diagonal Joints.  Strength of
Boilers.  Resistance to Collapse.  Provision for Expansion and
Contraction.  Flues.  Hydraulic Test.  Strength of Cylindrical
Boiler without Flues. Strength of Boiler with Flues. Tubular
Boiler.  Firebox Stays. Butt and Lap Joints.  Welded Joints.
Resistance of Tubes to collapse. Fairbairn's Experiments. Elliptical
Tubes. Effect of Length on Resistance to internal Pressure.  Small
Tube Boilers.241-270
CHAPTER XVII.
ON STRUCTURES SUBJECT TO INTERNAL PRESSURE.
Cast Iron Pipes. Pipes for High Pressures. Iron Tanks. Strengthening by Wrought Iron Hoops. Thick Cylinders. Barlow's Theory.
Gun Structures. American System of Casting Guns. Strength as
depending on Exterior Form. Hydraulic Press Cylinders. Chilled
Shot. Built-up Guns. Construction of Hoops. Armstrong's System. Longitudinal Strain on Guns. Need of Mass in the Breech.
Strengthening of Cast Iron Guns. Introduction of a Lining. General
Conclusions.                                      271I-296
INDEX.......                      297-302








PART I.
ONV THE STRENGTH OF MA TERIALS EMPLOYED
IN CONSTRUCTION.
-4 —CHAPTER I.
ON SOME OF THE PHYSICAL PROPERTIES OF MATERIALS.
IN ORDER to understand that branch of applied mechanics
which treats of the strength of materials, it is first of all
necessary that the student should possess a precise knowledge of those physical properties, on which the constructive
value of a material and its adaptation for given circumstances
depend. The fitness and reliability of materials for special
purposes is by no means, exclusively, a question of strength,
but is contingent on hardness, stiffness, toughness, malleability, and other inherent properties, which result from the
conditions of freedom or restraint existing amongst their
constituent molecules. In judging of the suitability of a
material for special duties, it is further necessary to know
its powers of endurance under the action either of forces
tending to abrade it, or of frequently repeated loading, or
of vibration and impact, or of varied changes of temperature, as the case may be.
Our knowledge of the different kinds of materials now
chiefly employed in connection with practical operations,
has been extended by the publication of a large mass
of valuable data, obtained in the numerous experimental
B




2           On the Strength of Materials.
investigations of scientific men, during the last hundred
years, and particularly during the last quarter of a century;
more especially is this the case with the metals, cast iron,
wrought iron, steel, copper, and brass. Hence, notwithstanding many discrepancies which occasionally present
themselves, there are a great number of well ascertained
facts and many definite laws for the guidance of those
engaged in construction.
To the young beginner, almost every kind of material
with which he comes into contact appears to present a
different appearance and character from that which it really
possesses, or which it will appear to possess when he knows
more about it. Thus, it may happen that a substance
which, at first, seemed a hard, solid, rigid, continuous
mass, is, in reality, soft. and porous, and even movable
in its internal structure; that every description of metal
or wood is yielding and elastic; that there is no such condition as absolute permanence of form, but that every material body suffers displacement of its parts under the application of external forces, being distorted or deformed, being
extended, or compressed, or deflected, or twisted. The
constituent molecules of some solids may even be made to
slide over or amongst each other, may be spread out into
a flat web, may be gathered into a more compact form, or
may be crushed, to an extent dependent only on the straining force which is applied.
The apparent fixity and rigidity of solids is unreal, and
this fact lies at the foundation of the whole subject; hence
one of the most remarkable features for the student to
observe, at the outset, is the unfailing mobility of the internal structure of materials on the application of sufficient
stress. The naked eye may not be able to detect the movement, because our perceptions are not sufficiently acute to
notice the minute change that takes place, but there is'a
change following every application of force. Hence, it is
found necessary to resort to testing instruments of various




On some of the Physical Properties of Materials.  3
kinds, in order to apply great straining forces and to make
the precise movements of the mass visible. For example, by
the use of a testing machine, it is found that a bar of
wrought iron, one square inch in section, will be elongated
x — oth part of its length by the weight of a single ton, and
will continue to stretch with every additional ton, until
rupture takes place; or, to give another example of
mobility, the change of temperature between the extremes
of summer and winter is sufficient to expand, or contract,
a wrought iron bar to the extent of ICth of its length.
At the present time, there is practically no exact theory
known which can be said to express the extension or
contraction of various materials, nor even of different
samples of the same kind of material, because the physical
conditions of the several pieces are not exactly alike; hardness or softness and other conditions, due to previous
treatment of the constituents of the specimen, step in to
modify the result.  The present object is, to show to the
student the practical aspect of the case, and to explain
the properties of some common materials, as they are met
with in the workshop; but in the latter part of the book,
when considering the principles which determine the form
and arrangement to be given to structures, it will often
become necessary to fix arbitrary rules, founded on the
average result of numerous experiments, and in the application of which the judgment will have to be exercised.
That property by which bodies tend to occupy a determinate bulk, or in the case of solid bodies a determinate
bulk and figure, under given pressures and at a given temperature, is termed their elasticity. By virtue of this property,
they oppose a resistance to forces tending to change their
volume or figure; and, if they have been deformed by the
application of such forces, they return more or less completely
to their original bulk and figure when those forces cease to
act. A body is said to be perfectly elastic, which returns
B2




4            On the Strength of Materials.
exactly to its original bulk, or, if a solid, to its original
bulk and figure, after such distortion or straining, when again
placed in the same conditions of pressure and temperature
as at first. A body which imperfectly returns to its original
condition after straining is said to be imperfectly elastic.
For all solid bodies, there are known limits to the amount
of straining force which can be applied, without producing
a definite and measurable change of figure or permanent set.
The limit of straining force which can be applied, without
producing any measurable permanent set, is termed the limit
of perfect elasticity or, more simply, the limit of elasticity
for the given material under the given kind of straining
force. For less straining forces the body is, for practical
purposes, sensibly perfect in its elasticity.  For greater
straining forces it is sensibly imperfect in its elasticity. It is
found by experiment that, up to the limit of elasticity, the
displacements suffered by the molecules of the body are sensibly proportional to the stresses which cause them, so that
a double displacement is caused by a double straining force;
a triple displacement by a triple straining force; and so on.
This property of elasticity is the more perplexing, because
the deformations of solid bodies are so minute, and often
so difficult to detect or to measure. Wrought iron is usually
considered as one of the most elastic metals within certain
limits, still its elasticity is far from perfect, except with a very
slight stress, and most probably it may yet be found to be
imperfect, even with the smallest load. Nevertheless, for
practical purposes, Hooke's law, which supposes all bodies
to be elastic within certain limits, may be accepted as sufficiently near to the truth.
The elasticity of materials, when considered as a practical question and as affecting the mechanical arts, is an
important physical property. It is a property which is
largely taken advantage of for most purposes, and for some
other purposes it has to be carefully counteracted or guarded
against.




On some of the Physical Properties of Materials.  5
The perfect elasticity of some solids within certain limits
of straining force, is proved by this circumstance, that many
such bodies suffer an innumerable number of repetitions of
straining action, without being sensibly altered. A steel
watchspring will work on for a century and give no marked
symptom of change, and a twisted steel or even iron rod will
shut a door for an unlimited number of times, and a chain
or rod of iron will lift weights and submit to be lengthened
and shortened constantly, for a long period, and apparently
will recover its normal shape when the force is removed.
But, most probably, this is so only because our means of
observation are incapable of detecting the change which
is gradually taking place.
Those other materials, which do not exhibit this property
to the same extent, or which readily retain any new form
which may be imparted to them, are usually considered
as non-elastic, and indeed justly, for practical purposes,
although, strictly speaking, they are only very imperfectly
elastic, as is evident when they are subjected to careful experiment: such materials are often termed plastic materials,
as for instance, soft lead, putty, clay, and similar substances.
Aeriform bodies, such as steam, air, and other gases, have
no elasticity of figure, but are perfectly elastic as regards
volume, for they may be expanded or compressed to an extraordinary extent, the expansion or contraction under constant temperature being sensibly proportional to the force
employed to act upon them.
A very common error exists respecting the elasticity of
liquids, such as water: they are frequently treated as if they
were without elasticity; but this is not strictly the case, except as regards elasticity of figure. Water does not seem to
admit of being drawn out beyond the volume which is due
to the pressure with which it is surrounded, and which in an
open vessel is determined by the pressure of the atmosphere
and its own depth. But when confined, water admits of a
limited amount of compression, and when the pressure is




6             On the Strength of Materials.
removed, it instantly returns to its original volume, and so
far is perfectly elastic. From the result of some careful experiments it was found that the pressure of one atmosphere
reduced the bulk of water by nearly the 2  o ooth of its
original volume, an amount so small as to be scarcely appreciable in practical operations, and which, as a rule, is disregarded. Even in the action of the hydrostatic press, working often with a pressure of 500 atmospheres, it may be
neglected with no practical disadvantage; nor is the elasticity usually observable, in consequence of the greater
movement of the surrounding metal interfering.
It is due to the great compressibility of elastic gases, such
as air, and the incompressibility of liquids such as water,
that machinery for working with the former can be driven
at a much higher velocity than similar machinery working
with the latter. From the greater compressibility of air or
steam, and the work absorbed in compression, there is not
that perceptible shock which occurs with water or with
solids under similar circumstances. It is this compressibility
which imparts to aeriform fluids such a degree of yielding
softness when they are acted upon by mechanical apparatus.
The elasticity of solids is of a different character to that
shown by gases or liquids, although many solids partake of the
properties of the former in some degree. In solids, the elastic
property is mostly shown in the more or less complete
recovery of the original form, rather than in the recovery of
volume, after compression or extension, and although within
certain limits the elasticity of solids, such as the metals,
may seem practically to be nearly perfect, still it is doubtful
if they are entirely so. By very careful testing of long bars,
it was found that they did not return to precisely the original
length.  This change is not easily observed in making
experiments with short specimens, the alteration being so
minute.
Another notable feature is the circumstance that the
stretching of a rod, or chain, or bolt, when subjected to ten



On some of the Physical Properties of Materials.  7
sional strain, is not uniform unless the substance or strength
of the bolt or chain is uniform.  The greatest extension
takes place at the weakest point, which is a great disadvantage; hence rods, bolts, or chains are frequently broken
with less force, tear, and wear than would be inferred, if this
fact were neglected.
The elastic property of solid materials takes a much more
important part, in the stability of structures, than is usually
apparent to the casual observer. A familiarity with the testing machine and the lessons which it teaches, cannot fail to
render tile constant presence of elasticity a living reality to the
thinking mind, and this should never be lost sight of, becauseany variation in the length of parts that have to support
each cther introduces an element of weakness. The long
bolt, or the long stay, may not, on this account, take a fair
share of the work; and even in a single part, where the
strain enters upon it by flanges, asin pillars, or by joints, as
in tie-iods, unless we assume absolute perfection in the mechanical fitting, some portion may have more than its share
of duty; hence it is that such end parts, so exposed, require
to be made stronger than the main body.
Notwithstanding this, the elastic property, which is so
prominent a feature in all the materials which are employed
in the workshop, is of inestimable value, as without it the
iron could not be hammered or otherwise roughly manipulated The great freedom of treatment to which the metals
maybe subjected, in reference to their malleable and other
properties, is greatly due to their elasticity.
When a rod of iron, or a beam, is in a normal or quiescent condition, its capacity for rigid duty is not equal to
that of a similar rod or beam in which the elasticity has
been partially used up, by bending it into such a curve as will
nat injure its future stability.
By using up, or calling into active exercise, the elasticity
of a beam or other article, the strength, or rather the stiffness, is considerably increased, and great ingenuity is fre



8           On the Strength of Materials.
quently shown by practical men in such adaptations, whereby
a comparatively flexible beam, by being subjected to a bending
force, is thereby enabled to afford an amount of rigidity considerably greater than would be obtained from a much
stronger beam, when left in its natural state. A beam fixed
between the walls of a building, on which are to be erected
the supporting columns of a steam-engine, will have a considerable amount of spring and vibration, unless the beam
is inordinately strong; but if this beam is firmly propped in
the centre, and then bent down at the ends, by means of
wedges within the wall boxes, the elasticity of the beam
will be used up and the stiffness very greatly increased.
Another example is afforded in the case of long horizontal tie-rods. The elasticity of the rod permits it to bend
into a curve, due to gravitation. But by first ascertaining
how far the rod will bend, and bending it previously to
the same extent before it is erected, and then fixing it in a
position in which the camber is upwards, the after-bending
will bring the rod down to a straight line. The rod under
such conditions is not only more pleasing to the eye but is
in all respects more rigid, from the elasticity being absorbed
by the arrangement.
In the familiar case of the springs of railway carriages,.
it is found advantageous to use up a considerable portion of
their elasticity in the primary adjustment, by means of a
rigid bar to which they are bound. By this means, the carriage does not sink as the passengers enter, and the springs
are so arranged that the remaining elasticity comes into Dlay
when the carriage is nearly full.
It may appear inconsistent, to speak of a beam or rod
being thus stronger by using up so much of its strength, but
such is the case, for the class of purposes here indicated;
they are rendered stiffer with a given quantity of material,
and it takes a greater stress to give further movement to the
beam or rod.
In the application of wrought iron, steel, and all the




On some of the Physical Properties of Materials.  9
highly elastic materials, although they may be safely loaded
up to near their limit of elasticity, still, owing to the uncertainty of perfection of the material, engineers seldom
venture far beyond the half of the amount of stress, which
the apparent limit of elasticity appears to warrant.
CHAPTER II.
ON THE EXPERIMENTAL TESTING OF MATERIALS.
BEFORE describing the nature and manipulation of a mechanical testing apparatus, it is necessary to make a few preliminary remarks, both in regard to the mode of carrying
out experiments that are to be reliable, and to the care,
precision, and accuracy which are required, and to draw
attention to some of the many contingencies that may prevent the attainment of true results, or bias the mind of the
operator, even where good instruments are employed, and
with every desire to avoid error.
In making important experiments with solid materials, it
is not advisable, and can never be satisfactory, to depend
upon the result of an experiment upon a single specimen,
however good it may appear to be, for the smallest defect
or scratch upon the outside, or even a hidden fault in the interior, may modify its behaviour to an unknown degree. The
stress which is applied will mostly expend itself upon the
specimen at its weakest point, instead of being distributed
throughout the mass operated upon. Three specimens at
least should be tested, and the diameter, length, and quality
of each specimen should be uniform, as far as may be
practicable.
It is found, in the testing of short and long specimens,
even of the same kind of material, that in general the short
and long pieces do not all stretch alike, or in the proportion
due to their respective lengths: the long bars stretch more




10          On the Strength of Materials.
than the short ones.  The cause of this discrepancy is
rather obscure, and it is doubtful whether it is owing to the
long bar, from its mere length, having a greater proportionate risk of invisible defects, or to some other unknown
cause; but practically it is found that greater uniformity is
attainable with short specimens, possibly in consequence
of the closer scrutiny to which they may be subjected.
Hence it may be inferred, that in testing materials intended
for a particular structure, the nearer the test bars approximate in their dimensions to that structure, the more reliable
will be the average result for the particular duty. At the
same time, the length does not affect the strength, for
with the exception of weakness due to hidden defects, the
actual ultimate tensile strength of any portion of a specimen
is not much, if at all, affected by the length of the bar; each
portion has to act independently and for itself, irrespective
of the other parts, its behaviour being exactly according to
the stress to which it is exposed, and this is not inconsistent
with the disproportionate rate of extension of long and short
specimens, referred to previously.
In testing bars or strips of metal, whether the pieces are
long or short, but more especially in the latter case, it is of
much importance that the stress should be applied to the
specimen in a line coincident with the axis of the specimen;
if it is not, the result will be erroneous, because, the stress
not being uniformly distributed on the cross sections, one
side will have to yield prematurely, and thus the resistance
of the bar will be overcome'in detail: for want of attention
to this particular, many experiments made with rough apparatus do not afford reliable results.
Time is an important element in the movement of the
molecules of the specimen. In carrying out experiments
with metals in a testing machine, it is evident that the
period of time during which the specimen is exposed to the
stress must, as a rule, be limited. With a short duration of
time and a short specimen no'permanent set' can be




On the Experimental Testing of Materials,    II
detected, if the load does not exceed a certain limit. To
that special limit attention is directed, as it is a most important one for the student to understand thoroughly; it is
usually termed the limit of elasticity, and is as varied for
different materials as the point of ultimate rupture. But the
fact that the testing machine does not show permanent set
until the apparent limit of elasticity has been exceeded,
does not prove much, for it is quite possible that a long continued permanent load, or an often repeated application of
a load not exceeding that limit, might ultimately fix the molecules in a new position; on this point, however, a difference
of opinion exists, though the experience derived from
thousands of carefully performed experiments, with short
specimens, in an accurate machine, would lead to the conclusion, that within a certain range every metal returns to its
original dimensions, as near as can be measured with moderately refined instruments.
It is more than probable, however, that the first application of a load, much less than that commonly supposed to
mark the limit of elasticity, does produce a minute permanent set. From the experiments made by the Commissioners appointed to enquire into the Application of Iron to
Railway Structures, it would appear that with the smallest
stress applied, namely,'56 ton per square inch, a minute,
but perceptible, amount of permanent set was produced
upon a bar 50 feet long. When the stress was equal to
ir69 ton, the set on the bar was equal to'0025 inch,
or the'ooo,oo4th of the length of the bar, an amount so
exceedingly small that it could not be measured in specimens only a few inches in length. For practical purposes,
the permanent set in such cases may be disregarded, being
inappreciable.
For the present purpose, it will be assumed that there is
a limit of stress up to which every specimen is perfectly
elastic, so far at least as our means of measurement permit
us to ascertain. In making experiments on the value of




I 2          On the Strength of Materials.
different materials, one great object is to determine this
limit of perfect elasticity. The operator must observe with
extreme care the indication of the slightest permanent set,
on the removal of the load. The limit of elasticity marks
the maximum stress which can be exerted on a material
without producing permanent deformation and therefore
danger of ultimate rupture.  But there still remains an
amount of strength beyond the elastic limit, and which is a
margin of safety in reserve. If a bar has been strained
beyond the elastic limit and has taken a permanent set, it is,
paradoxical as it may appear, in one sense stronger than it
was before; that is to say, it will require a greater strain to
be applied, in order to move the molecules a still greater
distance and to cause them to take an additional set. This
is an important fact, and it is at once observable, when
making experiments with short specimens, that no new load
which may be applied, less than that which produced a
given permanent set, will practically carry the effect farther
onward. Therefore, by taking a permanent set, the limit of
elasticity has in fact been raised, as the bar will not permanently stretch any more, without being subjected to a still
greater strain. Even this, however, is not strictly correct, if
the loads are much in excess of that which corresponds to
the elastic limit.
The elasticity of the several materials that are employed
in the workshop is extremely unequal, both in degree and'
in the manner in which the condition is shown. For example, some solids have elasticity combined with hardness
and brittleness in a high degree, and are similar in character
to glass; such are some kinds of cast iron, hard steel, and
certain mixtures of copper and tin, in which, tested by
impact, the elasticity seems as perfect as in an ivory ball,
yet, in the testing machine, the extent of movement is so
small that it can scarcely be measured by fine instruments,
and for any practical purpose cannot be taken advantage of;
while, on the other hand, there are some of the metals that




On the Experimental Testing of Materials.   I 3
seem to have the properties of india-rubber, but in a less
degree; such is wrought iron, in a certain state, and still
more so mild cast steel, when tempered in a particular manner, which developes the property of elastic flexibility in an
extraordinary degree.
In preparing various classes of specimens for the testing
machine, the strength of the machine itself must be first
taken into account, and this will determine the size of the
specimens according to their strength.  It is usual to have
the several specimens carefully prepared in a lathe, of
sizes nearly inversely as their strength, or in some proportion to their respective tenacity. It is extremely convenient,
and it simplifies the subsequent calculation, to make them of
such a diameter that their sectional area will be a convenient
multiple or fraction of a square inch; say, for instance, one
square inch, half a square inch, or one quarter of a square
inch. The Figures I, 2, and 3, show the form and relative
FIG. I.
FIG. 2.
FIG. 3.




14          On the Strength of Materials.
sizes of such specimens of cast iron, wrought iron, and
steel, as are commonly used in the testing machine now
to be described, and which has been used in the Royal
Arsenal at Woolwich, since I854. The drawing is made to
a scale of one-half of the usual size of the actual specimens,
when prepared for the testing machine, and the peculiar
shape which is given to these three specimens is that which
is used for experiments on tensile resistance, the middle
portion being the part which is to be tested, and which is
carefully turned in a lathe, so as to be perfectly parallel
throughout; the ends of the specimens are enlarged as shown,
in order to enable the machine to take a firm direct hold,
and the advantage derived from the round form is chiefly
this, that the specimens are produced in a lathe, with extreme accuracy, at a comparatively small cost for labour,
and are in every respect as convenient and serviceable as if
they were square, or of any other form.
Specimens for ascertaining the resistance to a compressive stress are generally made in the form of short solid
cylinders, of such dimensions as can be overcome by the
power of the testing machine, and therefore are so simple
in form that a further description of them is unnecessary.
In the construction of the earlier mechanical testing apparatus, the mechanism generally consisted of a simple lever,
which was mounted upon knife-edges at the several centres
of motion and suspension, the short end of the lever
laying hold of the specimen by a suitable bridle, and the
weights being applied at the other end of the lever in the
same manner as in a weighing beam. Great accuracy
might be arrived at with such a simple lever, and some
modern machines are fitted upon this principle and supplied
with every requisite for testing the strength of the weaker
class of materials, such as bricks, stones, mortar, Portland
cement, glue, the bite of nails in wood, or for similar
purposes; but they are not so convenient as those on the
compound-lever arrangement, when great straining forces
are required.




On the Experimental Testing of Materials.   I 5
In the modern construction of testing machines, intended
to operate upon very large or very strong specimens of
different kinds of metal, the lever arrangement is being
superseded by some modification of the hydraulic press.
The hydraulic press affords the most convenient means of
giving the necessary strain, but in the older machines, the
means of measuring the strain applied were imperfect. The
best kind of hydraulic machines are so contrived that the
precise force, which is exerted by the water, is shown by a
delicately adjusted steel-yard, or some other modification of
the lever principle. The lever is selected as more certain
and reliable than pressure gauges, because, however carefully
the latter may be constructed, they are liable to alteration
and error. In addition, when the load has to be calculated
from the pressure in the press cylinder, the friction of the
ram must be allowed for, and this cannot be done with any
great accuracy.
CHAPTER III.
ON A MACHINE FOR TESTING THE STRENGTH AND
ELASTICITY OF MATERIALS.
THE apparatus now to be described is intended to show the
strength and elasticity of materials in various ways. It is
probably one of the best and most correct machines which
has yet been made, for the purpose of testing small short
specimens, and for affording extreme accuracy in the results,
so far as is possible with short specimens. The machine is
shown in Figs. 4 and 5. This form of testing machine originated in the United States of America; the first one
was brought to England from that country in I854 by the
author: since then many similar machines have been made,
variously modified in size and arrangement, and have




z6          On the Strength of Materials.?                a              a, 
0,__  j~ j~                           _ _




Testing Machine.                1I 7
found their way to other countries. The machine in daily
use at Woolwich, with which, during the last fifteen years,
many thousand experiments have been made is also of
this construction; it is principally employed in testing
specimens to ascertain their tensile and compressive resistances.
The extensive application of machines on this principle is
chiefly due to their simplicity and compactness of construction, and the great convenience which their several arrangements afford for various classes of experiments, as well as
their extreme accuracy. They are provided with suitable
bridles, holders, and other apparatus, to test tensile, compressive, transverse, and torsional resistances, and are adapted
for experiments on the force required to punch or shear,
on the hardness or softness of bodies, and on the flow of
solids.
Figure 4 is a side and Fig. 5 an end elevation of this
machine; the former shows most of the details, and will
enable the general arrangements of the machine to be
understood, with a little explanation.  It consists of a
combination of two levers, a and b, which together give
a purchase of 200 to I; that is to say, I lb. applied to the
end of the upper lever at c will exert a stress of 200 lbs.
on the specimen at s, and as all the bearing points of the
entire lever apparatus are hard knife-edges, on hard smooth
surfaces, the friction is reduced to a minimum.
Where so much accuracy is aimed at, it will be necessary, before commencing an experiment, for the attendant to
see that the machine is nicely balanced, not only in regard
to its own members, but likewise with reference to the
various appliances which may have to be employed in
order to carry out the experiments, any of which may
disturb the equilibrium. This adjustment of the balance is
effected by the small weight d on the upper lever, which is
used in the steel-yard fashion, until the machine is accurately
adjusted; if the want of balance is the other way, the adjustc




8          On the Strength of Materials.
ment is made at the opposite end of the lever at e, by the
suspension of as much weight as is requisite to secure
perfect adjustment and to render the action of the machine
as delicate as that of a weighing-machine beam.
The testing weights used are, for convenience, of a peculiar form, to adapt them for being placed upon shelves on
the small rod of iron, which is shown suspended from the
end of the lever at c, each weight being accurately adjusted, so as to exert a definite stress upon the specimen
under operation. The plan of the weights is shown at w w;
they are round in form, with a slit extending from the outside to the middle, to enable the operator to slip them easily
into position upon the suspension rod.  Several sizes of
weights are used: the largest weighs 25 lbs., giving a
strain of 5,0oo  lbs. on the specimen-of these there are
nine; of the second size there are ten, each weighing 5 lbs.,
or giving a strain of I,ooo lbs. on the specimen; of the
third size there are also ten, each weighing I lb., or giving
a strain of Ioo lbs. on the specimen; thus making a total
of 56,ooo lbs., or 25 tons, which is the greatest stress that
can be safely exerted by the machine.
The operator is, by past experience, enabled to judge of
the effect which will be produced on the specimen, by the
respective weights as they are applied one after the other,
and so is able to load the specimen gradually up to the
required limit. As the critical point is being approached,
he uses smaller sized weights, until they are equal in effect
to a larger one; he then removes the smaller weights and
puts on a larger one as their equivalent, and continues
with the smaller size, until their aggregate weight is again
equal to that of a larger, and so on until the end is attained. When extreme accuracy is necessary, the utmost
care has to be observed in the application of the weights,
so as to avoid all rashness in the mode of carrying out the
experiments.
When a specimen is subjected to a strain, it immediately




Testing Machine.                 19
commences to stretch, and as the leverage of the machine
is 200 to I, will the stretch be magnified in the same
proportion; that is to say, a stretch of -,th of an inch in
the specimen will cause the end of the lever to drop and
the weights to sink 2 inches. Such a condition would be
inconvenient, and therefore has to be provided for, and in
this machine the effect of the stretching is compensated by
the arrangements of the machine, which admits of adjustment at the point of suspension, the fulcrum of the upper
lever being raised or lowered by means of a screw. As the
specimen stretches, the fulcrum is proportionately raised,
and this raising of the fulcrum is continued simultaneously
with the stretching, and therefore the upper lever is constantly being raised and kept in a horizontal position during
the operation. As it requires considerable power to manipulate the screw, a train of bevil-wheels and spur-gear is
employed as an auxiliary; this part is seen more clearly in
the end elevation at g. When the screw is running down,
or when the strain is not great, the spur-pinion is dispensed
with, and a handle is slipped into the spur-wheel, then the
pinion is thrown out of gear, and thus the operation of
tracing the limit of elasticity is greatly facilitated, by the
promptitude with which the change in position of the levers
may be effected.
If there were no provision for the jerk, with which the
upper lever would necessarily go down, when final rupture
takes place, it would be liable to be broken or bent at the
termination of each experiment; this is prevented by means
of a sliding stopper hI, which is made of wood and is constructed with a slit or opening through which the end of the
lever is passed, with just sufficient room to give the lever
freedom to play, and in order to have this:sliding board
always in a proper position of readiness for the fall of the
lever, this sliding stopper is moved up or down by the
same screw movement which raises the fulcrum.  Corresponding racks are placed at each end or side of the machine,
C 2




20          On the Strength of Materials.
with a horizontal spindle or shaft, geared at ii; hence the
movements of both the fulcrum and board are simultaneous,
FIG. 6.
FIG. 7.
FIG. 8.
and the lever is caught by the wooden stop, before it has
fallen a quarter of an inch.
In the Figs. 4, 5, the testing machine is shown as it
would be arranged for ascertaining the tensile properties of
materials. By referring to Figs. I, 2, 3, the form of specimen
will be seen; and in Fig. 6 is shown an enlarged view of the
holder for tensile experiments. The ends of the specimen are
held in carefully made split sockets, that fit the ends exactly,
the two halves of the socket being kept together by rings
or collars, which are slipped over and embrace them firmly
without any effort or adjustment, and which may be easily




Testing Machine.                 21
removed by simply tapping them with a wooden mallet.
Thus the change from one specimen to another is made
without difficulty.
The machine being ready for an experiment, and the
specimen'in place, we may suppose that a weight is
applied. If the specimen is of wrought iron, and is subjected to a stress of one ton per square inch, the middle
or parallel portion of the specimen will perceptibly elongate, a distance equal to the   o-,,th part of its length,
and the end of the upper lever will sink 200 times that
amount. If a proper measuring gauge is pushed between
the shoulders of the specimen and accurately applied, it
will be shown that such is the case, and, so long as the stress
is continued, it will remain thus stretched; but if the strain
is now removed, it will in time return again to the original
length, thus showing that the material is so far elastic. On
reapplying the stress, it will again stretch, and with every
addition of one ton it will take an increment of extension
in about the same proportion, and again and again return,
if the weights are removed, nearly to its original place.
This will continue up to a certain limit of load, of about
io tons per square inch for wrought iron; under that point
the specimen will return almost to its former dimensions,
but beyond it the return will not be nearly so perfect.
It has to be explained, however, that the statement in the
preceding paragraph, that each ton of stress applied to a
square inch of wrought iron will cause an elongation of
-To —th of the length, is only approximately correct.  This
amount of elongation is rather over than under the usual.
amount of extension; but it is so near that, for all practical
purposes, it may be accepted as correct. For each additional ton of strain, the bar will stretch another   I  th,
until the limit of elasticity is reached, which as a rule is found
between 8 and I 2 tons, according to the quality of the wrought
iron tested.  Hence, io tons is usually considered to be the
average limit of elasticity of moderately good wrought iron,




22          On the Strength of Materials.
and the total stretch, up to that point and with that load,
amounts to very nearly T-01 0Tth of the length of the part operated upon. It is nevertheless very difficult to ascertain the
true limit of elasticity, and published results often show great
discrepancies as to the limit at which permanent set was
first observed. In such cases, a judgment must be formed
as to the value of the results, which depends on the accuracy
of the testing machine employed, and the care and skill
of the experimenter.
Some very careful experiments were carried out by the
Commissioners, who were appointed to enquire into the
Application of Iron to Railway Structures, with especial
reference to this point, and the results are embodied in
a Table at page 58. If such a course of experiments were
repeated a few hundred times, so as to confirm the result,
the natural law might probably be inferred, and, as will be
seen by a reference to the facts in the Table, the minute
results therein shown are consistent with the more general or
approximate data given above. Still, that valuable Table only
serves for the one or two specimen bars which were experimented upon; and those who have made many experiments are always impressed with the extreme variation of
almost all the properties of materials, in different specimens,
and especially of the limit of elasticity and the point where
permanent set commences. In some specimens which were
cut transversely from a large mass, the elastic limit was
found to be under four tons of strain per square inch, while
in other specimens of the same iron, but in the form of good
rolled bar, of smaller sizes, it was found to have risen to
about I2 tons per square inch, and generally that particular
iron had a limit of elasticity ranging from 8 to I2 tons.
When the elastic limit is reached by the operator, then the
future behaviour of the bar under trial will altogether depend
on the precise nature of the iron. If it is soft and ductile,
the iron will be drawn out to a much smaller diameter in
the neighbourhood of the point of fracture, before the final




Testing Machine.                 23
rupture takes place. Although under such conditions it is
usual to consider the breaking load as so much per square
inch, calculated from the original dimensions, still, in point of
fact, the ultimate strength is really more than that; because,
from the altered diameter of the specimen at the moment of
fracture, its area may have been reduced to -ths of the original
area. This peculiarity is sometimes termed toughness; such
iron will afford good warning before breaking, and is consequently preferred for purposes where repeated tension has to
be exerted.
During the performance of the foregoing class of experiments, tle operator has to watch carefully the behaviour of
the specmen, in order to note its general character, for by
continuing to increase the weight, gradually, upon the end of
the level, the whole of the characteristics of the specimen
develop, themselves, more or less clearly, and the appearances observed will much depend on its own inner nature.
In sore, the metal will flow, or be drawn in the heart of the
bar one, thus leaving a corrugated exterior surface from the
crumping of the outer skin; in other specimens the flow
is mo'e uniform, and the outside is comparatively smooth.
If, or the other hand, it is hard and rigid, it may not be
drawm out to any great extent, but may break, with very little
rediction of sectional area, and exhibit a high tenacity. If,
on the contrary, it is of a soft and fluent nature, it will flow
fredy and be drawn out to a considerably smaller section,
ani then will break at the point where the diameter is most
rejuced. It may even now give a total strength varying
from 20 to 25 tons per square inch of the original dimensins. The testing machine is equally suited for any other
lknd of metal; and, in dealing with familiar materials, such
Us cast iron, steel, copper, bronze, or other alloys, in order;o arrive at their tensile properties, the same course is pursued as with wrought iron.
In arranging the machine to test compressive resistances,
the shackles which hold the specimens for tenacity are re



24          On the Strength of Materials.
moved, and another description of instrument is put in the
same position in the machine. This instrument is shown
in Fig. 7.  It consists of two parts, a and b, the one
sliding within the other, one of the parts being attached
to the lever, and the other part to the framing of the
machine. The specimen for this purpose is in the form of a
small cylinder; weights are applied to the end of:he upper
lever, producing a stress 200 times as great on the specimen, in consequence of the leverage, which is the agency
employed in compression. As movement takes plaze, either
from the elasticity or the permanent set of the material,
the fulcrum of the lever has to be moved, so as to keep
the proper position; this is accomplished by turfing the
handle, which gives motion to the vertical screw, shown
at g in Fig. 4.
A specimen-holder, nearly similar to that used fcr compression, is likewise used for other purposes, sich as
punching, shearing, or indenting, and for testing the hardness or softness of materials. Such a specimen-holer is
shown at Fig. 8. It is here arranged for testing theforce
required, to produce a certain amount of indentation, ad is
applied to the machine in the same manner as the specinenholder for compression.
In Fig. 4, in the side elevation of the testing machne,
are shown a row of points marked k, k, k, k, k-these tre
knife-edges firmly secured, and are used in testing tralsverse resistances. In Fig. 9 is exhibited one of the usal
arrangements of the apparatus, when employed to tet
tile transverse strength of materials.  The bar, or ro,
or small girder, is held up against two knife-edges, k, k
and the load is applied at the centre; the points of support
are some definite distance apart, which is easily measured
in this case. The distance shown is io inches; but by
looking to the points k in the machine, it will be seen that
provision is made for increasing the distance to 20 inches
or 30 inches. The bar is kept up to these knife-edges by




Testing Machine.                 25
the knife-edge contained in the holder at 7, which is so
constructed as to embrace the bar, freely, during the experiment. In this instance, as in all other cases, the weights
FIG. 9.
are applied gradually to the end of the lever; at the same
time the behaviour of the specimen is observed, in regard to
its strength, elasticity, buckling, set, &c. Experiment shows
that the strength of rectangular bars, supported at the ends
and loaded at the centre, is inversely as the distance between
the supports, and directly as the width or thickness of the
specimen, and as the square of the depth. The width of the
specimen is that dimension which is perpendicular to the
plane of flexure, and the depth is the dimension in that plane.
When a specimen is loaded transversely, it immediately




26          On the Strength of Materials.
commences to bend in a curve, which, in the case of
wrought iron or soft steel, from the change of form in the
cross section, indicates considerable movement to have
taken place, amongst the molecules composing the part
mostaffected. The respective parts of the bar under tension
and compression seem to meet, or run into each other,
but not in a line that would be indicated by a previous
knowledge of the tenacity or compressibility; this is an
element that should be taken into account in any calculation of the strength of structures, when built up of the
flowing metals.
Again, looking to Figs. 4 and 5, at the point m, there is
shown the end of a specimen which is secured to the frame
of the machine by means of two cotters; and the object of
this arrangement is to ascertain the resistance of the specimen to torsion, or to a twisting strain, like that which is
developed in the case of a shaft employed for conveying
motive power. Any form of bar may be secured by the
cotters. The bar is fixed securely at one end, or it may be
at both ends, but by having a vacant space between the fixings, sufficient room is left for a lever to be firmly secured in
the middle of the specimen. The position of this torsion
lever is shown at n; and, as will be seen, the lever terminates with the outer end formed into a segment of a circle of
some considerable range. To the lowest point of the segment, there is fixed a suitable pitch-chain, which is carried
round the segment and upwards, to be hooked on to the
lower end of the suspension rod r, by which it is connected
with the lever a, in the same manner as for tension and compression.
It will be observed that the balance of the levers may
be a good deal disturbed by this apparatus; the machine
has therefore to be adjusted by the weights d or e, after
which the testing may be proceeded with.
The point most necessary to be determined is the limit of
torsional elasticity, which will be referred to in the Chapter




Testing Machine.                 27
relating to Torsion. Experiments with bars of different sizes
show that the torsional strengths of shafts of different sizes
are to each other nearly as the cubes of their diameters,
any departure from that ratio being probably due to some
accidental cause.
In making experiments, it is instructive to observe the
deformation of ductile materials such as wrought iron and
the softer steels, and to consider the action of the molecules composing the specimen, both when under tension
and compression. The change of form which is observed
can only be readily understood, by considering the metal as
a fluid, the iron behaving in a manner similar to that of
water passing through a tube or channel of any form. When
a bar is drawn out, the principal flow of the, apparently, solid
metal, is in the middle of the stream; and hence the peculiar
sectional form which is assumed either by a round or
square bar, or one of any other shape, showing that the
farther the molecules of the material are removed from the
centre of the flowing current, so much the less are they
affected by the influence of the general movement. This
unequal flowing of the molecules, may partly account for
the apparent weakness of thin plates as compared with
round bars of the same sectional area.
With a flowing, malleable, or ductile metal, the round
bar when under tension is drawn out to a small diameter, uniformly, all round, but the metal goes in the middle
chiefly, and the outside is shrivelled; while, with a rectangular or square bar, the flat surfaces are slightly hollowed,
to an extent proportionate to their distances from the centre
of the flow. Thus the corners become more prominent than
they previously were.  When cast iron is treated in the
same manner there is no perceptible change of form,
volume, or specific gravity, whereas with the flowing
metals both volume and specific gravity are altered, the
volume being increased and the specific gravity diminished.
The strength is in some small measure interfered with by




28           On the Strenzgth of Materials.
changes of form to which the material has been subjected
during its manufacture. Good fibrous wrought iron is generally a little stronger in the direction of the fibre than transverselyto the fibre, but the difference in any case is very small,
and the strength is commonly assumed to be practically
the same in both directions. The behaviour of a piece of
iron, in this respect, is quite different to that of a piece of
wood, which is much stronger in the direction of the fibre
than it is when fractured across the grain, or in any other
direction. A bar of wrought iron when it leaves the rolls
is in a condition of great restraint; the exterior is not in
perfect equilibrium  with the interior of the bar, which at
the commencement of an experiment affects the elongation
and permanent set. The first effect of the application of a
load is to liberate the constrained surface, and true conditions on which to form an opinion do not exist until equilibrium is established in the bar itself. Previous to that
the result is deceptive; hence the advantage of carefully
turned specimens.
The structure of a rope or a bundle of fine wires does
not accurately represent the condition of materials, such as
wrought iron, or even wood. It will lead to a false conclusion, if we reason on the assumption that, in materials like
wrought iron, the fibres of which it appears to be composed
are detached and independent of each other, or that they
slide with freedom as in a rope or bundle of wires. On
the contrary, they are firmly joined together, side by side,
with a force nearly, if not altogether, equal to their general
tenacity. The idea of a flowing stream, in which the velocity of the different parts depends on the stress applied and
on their mutual adhesion and friction, represents much more
truly the condition of a ductile bar under strain.
It is very interesting and instructive to take a square
bar of iron, the larger the better, and carefully bend
it round a large mandrel, into a circle, and then to observe
the alteration of sectional form which ensues, the thinning




Testing Machine.                 29
away of the outer side and the increase of thickness at the
interior, and the curved lines that gradually form and shade
away from the one side or corner to the other. We can
scarcely realise the changes that have taken place in the
interior of such a bar, but they are strange and wonderful,
and instructively illustrate many of the foregoing remarks,
on the behaviour of a malleable, ductile, or flowing metal.
The testing machine here described may appear complicated, but it is really a simple machine; any appearance of
complication arises from the smallness of the diagram, or an
imperfect appreciation of its mechanism and manipulation.
It is easily used, little besides care and patience is required
to arrive at accurate results.
By means of the various contrivances shown in the
diagrams, combined with other expedients, almost any of
the physical properties of materials may be ascertained
by the testing machine, with sufficient accuracy for the
guidance of those who have to apply them in the operations
of daily life. Next to an acquaintance with the natural
laws of mechanics, and to being familiar with the contrivances that have been devised by men in all ages, for turning the natural laws to account, a clear perception of the
several properties of the materials that are to be employed
in construction will be found most useful, indeed the value
of such knowledge can scarcely be over-estimated.




30          On the Strength of Materials.
CHAPTER IV.
CAST IRON.
THE usefulness of the different materials by which we are
surrounded, measured by the extent of their application in
the mechanical arts, has varied greatly in course of time.
In past ages wood occupied a much more prominent position as a material of construction than it does now, having
been superseded by metal for many of those purposes for
which it was, formerly, exclusively employed. This substitution of metal for wood would even have been still more
complete, if the former possessed some of the peculiar properties which render the latter extremely valuable for certain
purposes. Amongst the metals, iron-either as cast iron,
wrought iron, or steel-occupies now the foremost place
among the materials at the disposal of the engineer. It
is not strange, therefore, that, from their great importance
in the arts, iron and steel should have been the subject
of more experimental research than has been bestowed upon
any other material. At the same time our knowledge is
far from perfect, for there yet remain many obscure points,
of great importance, which require further and frequently
repeated investigation.
Cast iron is the crude metal derived from the smelting
furnace.  The ore and fuel are thrown into the furnace
together, an intense heat is generated by means of a strong
blast of air, the refractory ore is thereby reduced, and the
iron gradually melts and runs down to the bottom by gravity.
Iron ore is very refractory, and in general is found mixed
with earthy materials. Hence, it cannot be reduced by the
carbon of the fuel alone, but requires the addition of fluxes,
capable of combining with the earthy materials of the ore
and of facilitating their fusion. If the ore is argillaceous,
or contains clay, then the flux employed has to be of a




Cast Iron.                      3 I
calcareous nature; whereas, if the ore is calcareous then
clay is required as a flux, or, what comes to the same thing,
the two sorts of ore may be mixed in suitable proportions,
so that the one acts as a flux to the other.'When the flux
or third material is required, it is thrown into the furnace
along with the ore and fuel, and at a high temperature it
unites with the earthy matter of the ore and becomes slag,
setting the greater part of the iron free.
It will thus be seen that, at the very threshold of the iron
manufacture, there are several causes in operation which
may seriously affect the quality, as well as the cost, of the iron
produced. The liquid iron having to be in such intimate
contact with the fuel and flux and their impurities, its
quality is necessarily exposed to danger and may be materially affected by contamination with sulphur, phosphorus,
or other injurious substances, present along with it in the
smelting furnace.
During this preliminary smelting process, the cast or liquid
iron has ample opportunity of combining with and absorbing a considerable quantity of carbon; this absorption of
carbon in cast iron, whether in combination with the iron
or not, is its distinguishing feature and determines its behaviour in most respects.  It is the presence of carbon
which gives to it its fusibility and enables it to be remelted
again and again, and thus renders it suitable for the founder,
the degree of fusibility depending on the quantity of carbon
which it contains.
The presence of carbon renders the iron more liquid
when in the fluid state, and softer and tougher when in
the solid state.  Still, when the iron has an excess of
carbon, it is not so strong as iron with a less proportion
of carbon; hence the practical knowledge and judgment of
the founder requires to be exercised, in the employment of
the different sorts and in mixing those of different qualities,
in order to obtain a metal with the requisite hardness, softness, closeness of grain, strength, and toughness, for the different kinds of castings which he has to produce.




32          On the Strength of Materials.
When iron contains carbon in great excess, a portion of
it may be in an uncombined condition, and this influences the
quality in the direction of fluidity, softness, and weakness.
The proportion of carbon in cast iron varies from 5 per cent.
to 2 per cent. Cast iron may be poured into moulds of any
form, and when carefully treated has considerable tenacity
and even toughness and compressibility. But at the best it
is comparatively an uncertain metal, and gives little, or,
indeed, no warning previous to ultimate fracture, which is a
radical defect.
In the preparation of cast-iron guns, where great strains
are to be resisted, the conditions to be aimed at are rather
contradictory. It might be inferred that the strongest sorts
of iron would be the best for this purpose; but guns made
of such iron fail at proof from their brittleness, and a softer
mixture stands the proof much better. On the other hand,
the interior surface of the bore when made of soft iron is
not sufficiently close in the grain. The opposite conditions
thus indicated to be desirable, can only be obtained by a
compromise, the iron used being hard enough not to be
spongy, and soft enough not to be brittle.
The best results, both in regard to elasticity and strength,
are obtained by mixing a number of different kinds of cast
iron, all carefully selected.  Such a combination gives a
higher result at the testing machine than the average of the
different samples, when cast separately, yet the strength of
the mass has seldom an ultimate tenacity exceeding 9 tons
per square inch, and much oftener it is found nearer to 8
tons. Sometimes the tenacity reaches 14 tons, and cast iron
has been produced with a tensile resistance of I5 tons, but
such a tenacity is rarely attained. The average ultimate
tenacity of ordinary cast iron is about 7 tons, and in inferior
qualities only 5 tons, or even less, but these are exceptional
qualities, and are of no value for purposes where strength
is of consequence. The quality used for guns should have a
tenacity not less than Io tons per square inch.
In submitting cast iron to the testing machine, its limit of




Cast Iron.                    33
elasticity, as shown by short specimens of common quality,
is found to be rather low, or about one-third of its ultimate
tenacity, but rather over than under; hence, in works of
construction, it is not considered safe to strain ordinary cast
iron, in tension, above 2 tons to the square inch, and even
with the higher qualities, the greatest working tensile stress
should never exceed 3 tons. For structures exposed to impact, the limiting stress should be much less, say I ton. It
must be clearly understood, however, that these figures are
only approximate.  The. student should closely study the
Tables given at page 39, containing experiments on castiron bars of Io feet in length, from which it would appear that
the limit of elasticity is much less than we have assumed
above, being under I th of the ultimate strength, and in
which a permanent set was produced with a stress of 3 of a
ton; probably it would have been detected earlier, if the rod
had been Ioo feet instead of io feet long. Still, for practical
purposes, it would be inconvenient to be governed by
such minutiae of measurement, and the approximate figures
given in Table I. are sufficiently accurate for common purposes.
The following Table gives the extensions and elasticity of
good cast iron, when the ultimate strength is about io tons
per square inch of sectional area, which is considerably above
the average for cast iron of commerce; as before stated, it
may be considered as the lowest quality admissible for guns.
The first column shows the weights applied in lbs.; the
second column gives the load on the bar per square inch of
its section, in tons; the third column gives the visible measurable stretching while the stress is acting, within one
minute of the application of the last increment of load; the
fourth column gives the permanent set when the weight is
removed, or shortly afterwards; the fifth column shows what
may be considered to be the average elastic extension of a
good specimen of cast iron.
By looking down the columns it will be seen that the
D




34            On the Strength of Materials.
elasticity is apparently perfect with 4 tons, but that with
5'2 tons there is a perceptible permanent set, equal to
the quarter of a thousandth part of an inch, and although
the elasticity continues to the end of the experiment, still
the limit of perfect elasticity has been reached at some unknown point between 4 and 5 tons, or well up to half the
ultimate strength, which is higher in proportion than with
the common cast iron of commerce.
Table showing the properties of cast iron, of the lowest
quality suitable for guns:TABLE I.
Visible stretch             Difference
Weight   Stress in tons under thetretss Permanen. set between the 3rd
applied in  per square inch  at end of half  when the load and 4th columns, 
lbs.    of section.              was removed. showing the elastic extension.
4000      2'0'     0005           -'0005
6000      3-0'001'*00I
7000      3'5          0015          --'00I 5
8000      4'0'002            -           002
10400      5'2'0025'00025'00225
I 60oo     5'8'003'      0005'0025
I2500      6'25'0035' 0075'00275
14000      7'0'004'001'003
14800      7'4'0045'00125'00325
I6ooo      8'0'0055'00175'00375
17100      8'55         007'0025'0045
17900      8'95       0oo8'003         *005
I88oo      9'4'o'0045'055
19600      9'8'0155          -
20500    Io025            -'014         Broke
From the circumstance that the ultimate tenacity or tensile strength of castings of cast iron may vary between I 5 tons
and 5 tons per square inch of section, much attention has
been given to the subject, both on the part of makers and
of purchasers of castings, in order to secure qualities suitable for particular duties-more especially has this been
the case where strength alone was the chief consideration.
It may be said that, for the majority of ornamental castings,




Cast Iron.                    3 5
strength is of less importance, as for them the chief consideration is that the metal employed shall be extremely fluid,
so as to enable it to flow like water, and fill up every ramification of the mould. In a less degree, the makers of the
great majority of castings, for lathes or machine tools, aim
rather at securing closeness of structure with a moderate
degree of hardness than great strength, because mass for
its own sake is of value in such articles, and this involves
the presence of a quantity of material which renders the use
of strong cast iron of less importance.
For beams and girders, strength and rigidity are the first
considerations. Some founders, who are very careful in
regard to the strength and practical goodness of their cast
iron for such purposes, go to the trouble and expense of first
melting the several sorts of iron, collected to form the intended mixture, and running the mixture into pigs. These
are afterwards broken into fragments, which are examined
one by one and carefully selected; those pieces which present the best fracture are laid aside for important castings,
the other pieces being kept for castings of less importance.
It is also usual, with such careful founders, to cast test bars,>
in order to ascertain, in a rough and ready manner, the
approximate strength of the metal they use for their own
guidance.
A common method is to cast in a dry mould a bar I inch
square by 54 inches in length, which is laid upon supports
48 inches apart, and weights are suspended in the middle
until it deflects - of an inch. Some sorts of cast iron are
found too rigid to deflect much; but good tough iron will do
so invariably, and return again apparently uninjured. From
hundreds of experiments made as above, it was found that a
good mixture, properly cast, will sustain a load of 620lbs.
with a deflection of half an inch, while some have gone down
the same distance with 240lbs. Founders have ascertained
that some sorts of cast iron, which are comparatively weak
by themselves, will have their strength greatly increased by
D 2




36          On the Strength of Materials.
being mixed with another sort of iron.  Hoematite iron,
for example, sustained, as above arranged, 480 lbs.; yet,
when mixed with some Welsh iron, not of a stronger
character, it carried a load of above 500 lbs.; thus showing
that the question of mixtures of cast iron is an important
subject in itself, which opens a wide field for the scientific
founder.
Other founders, who contract for castings, are less careful,
and have to be checked; hence, some engineers in their
contracts for castings insist upon specimen bars being cast
and tested, both for deflection and tensile strength. Such
bars, of a section 2 inches by I inch, with a distance of 3 feet
between the supports, are required to sustain a load, in the
middle, of 30 cwts., and to deflect, before fracture takes
place, at least'29 of an inch. A bar I inch square is
required to sustain a tensile stress of II- tons per square
inch, which is a very high tenacity.  Such a course is highly
to be commended, and helps to stem the downward course
to the cheap and worthless; the student will do well to
note the above, and he will find that, in the long run, it is the
course which will best answer his purpose-weak and cheap
material may do for a time, but its employment invariably
brings its own punishment.
It will be evident, that the mere casting of test bars does
not afford absolute security that the castings shall be of the
same quality. Some engineers, contracting for a large number of beams or girders, on which the stability of an important structure has to depend, have experienced considerable difficulty in obtaining definite security that the proper
quality of iron has been employed by the founder. In the
case of cast-iron beams of the usual form, namely, with a
vertical rib, and having the upper and lower flange in the
proper inverse proportion to the tensile and compressive
strength of the iron, they bend each girder separately by
hydraulic pressure until they severally deflect the Toth part
of their length. The pressure applied, as shown by the water




Cast Iron.                   37
gauge, is said to be about the half of the force required to
break them; such a ratio, however, must depend on many
conditions. A safer course, which is resorted to by some,
is to cast an extra beam to every score that are required,
or some other proportionate number, and then to test such
a proportion of the beams cast, selected at random from the
whole number, until actual rupture ensues; if the test beams
break under a given load, the whole number are rejected.
Tables II. and III. refer to cast iron, and are compiled
from valuable experiments made by the Commissioners
appointed to enquire into the Application of Iron to Railway Structures. The first Table has reference to the behaviour of the bars under tension, and the second Table
to their behaviour under compression, the bars being I inch
square and io feet long.
From the length of these bars, and from the care bestowed
upon the experiments, certain important data are furnished
by them, especially in regard to the elongation and compression, and the accompanying permanent set, with loads for
which they are inappreciable in short specimens. The Tables
would have been still more perfect, if the temperature had
been noted.
The actual measure of the permanent stability of any
material is the point at which permanent elongation commences, and that point should have the chief attention.
Still, if we are to trust these experiments, some judgment
will be required, because, as Table No. II. shows, this point
of commencing permanent set was only equal to I-Ioth of the
ultimate stress required to produce fracture, it is therefore
necessary to take into account the ultimate strength as well,
in order to know the margin of strength that lies beyond
the elastic limits. With short cast specimens, this elastic
limit, if in any degree observable, would appear to stand
at a much higher point than that shown by these experiments, and, as before stated, it is usually considered to lie
between the third and the half of the ultimate strength.




38          On the Strength of Materials.
As this Table shows, the iron is in some measure elastic
even to the end, and yet can scarcely be said to be perfectly
elastic even at the beginning of the experiment. Only a
few experiments were made with long bars. If a similar
course could be gone through with every description of cast
iron-hard, soft, weak, and strong, in all their varieties-engineers would then be furnished with more clearly defined
knowledge on these points than exists at present. Such an
enquiry should be made, with special regard to the relative
fitness of different sorts of cast iron for different purposes.
Precise data thus obtained would be of great advantage to
practical men.
From several hundred experiments made at Woolwich
with selected specimens of the higher qualities of cast iron,
the ultimate tenacity was found to range from Io,866 lbs.
up to 31,480 lbs., the average tenacity being 2I,I73 lbs. per
square inch. This average result, although high, is under
Io tons, and lower than the average of 850 samples, sent
in for competition, the tenacity of which ranged from
9,4I7 lbs. up to 34,279 lbs. per square inch.
It will be observed that the best of these specimens
had a tenacity over 15 tons, and the worst a little over
4 tons per square inch.  Similar experiments, carried out
on a number of specimens of the ordinary cast irons of
commerce, gave an average of 12,912 lbs., or a little over
6 tons, and some specimens of Nova Scotia iron gave an
average of I5,821 lbs., or a little over 7 tons per square
inch. The foregoing were the ultimate breaking strains,
and the specimens were only 2 inches in length. They
gave no warning of fracture, that was perceptible to the
senses with ordinary instruments; but no doubt there was
movement, if it could have been measured by suitable means,
with verniers, for instance, that would read to the Ioo,oooth
part of an inch.
Table No. II. shows the elongations and amount of
permanent set with a given load of cast-iron bars one
inch square and io feet long; these bars are of the same




Cast Iron.                          39
size as the wrought-iron bars of Table VII. A comparison
of these Tables will show a great difference, irrespective of
strength, between the behaviour of the cast and the wrought
iron; the relation of weight to extension is not nearly so
uniform in the cast iron.
Table No. II. also shows that a permanent set took place
Pyith a load of'7 of a ton, or less than Iloth of the breaking
stress, which is not in accordance with the general notion;
the set, however, is so small that it could not be measured
in a short specimen. The sixth column is the most instructive, it shows the relation of weight to extension, which
i'- far from being constant.
Synopsis of experiments on the extension of bars of cast
iron, I inch square and Io feet long:TAB3LE II.
Extensions in
Weights in                                      Ratio of
Sets in    weight to
Fractions  Fractional fractions of  extension
of an inch   parts of  of an inch.  g
entire
lbs. =w    tons.     =e.      length. 
1053'77'47'0090    -3    set not visible.   17086
I580-65'70'OI37      87.9'00022      II5I3
2107'54'94       *oi86       1'0o00545    II113309
3I6I-31  I'4I'0287      4181'00Io7      10II50
4215o08   I88'0391'0075       107803
526885    2 35       0500'00265     105377
6322-62   2-82'0613       1'00372     I03I42
7376'39   3'29'0734      1634      00517      00496
8430'16   3'76     o'0859      1396'oo664         98139
9483 94   4-23'0995     1206      00844      95316
Io053771   4-70.1136      -5~-'o1062      92762
II159148   517        I283       0       01306       90347
12646'25   5'64       I448      8        oI609       87329
13699'83   6'II'1668       7'02097      82133
14793'1    6'60'I859               024IO       79576
16664'00    743   mean breaking weight.
Apparent Limit of elasticity, L of breaking weight.
Although the property of tenacity is that which receives




40          On the Strength of Materials.
the greatest attention, for many practical purposes the
resistance which is offered to compression is also of great
importance.
Cast iron offers a considerably greater resistance to
compression than it does to extension, and the same
may be said of the other metals. In a series of experiments
made with a variety of metals, to test their properties
in this respect, the several kinds did not differ so much
from each other as might have been expected; the metals
employed were cast iron, wrought iron, and steel, and the
specimens were in the form of short cylinders. The object
for which the experiments were made was, to ascertain the
force required to shorten a specimen, permanently, to the
extentf Too th of an inch in the direction of the length. To
produce this amount of shortening required a stress which
was found to range between 30,500 lbs. and 40,700 lbs.
per square inch of area, the length of the specimen operated upon, in each case, being I inch and the diameter'533 inch. This was not the force required to crush the
specimen into a cake, or into fragments, which is not the
point which it is of greatest practical importance to ascertain. For practical purposes, we require to know the pressure at which the surface of the specimen begins to yield
or give way, and this pressure or stress is termed the elastic
limit in compression; caleris paribus, that is the best material which requires the greatest pressure to produce the
result.
Of ten specimens, cut from cast-iron guns of high quality,
the softest was found to yield with 30,000 lbs., while the
hardest specimen was able to sustain 40,300 lbs., and the
average resistance of the ten specimens was about 35,000
lbs. Of ten specimens of wrought iron, cut from forgings of
high quality, the softest began to yield with 22,800 lbs.,
and the hardest with 31,000 lbs., the average being 26,900
lbs. In each case weight was added until the specimen
became shorter, by the — th of an inch.
Table No. III. gives the compression and'sets' with Io




Cast Iron.                         41
feet bars of cast iron; the bars were prevented from bending
or buckling, by means of a case or mould in which they
were contained.
Experiments on  compression of cast-iron  bars  I inch
square and   o feet in length:TABLE III.
Compressions                 Ratio of
Weights                              Sets in    weight to
fractional  compression
Fractions of Fractional parts of an  w
r ran inch=c.  parts of    inch.
in lbs. =w.   in tons.  ncc. the length. 
2064'74     9217'01875    — 3-'00047      II0120
4129'49    I'84'03878     310'00226      I06485
6194'24   2-76       05978     ooW        o *004    I03617
8258'98'368      0.7879    1523        00645      104823
10323 73   460'09944     1         *oo847     I03819
12388'48   5 52      I12030     91        *OI0875   I02980
14453'22   6'44'14163'O405      I02049
I6517 97   7'36'I6338      4        OI712      101102
18582'71    828       i8505.-       02051      I00420
20647 46   9-21'20624'02484      I00114
24776'95  II04'2496I      48'0322        99263
28906'45  I2-88'29699      404'043         97331
33030'80  I4'72'3534I      339      o6096       93463
37I59'65   6'56'4II49      291       08421      90304
Bar much
undulated
Limit of elasticity, I of weight which permanently injured the bar.
Table No. IV. is also from the Report of the Commissioners; and from the Tables here given, together with the
other results of experiments referred to in this chapter, the
nature of cast iron may be approximately inferred, at least so
far as regards its more prominent characteristics.




TABLE IV.
SYNOPSIS OF EXPERIMENTS ON EXTENSION OF CAST IRON.
Form: Round rods or bars 50 feet long.  I square inch area of section.
Weights laid on per square inch, with their corresponding
extensions and sets.
Number                                                       Mean breaking        Mean
Name of Iron.       of                            Extensions        Sets in   weight per square    ultimate
Experi-  Weights  Weights infractions in        fractions of    inch.          extension.
ments.  in lbs.  intons.  ofaninch.  parts of   an inch.
_    of n ich.entire length            ic
21I7'94'0950     6315'00345
6352      2'83'3115'025      I6408 lbs.     I'085 inch, or
Lowmoor, No. 2.   2          105862     4 72'5740      1 4      o625   - 725 tons.           of the length.
14821     6 6I       9I47       12'12775
2096'94'09422        T'00268
N  6289  287 9    3065       1'       o01675    14675 lbs.'9325 inch, or
Blaenavon, No. 2. (  2        10482      4'67'5770       1039'0575    =6'551 tons.  8-~ of the length.
13627     6-83'8370         1'I 475
2109'94'09225     6368       00I
6328     2-82    33177        1888     I045       I6951 lbs.     1I167 inch, or
105Gartsherrie, No. 3,    2  547    647598   5862  1023   0475    -7'567 tons.  514 of the length.
14766     6'59'9452                  113251
15820    7'o6   1'0487'13812
Mixture.                         2I07'94 -0914          _5_I4'00376
Miu N ro. 3, Leeswood G~ No., Gen6322     2'82'2967                  0 1oi823    14812 lbs.'8095 inch, or
No. 3, Glengarnock| 12536                 470    85 349'04321  =6'6125 tons.        of the length.
g  12643   5'64'6702       3loa85'o64I7




Cast Iron.                            43
The following Table refers to experiments, made at Woolwich, with short cylinders of soft cast iron under compression:
TABLE V.
Length of specimen, I inch; diameter,'533 of an inch.
Compression in decimals of an inch.  Elasticity as
Weight applied                                   shown by differper square inch                                  ence between the
into tons.       Visible.       Permanent.   visible and permanent compression.
3'8'003'      0005'0025
I 1'0            004            o00I             -003
17 6.0056            002'004
20.6'oo7'0025'0045
50'0          crushed
The stress required to produce a complete crushing of
the specimens is shown by the two following Tables:TABLE VI.
Experiments on the crushing strength of Cylinders of Cast Iron, made
by Eaton Hodgkinson, Esq., for the Commissioners.
Description  Diameter  HMean crushing    No. of
Description    Diameter       Height      weight per square  Experiof        of specimen    of specimen    inch of section    ExperiIron.       in inches.    in inches.      in tons.       nients.
i6 various'75'75            39'37         48
sorts'75           1 5           38'28          48
By reducing a number of other experiments to a common
form, we obtain the following results:



44            On the Strength of Materials.
Abstract of various Experiments on the Crushing Strength of Short
Cylinders of Cast Iron.
Mean crushing
Authority.            Nature of Iron.        weight weight
per square inch.
Hodgkinson for    From various parts of the kingCommissioners  dom.38 8
Fairbairn..   Carron, hot blast, No. 2..  541'5
Do...   Carron, cold blast, No. 2..  55'975  o
No. I Iron, from various parts
of the kingdom.....  40o  8 8
No. 2,   do.    do.   do.    42'68 
No. 3,   do.    do.    do.    58'675      |
The following abstract, from the researches of Sir William
Fairbairn, C.E., Bart., shows the gain or loss of strength from
remelting cast iron:-'Abstract of experiments on the crushing strength of the
same iron, after successive remeltings:' Decreased in strength-from 44 to 40'17 tons, by remelting four times, then gradually increased from 40-17 at the
fourth remelting to 95'9 tons at the fourteenth remelting.'' Abstract of experiments on the transverse strength of the
same iron, after successive remeltings:'Increased in strength and elasticity up to the twelfth
remelting, and then gradually decreased in both properties
from the twelfth remelting until, after being remelted eighteen
times, it only possessed`ths of its original strength, and the
ultimate deflection of the bars had decreased from I'44 to'476 inch.'
The following Table contains results of experiments, on
the transverse strength of cast iron, at various temperatures,
by Sir W. Fairbairn. It may be useful to note the I Ith experiment, such exceptional results are constantly met with
in practice:




Cast Iron.                        45
TABLE VII.
These results are reduced to I inch square, and 2 feet 3 inches between
the supports.
Nature of Iron. | Temperature.    |Breaking weight Relative power of
in lbs.   resisting impact.
270             874'0           597'7
32              949-6           382'4
Cold blast         1I3              812'9          273'I
No. 2.           I92              743 I           223'7
red in dark         723'1 
red by daylight       663'3
200             8I I *7         325 0
32              919'7          395'0
Hot blast          84               877'5          369'4
No. 2.           I36               875'7          340'6
187              638'8          229'3
I 88             823'6          298'9
red in dark         829'7
Hot blast         2I20              814'4
No. 3.           600               875'8
Cold blast        2120              924'5
No. 3.           6o0              I0330'o
These experiments teach us, first, that No. 2 cast iron is
stronger at the freezing point to resist transverse strain and
impact, than at any other temperature, and that when the
temperature is raised from 320 to II30, the cold blast iron
loses I4'4 per cent. of its transverse strength, and 28'5 per
cent. of its power to resist impact. Secondly, that the No. 3,
or hard irons, are stronger at high temperatures than at
lower ones, apparently the reverse of the result with the
No. 2 iron; this, however, is probably owing to the increased
ductility of these irons at the higher temperatures.




46          On the Strength of Materials.
CHAPTER V.
WROUGHT IRON.
THE material most extensively used in the arts is the malleable, ductile, tough, fibrous, weldable material, usually
termed malleable or wrought iron. This variety of iron, at
the present time, is made directly from cast iron, by a process of elimination, whereby the greater portion of its carbon, as well as any sulphur, silicon, phosphorus, and other
impurities are got rid of, as far as may be practicable, the
change being effected by subjecting the iron, while in a hot
or liquid state, to the oxidising influence of a powerful
flame, by which the impurities, as well as the carbon, are
carried away in the form of gas or combine with the slags
in the furnace
This purification of cast iron gives the mass an entirely
different nature and new characteristics. By the process, it
becomes considerably stronger, it acquires a great degree of
toughness, yet, unfortunately, it loses the capability of being
cast in moulds.  As a compensation, it acquires a new
property, namely, the quality of assuming the viscous or
sticky condition, so that when two or more pieces are
brought together at the proper temperature, they may be
united by the welding process, as it is termed, either by
the blows of a hammer, by pressure, or otherwise.
When cast iron is thus acted upon by an oxidising
flame, and every part is exposed to its influence in the
puddling process, the newly-converted mass of viscous iron,
when removed from the furnace, may be compared to a




Wrought Iron.                   47
dirty iron sponge full of impurities, the doughy mass has
to be put under a steam hammer, or some form of
squeezer, in order to drive or wring out the mechanical
impurities which it contains. It is then elongated, by means
of rolls, into a rough bar and cut into short pieces; these
pieces are piled up into a bundle, which is reheated to the
welding point, and again rolled, so as to cleanse the iron
thoroughly. Indeed, for the better descriptions of wrought
iron, the processes of piling, reheating, and rolling are sometimes repeated several times, until the proper quality is
attained.
The quality of wrought iron thus treated depends, to a
great extent, on the original selection of the mixture of cast
iron from which it has been made, as well as on the purity
of the fuel used for the converting process. Iron has an
inherent and great affinity for sulphur, phosphorus, or other
impurities, which it has the opportunity of taking up; hence
wrought iron of high quality can only be obtained by extreme
care in every stage of its manufacture.
In the manufacture of those qualities of wrought iron
in which purity is the most essential condition, mineral fuel is
dispensed with, and charcoal made from wood is employed
as a substitute. Such wrought iron is chiefly used for subsequent conversion into the better qualities of steel, by
means of the process of cementation.
By whatever process the change from cast iron into
wrought iron is effected, the decided alteration of the whole
character for the better is unmistakable, and that the ultimate tensile strength should be increased to an average
of 25 tons per square inch, and the limit of elasticity
to Io tons per square inch, by the elimination of carbon and other impurities, are very remarkable facts.
Wrought iron is found to differ almost as much in quality
as cast iron, and this is partly due to the fact, that it
is seldom or ever entirely free from carbon or other ingredients. The amount of carbon varies from an impercepti



48          On the Strength of Materials.
bly small quantity up to 4 per cent., wrought iron containing the latter proportion being almost equal to mild steel.
The presence of a small quantity of carbon, while serving
to increase the strength, rigidity, and hardness of the material, at the same time greatly interferes with the welding
property; for this reason, it is much more difficult to weld
the stronger or more steely kinds of wrought iron, than the
softer, weaker, and less steely varieties.  Pieces of soft
iron go together, and unite into a homogeneous mass much
more kindly than harder and less pure pieces, when
raised to the welding temperature. The quality in this
respect is easily ascertained, by making the iron red hot and
plunging it into cold water, when the soft iron is found to
retain its softness, but the hard iron becomes still harder,
in a manner similar to the behaviour of steel, though in a
less degree.
The production of large masses of cast iron, by melting the
metal, is much more easily accomplished than the process
of welding similar masses of wrought iron; in the former
case the founder has simply to prepare an earthy or other
refractory mould, to provide due means for its ventilation
at the moment of casting, and then to pour out the fluid
metal, which finds its level and fills up the empty space,
in the same manner as water, when poured into a vessel.
But, in the formation of large masses of wrought iron by
forging or welding, the operations are of a slower and
more expensive nature than those of the founder.  The
process of forging is a kind of gradual building up, bit
by bit, variously conducted, to suit the individual circumstances of each particular case.  In a general way, the
operation of building up large masses of wrought iron, is
accomplished by uniting an innumerable number of small
pieces into blooms; then a number of blooms are united into
slabs and smaller slabs into larger ones, until at length the
ultimate dimensions of the required forging are attained.
When it is borne in mind that a sound weld depends not




Wrought Iroa.                    49
only upon the purity and equal temperature of the surfaces,
but likewise upon the absence of all vitrified oxide, dirt, or
impurity of any description between the parts to be joined,
it is manifest that the forging of immense masses of wrought
iron is an operation surrounded with many practical difficulties, and its application, for purposes where perfect homogeneity of mass is essential, is therefore limited.  It is, also,
too often found that the excess of heating, in proportion to
the hammering or working that can be given to the mass,
is injurious, and that, consequently, the iron of the heavy
forging is reduced in strength, when compared with the
original iron, in the condition of rolled bar.
RESISTANCE OF WROUGHT IRON IN TENSION.
The ultimate tensile strength of wrought iron is usually set
down as 25 tons per square inch, but this is above the
present general average, 23 tons being nearer as an approximate round number. At the same time, hard steely
wrought iron frequently has a tenacity of 30 tons, and inferior
kinds of only I9 tons. As a rule, square or round bars are
stronger than plates, by at least 3 tons to the square inch,
but weaker iron is not inferior for some purposes; such
iron frequently has the welding property in a marked degree, and is preferred in consequence, wherever welding
has to be extensively resorted to, in converting wrought
iron into the required articles.
Entire specimens, cut from a rough bar of wrought iron
as it leaves the rolls, are generally found to be weaker
than a portion of the same bar, which has been turned in a
lathe before testing.  If a portion of the same rough bar is
hardened by any mechanical process, such as cold rolling, or
swaging or hammering, the strength and hardness are both
increased, but by annealing the specimen, the original conditions of strength and softness are fully restored.
In the process of rolling iron into bars or plates, the
molecules and aggregates of molecules of the iron are elonE




50          On the Strength of Materials.
gated into what is usually termed fibre. It might, therefore, be expected that the bar or plate -would be stronger
vhen drawn asunder, with the tension in the direction of the
fibre, than when drawn asunder, with the tension at right
angles to the fibre.  There is, however, comparatively
little difference, unless the mass is thin, as in plates; the
chief difference, shown in experiments, is that the elongation
of the bar by the strain is greater in the former case than in
the latter, and an earlier warning is given of the impending
fracture.
The strength of wrought iron, as given in the older tables,
ranges higher than would be the case if similar tables were
now to be compiled, from specimens taken at random from
the iron of commerce. Wrought iron has been quoted to
have a tenacity of 34 tons, but unless the experiments giving
that tenacity were inaccurately carried out, the iron must
have been hard and steely; we still find wrought iron occasionally with a tenacity of 30 tons, and the writer has some
specimens of 32 tons, but such iron is hard and almost
unweldable, and is much more brittle than iron of a lower
tenacity; hence, in selecting iron for any particular purpose,
the peculiar strains to which it will be exposed must be taken
into consideration, before determining the quality which
should be employed.
The strength of wrought iron is not much affected by
variations in temperature, when under 350~ Fahr.; above that
temperature, it begins to lose strength, and as it approaches
to a dull red heat, the ductility greatly increases, and the
flowing property comes into play and reduces the resistance
fully one-half; hence the opportunity for the smith to'strike
while the iron is hot.'
There has been considerable difference of opinion, in
regard to the strength of wrought iron when exposed to severe
frost, but, from recent investigation, it would appear that practically it is not much, if in any appreciable degree, affected
by the lowest temperature of an English winter. Never



Wrought Iron.                  5I
theless, there exists a popular notion that iron and steel are
greatly affected by frost, and thereby rendered more brittle
than when at an ordinary temperature, and this notion receives some support from the fact, that the fracture of rails
and railway axles is most frequent in winter. But the fact
is susceptible of another and more probable explanation, for
in winter the roads are hard and rigid, causing great jar.
The notion that the strength of iron is less in cold weather is
not borne out by the general average of the experiments made
in summer compared with the average of those made in
winter; these experiments, however, are performed within a
building.
That wrought iron is to some extent more ductile in
warm weather, than in extremely cold weather, seems probable, even, although the tenacity is not much affected.
This effect of the presence of heat may also, in some degree,
justify the general belief which exists amongst practical
men, of the greater liability to fracture in winter than in
summer. It is frequently found, when guns are being proved
to destruction by continued firing, that the first round in
the morning, when recommencing the experiments with a
cold gun, proves fatal to its endurance.
The appearance which is presented by a fracture of
wrought iron depends greatly on the mode in which the
rupture has been effected. If it is accomplished suddenly,
the fibre is actually broken short, and the crystalline texture apparently predominates; whereas, if the fracture is
produced by a slower process, the fibre of the bar is then
made conspicuous, because the element of time is essential
to enable the fibres to be drawn out from each other. All
such appearances, however, are greatly modified by the
quality of the bar which is operated upon; some judgment,
as well as experience, being necessary to arrive at a just
conclusion, and opinions formed from the fracture only are
not always to be depended upon, even when the examination is made by an expert.
E 2




52          On the Strength of Materials.
From many fractures of wrought-iron axles, chains, and
other pieces, which have been in daily work for a number of
years, the idea has become general, that with sufficient
intensity of jar, repeated an indefinite number of times, a
change takes place in the structure of the iron. This is not
inconsistent with the results obtained by the testing machine.
It is satisfactory to know, however, that such a result may
in a great measure be anticipated and prevented by simply
subjecting the axle, chain, or hook, from time to time, to
the process of annealing, by which its original condition
is practically restored.
At the same time, it may be stated that this is an obscure
subject, for there is not much positive evidence of the fact,
or explanation of the cause of such deterioration in wrought
iron, after being exposed to long-continued working, as in
the axles of railway carriages and in crane chains; still,
if such is the case, it is now generally considered to be due
to the effect of vibrations often repeated, when the part
affected is for the moment loaded up to or beyond the
elastic limit. For it has to be remembered that, in the case
of the crane when surging, or in that of a carriage jolting,
this element of motion suddenly checked produces a stress
which has to be added to the ordinary stress existing
when the several parts are in a state of rest. Cranes that
are usually worked with less straining than that for which
they were originally intended, seem to maintain their efficiency for an indefinite period, whereas other cranes, which
are much employed up to their maximum ability, do sometimes give way unexpectedly, and with a less load than that
to which they have been usually subjected. In some cases
there is an excess of deflection on a particular part, which,
being often repeated, has ultimately caused fracture.
Whatever may be the explanation, the phenomenon
is now' familiar to those who are engaged in practical
operations, that when a piece of wrought iron is thus subjected to a long-continued series of blows, or violent jars,




Wronzght Iron.                 53
of sufficient intensity, or that call the full elasticity of the
material into active exercise, rupture will sometimes take
place prematurely, and must be expected sooner or later.
The change to rigidity, which overtakes iron when worked
cold, may partly account for some of the frequent fractures
of the chains of cranes or other iron-work similarly exposed, and this view is in some measure supported by
the fact that when such chains are annealed, at stated intervals, say annually, the liability to accident is greatly
diminished.
Considerable difference of opinion and practice exists
with regard to the safe load which may be put upon ordinary
wrought iron; this of course must depend greatly on the
nature of the structure, and the kind of stresses to which it
is exposed. A crane, for example, with the risk of a load
being suddenly checked, during lowering, by a rash attendant,
must necessarily have a larger margin of safety than would
be required for a structure with a steady load; for the latter,
a strain of 5 tons might be ventured with safety, while for the
former 2- tons would hardly afford the same measure of
security; some judgment has therefore to be exercised in
determining the strain that wrought iron may be entrusted
with, but in no case should it be burdened with a load'producing a strain of more than 5 tons per square inch of
~sectional area.
Wrought iron is frequently found to present peculiar
phases of character, which must not be overlooked, as the
conditions affect its strength; some descriptions of wrought
iron, otherwise good, are found to be intractable and apparently brittle when red hot, yet are perfectly pliable when
cold-this peculiarity is usually considered to be due to the
presence of a small trace of sulphur in the iron. Then
there are other descriptions of wrought iron having the opposite quality, being comparatively brittle when in the cold
state, and perfectly pliable and workable when red hot. This
quality is said to be due to the presence of silicon or phos



5 4         On the Strength of Materials.
phorus. These conditions are so marked as to suggest the
idea that the gas of the impurity forms a coating, which
envelopes or separates the molecules of the iron, and they
greatly diminish the value and usefulness of the iron in
which they exist, for many practical purposes. These defects
are mostly due to the impurity of the original ore, and the
ironworker usually corrects them in the course of manufacture, by combining ores of opposite nature in suitable proportions, so as to obtain an average quality; of these two
defects the'red shortness,' as it is commonly termed, is
the less objectionable.
In wrought iron, and especially that of high quality,
there is no fixed point at which permanent set begins to be
observed; for although we speak of the limit of elasticity as
somewhere in the vicinity of 28,0ooo0 lbs. per sq. in., or about
half the ultimate tenacity, yet, at the same time, it has to be
clearly realised by the student that such a statement is only
approximate; and although round numbers may be easily
remembered, and are sufficiently correct for most practical
purposes, still they must be considered as approximate
only, and taken with reference to the exact truth contained
in the following Tables. The student is referred more especially to Table VIII., which throws more precise light on
the facts than any general statements.
It has also to be observed that in making experiments,
whether with long or short specimens, there is a marked
difference between the behaviour of cast iron and wrought
iron. It has been already stated that the former has comparatively little perceptible elongation, unless the bar is of
considerable length, while the elongation of the latter can be
seen distinctly, and increases even to the last moment. The
final stress which is required to produce rupture must not
be calculated from the original dimensions of the specimen.
The final stress is the actual strength of the reduced area
after stretching. This stretching is one of the distinguishing
features of wrought iron as compared with cast iron, and is




TWroulght Iron.                  55
one of the special virtues of good wrought iron and mild
steel.
When wrought iron is wire-drawn and its section reduced,
the strength of the part so elongated is increased by
the process. Thus, an iron wire, when made with iron of
a strength of 25 tons, will have its tenacity increased to
35 tons per inch of sectional area by the mere process of
drawing; and the most remarkable feature of all is this, that
the specific gravity is actually reduced at the same time, showing that the conditions which give strength are not solelydependent on the closeness of the molecules.
In the former part of this chapter attention has been drawn
to different qualities or natures of wrought iron. When a
variety of specimens of different kinds of wrought iron are
under great tension, their respective behaviour will greatly
depend on their own inherent hardness or softness; a
hard specimen will be found to elongate very little, and
will ultimately fracture without reduction of diameter, while
a softer specimen will be drawn out considerably, the
middle part becoming gradually smaller, and fracture will
ultimately take place at the smallest section, most probably
at a lower strain than was the case with the harder iron. It
might then be inferred, that the hard iron was the better of
the two; but such an inference might not be correct, as
the latter would, practically considered, be much more
reliable'when subjected to jar or sudden strain.  At the
same time, in estimating the original strength of the elongated specimen, it will be objectionable to take the reduction of diameter into consideration because for practical
purposes the elongation should never be called into exercise. The advantage of soft iron is, rather that it can be used
with greater safety, than that a higher stress can be permitted
where it is employed, for the softer iron is more likely to
be drawn out than to be broken asunder, and gives ample
warning before ultimate fracture.
In experiments made at Woolwich with ten short speci



56          On the Strength of Materials.
mens, cut from heavy wrought-iron forgings, the average
apparent limit of elasticity was only 23,760 lbs., while the
average point of ultimate fracture by tension was equal to
48, 60 lbs. The forgings from which the specimens were cut
were all intended for gun purposes, and consequently were
of high quality, or intended to be so. Similar experiments
were made with rolled bars of the same high quality, both in
the state of bars and when wound into spiral coils, and then
subjected to the welding process, in order to form them
into entire cylinders for gun manufacture. All the specimens
were found to have suffered from forging, to the extent of
3,48i lbs. per square inch, on an average. The results were
as follows:
Limit of elasticity CylindBar. 3,000 lbs.
VCylinder..   27,852,,
Ultimate nrupture CyindeBar.58,986
Cylinder... 55,500,,
No doubt, the loss of strengtn here shown was owing to
the absence of sufficient working, to counteract the weakening tendency of excessive heating.
From some interesting experiments, that were made to
ascertain the tenacity of the welds of different qualities of
iron, and with differently formed welding surfaces of contact,
it was found that with soft iron of the finest quality; heated
in a clear fire, with scarfed joints, and both surfaces slightly
rounded, the strength of the welds was equal to that of the
original bar, or in round numbers about 25 tons per square
inch; while with other descriptions of harder iron, variously
selected, the strength of the welds was always less than that
of the bars, the lowest and worst example, which was an
exceedingly hard and steely specimen, having a tenacity at
the weld of only 12,000 lbs. per square inch. A still lower
average result was obtained in all attempts at butt-welding,
with similar qualities of iron, even when the surfaces were
rounded to give a clear way of escape to the vitrified oxide 




Wrought Iron.                   57
the average ultimate tenacity was 32,I4olbs., or a little over
half of the strength of the iron. With turned butt surfaces,
when heated in a furnace flame, it was a little higher than
the above.
Some experiments were made with bar iron of a very hard
steely quality, its ultimate strength being about 30 tons,
in the specimens cut from the bar. It was, of course,
difficult to weld such iron, and impossible to secure butt
welds with a tenacity greater than Io,ooo lbs. per square
inch, the steely property, due to the presence of carbon,
being the barrier to a ready union of the pieces, when
manipulated by the smith in the ordinary manner. It will
thus be seen that the strength of a wrought-iron structure
depends on many conditions, all of which have to be taken
into consideration in estimating the reliance to be placed
on it, or in calculating the requisite quality or quantity of
material to be employed.
Table VIII. is a synopsis of the results of an experiment
made to show the extension and permanent set of a rod
of wrought iron 50 feet long, and which have been reduced
in this Table so as to compare with the other experiments,
of a similar nature, but made upon a rod of cast iron io feet
long, as given in Tables II. and III.
Column No. I shows the relative value of the successive
weights that were used in making the experiment, and is
inserted for comparison with the relative value of the extensions that were produced by those weights, as shown in
column No. 4; by means of this comparison will be seen
the departure from a constant uniformity which might be
expected.
Column No. 2 gives the actual weights in lbs. which were
applied per square inch of section of the rod.
Column No. 3 gives the actual weights applied per square
inch of section of the rod in tons.
Column No. 4 shows the relative value of the extensions
produced by the successive weights, and teaches when




TABLE  VIII.                                           -          m
00
I.          - 2.      3.         4.               5.              6.         7.             8.
Relative'                     Relative            Extensions
value of the    Weight applied per   value of the                             Permanent    Ratio of the
weights used, square inch of section   extensions                                sets on   weight applied     Q
when the                          produced                           In      io feet of   per square inch    X
smallest is                       when the         On Io feet      fractional  length in      to the
represented                       smallest is      of length in     parts of   inches.      extension.
by unity.    in lbs.    in tons.  represented       inches.        parts of      hes.      extengthsion.     %
by unity.                      the length.
I261 I8     *56       I'       0052            23076   perceptible   242665'4
2         2523'7'I2        2'2'OII5            10434                 219453'9
3         3785'6'69 369    32'o69                1:0005       223998'2
4         5047 4    2'25        4 3'0224                       oQ'0006    225331'7
5         6309'3    2 81         53'02772           43'   ~    0005       227607'8
6   757'I    338           6'3'03298           a'o?'00045  229567 9
7         8833'0    3'94         7 2'0379            3-16'0005       233060'9
8        Ioo0094'8    4-50       82'043               790      000oo5     2347642 
9        II356'7    5'07         9'3'04854            -4                   233966'2  0 
10        I26I8'6    563        o103'0537             2048'0007       234982'8  O0;'
I I       Ix3880o4    6'I9      I'4'0595           -0                    2332847  
12        15142 3    6'76        12'4'      0648            1851                  233677'5   r
13        16404-I    7'36       13-4'o698             1719                 235016'5   w
14        17666 0    7'88        14-4'       0753              593'ooI3        234608'4 
15        I8927-9    8'44       15'7'0817              -1_                 231675 2
I6        20189 7    9'00       i6 8'0874             3'0027      23 003'8
1 7       21451 6    9'62        17'9'0931             128                   2304145




i8      22713'4  1O'1 I9 0'0992           1209       0041    228966'I
19      23975'3  10~70        20'3           1o57           1-         -        226824
20      25237'1  11'26        2I'6            1125          1'00oo68    2243303
21      264990   I I 82       231'1204                              22009'5
22      27760'8   12'38       24'7'I288            I        O012     2I5534 7
23       29022'7   I 3'04     27'9'I45                        _       200156'7
24       30284'6  13'52       38'3           199I             1           
-       13 52     38'6'2007      1
after 5 minutes  f                       1508948
I 3 52     388'28 0                218    0736
after 10 minutes                 736
13'52      395            2054             1        0774 
after 15 minutes                 774
same weight 13 52       40-0'208                               0796        -
repeated                       after 20 minutes      577       f796
13'52      403            2096            1
after i hour   3    572 
- -.2366                       -         _sameweight I3.52        45 5    after bearing the                082 
repeated                        weight 17 hoursf I
after 5 minutes'
same weight 14'08       47'I'2449           489'III 
repeated                       after 5 minutes  J
26       32808o   14.72   1.059'5506, 17'4141
-  sameweight I4'72       135'7024            1
repeated                       after 5 minutes       170'5635    46709,,     14'72      153'I'7966             1'6558
after Iominutes  j    150
14'72     195           1 014             1        *866
after 15 minutes f    -7 




r.          5 1 2.                4-               5-        u                 7-            o.
Relative                          Relative             Extensions
valueight applied                 value of the                                 Permanent    Ratio of the
aeigh ofh  per square inch of    extensions                                  sets on    weight applied
whed              section          produced                            In     io feet of   per square inch    Q(
smallest is                        when the         On Io feet      fractional  length in     to the          I
represented                        smallest is     of length in      parts of   inches.      extension.
by unity.    in lbs.    in tons  representedy.      inches.        the length.
27        34070      15'20      258'8             I' 346    }                                            " 
after I minute         -89 
-  -'20  269 2           I4  
after 2 minutes   J          -8    _ 
I5'20     307'7            I6                  L       I 44          21I294
sameweight  I5'20       3173              1.65               I 
repeated                          after I minute           7'-  -         -         343'4           I1786        }                628
after I hour.
28        05332       15'76     392'3            2'04                         874           320
after 5 minutes1
sameweight  15'76       4192            2'18                          - 201
repeated                          after 5 minutes          55
-        rI5 576    433 4            2-254                1    2 08
29        36593'8    16 33      4884             2'54
after 6 minutes         474407
30        37856       6 90      556 5            2894               41    loop broke       13081




Wrought Iron.                   6r
compared with column No. I-first, that the elasticity of
wrought iron as here shown is not so perfect as is generally
assumed, even when the strains to which it has been
subjected are small. If the material had been perfectly
elastic, the relative value of the extension would have
corresponded, exactly, with the relative value of the weight
applied, that is to say, twice the first weight applied would
have produced twice the extension caused by that weight;
three times the weight, three times the extension, and so
on; but this is not the case.  Second, that there is only
a slight and gradual increase in the relative value of the
extensions over that of the weight applied, until that weight
exceeded twenty times the first weight applied, and when
the material was subjected to a stress of II'26 tons per
square inch, or about one-half the ultimate strength. Third,
that after this point is reached, the value of the extension
produced exceeds the relative weights very rapidly, until
when the rod is subjected to a stress of I6'33 tons per
square inch for six minutes, it is stretched seventeen times
more than it would have been if the material had been
perfectly elastic.
Column No. 5 gives the actual extensions, and shows very
clearly-first, that the length of time during which the material is subjected to a stress, has a considerable effect upon
the result. To take an example from the Table, with a weight
of I3'52 tons, or rather more than one-half the breaking
weight of good wrought iron, the extension on io feet of
length increased from'I99I inch, when the weight was first
applied, to'2366 inch, after the weight had acted upon the
rod for seventeen hours. Second, that by removing and
replacing the same weight the extension is increased. (This
is not observable to the same extent in testing short specimens.)
NOTE.-In each of the above cases it will be observed that
the weight is more than one-half the ultimate breaking load.
Column No. 6 gives the extensions produced in fractional




62         On the Strength of Materials.
parts of the length, and will be found to be useful for ascertaining the extension of wrought iron of any length by the
application of any of the weights, per square inch, given in
the Table.
Column _No. 7 gives the amount of permanent set produced by the successive weights, and teaches-first, that
with so small a weight as half a ton per square inch, a perceptible set was given to wrought iron; second, that the
amount of set was very small, until the material had been
strained up to one-half its ultimate strength, or about i I-26
tons per square inch, at which point the set was 0oo68 or
2 —rds of the extension; third, that the set rapidly increased
after the stress had reached II 26 tons per square inch, or
one-half the ultimate strength, and when it reached 14'o8
tons or about 5ths of the ultimate strength, the set amounted
to'Jo83 inch or +ths of the extension, and when strained
to 15'76 tons per square inch, or less than 4 of the ultimate
strength, the set was nearly 9 fths of the extension produced
by that weight.
Column No. 8 only shows a comparative result, and the
numbers there given are obtained, by finding the ratio of
the weight applied in lbs. to the extension produced in
inches, and had the numbers in this column been equal,
then the material would have been perfectly elastic; but
although not equal, yet the numbers in the first twenty observations are nearly so, and therefore show that the material
was near to perfect elasticity under strains less than II'26
tons per square inch; hence, we infer that the usual expression of' the limit of elasticity' is admissible, and in the case
of wrought iron is almost correct.
The mean of these twenty numbers is 230,760, and consequently the extension of this bar of wrought iron in inches
may be taken at Ro-o-, th of the weight applied in lbs. to
stretch it, and the weight applied in lbs. to produce any
given extension would have to be 230,760 times the extension in inches. Now the modulus of statical elasticity




TWrought I-olz.                63
is generally understood to be the weight or force in lbs.
which would stretch a bar to double its length, if its elasticity
remained perfect, and as this Table shows the extension on a
bar I20 inches long, we obtain the modulus when we find
the weight required to stretch this bar 20o inches, namely,
230,760 X I 20= 2 7,691,200 lbs.
NTE. —This number is variously stated in books upon
the subject, and would be greater or less than that given
above according as the material employed was more or less
elastic.
It would have added to the value, and probably would
have modified the results, if the expansion and contraction,
due to the differences of temperature, had been taken into
account, which does not appear to have been the case,
judging from the description given in the Blue-book in
which they are recorded.
RESISTANCE OF WROUGHT IRON IN COMPRESSION.
In the manifold structures for which wrought iron is now
so extensively applied, the property of resisting compression
is frequently called into active exercise. Whenever wrought
iron is thus subjected to compression, in either large or
small structures, a permanent deformation is produced, if
the load exceeds a certain value. It may be seldom that
the liability of this metal to give way by compression is
observable, as compared with failure in tension, because
the failure in the former case is more likely to occur
from flexure or want of stiffness, which again may be due
to want of fixedness.  In structures subjected to thrust,
much depends on the mode of attaching the several parts,
such as struts or stays, to the adjoining parts. A great
improvement has taken place of late years in this respect;
the extensive employment of the lathe, planing-machine, and
other refined t6ols, extensively used in modern workshops,
permits the parts which have to be joined to be truly faced at
the ends, so as to take the correct position firmly; and this




64           On the Strength of Materials.
can be done at such a moderate cost as to render it commercially practicable. The sound and uniform bearing at
the connection, thus obtained, adds greatly to the strength
of the whole structure. By secure fixing, the work to be
done on the material, in order to cause it to deflect, is
considerably more than doubled, without increasing the
weight.
Referring to page 40, there are quoted certain experiments
with wrought iron and cast iron as agents to resist compression, from which it will be seen that the stress necessary
to shorten certain specimens of good wrought iron an
amount equal to _x3, ths of an inch was, for the softest
22,800 lbs., the hardest 31,000 lbs., the average being
26,9oo lbs.
The foregoing experiment was made with specimens cut
from forgings. From experiments made with ten other specimens taken from rolled bar iron of high quality, the specimens having been reduced in a lathe from 3-inch bars, the
softest specimen required 31,000 lbs., and the hardest
35,000 lbs., or an average of 33,000 lbs. It will be observed, that the iron of the bar was considerably superior
to the iron cut from the forging, although the two were of
similar quality originally, thus showing the effect of better
working.
Table IX. contains the results of a number of experiments carried out at Woolwich with short cylinders of
wrought iron under compression.
Structures scarcely ever fail practically from the actual
crushing of the material; failure is more often due to the
alteration of form which takes place, disturbing its fitness for the particular purpose for which it is intended.
When the pillar, strut, or frame, is long, it generally
yields by flexure rather than actual crushing.  If properly stayed or trussed at intervals, a long rod will act
as a pillar, taking the stress through the entire length, the
elasticity being uniform throughout. But in order to save




Wrought Iron.                        65
TABLE IX.
Diameter of specimen,'533 inch; length, I inch.
Elasticity as
Compression     shown by the
Weight    in decimals of an inch.   difference
Nature of Iron.    applied in                   between the
tons per                       visible and
square inch.   Visible.   Permanent.  permanent
compression.
7'8'0015                 I 0015
I 6'0025      -'oo0025
13'4'0045       00oo3     oo 15
34        0055       0035'002
15'8       0075.0055'002
500         -         245         _
I I 0      002        -          -
12'4      003'0005'0025
14'~'0045'002'0025
Specimens cut from    15o0       0065      0035       0025
bar iron, varying    i6o       *0075     005        0025
50.0        -        -26         -
from' to 2~
2inch0     005                  oo 0015
inches square.      13.2'0025      0005'002
14'o'004  oo      I 5'0025
15*0       *oo6'0035'0025
I6'o      o008       *oo6   - 002
50.0        -  -245              -
14'4'0025       00oo5'002
14'8       -'007
500         --        259
II*8                 003
I 2'0'0035
12-6.         004
13'2                  0045
50'0'2655
Marshall and Mill's   14'4.0005
Irons, soft and of    1556    not noted  0025
15'6'       004
fine quality.      i6' o                 0055
50'0'         2615
I0'0'         0005
I I'O'0025
I 6                  oo45
50o0        -         286
F




66             On the Strength of Materials.
TABLE IX. —conttinued.
Elasticity as
Compression     shown by the
Weight    in decimals of an inch.    difference
Nature of Iron.   applied in                        between the
tons per                         visible and
square inch.
Visible.   Permanent.  permanent
compression.
I I 8'002
I 2 4'0035
13'2                  0055
Taylor Brothers'      50-0'2775
Yorkshire Iron;    130       not noted     015
in the direction    140                    004
50'0                   287
of the fibre.
14'o0'001i 5
14'5:004
50-0.249
I0'0'00I
I 1I 4'004
50'0'         318
I0'0'00I
II'2'004
50'0.         3105
S. C.  Iron.
II 8                   oo005
Specimen  cut       12'8     not noted'004
from  a welded      50-0                   *288
coil.               12'0                  -0005
12-6'0035
50'0'2845
12'6'0005
I4'2'003
14'8                   0045
500'2565
the expense and trouble of trussing or encasing long bars,
when making experiments to ascertain the resistance to a




Wrought Iron.                   67
crushing force, it is more convenient to deal with short
specimens, otherwise deflection will commence before the
molecules begin to flow.
By increasing the stress upon these short cylinders of
wrought iron or soft steel, they are found to shorten gradually by bulging outwvards in the middle. The effect of this
change of form  is to slightly stiffen the metal, and this
affects the malleable or flowing property; unless the specimen is extremely soft, it will soon show symptoms of slight
fissures or cracks at the part which is bulging. To prevent
this, the annealing process must be resorted to, and with
care the pillar can be flattened down to a thin disc, gradually presenting a larger surface for the machine to act
upon.  Reckoning the intensity of the ultimate pressure
from the original dimensions, a stress of upwards of 1oo
tons per square inch is necessary to actually flatten down
wrought iron.
WVhen wrought iron or steel is flattened by compression,
it might be supposed that the specific gravity would be increased; but such does not appear to be the case to any
appreciable extent. The specimen either cracks and splinters, or finds relief by lateral yielding, the enlargement
commencing in the middle of the cylinder.
1 2




68         On the Strength of Materials.
CHAPTER VI.
STEEL.
THE material termed steel is a nearly pure alloy of iron with
a small portion of carbon. As a metal, it is closely allied
both to cast iron and wrought iron, and may be made from
either.
The most common mode of manufacture is to convert
pure wrought iron into steel, by measuring out a definite
portion of carbon for the iron to absorb; or, it may be made
from cast iron by a reverse proceeding, namely, by the elimination of the carbon and impurities, allowing so much
carbon to remain as will give it the required steely qualities.
Although steel thus forms a connecting link between
wrought iron and cast iron, it differs greatly from the latter,
cast iron being a crude, indefinite, and impure alloy, while
steel is a purified alloy, containing a definite percentage of
carbon.
Steels differ from each other in many respects, but more
especially in regard to the degree of steeliness, and, within
certain limits, any degree of steeliness may be obtained.
In making fine steel by the cementation or common process,
bars of pure wrought iron are enclosed in a fire-clay box,
and surrounded with powdered charcoal. The whole mass
is then kept at a red heat for a certain period; the porous
molecular structure of the iron is opened, and the carbon vapour finds its way into the body of the iron. The
bars of iron are subjected to this process for a period generally ranging from five days to a fortnight, or even longer,
according to the quality required; the longer the process is
continued, the more steely does the bar become.




Steel                      69
As some of the bars have a greater opportunity of
absorbing carbon than others, and from other contingencies,
the steeliness of the batch varies. and uniform quality can
only be obtained by breaking the bars into small pieces,
sorting these fragments into lots, judging by the appearance
of the fracture, and then melting the lots in a crucible.
The liquid steel is cast into an ingot, and hammered or
rolled into the cast steel of commerce. In casting very large
masses, a great number of crucibles are necessary, and as
these must all be ready for pouring at the same time, and
must be emptied into the mould consecutively, the successful
casting of heavy ingots requires a very good organisation.
In the arts, cast steel is required of all degrees of steeliness. The milder sorts of steel are only a little more steely
than the harder varieties of wrought iron, and the mildest
quality may contain about ~ per cent. by weight of carbon.
The highest qualities contain as much as I per cent. of
carbon, and there are many intermediate qualities. As compared with hard wrought iron, mild steel, while not containing much more carbon, is yet more perfectly homogeneous
in its granular structure, and is superior to it both in
strength, and in almost every other good quality.
A. good serviceable quality of steel, for many purposes,
is now extensively made by the' Bessemer' process, which
appears to be, at first sight, a reverse method to the cementation system described above. The Bessemer process commences with cast iron in the crude state, which is melted
and poured into a vessel, and, while liquid, a strong current
of air is passed through it.  The carbon in the iron is
burned out, by the oxygen contained in the air passing through
the molten mass, and the other impurities are gradually
eliminated, until at length the iron is in condition of comparative purity, and chemically similar to wrought iron. In
order to make it into steel, there is added to this purified
metal, a measured portion of a pure cast iron, generally that
called'sfiegeleisen,' or'looking-glass iron,' containing a




70             Strength of Materials.
large and definite quantity of carbon, which converts the
pure iron into a steel sufficiently good for an immense
variety of applications. From the circumstance that steel
may be produced by this process at a less cost than by the
former method, it is, to a large extent, taking the place of
wrought iron.
Steel, like wrought iron, is much improved in its nature
by being thoroughly worked either under the hammer or by
rolling. Like most metals, steel is found to be more or less
porous after casting, which is due to small air or gas bubbles
which have not been able to find their way to the surface;
the effect of hammering or rolling is to consolidate the
mass, and to render the grains of the metal finer, denser,
and stronger.
During the last few years, Sir Joseph Whitworth has been
engaged upon a course of most valuable although most
expensive experiments, conducted with powerful apparatus.
These experiments have for their object, the attainment of
steel of great density and general goodness, by combining
the best materials and the most thorough working, and using
every care that the best skill can devise. In Sir Joseph
Whitworth's system, after the liquid steel is poured into a
metal mould of sufficient strength, a piston or plug is inserted, upon the top of which there is at once brought to
bear the full force of 800ooo tons of hydraulic pressure. The
effect is at once perceptible, a corresponding pressure pervades the liquid steel, the porosity is overcome, and the
metal shrinks rapidly by the sudden closing up of its pores.
The shrinking continues for some time at a rate perceptible
to the eye, and afterwards more slowly, for a period of
nearly a quarter of an hour. Some of the shrinkage ob-.served is due to the decrease of temperature, but it is
mainly owing to the hydraulic pressure.
In considering the effect of pressure, as compared with a
blow, in the consolidation of a mass of steel or iron, it would
appear that the former should be the most effective, because




Steel.                      71
the metal acted upon has not an opportunity of lateral yielding, whereas in either hammering or rolling, the metal flows
away from the point of impact of the hammer, or squeeze of
the rolls. Such is the view taken by Sir J. Whitworth. Hence
when the ingot, which has been compressed in the liquid state,
is taken out of the mould, it is brought to a working temperature, and is then subjected to a series of squeezing operations
by hydraulic pressure, squeeze after squeeze being applied
until the required dimensions are arrived at; this hydraulic
pressure does not act without control, a dial and pointer
shows the amount of compression of the mass, and so prevents any risk of over squeezing and making the article too
small. The whole process is carried out with such ease,
and is so gentle in its action upon the tools that are employed, that on seeing the process performed, it is impossible not to feel that it is a great step in the right direction,
and it is encouraging to find a man with the courage to go
into such gigantic enterprises, for the purpose of advancing
practical science.
The goodness of steel may be said to depend on three
things: first, the materials selected; second, the nature of
the working to which it is subjected; and, third, the care
taken by the makers. These three conditions apply to all
descriptions of steel, from mild to high, and each quality is
equally good for its own special applications.
The quality of fine cast steel suitable for cutting tools,
contains a larger percentage of carbon than either the milder
varieties of cast steel, or the varieties of Bessemer steel,
which are taking the place of wrought iron, or even the fine
cast steel that is used for the lining of guns. The gun steel
contains about'03 per cent. of carbon, and in its natural
state has an ultimate tenacity of from 30 to 35 tons, but
when made red hot and cooled in oil, its ultimate strength
rises to 40 or 45 tons per square inch. The apparent elastic limit of short specimens rises in an equally remarkable
degree, being for untempered steel specimens 13 to 15 tons,




72          On the Strength of Materials.
and for steel tempered in oil, 28 to 32 tons per square inch.
The toughness is also increased, so that the tempered steel
may be bent, twisted, or drawn out to a degree, far beyond
that to which it would submit in the untempered condition.
It will thus be seen that the limit of elasticity of mild cast
steel, when tempered in oil, is fully equal to three times that
of wrought iron. This is an important consideration, which
will in time determine the extensive use of that material, and,
more especially, because it stretches so much before final
rupture takes place, which is a very valuable and important
property.
The marked increase in the tenacity of cast or Bessemer
steel, as compared with wrought iron, has already led to the
application of the latter for many purposes where iron was
formerly used, and wherever strength requires to be combined with lightness. In the case of rough structures, such
as girders or bridges, the use of steel will gradually advance,
the chief difficulty in the way of its more general application
being, not so much the difference of cost, as the fact that
the engineer is unable to determine whether good steel has
been really employed by the contractor; testing would settle
the point, but the trouble and expense of testing every bar
or plate is a serious practical barrier to the adoption of this
plan. It has been suggested that the specific gravity of a
cutting from each part of a structure might be taken, steel
being o'I per cent. heavier than wrought iron, but the difference is so small that the distinction of the quality by that
test is rather too delicate for practical purposes.  It has also
been proposed to take some of the punchings from the
plates, to draw them out to a small bar at a smith's forge,
and to test them either for tenacity, or by tempering with
fire and water in the usual manner. All such testing, however, is not in accordance with the usual practical notions of
the workshop at the present stage of our progress.
In a wide range of experiments made with ordinary cast
steel, when in its natural or untempered state, the ultimate




Steel.                       73
tenacity was found to vary from 114,000 lbs. down to 67,000
lbs., the highest being a little over 50 tons, and the lowest
a little under 30 tons per square inch; but specimens of
steel are often met with both of greater and of less strength
than the foregoing. A cast-steel specimen of extreme softness, cut from a Krupp gun, gave an ultimate tenacity of
72,000 lbs., which is very remarkable when the extreme
softness of the specimen is borne in mind.
It might be inferred that the strongest quality of steel
was always the best, but it is not so, the amount of tenacity which is desirable depends altogether upon the purpose for which it is to be used, the weaker and softer, or
less steely qualities, being more tough, are preferred for
many purposes, more especially for structures exposed to
vibration.
The effect of tempering upon steel is to increase its
strength, and, when the chill is rapid, to render it harder
and more brittle. In tempering ordinary articles they are
first heated to redness and then cooled in water, and thereby made over strong and over hard, and unfortunately very
brittle; these defects are then subdued by the application of
a gentle heat, which is continued or increased until the
required degree of temper is attained.
The strength is determined by two things: first, the
steeliness of the steel, that is, the proportion of carbon
which it contains; and second, the rate of cooling.  The
highest degree of strength is obtained by selecting a high
steel, heating it to a dull red, and then chilling it rapidly.
These two conditions give a degree of strength combined
with toughness, far beyond that which is obtained by giving
a higher degree of heat, then chilling it suddenly, and
afterwards reducing the temper.
When a block of mild cast steel is prepared for the
interior barrel of a gun, it is desirable that all its properties
should be known before it is put into the gun. Accordingly, two sets of specimens are cut from the block of steel,




74          On the Strengthz of llaterials.
each set of specimens consisting of three pieces: the one
set is used to find the best tempering heat for strength,
elasticity, and ductility; the other set of specimens is used
to find the best heat for toughness. One piece of each
set is tried in its natural or untempered state; one of each
is tried after being raised to a high temperature and dipped
into oil, and one of each is tried after being raised to a
less temperature and dipped into oil.  The three specimens of the first set, prepared as above, are turned in a lathe
for the testing machine, and their several properties carefully
noted. Then the three pieces of the second set are taken to
suitable apparatus, and bent backwards and forwards, in
order to ascertain the toughness of each of the three pieces.
When all is concluded and the results noted, a careful consideration is given to the whole; and after the advantages
or disadvantages are summed up, the heat to be given to
the gun-block is determined upon, and it is thereby brought
to the highest conditions of strength, elasticity, ductility,
and toughness, of which it is susceptible. Some specimens
after tempering are found to stretch from I5 to 25 per cent.
of their length, thus showing a degree of ductility which is
surprising, and combining the qualities of steel and wrought
iron in the gun-barrel.
The next Table shows the behaviour of gun steel in the
testing machine.




Tle following Table shows the mean results of a number of experiments carried out at' Woolwich
with cast steel of mild quality:TABLE X.
Elongation          Elasticity as
Length       in decimals of an     shown by the      Breaking
Weight       Diameter      of               inch.              difference     weight per
Nature of Material.  applied in tons  of      specimen                              between the    square inch
per square inch   specimen.    in                                visible and     of section
of section.                inches.                              permanent       in tons.
Visible.    Permanent.    elongation.
I2'39'003         -             003
Soft cast Steel in4                                                      00 
I3'98'004'ooI'003
natural state.'5 46533                   2 
34'79                                  -'372           -
15'OI'0025         -'0025
Cast Steel ternm-      978'003'oo3
20-46'00325'00325
pered at a high      25'01'0035.o0035
25'92'533         2         004          -             004           53 78
heat by immer-       32 o6                                 005 o0005.0045
sion in oil.         32'29                                oo'006  oo   015'0045
33'66'0085         0035
53 78                                  _
-- — ~~~~~~~~~~~~~~~~~




76          On the Strengthz of Materials.
As the result of a long course of valuable experiments
made by Mr. G. Berkley, C. E., he draws the following
general conclusions:'First, that Bessemer steel will bear
before rupture a minimum tensile strain of 33 tons per
square inch of section, and stretch about I inch in 12
inches of its length; second, that the same material will
bear, either in tension or in compression, a minimum stress
of I7 tons, before the extensions or compressions per unit
of stress become irregular Or excessive, as compared with
those which have preceded them, in other words, before
the yielding point of the material is reached; third, that
this material will probably contain about'45 per cent. of
carbon, chemically combined with the iron; and, fourth,
that this description of steel, if properly made and annealed,
is as uniform in quality as wrought iron, and may therefore
be employed (precautions being taken to test its quality) as
a substitute for wrought iron, while allowing an increase of
strain of 50 per cent. to be imposed upon it.'
Different qualities of steel have varying ability to resist
compression; with ten specimens of highly converted cast
steel, of a quality suitable for cutting instruments, an average
of 76,oo00 lbs. was required to compress the specimen
— 6eth of an inch.  With ten specimens of soft mild cast
steel of the finest quality, the softest required 25,000 lbs.,
the hardest 46,000 lbs., or an average of 35,500 lbs. to produce
the same impression; while two specimens, cut from  a
Krupp gun, required an average force of 25,300 lbs. One
of these specimens was afterwards flattened down into a thin
disc without cracking at the edges, thus showing a remarkable degree of malleability in the cold state. In all these
experiments the specimens at the commencement were I inch
in length and'533 inch in diameter.




77
CHAPTER VII.
ON COPPER AND OTHER METALS, AND THEIR ALLOYS.
Copper.
THE reddish-brown sonorous metal termed copper is possessed of considerable strength and elasticity. It is a metal
which is both malleable and ductile, but it possesses the
former property in a higher degree than the latter; hence,
it is much better fitted to be hammered, rolled, or worked
into thin hemispherical pans, or sheets, or other forms, than
to be elongated into a fine wire by pulling it through a drawplate. It is likewise to be noted that both properties, malleability and ductility, depend upon, and are mostly due to
the purity of the metal.
Next to iron, in its various conditions, copper is one of the
most important and useful metals found in the workshop,
and is extensively employed in the mechanical arts. Its ores
are widely distributed, and are found more or less abundantly
in almost every country in the world. Copper probably has
been longer in common use than any other metal.
As will be seen by the Tables in this chapter, its tenacity
is somewhat uncertain, and is not at any time equal to that
of either wrought iron or steel; but still as it is superior in
strength to gold, platinum, or silver, and indeed to all the
softer metals, and possesses many of their good properties
besides, it is held in high estimation.
A square inch of good wrought copper will break with a
tensile strain of about I5 tons; but it is not so strong in the
ingot or cast condition, for in that state it will often break
with less than the half of the above tension. The effect of
working upon copper is remarkable, both as regards its
malleable and its ductile properties; even a piece of good




78          On the Strength of Materials.
copper wire, I th of an inch in diameter, may be worked up
to such a condition, that it will require a strain of 300 lbs.
to pull it asunder.
At page 120 of Mr. Bloxam's treatise on Metals, there will
be found much useful information respecting the quality of
copper, as depending upon the refining or toughening processes of its manufacture, by which it will be seen that it
is altered, from a brittle state, into one which is soft, malleable, and ductile; but in the hands of the unskilful founder
or furnace manipulator, the malleable copper, from mismanagement, may again become brittle by the absorption
of charcoal, and, when such is the case, it has to be again
refined by the action of air, while melted in the crucible
of the founder.
It may be observed that, in the working of copper, either
by drawing, rolling, or hammering, it is altered in the same
way as wrought iron or steel; under the operation it becomes rigid, stiff, hard, and liable to crack, or even to disintegrate, and it can only be restored to its normal quality
by a course of annealing, which may sometimes require to
be several times repeated. This change is mechanical, and
is quite distinct from the chemical change to brittleness
before referred to.
Copper, even when employed by itself, is extensively used
for manufacturing purposes, but the articles made of pure
copper are few in number, when compared with the manifold forms in which it appears when combined with other
metals. It is also the hardest as well as the most tenacious
of all the workshop metals, except iron and steel, and it
may be worked by the smith either cold or hot. When
heated to redness, it can be forged, drawn down, or upset
much in the same manner as wrought iron, but when it is
heated to fusion or even to redness, and at the same time
exposed to the atmosphere, it is found that the exterior
surface is rapidly converted into black scales of peroxide.
This may be roughly removed by heating and plunging into




Copper and other Metals, and their Alloys.   79
cold water, or the pure metallic surface may be laid bare by
immersion in a solution of ammonia; but by repetition of
the above process the copper may be entirely wasted.
In ordinary use it is very liable to corrosion, by the mere
exposure of the bright parts of the metal to a damp atmosphere. The green oxide of copper so produced being poisonous, it is of importance that any copper vessels used for
culinary purposes should be kept perfectly clean; the usual
precaution is, to cover the interior surface of the copper with
a coating of tin, but this does not afford perfect security, as,
from its softness, it is liable to be scraped or otherwise worn
off, by the tear and wear of daily use.
Copper has been used to a considerable extent for the
construction of steam boilers, more especially for marine
purposes, or where salt water has to be evaporated. But,
of late years, it has been entirely superseded by iron or
steel. Although copper does not ordinarily rust to the same
extent by the action of salt water, still it is more rapidly
damaged in the furnaces by the use of sulphurous coal, and
when the boiler does happen to leak from any cause, the
leakage does not take up so readily as with iron, and the salt
incrustation is found to deteriorate the metal more, in the
vicinity of the leakage; besides, copper is sooner reduced in
tenacity by over-heating. Iron does not become perceptibly
weaker up to a temperature of 570~, after that temperature
is reached, the strength gradually decreases; whereas copper
is in its best state when cold, and loses tenacity by every
increase of temperature. With so many disadvantages, and
having a tenacity of only 15 tons, it is now disused, and
iron or steel is now almost invariably employed for steamboilers, except in some special cases.
The specific gravity of copper has a considerable range;
it varies from 8'78 in the crude state to 9go, after rolling or
hammering. An increase of density was supposed to arise
from the mere condensation of the particles of the mass, but
it is now suggested that when copper is melted in contact




80          On the Strength of Materials.
with the atmosphere, it absorbs oxygen, which does not
afterwards find the means of escape, and consequently the
metal becomes slightly porous; this absorption of oxygen is
in a measure prevented by careful fusion under common
salt. The density of the metal, so fused, is nearly equal to
that of worked copper, namely 8'92I, and after being subjected to a pressure of 300,000 lbs. per square inch, it increases to 8'930; this difference however is small, and it
has been suggested that the change may be more owing to
a diminution of the empty spaces still remaining, rather than
to the approximation of the molecules.
Articles which, from their form, require to be cast in a
mould, are seldom made of pure copper, because it is too
soft for the generality of purposes; it is likewise very porous
after casting, and has great tendency to unsoundness. Copper
melts at 2,0000~, which is higher -than the melting-point of
most of its alloys, as given in Table XVII.
The ultimate tenacity of copper when in the cast state
ranges from 19,000 to 26,000 lbs. per square inch of section; but the strength of this copper may be considerably
increased by working, wire-drawing, hammering, or consolidating. When carefully drawn into wire, copper has a
tenacity as high as 6o,ooo lbs. per square inch; ordinary
copper bolts a tenacity of 33,000 lbs.; while several other
specimens, of pure wrought copper bolts, gave an average
ultimate tenacity of 36,o000o lbs. per square inch.
Copper, in castings, is generally mixed with other metals;
such mixtures of metals are called alloys, and those alloys in
which copper predominates are the most numerous; tin,
zinc, lead, phosphorus, aluminium, iron, and other metals
are all employed to form alloys with copper.
The fluidity and tenacity of copper may both be considerably increased by the addition of a small percentage
of phosphorus.  In several experiments that were made,
this alloy was remarkable for its density and tenacity, and
when broken showed a regular, sound, and uniform frac



Copper and other Metals, and their Alloys.          8 I
ture Mr. Abel found that by an addition of from 2 to
4 per cent. of phosphorus, a metal was obtained more uniform than bronze, and having an ultimate tenacity of from
48,000 to 50,000 lbs. per square inch. Mr. Overman states
that by the addition of phosphorus, copper may be rendered
as hard as steel.   The combination  of phosphorus with
copper increases the tendency to corrosion, unless another
metal, such as tin, is also added. By adding phosphorus to
bronze, its homogeneity is materially increased, and its tendency to oxidise by exposure to the atmosphere completely
neutralised.
The two following Tables show the results of experiments
with exceptionally good copper, and with copper containing
a small percentage of phosphorus.
Results of experiments made at Woolwich, to ascertain
the tensile strength of copper:TABLE XI.
Length of specimen under test, 2 inches.
Breaking weight
in tons per square  Specific       Remarks.
inch of section.   gravity.
I5 39
14'78                       Cast copper
14'89                       (very pure).
II'79       8'688
Specimens cut from virgin
15'75               copper bolts. The bolts were
I6'25                21, 2, I, and j inch diameter
15 9                respectively, and the diameter
I6-25                of the specimens cut from the
bolts, II, Itl,'6",'6".
The above results, especially those on cast copper, indicate a better quality of copper than the average.
G




82            On the Strength of 3Materials.
Results of experiments made at Woolwich to ascertain the
tensile strength of phosphorised copper:TABLE XII.
Length of specimen under test, 2 inches.
Breaking weight in tons  Specifi    Percentage    Diameter
per square inch       pecifc        of       of specimen in
of section.        gravity.    Phosphorus.    inches.
6'23                            I           1'13
7'56             8'202          I           1'13
I6'47             8'592         I            1'13
17'13             8'876         I            1'13
909'                            I            1'13
20'25             8'614         2            1'13
20 34              -I I                      1'I 3
20'41             8 580         2'98
2I'27             8'6I5         2            I'I3
2I'38            8'422         3            II2
21'5
The above nine specimens show  an average tenacity of
about 20 tons per square inch. The first two specimens are
not included.
BE-onze and Gun-metal.
The alloy of copper and tin, usually termed bronze, but
sometimes gun-metal, has been of great value and importance
in the arts, from time immemorial.  When this alloy is used
for artillery purposes, it generally consists of from  90 to 9012
parts of copper, and from  9- to io parts of tin.  A small
fraction of the tin, however, is invariably lost in the melting
process, even if performed in crucibles, still more is lost
when a reverberatory furnace is employed, and most of all
if the fusion is made in a cupola. When copper and tin are
alloyed in proper proportions, a harder metal than either is
produced, with a density greater than the mean density of
the constituents. The alloy is more fusible than copper,
and less liable to corrosion.
Copper and tin mix well in almost all proportions, a




Copper and other ~Metals, and their Alloys.    83
small percentage of the latter rendering the alloy both hard
and tenacious, and by changing the proportions of tin to
copper, alloys are formed, varying extremely in colour, hardness and soundness, and also in regard to their capability
of yielding a sonorous sound when struck, as when used for
bells.
By the addition of tin, hardness is increased.  With a
proportion of Ith of tin to -ths of copper, by weight, the metal
assumes its maximum hardness for the purposes of the
engineer, or for any application in which it requires turning
or planing. The founder sometimes resorts to a mixture
containing from ~th to ~rd of tin, it then becomes highly brittle
and elastic, like glass, and the sonorous property is improved in a high degree; at the same time its brittleness
rather than its hardness is the prominent feature. A mixture, consisting of 2 parts of copper and I of tin, forms an
alloy so hard that it cannot be cut with steel tools, and
has a highly crystalline structure.  At this stage almost
every characteristic for which tin and copper are distinguished seems to be entirely changed.
The specific gravity of ordinary bronze varies from 8'4
to 8'94, according to the nature of the mixture and the way
in which it has been treated by the founder.  From Muschenbroek's experiments, it appears, that the density of the
alloy becomes greater, as the proportion of tin is increased:
with io parts of copper and I of tin, the specific gravity was
8'35; with 8 parts of copper and I of tin, it was 8'392; with
6 parts of copper and I of tin, it was 8'707, and with 4 parts
of copper and i of tin, it was 8-7 23. The results obtained in
experiments made atWoolwich showeda rather lower specific
gravity than the foregoing, and Major Wade states that,
judging from an examination of the specimens obtained from
the heads of all the guns cast at an American foundry, the
density varied from 8-3o8, to 8'756, thus showing a difference
of weight between the lightest and heaviest specimens of
28 lbs. in the cubic foot.
G2




84          On the Strength of Materials.
Bronze, melting at I,90oo Fahr., is more fusible than
copper, but much less fusible than tin. Indeed, the difference of fusibility in these two metals is so great, as to render
it exceedingly difficult for the founder to obtain a perfectly
homogeneous bronze alloy, when the casting is in great mass;
upon examination it is often found that specimens of large
castings show a great want of regularity, and contain patches
or spots of the appearance of tin, mechanically interposed
between the particles of alloy. These spots, although apparently of tin, seldom contain more than 25 per cent. of that
metal.
In forming bronze, great care must be exercised by the
founder, when mixing the metal in the furnace. The more
refractory metal should be melted first, and the more fusible
metal added. The alloy should be well stirred and then
cast, and should be cooled as rapidly as possible, in order
to obtain uniformity and compactness, and to obviate the
tendency to separation in the process of cooling, the denser
metal being generally found at the lower part of the casting,
and the lighter one at the top, if the cooling is long protracted.
When mixing tin with copper in tne furnace, it is of the
utmost. importance that the tin should not be long exposed
to the influence of the air, because, when it is heated to
redness, its affinity for oxygen is so great that it will be
rapidly oxidised and converted into the peroxide or putty
of tin (the putty-powder of commerce). This is very disadvantageous, for when a reverberatory furnace works
slowly, the metal is found to contain innumerable particles
of the putty-powder, and to such an extent is this sometimes
the case, as to render the turning or boring of the gun a work
of extreme difficulty, because the cutting instruments are
blunted by the hard oxide. Hence it is that the crucible
system of melting has a great advantage over the large
reverberatory furnace, wherever it is applicable, the chief
barrier to the use of the crucible being the difference in cost.




Copper and other M~etals, and their Alloys.    85
In mixing copper and tin to form gun-metal, the best
arrangement is to alloy them first in the proportion of 2 to
i, obtaining a white hard crystalline silvery speculum
metal, and then by another melting to mix this hard alloy
with the requisite quantity of copper to give the required
quality of bronze. In this second melting, as well as in
the first, the copper is first melted; then, shortly before
the time of casting, the 2 to I alloy is added, and the whole
well stirred to secure a good mixing. It is afterwards immediately cast in order to prevent the oxidation of the tin.
A small quantity of zinc is sometimes added to common
bronze, for the purpose of making it mix better. Zinc increases the malleability without materially reducing the
hardness, but it is seldom used in gun-metal.
The investigation bestowed upon the strength and other
properties of bronze, during the last hundred years, has been
most thorough; many experiments have been made, but the
results are so discordant as to render it difficult to give an
account of them without inserting such a number of tables
as would be incompatible with the size of this volume.
The late Mr. Rennie came to the conclusion, that the
average ultimate tenacity of gun-metal is about 36,333 lbs.
per square inch, that is, when the mixture consists of copper
and tin only, and in the usual proportions.  Musdhenbroeck's experiments go to show that with a mixture of 6
parts of copper and I part of tin, the ultiniate tenacity is
equal to 44,000 lbs. per square inch. From similar experiments made at Woolwich, the following results were obtained, with a mixture ofTenacity.
12 parts of copper and I of tin, 29,ooo lbs. per sq. inch.
II,,,,      30,700,,,
10,,,,       33,000,,,,
9,,,,      38,000,,,,
The above results are not quite uniform, but they nearly
agree with those obtained both by Rennie and Muschen



86         On the Strenzgth of Materials.
Droeck, and are perhaps not far from the average of the
bronze usually met with in the arts.
Judging from an immense number of experiments made
at Woolwich, at different times, during the past fifteen years,
and without having regard to many minor points, or to the
shades of proportion of mixtures and different modes of
casting, the average tenacity of bronze is 31,280 lbs.,
varying from 22,500 lbs. to 41,000 lbs. per square inch.
As a rule, it begins to yield visibly, and to take permanent
set at I5,I64 lbs., and the average ultimate elongation per
inch in length is'290 of an inch.
Some experiments are recorded, both much lower and
much higher than the above, even ranging from I 7,698 lbs.
upwards to 56,786 lbs., but these are exceptional, and
33,000 lbs., which is between I4 and 15 tons, may be considered as the general average of good bronze.
Major Wade, of the United States Army, has given great
attention to this point, and he has found the ultimate
tenacity to vary from 23,929 lbs. to 35,4841bs. per square
inch, a difference in the ratio of 2 to 3. These differences
occurred in samples taken from the same part of different
castings of gun-heads, where the materials used were apparently all of the same quality, and were melted, cast and
cooled in the same manner, and every effort was used to
have them similarly treated in all respects; but the causes
of such irregular and unequal results, when the materials
used and the treatment of them were apparently the same,
are very obscure and perplexing.
Benton states that the quality of bronze depends much
upon the nature of the furnace treatment of the melted metal,
and that by extreme care, and by this alone, the tenacity
of bronze made at Washington Navy Yard Foundry has been
raised as high as 60,00ooo lbs. per square inch, so as to equal
in tenacity good wrought iron, which is worth noting.
The following results have been obtained by analysing a
number of tables which show the tensile strength of various




Copper and other Metals, and their Alloys.    87
specimens taken from different parts of bronze guns, some
of the specimens being cast in open moulds with both green
sand and dry sand; others in closed moulds with green sand,
and the remainder in iron moulds, forming large chills to
cool the metal rapidly.
The average weight in lbs. per square inch, required to
overcome the elasticity of the metal was 14,694 lbs., and
the point of fracture 27,305 lbs. per square inch, the average
elongation per inch in length being'144 inch. Each specimen was I-o66 inch in diameter, and the length of breaking
part 2 inches.
In sixteen specimens, each'533 of an inch in diameter by I
inch in length, the compressive stress per square inch required
to produce a permanent set of'ooI ranged from 5'4 tons
to 7 tons, and was usually near to a mean between these
two extremes. A stress of 7 tons applied to each specimen
produced a permanent set ranging from'ooI5 to'0005 the
most elastic having been cast in an iron mould which cooled
the metal rapidly. Taken as a whole, the bronze when in
compression, showed a permanent set of'ooi of an inch with
an average compressive load of 13,843 lbs. per square inch,
and'384 with 50 tons per square inch, with which load the
specimens were fractured. When compared with steel, or
even iron, bronze offers a small resistance to compression;
but its compressibility is found to depend greatly upon the
perfection of the alloy, and the amount of fluid pressure
induced by the height of the deadhead, and the rate of
cooling employed to prevent the separation of the tin.
Brass.
Of the many useful alloys known to the mechanical world,
brass is perhaps the one which is most extensively used,
being easily worked, and of a fine yellow colour. In this
alloy as in nearly all the others used in construction, copper
is the most prominent metal.




88          On the Strength of Materials.
In the melting and mixing of copper and zinc, great care
has to be exercised to prevent the zinc from passing away in
vapour, which is usually effected by covering the crucibles
with charcoal powder and a close lid of clay. Brass is more
malleable than copper when in the cold state, but it will
not submit to be forged at a red heat, on account of the
low melting point of zinc; even a small addition of zinc to
copper will affect it in this important respect, which is a
disadvantage.
Tin and lead are sometimes added to copper and zinc in
making brass. The quantity of copper varies from 6o to 92
per cent., and it is not uncommon to add from I to 3 per
cent. of lead, and from 4 to 3 per cent. of tin, according to the
nature of the work for which the alloy is required. The best
proportion for fine or yellow brass appears to be copper
2 parts and zinc I part.
The specific gravity of brass ranges from 7'82 to 8'5, and
is thus greater than the mean of its constituents, that of copper being 8'78 and of zinc 6'86, which is probably due to the
zinc finding empty spaces, into which it enters.
The melting point of brass is lower than the mean of its
constituents would indicate, being from 1689~ Fahr. to 900oo
Fahr., the fusibility entirely depending upon the quantity of
zinc which has entered into combination with the copper;
and the fact of the fusibility of the alloy increasing with the
further addition of zinc furnishes strong evidence that the
change in their properties, which the metals undergo, arises
from chemical affinity. The liquidity of this alloy may be
very much increased by adding a small portion, say half an
ounce, of dry phosphorus in the crucible, and then stirring
the metal before running it into the mould; by this means
the alloy is rendered so liquid that very thin and sound castings may be obtained without difficulty.
The addition of a little lead causes brass to be more ductile and better adapted for turning in a lathe than common
brass, while a large addition of lead renders it very brittle,




Copper and other Metals, and their Alloys.    89
and if the proportion of lead amounts to nearly one half,
then a partial separation takes place in the act of cooling.
The tenacity of fine brass (as stated by Rennie) is only
about I8,ooo lbs. per square inch; but the average ultimate
tenacity of the best quality of brass, when composed of two
parts copper and one part zinc, is much higher, being
28,900 lbs. per square inch.
Muntz-metal.
The alloy of 6o parts copper and 40 parts zinc, termed'Muntz-metal,' is a mixture which has been much used for
the sheathing of ships, and for the bolts of marine engines
liable to rust, and for similar purposes. Its tenacity is
about 22 tons per square inch, or nearly equal to that of
good wrought iron, while its endurance in salt water is
nearly equal to that of bronze or brass.
In consequence of the high tenacity of this metal, it was
thought desirable to cast guns of it, but it was found that,
during the increase of temperature due to rapid firing, the
presence of zinc affected the tenacity of the metal to a
considerably greater extent than is the case with ordinary
gun-metal, and, notwithstanding its great strength, it was
found unsuitable.
The following Table shows the results of various experiments made at Woolwich to ascertain the tensile strength of
brass, &c.:TABLE XIII.
Length of specimen under test, 2 inches.
Breaking weight   Specifi  Diameter
in tons per square  gravity.   of specimen in  Mixture.
inch of section.           inches.
603'2       8}8                   Copper, 7 lbs.,
x6 o2       8534699                  zinc, 3 lbs. 8 ozs.
io'68
Ilo9I        -
3'I2       70~49. I'3     Copper Io parts, iron
3'22       6-9I5                  IO parts, zinc 80 parts.




90o-        On the Strengthl of Materials.
The average ultimate tenacity of the first 6 specimens
was I2'92 tons or 28,940 lbs. per square inch. The result
of the last two experiments is instructive.
Sterro-metal.'Sterro-metal' is a new alloy of copper, zinc, tin, and
wrought or pure iron.  One of the principal objects for
which this particular kind of metal was first introduced
was, to supersede the use of cast and wrought iron, and
bronze, in the manufacture of heavy ordnance. Cast iron
is objectionable on account of its low tenacity, elasticity
and ductility, and has proved not altogether suitable for the
purpose. Wrought iron is rather too soft for the interior
of guns; necessitates, in its manufacture into guns, a great
amount of expensive working, and, even when the greatest
care and skill is exercised, its perfect soundness cannot be
altogether depended upon. Bronze, when of the proper
quality, is too soft; its ultimate tenacity is uncertain, being
generally not much greater than that of good cast iron; it
is more easily stretched to the elastic limit than is the case
with good wrought iron; and, in addition to all these objections, it is more costly.
For the foregoing, as well as for other reasons, many
efforts have been made to obtain a homogeneous and dense
metal, which could be cast into large masses, and which
combines all the good properties of hardness, great tenacity,
elasticity, and soundness.
Sterro, or firm metal, was first introduced to the arts in
Vienna, a few years ago, by Baron de Rosthorn, and was
rapidly seized hold of by experimentalists in various parts of
the world. At Woolwich a series of interesting experiments
were made, with specimens containing different proportions
of the various metals, prepared in the Royal gun factories,
as well as with specimens of the metal obtained from Austria.
Since that time additional experiments have been made,
and the results are given in the following Tables.




Copper and other Metals, and their Alloys.    9t
From these it will be seen that this alloy has some most
valuable properties, especially stiffness, as its name implies,
together with great tenacity and power of resisting compression, with considerable hardness, which is very desirable in
any gun-metal, to resist the abrasive effect of the projectile.
Although the sterro-metal possesses the most prominent
of these qualities in a high degree, yet, even with this metal,
it is very difficult to ensure a perfectly uniform and sound
casting with any degree of certainty, several fractured specimens showing the mixture of metals to have been very
incomplete.
From the following Tables it will be seen that the ultimate
tenacity varies from 43,000 lbs. to 85,000 lbs. per square
inch, or an average of 60,480 lbs., per square inch; it also
required a strain of 30,000 lbs. to produce a permanent
elongation of 002 of an inch per inch of length, while the
ultimate elongation with the average strain of about 60,480
lbs. per square inch was o'0675 of an inch per inch of length.
Whereas, bronze begins to yield at 15,000 lbs., and.is fractured at 33,000 lbs. per square inch, and shows an ultimate
elongation of'290 of an inch per inch of length.
The recent experiments made with steel, and especially
with steel tempered in oil, and the practical success which
has attended it for gun purposes, has probably barred the way
to further experiments being made with sterro-metal, at least
for the present; but it will be evident that the subject opens
up a wide field for other experiments with mixtures of iron
and other metals, with a view to discover other combinations
which will give strength with other good qualities.
Results of experiments made at Woolwich to ascertain the
tensile strength of sterro-metal:



92             On thae Strength of Materials.
TABLE XIV.
Diameter of specimen,'707 inch; length of part of specimen
under test, 2 inches.
Strain at
Breaking permanent Ultimate
weight in'elongation elongation
tons per of'oo002 in. at breaking    Treatment.   Mixture.
square inch! per inch  point in
of section. of length  inches.
in tons.
26'75     675       i      as received.     Austrian.
21-5      Io       05                       Copper 60, zinc 39,
cast in sand.    iron 3, tin I'5.
19'25    13'75     OI5   I'
24'25    1 I7'25   *OI6    cast in iron.
cast in iron and   Copper 60, zinc 44,
2325    I5'25      02       annealed.         iron 4, tin 2.
28-0     17-0'045    forged red hot.
j3161     -         -       cast in iron and
32'52     -               j forged red hot.
34'0                              -         Copper 60, zinc 37,
iron 2, tin I.
38So                                        Copper 6o, zinc 35,
iron 3, tin 2.
270       -         -      after simple
27 o  afusion.  Copper 55'04, spelter
-40   i             -     forged red hot.    tin  3. iron   77,,8                        drawn cold.
280                 -      after simple 
fusion.
3240                -      forged red hot.   C
drawn cold and  Copper 763, spelter
reduced  from    40-22,  iron  1 86,
37'0I  00 to 77 trans-   tin I5.
verse sectional
area.
The next Table shows the results of experiments made at
Woolwich to ascertain the extension and tenacity of sterrometal:




Copper and other Metals, and their Alloys.                93
TABLE XV.
Diameter of specimen,'707 inch; length of part under test, 2 inches.
I       2            3             4            5      6
Permanent  Permanent
Breaking    Weights   Elongations in   Permanent   Permanent
weight in   applied in   decimals of an  elongations in  elongations
tons per    tons per  inch due to the  decimals of an per inch of Treatsquare inch  square inch weights applied                     ment.
of section.   of section.  in column 2.   weights applied decials of
in column 2.  an inch.
2'27'00
3-86'002
6-02'003
7'27      o004
8'98        oo05'ooI        0005
Io'1040      oo6           oo I 5      ooo75
I'03'007'002'00I
12 50       o008           oo35'00175
13'07       0oo9'004       *002
20'35'074
4 09        oo I
5 68        002
6-48        003           0005        00025
6 93       o004           oo0 I       0005
7'73        005'*0025       *00125
830'006'003'oo I 5
9'21' oo8               *005'0025
I9'78'227.~
5'               I I2                           d
7'16'004'00I'005
8'07'005'0025'oo025
841         *0oo6         003         ooI 5
8'75        007            00oo4       002
9'32'o09           0055'00275
20'92'282
4'54    - 002'0005'0025
6'36'004          0oo0 I        005
IO'  2       -o'006'0025'oo 00125
19'21      I2'50'009'0045'*00225
7'5'0oo3         0005'0025
I I'37'005'001o        0005
13I 9'007          *002'00 I
20'35      I4-66'009'004'002




94              On the Strength of MHaterials.
TABLE XV.- conztinuea.
I          2            3              4           5'6
Permanent   Permanent
Breaking    Weights   Elongations in   Permanent   Permanent
elongations in  elongations Treatweight in   applied in  decimals of ann  elonats Treattons per    tons per  inch due to th  decinals of an  per inch of ment.
inch due to the  length in
square inch  square inch  weights applied weights applied   decimal of
weights applied decimals of
of section.   of section.   in column 2.   in column 
in column 2.   an inch.
5'23'002
8-64'004 
4 O oo06      *'0005  50002
15'46'   007           ooI         05.0
17.05        *008.     0025       o00125
I8'53'oI'003         ooI 5
32'52'I67'0835'002
5'57         02 
9 66'004
13'75    -006'aOl.         E
I 375  *oo6         *OOI         0005 
16-26'008'0025'00125
I6'94'oI             0035.00I75    U
30'02'I67'0835
4'32         OOI
6'25'002
8-07'oo3
9 21         004
1 O'23       0055          oo 15        000ooo75 
IO091'0065'0025'00I25
I  11I4      007'003        0ooI 5    |t
I 1'82'008'0045'00225
12'28'OI'005'0025 
20'35'132 
5'23'00I                                 0
8 52'003
10'57'005           001         0005      -
1 I ~ 14     00o6'002         00 1
12'05'008.0045'00225    V
12'50       0oI'005'0025
2126'156




Copper and other Me/tals, and their Alloys.             95
TABLE XV.-continued.
2           3             4           5       6
Breaking    Weights   Elongations in   elongations in  elongationsent
weight in   applied in  decimals of an  elongations in  elongations
tons per    tons per  inch due to     decimals of an per inch of Treatsquare inch  square inch weights applied inch due to the  length in  ment.
of section.  of section.   in column 2.  weights applied decimals of
in column 2.   an inch.
4'20        ~I 0
6 -02'002
8'52'003
10o'34'004
12.73' 005
14'32       o006           0005       00025
14'78' 007              00I        0005
728          009           0025       001oo25
27'06      1876'012'0045       00225
6'36        002
9'21'004          0'05        00025
IO-57       *oo6.     0025'001I25
I I 82'009'0045'00225! 
24'56.275'I 375
7'84        -003
9'55'004
I3'I9        005           0005'00025
14.78        oo006'    0005'00025
15'46'007           ooI         0005
17'39        008'0025'00125
I8'53       Io05          0035       ooI 75
25'69'037'o185
3'52'002'      
6'82'004
Io091        *oo6                                0
I2'28        007
I'92      *008           0005'00025    7
I 6-37' 009'0007        00035
17'28        OI            00I        0005       0
19.33        012           0025'00125
2I'83'0155         *005         0025 
32'29              1                I  2'06'00~25' V




96                 On the Strength of Mietals.
TABLE XV. -continued.
I           2              3              4             5       6
Breaking    Weights   Elongations in    Permanent    Permanent
weight in   applied in   decimals of an   elongations in  elongations
tons per   inch due to the  decimals of an  per inch of Treattolls per tons per             inch due to the
inch due to the  length in  ment
square inch  square inch weights applied  weihts ppto the length in  ment.
of section.   of section.   in column 2.                  al
in column 2.    an inch.
3.18    i     002
568'004
8-87          oo006
9'78         007                                     PC
1205.008.0005. 00025    a
12.50         009             0007         00035 0
13 30         O1I             001oo 0005
I15 0I         012.     0025. 00125
17'05'0155'0045         00225 
2751'II'055
Results of experiments made at Woolwich to ascertain the
transverse strength of mixtures of wrought iron and gunmetal:
TABLE XVI.
Size of Specimen.      Distance             Breaking
between  Deflection   weight
Length  Breadth    Depth        ports    in        in tons    Mixture.
in  in             in         in     inches.   at centre
feet.   inches.   inc      inches.              of beam.
feet.   inches.   inches.
2     1'99       2          20'142       2'8      equalpro
2 2o03    2           20                           equal pro99 2 20   274   portions.
2     1'99       2          20 ~      11O       2'24
2     2'03       2'05       20'IIO      2'56
2     2'02       2'02       20'I05      26        gun-metal I,
2     2'02       2'05       20         --         31       iron 3
2     2'04       2'03       20'140       2      o         t
2'01            2 o6       20        I 33.      83 3-etal i,
2     2'01      2'05      20'120      2I         Iron 7
It is right to state that the fracture showed the mixture of
the  two  metals to  be  very incomplete; nevertheless, the
result is worth noting, as it is probably the only Table of the
kind.




Copper and other Metals, and their Alloys.   97
Aluminiumn Bronze.
Aluminium Bronze is an alloy of copper with aluminium,
which seems to promise great results. It consists usually
of go parts of copper to io of aluminium, these proportions give an alloy which may be forged either cold or hot,
in the same manner as wrought iron, but which does not
weld. It is highly malleable and ductile, possessed of great
stiffness and elasticity, its specific gravity being nearly the
same as that of wrought iron; and it does not readily
tarnish by exposure to the atmosphere.
From  a number of experiments made at Woolwich, its
average tenacity was found to be 73,I85 lbs. per square
inch, and its maximum tenacity 96,320 lbs., thus its strength
was more than double that of bronze, greater than that
of wrought iron, or even of many of the mild qualities
of cast steel, and at the same time its resistance to compression exceeded that of ordinary cast iron.  Colonel
Strange, who has given much attention to this alloy, states
that its rigidity is three times that of bronze, and many
times greater than that of brass, and that it is less affected
by changes of temperature than either of those alloys. In
the liquid state, it can be cast into any form without difficulty, it works nicely under the file or in a lathe, and its
other advantages are numerous.
In making this alloy, extremely pure copper has to be
used, as with impure copper the alloy is much deteriorated
in all its good properties; and to produce the best results,
it requires to be remelted several times; the fact of remelting
being required, in order to develope its greatest strength
and stiffness, is not peculiar to this alloy, but is the case
with cast iron and some of the other alloys.
The chief barrier to the extensive use of aluminium
bronze is the cost of the aluminium, which makes the price
of the bronze at least four times that of ordinary gun-metal.
H




98          On the Strength of Materials.
Hence this metal is chiefly used for purposes where strength
and stiffness, combined with lightness and non-liability to
rust, aret he chief objects, as for surveying instruments, which
have to be light and strong, and not liable to injury when
carried from place to place, and used in tropical climates.
Other Alloys.
Bells are usually made of copper and tin of various mixtures, but generally in about the same proportions as for
gun-metal. The smaller class of bells have a greater proportion of tin, to give hardness, and sometimes a little zinc;
even silver is occasionally added, in order to improve the
tone.
For such purposes, the question of strength is unimportant. The same remark applies to the usual bronze
for statues, which is a similar mixture to gun-metal, but
analysis of different statues shows considerable variation in
the proportions of the two metals. The addition of a small
portion of phosphorus would have the effect of preserving
the surface of the metal from the influence of the atmosphere,
which fact should be noted.
Babbitt's metal for machinery bearings, which consists of
about 4 parts of copper and 8 lbs. of regulus of antimony
and 96 lbs. of tin, is not a strong alloy, but it is one of
the best reducers of friction, and its want of stiffness has to
be provided for in the iron casing in which it is contained
when in use, to form a bearing for a moving spindle. The
casing is required in order to prevent it from spreading outwards under the pressure to which it is exposed. In the
use of Babbitt's metal for bearings, it is necessary to exercise great care to prevent the heating of the journal. If
this takes place, the Babbitt's metal melts and runs out.




Copper and other Metals, and their Alloys.    99
Table showing the average melting point of most of the
metals and alloys referred to in this chapter.
TABLE  XVII.
Melting point
Metals and Alloys.              in degrees
Fahrenheit.
Cast iron, rich in carbon..                2786,,    2l to 3 per cent. of carbon...    2600
Wrought iron...   3280!,,  freed from silicon..           3500
Cast steel, mild,'28 per cent. of carbon..   3300oo,,   ordinary,'56 per cent. of carbon.     3000,,   high,'75 to *8 per cent. of carbon.  2850
Copper..                                       2000
Tin...                                          442
Zinc... 758
Gun-metal I..90oo
Brass..    847
Lead.......                         600
Antimony..                           800
Bismuth....483
Silver.                                         1 873
Platinum.......  3280
Gold..                                        2016
I 2




o00                  On the Strength of Materials.
ADDITIONAL EXPERIMENTS.
The following Tables show the result of certain experiments on the
resistance of metals.
Resistance of Metals to Tension.
Stress in tons per
square inch at       Elongation
materialf 
Size of specimen.   material. 
Yielding. Breaking. per inch.  per cent.  ~
Homogeneous           8    3 9          23,3
Diameter,'533   steel, soft,        12'8     319        123        23
area'223 sq. in. Do. tempered  
in  oil at a  29-7    47'    9        I29      129 
Length of break-  low heat,    J'
ing part 2 inches Do. tempered /
in  oil at al.299    48 5              s08     8
high heat,   J
Diaeter  Best gun-iron, ),.22 2     I. 13 
Diameter'754"  soft,                                    34        34
area'4465 sq.                  f      5
inch
inch  Bronze.     6'9    15.7' I62           16'2
Length of break- I
ing part 2 inches Copper.     -'5
Resistance of Metals to Com-ression.
Total compression
Yielding    with load of 50 tons
lYielding   allowed to remain on
Nature of    strain in tons    for 5 minutes    AuSize of specimen.     material,     per square                      thority.
inch.
per inch.  per cent.
Soft steel.       I2'47'218      21'8.
~Lenth  1 tTempered steel   28 98'054        5'4.' [
e  Wrought iron,         22        274       274
Length                o.           75         33   323
_t         CsBronze.        7'5'3423    34'23      0
Cast iron. 159'0308     3'08 
_ _ _ _ _ _ _ _ _ _ _ _ _   _ _ _ _   _ _ _ _  _ _ _




IOT
CHAPTER VIII.
TIMBER.
ALTHOUGH it is of less importance to investigate the strength
of timber at the present time than it was formerly, in consequence of the diminished use of that material in permanent structures, and the more general employment of
iron, still it will always be a very valuable material for certain purposes, and ought not to be neglected. Timber is
variously used, even now, in permanent works, and is
applied much more extensively in temporary structuressuch as centerings and scaffolding. Hence its properties
are well worthy of careful attention; and the student should
be familiar not only with the external appearance of the
principal kinds of wood, but also with their relative strength,
stiffness, toughness, and durability.
One of the most obvious inferences to be drawn from the
experiments recorded in the previous chapters is that very
wide variations exist in the strength and other elastic properties of different metals, and even of different specimens
of the same metal. If we could investigate the properties
of timber with the same care which has been bestowed on
the metals, we should find that there is an even greater
variation in the properties of different kinds of wood. This
arises in part from the fact that timber is much affected by
a number of external and internal conditions, during its
growth and seasoning, and in its subsequent treatment,
which gradually modify and change its properties.
It will only be necessary in this chapter, to treat of tht
powers of resistance of a few among the many kinds of
wood now employed in the mechanical arts. The greater
number of the varieties of wood owe their commercial value




102         On the Strength of Materials.
to special characteristics, such as beauty of grain and capability of being polished-the description of which does not
fall within the scope of the present treatise.
As a general rule, we may judge of the hardness of a
wood by its specific gravity, if it is in its natural state. But
the density may be increased by artificial compression (as
in the manufacture of trenails) and this increase of density
is generally accompanied by increase of strength. Some
varieties of wood, as, for instance, lignum vilt, are so dense
that they sink in water, while some of the softer woods have
not half the density of that fluid. The presence of gum or
resin in any wood adds both to its strength and durability.
Many woods will last a long time, if kept constantly under
water, but scarcely any wood is very durable when allowed
to become wet and dry alternately.
The strength of a piece of timber depends upon the part
of the tree from which it is taken. Up to a certain age, the
heart of the tree is the best; after that period, it begins to
fail gradually. The worst part of a tree is the sap-wood,
which is next the bark.  It is softer than the other parts of
the wood, and is liable to premature decay. The deleterious
component of the sap-wood is absorbed, if the tree is allowed
to grow for a longer period, and in time the old sap-wood
becomes proper timber fibre similar to heart-wood. Hence,
the goodness of a tree, for timber purposes, depends on the
age at which the tree was cut down. When young, the heartwood is the best; at maturity, with the exception of the sapwood, the trunk is equally good throughout; and when the
tree is allowed to grow too long, the heart-wood is the first
to show symptoms of weakness, and deteriorates gradually.
The best timber is secured by felling the tree at the age
of maturity, which depends on its nature as well as on the
soil and climate. The ash, beech, elm, and fir are generally
considered at their best when of 70 or 80 years' growth,
and the oak is seldom at its best in less time than Ioo
years, but much depends on surrounding circumstances.




Timber.                     I03
As a rule, trees should not be cut before arriving at maturity,
because there is then too much sap-wood, and the durability
of the timber is much inferior to that of trees felled after
they have arrived at their full development.
The strength of many woods is nearly doubled by the
process of seasoning, hence it is very thriftless to use timber
in a green state, as it is not only weak, but is exposed to
continual change of bulk, form, and stability. After timber
is cut, aad before it is properly seasoned, the outside is
found to crack, and to split more than the inside of the
mass, because it is more exposed to the dessicating effect of
the surroindcing atmosphere, but as the outside dries, the
air gradually finds its way to the interior. If timber is cut
up by the saw when green and allowed to season or dry
in a gradtal manner, it is found to be the most durable.
In the arts, however, artificial drying is often resorted to,
as in the:ase of gun stocks. These are put into a dessicating chlnber, where a current of air at go~ or Ioo0 is
passed oxer them, at such a rate as to change the whole
volume of air in the chamber every three minutes, and it is
found tha a year of seasoning may thus be saved. The
walnut wc)d is as good after this process, as if the seasoning
had been accomplished by time and exposure, and works
more smothly under the cutting instruments of the stock
machinery
~Wood sill always warp after a fresh surface has been exposed, anI will likewise change its form by the presence of
any moistire, either from that contained in the atmosphere
or from netting the surface. The effect of moisture on dry
wood is ta cause the tubular fibres to swell, hence it is that
if a planl or board is wetted upon one side, the fibres there
will be ditended, and the plank in consequence must bend.
The naural law that governs the shrinking or contraction
of timberis most important to practical men, but it is too
often ovelooked.
The anount of the shrinkage of timber in length, when




Io4         On the Strenzgth of Materials.
seasoning, is so inconsiderable that it may in practice be
disregarded.  But the shrinkage in transverse directions is
much greater, and presents some peculiarities which can
only be explained by examining the structure of the wood,
as resulting from its mode of growth. An examination of
the end section of any exogenous tree, such as the beech or
oak, will show the general arrangement of its struc:ure. It
consists of a mass of longitudinal fibrous tubes, arJranged in
irregular circles, which are bound together by neans of
radial plates or rays, which have been variously named;
they are the'silver grain' of the carpenter, or the' medullary rays' of the botanist, and are in reality the sarme in their
nature as the pith. The radial direction of these plates or rays,
and the longitudinal disposition of the woody fbre, must
be considered, in order to understand the action of seasoning.
For the lateral contraction or collapsing of the longitudinal
fibrous or tubular part of the structure cannot ake place
without first tearing the medullary rays, hence theshrinking
of the woody bundles finds relief by splitting the timber in
radial lines from the centre parallel with the medulary rays,
thereby enabling the tree to maintain its full dianeter. If
the entire mass of tubular fibre composing the tre were to
contract bodily, then the medullary rays would of necessity
have to be crushed in the radial direction to enable it to
take place, and the timber would thus be as muc injured
in proportion as would be the case in crushing th, wood in
a longitudinal direction.
If an oak or beech tree is cut into four quarters, ty passing
the saw twice through the centre at right angles, bfore the
splitting and contracting has commenced, the line a c and
b c in Fig. IO would be of the same length, ancat right
angles to each other, or, in the technical language of the
workshop, they would be square, but after being stored in a
dry place, say for a year, a great change will be ound to
have taken place, both in the form and in som of the
dimensions. The lines a c and b c will still be of be same




Timber.                     Io5
length as before, but from a to b the wood will have contracted very considerably, and the two lines a c and b c will
not be at right angles to each other, the angle being diminished, by the portion shown in black in Fig. io. The
FIG. IO.
medullary rays are thus brought closer by the collapsing of
the vertical fibres.
But, supposing that six parallel saw cuts are passed
through the tree, so as to form it into seven planks, what
will be the behaviour of the several planks? Consider the
centre plank first. After due seasoning and contracting, it
will be found that the middle of the board still retains the
original thickness, from the resistance of the medullary
rays, while the thickness will be gradually reduced towards
the edges for want of support, and the entire breadth of
the plank will be the same as it was at first, for the foregoing reasons, and as shown in Fig. I I. Then, taking the
planks at each edge of the centre, by the same law their
change and behaviour will be quite different; they will still
retain their original thickness at the centre, but will be a
little reduced on each edge throughout, but the side next to
the heart of the tree will be pulled round or bent convex,
while the outside will be the reverse, or hollow, and the




I0o6        On the Strength of Materials.
plank will be considerably narrower throughout its entire
length, more especially on the surface of the hollow side.
Selecting the next two planks, they will be found to have
FIG. II.
lost none of their thickness at the centre, and very little of
their thickness at the edges, but very much of their breadth
as planks, and will be curved round on the heart side and
made hollow on the outside. Supposing some of these planks
to be cut up into square prisms when in the green state,
the shape that these prisms will assume after a period of
seasoning will entirely depend on the part of the tree to
which they belonged, the greatest alteration would be
FIG. 12.                  FIG. I3.
a  a                       c
/ b
perpendicular to the medullary rays. Thus, if the square
was originally near the outside, as seen in Fig. I 2, then the




Timber.                    o07
effect will be as shown in Fig. I3, namely, contraction in
the direction from a to b. After a year or two the square end
of the prism will become rhomboidal, the distance between
c and d being nearly the same as at first, but the other two
edges brought closer together by the amount of their contraction. By understanding this natural law, it is comparatively
easy to predict the future behaviour of a board or plank by
carefully examining the end wood, in order to ascertain the
part of the log from which it has been cut, as the angle of
the ring growths and the medullary rays will show this, as
in Figs. 14 and I5. If a plank has the appearance of the
FIG. 14.
FIG. 15.
former, it must have been cut from the outside, and for
many years it will gradually shrink in the breadth; while
the next plank,'shown in Fig. I5, must have been derived
from near the centre or heart of the tree, and it will not
shrink in the breadth but in thickness, with the full dimension in the middle, but tapering to the edges.
The foregoing remarks apply more especially to the
stronger exogenous woods, such as beech, oak, and the
stronger home firs. The softer woods, such as yellow
Canadian pine, are governed by the same law; but, in
virtue of their softness, another law comes into force, which
to some degree affects their behaviour, as the contracting
power of the tubular wood has sufficient strength to crush




Io8         On the Strength of Materials.
the softer medullary rays to some extent, and hence the
primary law is so far modified. But even with the softer
woods, such as are commonly used in the construction of
houses, if the law is carefully observed, the greater part of
the evils of shrinking would be obviated. Hence also, it is,
that when a round block, as a mast, is formed out of a tree,
it retains its roundness because it contracts uniformly or
nearly so, whereas if a round spar is formed out of a
quartering of the same tree it will become an oval, or otherwise contorted towards that shape.
It would not be in accordance with the object of this
book to enumerate all the woods that are employed in the
arts, therefore a few only are selected, or such as are commonly employed in the United Kingdom for purposes where
strength is the primary object, viz. ash, beech, elm, fir, hornbeam, mahogany, oak, and teak.
Ash.
Ash is a coarse wood, but possessed of considerable
strength, and is distinguished for its great toughness and
elasticity, and is usually employed where severe shocks and
wrenches have to be encountered, such as for agricultural
implements, the felloes and spokes of wheels, and the shafts
of carriages, for hammer shafts, and for spring purposes
generally wherever wood is employed for that purpose.
From its great flexibility it is seldom employed where
rigidity is a desideratum. The combination of strength
with flexibility is the characteristic of ash, and when the
wood is from a young tree, or a tree not too old, it is an
invaluable wood in many respects; but as the tree becomes
older, the change to brittleness sets in and soon renders it
less valuable. It is also remarkable for its endurance when
kept dry, but when exposed to damp or to wet it rapidly
decays. The numerical value of its properties, as will be
seen by the Tables, varies considerably, but in general terms
it may be stated that, as compared with oak, good ash has




Timber.                       109
frequently a still greater tenacity and likewise a greater
degree of toughness, but from its flexibility, especially when
young, it has considerably less stiffness, which unfits it for
many purposes.
Beech.
Beech has frequently considerable strength, and is chiefly
distinguished for its uniformity, its smoothness of surface,
and closeness of grain.  It likewise possesses no little
beauty, and takes a good polish, more especially when its
silver grain is skilfully exposed. When well seasoned and
not too old it is frequently used for the cogs of mill gearing,
and is usually considered by millwrights as next to hornbeam, both in strength, toughness and general suitability
for that purpose. It requires, however, to be kept very dry,
for in damp situations it quickly wears out, but, when beech
is immersed in water constantly, its endurance is considerable. The strength of beech is nearly the same as that of
oak; it is also tougher, but its stiffness is inferior to that
of oak, even to the extent of 25 per cent.
Elm.
Elm, although a cross-grained, rough wood, and mostly
used for rough purposes, is yet held in great estimation for its
toughness and non-liability to split by the driving of bolts.
It is much used in the construction of blocks for pulleytackle, for heavy naval gun-carriages, and for the naves of
carriage-wheels. It is a wood which is little affected by
constant immersion in water, but decays rapidly when alternately wet and dry, and consequently is not very durable
for purposes involving exposure to a wet climate. Its chief
dvefect in ordinary use is its great liability to warp and twist
and get out of form; and, as regards strength, toughness,
and rigidity, it is inferior to oak, as well as in almost every
other respect.




I IO        On the Strength of Materials.
Pine and Fir Wood.
The fir and pine woods are members of a large family,
and are of great variety, and differ much in most of their
properties. These classes of timber, in addition to being
employed for building purposes, are likewise the chief
materials that are used in great works, where the question of
strength combined with cost becomes the most prominent
consideration. The most durable varieties are the larch,
the pitch-pine, and the firs from Memel and Norway, and
are valued mostly on account of the large quantity of resin,
pitch, and turpentine which they contain. The Canadian
pine, variously termed white or yellow, is not a strong wood,
but it is much used by engineers for making patterns or
models, on account of its smoothness of surface, its nonliability to warp, its comparative freedom from knots, and
the facility with which it can be cut. The white or yellow
pine is not nearly so strong or so stiff as oak, yet sometimes
it is almost equal to it in its tenacity and toughness.
In such a large family as that of the resinous firs and pines,
there is almost an equal variation in their strength, toughness, and rigidity. Much of the European wood, such as
that from Memel and from some parts of Scotland, is often
superior to oak, while at other times it is inferior, more
especially in regard to stiffness, as will be seen by the Table.
Hornbeam.
Hornbeam is a wood which is comparatively little used,
except by engineers, for the teeth or cogs of wheels and for
mallets, for which purposes it is perhaps superior to all other
woods, and this is mostly due to its great toughness and remarkably stringy coherence of fibre. Its cohesive strength
and other properties depend much upon its age as a plank,
and still more on the age of the tree from which the plank
was taken. When in the most favourable condition, it is
fully equal to the average of oak (even when considered




Timber.                     I II
merely as a wood), but when cut from older trees, and when
over-seasoned, it is frequently found worthless, and has soon
to be renewed. When of proper age and quality, it has no
equal for its own special purposes.
Mahogany.
Mahogany is a beautiful close-grained wood, but is used
not so much on account of its strength, but more frequently
because of its non-liability to shrink, warp, or twist, and
from the peculiar property of taking a firm hold of glue. In
the last respect it is superior to any other wood. Mahogany
differs greatly in regard to its closeness, hardness, strength,
and beauty. That from Honduras, called'bay-wood,' is
much inferior to that called'Spanish' mahogany, which
comes from the West Indies; the former is much used in
the construction of light textile machinery, but chiefly on
account of its cheapness, and the latter is used for furniture
or for other ornamental purposes. As regards strength, this
wood is inferior to oak in all respects, and its great characteristic defect is unsuitability for exposure to the weather,
or indeed for any purpose where it is made alternately wet
and dry. When so subjected, it rapidly decays, and loses
all its good qualities.
Oak.
Oak, taken as a whole, is one of the strongest and most
durable of woods, and is especially adapted for exposure to
the weather of a damp climate, and is indeed suitable for
almost every purpose where the properties of strength, stiffness, and toughness, combined with endurance, are required.
Its value for ship-building is proverbial, and its employment
for the staves of casks, for trenails, for carriage-wheels, and
for all such purposes requiring lightness and strength in
combination it is equally useful. From time immemorial it
was esteemed the best timber for heavy roofs, and the con



II2         On the Strength of Materials.
dition in which some of these grand old roofs have reached
our era, fully attests the wisdom of the selection.
Oak is found of many degrees of quality, but probably
none, taking every property into account, is superior to that
which grows in England, and which is perhaps more durable
than any other. Some of the foreign oaks are as good in
some respects, but, as a whole, English is the best.
Teak.
Teak is a most valuable wood, and is fitted for most purposes where oak is employed. It is equally durable, but is
more liable to split by the driving of bolts, and is not so
well adapted for a sudden jar or wrench. Its great durability is partly due to the large quantity of oil which it contains; the oil likewise is found to prevent the rusting of the
iron bolts employed in framing it. For gun-carriages or for
similar purposes, its value is increased by the small amount
of shrinkage which takes place, even after long exposure to
a hot climate. In this respect it is superior to oak; and it
has another property, it is not much exposed to attack by
the worm of tropical countries, which is a great advantage.
Teak, as a rule, is superior to oak, both in regard to
tenacity and stiffness, but is inferior as regards toughness;
but when those qualifications are required in combination
with endurance in a tropical country, it has the superiority,
and is therefore preferred.
The foregoing remarks being of a general nature, can only
be considered as preliminary to a study of the Tables which
follow. In past times, an immense number of experiments
have been made upon wood, more especially in regard to
its tensile strength. The next Table contains the result of
experiments made at different times to ascertain the general
range of the ultimate cohesion or tensile strength of the
foregoing descriptions of timber.




Timber.                         II3
TABLE XVIII.
These results have been collected from various sources, but chiefly
from the experiments made by Barlow, Bevan, and Muschenbroek.
Timber.               Ultimate tenacity in lbs. per square
inch of section.
Ash..                from I9,600 to 15,784
Beech....,, 22,200,, 11,500
Elm..,, 4,400,, I3,489
Fir..,,  8,00,, 7,000,, American.               I2,000
Memel..            II,000,, Riga.                                  12,600
Mar Forest...       12,000
Larch..               0I, 200
Hornbeam.                            from 20,240 to 4,253
Mahogany..,, 2,800,, 8,ooo
Oak...,,  I9,800,, 9,000,, English..                     I5,000,, African.              14,400,,  Canadian.                             12,000
Dantzic..                      14, 500
Teak..                         from 15,000 to 8,200
The next Table contains the result of experiments made
to ascertain the average resistance to ultimate crushing of
the foregoing descriptions of timber in the direction of the
fibre, when in the form of short pillars.




I 14          On the Strength of Materials.
TABLE XIX.
These results are mostly derived from the experiments of Hodgkinson.
Timber.             Average resistance to crushing stress,
in lbs. per square inch of section.
Ash.....                           from 9,363 to 8,683
Beech.....,, 9,363,, 7,733
Elm......IO, 331
Fir.....                       from 6,819 to 5,375
Hornbeam..                    7,289,, 4,533
Mahogany                                    8,280
Oak, English                        from Io,o58 to 6,484,, Dantzic                             7,731
Teak                                       12, 101
This Table contains the result of experiments made by
Rennie, to ascertain the average resistance to crushing in
the direction of the fibre of short pillars of timber.
TABLE XX.
Size of specimens, i-inch cubes.
Timber.            Average resistance to crushing stress,
ill lbs. per square inch of section.
Elm....                                    1,284
American pine....,6o6
White deal....I,928
English oak...                  3,860
These experiments show  a considerably less result than
those made by Hodgkinson, and it is scarcely possible to
reconcile the discrepancy between the two Tables; but most
probably it could be easily explained if all the facts of the




Tinmber.                        I 15
experiments were fully known, as everything depends on
the amount or degree of the crushing action  that was
effected in the several experiments.
Timber is sometimes subjected to a crushing force in the
other direction; namely, at right angles to the fibre, as in
the case of a wedge, or when baulks of timber are used to
support great weights, or when two pieces are framed and
bolted together or otherwise. In regard to this important
point, however, there are comparatively few experiments
recorded, and, as a rule, those which are recorded are not
very definite. All are familiar with the fact, that a very
small force will be sufficient to indent wood to some appreciable extent. The precise force that will crush a cubic
inch perceptibly —say   3 ths of an inch-may be said to
range from 500 lbs. to I,500 lbs., according to the hardness
or softness of the specimen.
The following Table contains the results of experiments
made by Hatfield, to ascertain the force required to compress
timber to a depth of ]lth of an inch, in a direction at right
angles to the fibre:TABLE XXI.
Stress in lbs. per square inch
Timber.          required to crush fibres trans-    Specific
versely, and to produce an    gravity.
indentation of JL of an inch.
Spruce fir...            500'369
White pine.                     600'388
Mahogany, Bay-wood.,300'439,,    St. Domingo.          4,300'837
Oak....             I, 900               6  2
Ash....            2,300'5I7
Lignum vite..               5,800               I'282
I 2




i r6       On the Strength of Materials.
Shearing or Detrusion of WMood.
In the application of timber to form framings for machinery, in the construction of roofs of houses, and in
scaffoldings, too much reliance is frequently placed by the
carpenter, upon the support which is understood to be derived
from an abutment against a notch cut into the solid wood,
thereby bringing into play the end shear of a portion of the
material, in the direction of the fibre. From experiments
made by Barlow, the strength of timber thus strained is
only about 2Oath of the tenacity of the same wood in the
direction of its length.
The assumption that wood may be treated like metal in
this respect is wrong.  With metals, the resistance to
tension and detrusion are nearly alike both ways, the solid'joggles' left upon an iron casting give an abutment nearly
equal to the tensile strength of the metal; but with wood
similarly arranged it is only equal to the lth part of the
tensile strength. This deficiency is usually compensated
for by the extra length given to the part which has to be
detruded; still, the weakness of timber in this respect should
be noted.
The following Table refers to the transverse strength of
timber when used as a beam. The results are collected
from a variety of sources; and as the experiments were made
with specimens of varied dimensions, the whole have been
reduced to one standard, for the sake of comparison:



TABLE XXII.
Results of Experiments made to ascertain the Transverse Strength of YTimber.
The experimental beams were of various dimensions, and were also supported and loaded in different ways, their actual
breaking weights are shown in column 4, and the numbers in column 5 are the calculated weights required to fracture
beams of the same timber, I foot long and I inch square, when supported at each end and loaded at the centre.
I                 2                3            4         5              6                  7
The experimental beams.                   Calculated
breaking
weight in
Position of the  Breaking lbs. of a beam  Authority.       Remarks.
Timber.         Dimensions.       load and the    weight   I foot long
supports.    in lbs.   and I inch
square.
r 7'  long x 2" square  Supported at    772        675     P. Barlow.
4'2",,    2",,    bothendsand  I,304         679     P. W. Barlow.   Dry.
loaded at the   324
2,6',,   x I'/,,     loaded at the    324        8IO Tredgold.             From a young tree.
Ash..  | 2' 61',X x I,,   centre.         254        635     Ebbels.           Medium quality.
2' 6"    X I          Supported at     314        785     Tredgold.
2',,  X 2"deep   I one end and    326           652     P. Barlow.,, and I"wide J   loaded at the
5',,  x 2" square   other.            239        595    Peake & Barrallier.
Supported at
7'  long x 2" square  Suppoth ends and    593    518      P. Barlow.
2' 6",,  x I/,       loaded at the    271        677     Ebbels.           Medium quality.
Beech.                           centre.
2'       x 2"deep     Supported at
f/      one end and    352         704
de3  x 2" Sare       loaded at the                     P. Barlow.',, x 2square J other.      401        6o




TABI,E  XXII.-(continued.)
00
I                  2                 3              4           S               6                   7
The experimental beams.                      Calculated
breaking
-   weight in
Position of the  reaking lbs. of a beam   Authority.           Remarks.
Timber.    |    Dimension            load and the     weight   and f  inh
supports.      in lbs.    ar
7' long x 2" square  end  a. Barlow.
longx 2"          Supported at both    386         337     P. Barlow.
4' 2,,    2                cen66                    343                          the same tree.
Fir              4.  "  >~ 3"  "     at the centre.
(American)                         Supported at one                          Fincham.           Red pine.
2',,  x 3",,      end  and  loaded  I,753         59
___L~ _ _l~_~_  (lat the other.                                                        
Fir.       (   2' long x I" square  Supported at both    343        686      Tredgold.
F   7' long x 2" square  Supported at both    325        284     P. Barlow.
6                                        FSupported at one
(6Supported at one    129         322                         Very young wood.
5,, x 2,,         end  and  loaded    i62         405    Peake&Barrallier. Dry.
at the other.




Fir                              ( MSupported at both
Frest    7' long x 2" sqare  ends and loaded   436            381    P. Barlow.
ar  )  at the centre.
Fir.        ( 4' 2' long x 2an square  ed  and loaded I, Io8    577    P. W. Barlow.    Dry.
2(Memel)  6",     x Iat the centre      218      545    Tredgold.
7' long x 2" square Supported loadedt both  422  369    P. Barlow.
2t 6"/    x V        ends and loaded
Fir.       2' 6",, x 1'        at the centre.               530    Tredgold.
(Riga)        4',,      2,   Supported at one   2IO       420    Beaufoy.
5',, x 2",,    end and loaded   153          382,   Peake&Barrallier. Dry.
2',, x 3"        at the other.     1,974      584    Fincham.          Dry.
Oak..(                          (Supported at both                          --'     7' long x 2" square  ends and loaded   526      460    P. Barlow.
at the centre.
Oak,        (  4' long x 3" square ISupported at both  3863    572    Fincham
(Canadian) {          x 2"
2' long X 27 square  ends and loaded  1,643       855    P.W. Barlow.
at the centre.589    P. Barlow.
Supported at both
Oak.        4',, x 3",,    at the centre.       4,450     659    Fincham.
(Dantzic)           { 4,        Supported at one
4',, x 2",,     end and loaded   196        392    Beaufoy.
at the other.




TABLE  XXII.-(continued.)
The experimental beams.                     Calculated
breaking                                            0
weight in
Position of the  Breaking lbs. of a beam  Authority.  Remarks.
Position of the  Breaking  I foot long
Timber.         Dimensions.         load and the     weight   and f  inch
supports.     in lbs.    square.
square.
7' long x 2" square                     637        557    P. Barlow.
2',, x I'         Supported at both   482         964X  Tdold.              From a young tree.    a
2t,, x I           ends and loaded    218         436  1   redg             From an old tree.       A
Oak..   2' 6"      X",,       at the centre.      284        7Io     Ebbels.           Medium quality.
(English)     4' 2  x 2"                       999       520    P. W. Barlow,    Fast grown.,,        4 x2  Supported at one'66     B32,x 2",    end  and  loaded    266         532  Beaufoy.
4,,,10O                                420
at the other.
onduras) 2' 6 longx I" square  Supported at both    255             637Seasoned wood.
ahogany    1'                       ends and loaded                        Tredgold.         Seasoned wood.
(S2pani'sh)    6",, xt            at the centre.      170        425
Mahogany 
(New South      4'      x3",,                        4,119        6I0    Fincham.
Wales)                                               ___' 6",,
7' long x 2" square Supported tboth    938         20      P. Barlow.
4',, x 3"          ends and loaded  4,897         721    Fincham.           Malabar.
Teak.. 3',, x                            xi",,    at the centre.  1,23,075    Mayne.  Johore.
e' an    x 2. C    Supported at one
l5',, x 25" >,     end and  loaded    257        642    Peake&Barrallier. Old and dry.
{ at the other.




Timber.                        I21
The attention of the student is particularly directed to
column 5 of the foregoing Table, which will furnish him
with much useful knowledge on the transverse strength of
wooden beams, in a small compass, and will enable him to
calculate either the strength or required dimensions of any
other rectangular beam of any of the kinds of wood therein
specified.
In Chapter XII., upon the strength of structures, the
principles which regulate the strength of such beams are
fully stated. It will be sufficient here to remark, that by
taking the beam in column 5, which is I foot long and
i inch square, as a standard, the student can ascertain the
strength of any other rectangular beam, by simply multiplying the strength of the standard beam by the breadth,
and the depth squared, both in inches, and then dividing
the product by the length, in feet, of the given beam, the
strength of which he wishes to ascertain.
Table showing approximately the mean breaking weight
of beams of timber, I foot long and I inch square, supported
at both ends and loaded at the centre, deduced from the
experiments enumerated in Table XXII.:TABLE XXIII.
Mean breaking                 Mean breaking
Timber.     weight of bevm    Timber.      weight of beam
in'.s.                       in lbs.
Ash..  690     OakBeech..     625,,  Adriatic         460
Elm...    405,,  African.     855
Fir-American.    524,,  Canadian         580
Christiana.    574,,  Dantzic. j      3,, Larch..    44,,  English.     591,, Mar forest.  381     Mahogany.      557,, Memel.           56      Teak..      814,, Riga.        457
The above Table can only be considered as available for
approximate calculations.




122         On the Strength of Materials.
CHAPTER IX.
TRANSVERSE STRENGTH OF IRON AND RESISTANCE TO
IMPACT.
THE transverse strength of materials, more especially that
of cast iron, in the form of beams and girders, has been
closely investigated by a number of scientific men, particularly by Sir William Fairbairn, Mr. Hodgkinson, and the
Railway Commissioners.  The subject will be treated of in
Chapter XII.; but it will be convenient to give here an
abstract of the results of various experiments, on the transverse fracture of bars of different materials.
A bar laid horizontally upon supports and loaded at the
centre bends, and ultimately breaks, by transverse strain.
The two points most important to be observed are the
breaking weight and the deflection of the bar. Some results,
obtained in experiments of this kind, are given in the subsequent Tables.
The following Table contains a synopsis of the results of
a series of experiments, carried out under the direction of
the Commissioners appointed to enquire into the Application of Iron to Railway Structures. Their object was to
determine the transverse strength and ultimate deflection of
cast-iron bars, when subjected to a statical load placed at
the centre:




Transverse Strength &' Resistance to Impact.  123
TABLE XXIV.
Size of  Distance   Weight       Ultimate
Name of Iron.   bar in   between   of bar be-   weight  deflection
inches.  suppots  tween SU       in lbs.  in inches.
in feet.  ports in lbs.
I3'49      46I   I1796
Blaenavon      I inch      1        3'34       359    I'315
No. 2.       square.    4        I3'02       437   1'85
13'6      423    I'69 7
Mr.   Stirling's.
Calder No. I,                      III07    2,4II  32I3
with  No.i   2 inches          11441      1,837    2'227
with   20   per  square.  9       11'46    2,083   2'395
cent. of malle-square.                        2,364   2 
able iron scrap.                                           )
Io7'6     1,249   2'906
Blaenavon     2 inches           Io096       1,414   3'486
No. 2.       square.     9      10o8.      I,I21 2'527
1o8 o     1,097   2 498
358'29     2,698   4'863
Blaenavon     3 inches   I1       365'39     2,67I   43908
No. 2.       square.     3      367-29     3,389   5'024
365 o6    2,686   4'391
Abstract of the results of experiments made by the above
Commissioners to find the relative transverse strength of
cast-iron bars, of various sizes, but similarly proportioned
in all their dimensions:
Note. -The horizontal pressures in the following Table were computed from the vertical pressures, by taking into account the weight
of the bar itself, which would not have any tendency to weaken the
bar when pressed in the horizontal direction, but of course acted as so
much additional load when the bars were pressed vertically.




I24            On the Strength of AIaterials.
TABLE XXV.
Horizontal pressures
Vertical pressures.   computed from the
vertical pressures.
Size of bar.
Mean ul-              Mean ulMe imate de- timate destrength  flectioni  strength  ectionin
in bs.              lbs.
in lbs.  flection in  infl lection in
inches.           inches.
3 bars 4   feet span  and
I inch square.          440'79   i- 
6  bars 9 feet span  and   1,338   30035    1,394    3-26
2 inches square.   
4 bars  132 feet span and   2,861   4'667    3,043    4 966
3 inches square.
3 bars 63 feet span  and   6   7   I29I6   6207    I3II
3 inches square.
This Table shows, first, that the strength of the similar
bars, I inch, 2 inches, and 3 inches square, and 41 feet,
9 feet, and I3- feet span, to resist a horizontal pressure, are
respectively 447, I,394, and 3043 lbs.; but if the elasticity
of the bars had been perfect, their strengths should have
been as the squares of their linear dimensions, namely, in
the ratio of the numbers I, 4, and 9. Dividing the strengths
by these numbers, the quotients ought, on the supposition
of perfect elasticity, to be equal. We find, however,
447+I = 447 for I inch bars.
1,394 4  = 349 for 2 inch bars.
3,943 +9  = 338 for 3 inch bars.
The quotients being unequal, and the greatest deviation
being in the case of the smallest bar. Probably this is due




Transverse Strength &  Resistance to Impact.  125
to the more rapid cooling of the small bars, which tends to
increase the strength  and  hardness  of the  metal.   The
Table shows, also, that for square bars of constant length
between the supports, the transverse strength varies nearly
as the cube of the side of the square.
Abstract of experiments  made for the  Railway  Commissioners on the resistance of cast-iron bars to long-continued impact, from a ball striking horizontally against the
middle of the bar:
TABLE XXVI.
Distance Side o Weight Velocity of Assigned  Number
between squareof ball imp ftin  deflection    of  Effect.   Remarks.
supports inches. in lbs.  feeont per  in inches.  blows.
3    5I4  8'812     or I'5   1,085   broken.   slightly
3    I5I4   9'4899 1  or' 5   4,000 not broken.
3   603    3'6156   or I 5  4,000,,
I3 ft.   3   603    4-5754 1or I-875 1,35o   broken.   slightly
defective.
3    I5I4  13-78341 or 225    127         " 
3   603    5'0839  1or 225  3,026,,
2     753   6'2I32      or I'00  4,00o0 not broken.
2   603    I'807  1  or I'100  4,000,,
2    75a       1  I  or I-50    229   broken.  defective.
9 ft.         75    9'3204  1 or I'50  1,282,,      slightly
2                                                defective.
2   603    2'5284  i or I'50  3,695,,
2    75.  I'I693 ]  or 2 00    127        "
2   603    3'5872   or 2-00    474
I    751   2I'46I  3 or'50   4,000 not broken.
4 ft. I    75    3'0368  2 or'75   4,000,,
4. f.1     75    3'50I8 -    or'875 3,7o0   broken.   slightly
l     3,     1                            defective.




126           On the Strength of Materials.
One bar only, and that a small one, stood 4,oo00 blows,
each blow bending it through half its ultimate deflection with
a statical breaking weight; but all the bars, when sound, stood
that number of blows, each blow deflecting them through A-rd
of their ultimate deflection. A cast-iron bar will, however, be
bent through ~rd of its ultimate deflection with less than ~rd
of its breaking weight, laid on gradually, or {th of the breaking weight laid on at once-hence the prudence of making
beams capable of bearing more than six times the greatest
weight which will be laid upon them.
Abstract of results of experiments on vertical impacts,
upon unloaded and loaded beams of cast iron (Blaenavon,
No. 2), 13 ft. 6 ins. between supports and 3 ins. square; the
falling weight being 303 lbs.:TABLE XXVII.
Height of  Velocity of
Weight of                              fall necessary impact due
beam      Additional load on beam in lbs.   to break the  impactdue
in lbs.                                 beam in    tht
inches.    height.
3763                 nil.                 281
382            4 lbs. at centre.          33        I3'3
375k           28 lbs. at centre.         42        I5'0
387    I66 lbs. spread over the beam      48        i6'o
+ 4 lbs. at centre.
386    3894 lbs. spread over the beam     48        I6-o
+ 4 lbs. at centre.
379    391 lbs. spread over the beam      66        I8'8
+ 4 lbs. at centre.
382    956~ lbs. spread over the beam     60        17'9
+ 4 lbs. at centre.
It will be seen that loaded beams resisted better than
unloaded ones.
The deflections were found to vary nearly as the velocity
of impact.
The sets were very great, but did not appear to injure the
strength of the bars more than in ordinary cases, with an
unloaded beam.




Transverse Strength &  Resistance to Impact.   I27
Synopsis of experiments on the transverse flexure of five
bars of Blaenavon iron, No. 2, cast to be 3- inches x Ix
inch in section, and I3 feet 6 inches between supports,
the pressure being applied in the direction of their least
dimension:
TABLE XXVIII.
Weight applied   Mean deflection    Mean set after
acting horizontally  after 5 minutes  5 minutes in  Remarks.
in lbs.       in inches.       inches.
28              -18I'ooi6
56'3754'oo68
I2'7686'OI92
I68'I84              0468
224             I'632'0914
280            2'I05' 486     Limit of elasticity,
336            2'604'2266        1 of ultimate or
392            3'169            -3292       breaking weight.
448             3'756'4574
504            4'402'6078
560             5'035'7854
616             5'777          1'038
672             6'565          I1287
728             7'6I0          I 707
784            8'73            2'I86
840             9'887          2'69I      Nos. I,3,4, 5, broke.
896           Io'7 (No.2)      3 006
934              -          No. 2 broke.
Similar experiment with  bars 2 inches x  I inch, and
9 feet between supports:TABLE XXIX.
Weight applied Mean deflection after   Mean set after 5
aclly ing horizon- 5 minutes in inches.  minutes in inches.
Limit of elas28'235'002          ticity, I  of
485       mean breaking weight of the sound    ultimate  or
bars.                  breaking
strength.




I28         On the Strewngth of.MZaterials.
These Tables show the apparently very great want of
elasticity of cast iron: a permanent set being attained in one
case with I th of the ultimate breaking weight, and in another
case with Il7th of the ultimate breaking weight.
The following results are intended to show the effect produced by repeated deflection-the experiments having been
made for the Railway Commissioners.
Cam experiments on the deflection of cast iron bars, I3
feet 6 inches long, and 3 inches square:The step cam gradually deflected the bars to the required
amount, and then allowed them to spring back instantly to
their original position, so far as they would do so.
The rough cam was simply a toothed excentric, working
into a rack fixed upon the bar, it deflected and also'allowed
the bar to return to its original position, gradually, and
imparted, by its roughness, a highly vibratory motion to
the bar.
First Experiment.-Three bars of No. 2 Blaenavon iron
were subjected to 0o,ooo deflections by the rough cam, each
deflection amounting to that caused by ~rd of the statical
breaking weight. The permanent set at Ioo deflections was
found to be'2 inch,'I7 inch, and'I9 inch respectively, and
there was no increase in the permanent set after Ioo deflections. These bars required the same weight to break them
after the experiment, as similar bars which had not been so
treated, showing that they were uninjured by the experiment.
Second Experiment.-This was similar to the first experiment, but the deflections were made with a step cam. The
permanent sets at I50 deflections were'I6 inch, -I3 inch, and'I2 inch. No increase was observed after I50 deflections, and
the bars were apparently not weakened by the experiment.
Third Experiment.-Similar bars to the above were deflected to the amount caused by ~rd of the statical breaking
weight, by the step cam.




Transverse Strength & Resistance to Impact. I.29
No. I bar broke after 51,538 deflections-(fracture good)
-bar weakened by experiment.
No. 2 bar broke after 25,486 deflections-(flaw).
No. 3 bar not broken after Ioo,ooo deflections-bar not
weakened by experiment.
Permanent set at 150 deflectionsi'08 inch,'2 inch, 0o8
respectively-no further increase.
Fourth Experinment.-Two similar bars of Clyde iron, No.
3, deflected to the amount caused by a the statical breaking
weight by the rough cam.
No. I bar not broken after 30,ooo deflections —(bar not
weakened).
No. 2 bar broke after 28,602 deflections.
Permanent set at I,ooo deflections,'35 inch and'37 inch
respectively-no further increase.
Fifth Experiment.-Three bars of No. 2 Blaenavon iron,
deflected to the amount caused by 1 the statical breaking
weight by the step cam.
No. i bar broke after 617 deflections.
No. 2 bar broke after 490 deflections.
No. 3 bar broke after 900 deflections.
Permanent set at I50o deflections,'44 inch,'37 inch, and
37.inch, no further increase.
Sixth Experiment.-Bar of wrought iron, 9 feet long and
2 inches square, showing statical weight required to obtain
certain deflections:TABLE XXX.
Deflections, Weights, Permanent        Remarks.
inches.   lbs.     set.'333     507      O      After the bar had 1,950 lbs. on, it'666     926      o        suddenly gave way; it did not
*833    I, 121    0        break, but no further weight
I00ooo   1,364'054        could be applied with certainty.
1 8      1,950'86
K




130           On the Strength of Materials.
Seventh Experiment. —This was a valuable experiment; it
showed that a wrought iron bar, 9 feet long and 2 inches
square, deflected to'833 inch (about 5ths of the strain that
permanently injured a similar bar), withstood Ioo,ooo deflections: permanent set 1 5 inch; bar uninjured.
Esghth Experiment.-Five similar bars, deflected by the
step cam:TABLE XXXI.
Amount of      Number of
No. of bar.  deflection.    deflections.    Permanent set.
inches.
I'33          10'000           O
2             *66          I0'000           O
3             *83          I 0000           0
4            1*00          I 0o000           o06
5           2 00              Io'3
50'54
I00oo'69
150'84
200            98
2C>0'98
300           1'84
CHAPTER X.,
RESISTANCE TO TORSION AND SHEARING.
THE torsional resistance of different materials has not received so much investigation as the tenacity, compressive
resistance, and transverse resistance of materials, and the
results of the few recorded experiments are very discordant.
Still there are sufficient data to show the relative value of
the different metals, and their approximate resistance to
torsion.
It is chiefly in designing the shafting of mills and factories
that we require to know the torsional strength and stiffness
of bars. Such shafting consists almost always of cylindrical




Resistance to Torsion and Shearing.         131
bars of greater or less length, and it is the agent by which
the power of the steam-engine or other motor is distributed
to the machinery, spread frequently over a large area. But
the crank shafts of engines, the propeller shafts of screw
steam  vessels, the spindles or shafts of cranes, and many
other parts of machines are also subjected to torsion.
The most important result of the experiments on torsion,
which should be remembered in designing shafting, is this,
that the strength of cylindrical bars, when twisted, is proportional to the cubes of their diameters. Hence, if we know
the torsional strength of any one cylindrical bar, say of i
inch in diameter, when made of cast iron, wrought iron, or
steel, we can calculate from it the strength of shafts of other
dimensions of the same material. If the strength of such a
shaft or bar of I inch in diameter is represented by I, that
of a shaft 2 inches in diameter will be equal to 8, of 3 inches
to 27, and so on. In the case of hollow shafts, by simply
cubing the exterior diameter and then deducting the cube
of the interior diameter, the difference will give the relative
value of the shaft, as an agent to transmit power or motion.
The torsional strength of wrought iron has received more
attention than that-of any other material, because modern
shafting is chiefly made of that metal.  We may therefore
indicate, first of all, the results obtained in experiments with
wrought iron.  Many experiments on torsion have been
made on cylindrical bars I inch in diameter, the load being
applied by a lever, the length of which is I 2 inches measured
from the centre of the bar.  We may therefore express the
relative resistance to torsion of different bars by stating the
weight which twists them asunder when applied in the way
indicated. The amount of load which is required to produce rupture is usually reckoned at an average of I,ooo lbs.
on the end of the lever, but will necessarily depend on the
quality of the iron as well as other conditions.  In one
instance, where a bar or strip of iron was cut out of a welded
coil, across the welds, rupture took place with less than
K2




32        Oln the Strength of Materials.
400 lbs., no doubt in consequence of a defective weld. The
highest torsional strength here quoted, namely, I,ooo lbs.
acting at a leverage of one foot, is derived from experiments
made with specimens of an exceptionally good quality of
wrought iron; 700 lbs. to 8oo lbs. is probably nearer to the
average strength of good common wrought iron.
Wrought iron is better for shafting than cast iron, because
of its greater torsional strength. In consequence of the
greater strength of wrought iron, shafting made of that
material is lighter than cast-iron shafting, and this is a
matter of great importance, because the friction of shafts on
their bearings is directly proportional to their weight. As
will be shown, however, presently, the stiffness of wroughtiron shafting is not much greater than that of cast-iron
shafting, and permanent set commences with less stress in
the former case than in the latter.
Cast iron, even more than wrought iron, varies in quality.
From some experiments made with specimens of cast iron,
the torsional resistance was equal to 9oo lbs. on the end of
the lever; but the highest result obtained at Woolwich, with
a specimen which was considered a particularly good sample
of cast iron, gave only 670 lbs., and there are some kinds of
cast iron which would not afford a resistance equal to the
half of even that weight. Hence it may be convenient to
assume the torsional strength of a cylinder one inch in diameter of good cast iron as equal to from 650 lbs. to 750 lbs.,
according to the quality and the conditions of the casting.
Some experiments were made in America with cast iron
under torsion, the bar being 8 diameters long. The force
required to give to the bar a permanent set of 2 was equal
to 7 ths of the force required to twist it asunder; the same
amount of set is given to wrought iron with a less stress, or
about 6 ths of the ultimate twisting force.
From some experiments made by Sir W. Fairbairn with
cast iron, some specimens were higher, and some were lower,
bat the mean was 733 lbs. on the end of the 12-inch lever.




Resistance to Torsion and Shearing.          133
Cast steel, being superior to wrought iron in most other
respects, is also stronger in its resistance to torsional stress.
From some experiments which have been made with steel
of high quality, it was found that a cylinder one inch diameter required I,9oo lbs., acting at a leverage of one foot, to
twist it asunder. Such a high result, however, must be rare,
and, judging by the result of an experiment made with part
of the tempered steel lining of a gun, which only required
1,355 lbs., we may safely consider the value of the average
quality of steel, as much less than 1,500 lbs. on the I2-inch
lever; that of good Bessemer steel being I, 150 lbs.
Cylinders of wrought copper, one inch in diameter, require
a load varying between 400 lbs. and 450 lbs. on the end of
the 12-inch lever.
The foregoing remarks refer to the ultimate torsional or
breaking strength; but, as before remarked with regard to
the other properties, the limit of torsional elasticity is the
chief point for the engineer to keep in mind, regarding
which, however, there are not sufficient reliable data on
which to lay down simple rules. Hence, it is usual not to
load shafts with more than Ilth of the weight that would be
required to produce ultimate rupture; there are some, however. who are satisfied with the half of that margin.
The torsional stiffness of a long line of continuous shafting
is dependent on the length; but the length of the line affects
the amount of twist only in this way, that the total twist of
the entire length is the sum of the twists of each portion of
the length. The amount of twisting of a given length is
proportional to the twisting force, so long as that force does
not exceed the elastic limit of the material.  If it does, a
permanent twisting is produced, similar to the permanent
set in experiments on tension.
In the case of long shafting, the torsional stiffness is of
more importance, practically, than the torsional strength. If
the shaft is deficient in stiffness, so that when working it
is twisted through a large angle, it runs in a jerky manner,




I 134        On the Strength of Materials.
and the machines driven by it work badly. In long shafting,
it is usual to secure sufficient stiffness by restricting the
angle of torsion to some definite limit, say to 4~ per yard
length of the shaft, and to secure this amount of stiffness a
larger shaft is often required than would be needed, if
strength alone were considered. The resistance of shafts of
equal stiffness, in this sense, is proportional to the square
of the area-that is, a 2-inch shaft will transmit i6 times
the force which would be transmitted by a I-inch shaft
without being twisted through a greater angle.
In the shafts or spindles of cranes, torsion, and not stiffness, is the prominent point, on account of their shortness,
which does not give sufficient play to elasticity to make the
want of stiffness objectionable.
\When a tie-rod, or a strut, or a long shaft, is so constructed
that certain parts have a larger or smaller diameter than
other adjoining parts, it is evident that the strength cannot
be greater than that of the weakest portion. In practice,
however, it is found that if the small part of the shaft passes
abruptly into the larger part, it is even less strong than its
dimensions would indicate, or rather, it is more correct to
say, that fracture takes place with a less load than in an
uniform bar, of the same section as the smallest part of the
shaft. More especially is this the case with the bearings of
shafts, if the re-entrant angle of journals or corner of union
is left square, instead of being rounded off. This discrepancy
arises from two obvious causes: first, because the small part
is the weakest portion of the bar, and consequently has more
than its share of the elastic work to perform; and secondly,
because at the junction of the two diameters, the fibre or
molecules do not spread out to take hold of the larger
diameter. In practice, it is found that this defect is obviated
to some extent by rounding the corner, and the risk of
fracture is thus reduced.
In considering the diameter to be given to a shaft, the
pull or transverse strain which will come upon it in the




Resistance to Torsion and Shearing.       135
middle of the distance between the bearings must also be
taken into account. A comparatively small shaft may be
strong enough to transmit the necessary power, yet not
strorg enough to bear the stress of several tight straps,
causiag deflection in addition to twisting, and hence the
strength of the shaft has so far to be reckoned as a beam,
irrespective of torsion, and a suitable diameter provided
accorCingly.
The results of experiments made with round bars of i-inch
diameter, and with a lever of 12 inches in length, have been
given, mnd it has been stated that the strength of other bars
is to that of a I-inch bar as the cubes of the diameters. It
has alsoto be observed, that the resistance of the shaft to a
given Rad is inversely as the radius of the lever. With
a lever Cinches long, or half the standard length of 12 inches,
twice th load must be applied to break the bar. With a
lever 24 nches long, or double the standard length, only half
the load would be sufficient to break the bar. Keeping
this in vew, we may compare the torsion produced by
wheels orpulleys of different diameters.
In pracical operations, it is seldom that the engineer can
obey the iws of correct proportion as strictly as he would
desire, on ccount of other considerations, which step in to
modify his)roceedings. If, for example, a long shaft were to
be construted, of sectional area in exact proportion to the
strain anticpated at each point, the shaft ought to be largest
at the drivig end, and gradually diminished by tapering down
to the othe extremity. In practice, however, this would be
found incavenient, because the arrangement involves a
multifarious assortment of pulleys, with bores of different
diameters, Ir the reason that, in the conduct of a factory,
such pulley have frequently to be transposed from one
point of thehaft to another point, in order to suit the varying circumtances of the machinery to be driven. To
obviate thiSdifficulty, true proportion in the shaft is con



I 36        On the Strength of Materials.
stantly departed from, and the shaft usually made of the
proper strength at the driving end, to transmit the required
amount of power, and then reduced by stages, at, say every
hundred feet, or even less. This implies greater cos: for
shafting in the first instance, but the future facility which it
affords is more than an equivalent to the original outlay.
In the working of a long line of shafting, the elasticity is
observable, and even becomes inconvenient and proninent
when the shaft is made too light. The following inte'esting
experiment was made, in connection with the driving If such
a long line.  The shaft was originally driven frcn one
end only, and afterwards means were adopted t) drive
from both ends; but the irregularity of the motiak, with
the new arrangement, gave rise to a consideral1e disturbance in the middle of the shaft, which made he new
arrangement equally unsatisfactory with the previous one.
The shaft was then cut in the middle, and driven at he same
velocity from both ends. Long pointers were fasened to
each shaft at the point where they came together. in order
to show the divergence of their motion; the caue of the
previous disturbance was soon made evident, fo the two
bars, although running at the same speed in revoltions per
minute, did not keep true time in their twisting the two
pointers were seldom  together, being sometimrs to the
extent of half a revolution apart, sometimes oe, and at
other times the other, in advance, the cause of csturbance
being the ever-changing strain, due to variatios in the
stress of the work done at various points; stihess and
steadiness can only be attained by an increase osubstance
beyond that necessary for strength; the line wa evidently
deficient in stiffness, notwithstanding its abudance of
strength.
When a bar of iron is wrenched asunder by irsion, the
break or rupture must be considered as a shear. Where a
given area of metal has to be sheared, it is foufd that the
force required is nearly the same as that necestry to tear




Resistance to Torsion and Shearing.        I37
asunder a piece of the same material of equal area. The
shearing force, however, is rather less than the tenacity,
therefore, to shear a square inch bar would require a force
of between I8 and 24 tons, according to quality.  The
torsional strength of the bar may be calculated from this
value of the shearing force, but practically the question does
not present itself in this form, and it is more simple to consider torsion in connection with forces acting at the end of
a lever.
It will thus be seen that the strength of shafts to resist
torsion depends on four conditions: first, on the strength of
the material; second, on the diameter of the shaft; third,
on the length of the lever employed in twisting; and fourth,
upon the force or weight applied at the end of the lever.
The following Table shows the values assigned in the foregoing remarks as the ultimate strength of a bar I inch in
diameter, the weight being applied to a lever I2 inches
long.
TABLE XXXII.
High...        1,900 lbs.
Cast steel.    Ordinary..,500,,
Mild...       1,355,,
Bessemer...        I,150,,
Wrought iron..                   700 to I,ooo,,
Cast iron....   650 to  750,,
Wrought copper....      400 to  450,,
The following Table conveys some precise knowledge in
regard to torsion:



00
TABLE XXXIII.
Resistanice of Mfetals to  Torsion,.
Permanent set.
Weight on                                      Ultimate
Length  end of lever   Ultimate    I turn being = to I.   torsion 
Size of specimen.    Nature of material.     of    required to    break       ing                   I turn    Authority.
lever.  destroy elas-  weight on                         being
ticity.    end of lever. Weight on Proportion equal to I.,
end of    r
leverd    of set.
lever.'
lbs.         lbs.       lbs. 
Mild  homogeneous                  l
Diameter I inch.        cast  steel  tern-              552          1,355       785'058'602    Kircaldy.
pered in oil.        i2 ins..
Length  of twisted
part 4 diameters.
Strong cast iron.                -            670         536        003        --     Woolwich.




Resistance to Torsion and Shearing.         139
Table collected from various sources, showing the relative strength of metals to resist torsion, wrought iron
being i.TABLE XXXIV.
Wrought iron.                            Ioo
Cast iron..    90
Cast steel.                              I'95
Gun-metal.50
Brass.46
Copper..'43
Tin..4.'I4
Lead.I.o..... IO
The above relative values must be considered as approximate only, because the resistance of different specimens of
the same material varies more or less, according to its
quality.
A few examples of the strength of crane shafts will be
given, in the Chapter on Complex Structures. The nature of
the stress to which such shafts are subjected is more like
that which is produced in experiments, on the absolute
strength to resist torsion, than is the case in mill work, where
strength and stiffness become complicated with the question
of motion and velocity. The strength of a shaft required to
transmit power depends entirely upon the speed at which
it is driven; a shaft that will convey io horse-power at
a velocity of 50 revolutions per minute, will be strong enough
for 30 horse-power at I50 revolutions, and so on in proportion.
Example.-To find the horse-power that can be transmitted by a good wrought-iron shaft, 3 inches diameter,
driven by a wheel 2 feet in diameter, at a speed of 150
revolutions per minute. From the Table, we find that a
I-inch shaft will break with a stress of 8oo lbs. acting at
the end of a I2-inch lever, or at the pitch-line of a wheel
2 feet in diameter, and as the strength increases as the cube




I40         On the Strength of Materials.
of the diameter, a 3-inch shaft will break with 2I,6oo lbs.
acting at the periphery of the driving wheel.
Let it be assumed that we are not to strain the shaft to
above Ith of the breaking-weight, then we may have a constant load of 2  6~ —3~ =3,6oo lbs. acting upon the wheel.
The wheel, in making one revolution, travels over 2
x 3-I4I6=6'28 feet, and in running one minute 6'28 x I50
=942 feet.
Then the number of foot-pounds that can be transmitted
by the shaft will be the load of 3,600 lbs. multiplied by the
distance through which it travels-namely, 942 feet; thus,
3,600 x 942 =3,39I,2oo00, and this number divided by 33,000
— I02'76, which is the number of actual horses-power that
can be transmitted by the shaft, under the given conditions
of speed and factor of safety.
In the foregoing example, it is assumed that the load upon
the driving-wheel is constant, which is true for ordinary shafts,
but, in the case of an engine crank-shaft, such a condition
cannot hold, because, even when the pressure on the piston
is uniform, there are some points in the revolution at which
the twisting force is practically equal to the full load upon
the piston, while at other points the twisting force is nil,
although of course the horse-power exerted by the piston
and crank are equal.
The average pressure upon the crank is equal to the
pressure on the piston, multiplied by the distance which it
travels in making a double stroke, divided by the distance
through which the crank-pin travels in the same time; now,
the piston passes through a distance equal to twice the
diameter of the circle described by the crank-pin, whereas the
crank-pin travels over a distance equal to 3'1416 times that
diameter, and hence the mean strain on the piston equals
3.1416 = I57 times the driving pressure upon the crank.
The varying strain thrown upon the crank-shaft is made,
approximately, uniform by the fly-wheel, and hence second




Resistance to Torsion and Shearing.       I41
motion shafts may be said to be uniformly loaded throughout, but it is evident that the portion of the crank-shaft,
between the crank and the fly-wheel, has to withstand a
greater strain than the part qrn the Qther side. of the flywheel, in the ratio of I' 57 to I, and it has to be made stronger
to resist it.
In the case of engines working expansively, the horsepower is governed by the mean pressure upon the piston,
but the crank-shaft has to be of sufficient strength to withstand the greatest pressure on the piston.
For example, if we had an engine working with the full
pressure of steam during the whole of the stroke, and another of precisely equal power working with the same steam
pressure at the commencement of the stroke, but cut off at
- of the stroke, the mean pressure on the piston of the latter
engine would be'846 of that upon the other piston; and
hence to enable them both to perform the same amount of
work, the area of the piston of the latter would have to be
Ii 8 times greater than that of the former, and the maximum
strain on the crank-pin and crank-shaft of the expansive
engine would be I'I8 times that upon the crank-pin and
shaft of the other engine, although the horse-power transmitted through the two shafts is precisely equal in each
case.
It is necessary, therefore, in ascertaining the. proper size
for a crank-shaft, to find first the maximum pressure brought
to bear upon the shaft, then multiply that pressure by the
number of times the breaking strength is to exceed the
working strength, and find the size of shaft, as explained for
the ordinary shaft.
Example.-Required the diameter of the crank-shaft for a
horizontal engine, which is to be worked with a pressure of
45 lbs. per square inch throughout the stroke, the diameter
of the cylinder being 36 inches, and the stroke 5 feet, and
the breaking-strength to be six times the working-load.
I. The maximum pressure on the crank-pin, exclusive of




142       Onz the Strength of Materials.
friction, will be the pressure per square inch upon the piston
multiplied by its area, thus, 45 x 362 x'7854=45,804 lbs.
2. The breaking-strength =45,804 x 6=274,824 lbs. The
leverage at which this maximum pressure acts is -=2- feet,
and the weight required to break a i-inch bar when acting
at the end of a 2k-feet lever =-~%= 320 lbs.; but we have
to provide for a weight of 274,824 lbs., and hence we must
have 27-4244-=-859 times the resisting power of a I-inch bar,
and as the strength varies as the cube of the diameter, the
cube-root of 859, which is equal to 9'5, will be the diameter
in inches of shaft required.
To take another example. Let it be required to find the
diameter of the crank-shaft, for an engine equal in power to
that in the previous example, worked at the same pressure
at the commencement of the stroke, but with the steam cut
off at half stroke, the length of stroke of the piston being
equal in both cases.
It has been shown that the area of the piston, and hence
the maximum pressure upon the crank, would be I-I8 times
greater than in the former engine; therefore we must provide
for 274,824 X I'I8= 324,292 lbs. acting on the same leverage,
and the diameter of the shaft required will be the cube-root
of 323 —2,   which is equal very nearly to io- inches.
These few examples will serve to show that it is necessary
to know something more than the horse-power of an engine,
before we can determine the proper size of the crank-shaft,
for, although the horse-power is precisely the same in both
engines, the one requires a shaft -ths of an inch larger in
diameter than the other.
In each of the foregoing examples, the shafts have been
assumed to possess sufficient torsional stiffness, and they
would do so in the case of the crank-shafts which are short,
and the 3-inch shaft might be able to transmit the stated
number of horse-power, but it might twist through so many
degrees that it would be quite unserviceable for driving
machinery. There are likewise other points to be looked




Resistance to Torsion and Shearing.        143
at, beside the horse-power to be transmitted, even with
ordinary shafting, namely, the distance from the driven end
of the shaft to the point at which the power is taken off, for
a shaft might be quite strong and stiff enough to drive
certain machinery, provided the power was taken off near
the driver, but the case would be widely different if the same
power were taken off the shaft, say at 300 feet, instead of at
a comparatively short distance from the driver.
The torsional stiffness of wrought-iron shafts varies as the
fourth power of their diameter divided by the length, and it
is found that wrought-iron shafts above 42 inches diameter,
which are strong enough to resist the torsional strain, will
be found sufficiently stiff to do their work; but below that
diameter the stiffness of the shaft is the weak point, hence
deflection must be otherwise provided for, in order to afford
the necessary strength; therefore, a shaft of larger diameter
than is absolutely necessary to resist the torsional strain
must be used, in order that it may be sufficiently stiff for
the steady driving of the machinery, so that in determining
the proper size to be given to a long line of shafting, all
these important points should be taken into account.
Resisfance to Shcaring and Punczhig.
The resistance of materials to the action of shearing and
punching has not yet received so much attention as some of
the other more important modes of resistance. Still, sufficient is known of the amount of force necessary to shear or
punch wrought iron, to enable the student to form an approximate estimate from their relative tenacity, of the shearing and punching resistance of other metals, which have not
yet been operated upon to the same extent.
The force which is required to shear or punch, depends
on two conditions: first, upon the extent of surface which is
to be shorn or broken through; and secondly, upon the
nature or kind of material that has to be operated upon.




144        On the Strength of Materials.
The operation of shearing or punching is not strictly a
cutting, but a detruding action, and the whole work is done
almost at the commencement or immediately after the first
yielding.  That some time is occupied in shearing or
punching is largely due to the inherent elasticity of the
material, the punch, and the machine, but when the elasticity and slackness are used up, then the resistance offered
by the material is at once overcome, and the metal is detruded, broken, or pushed off, rather than cut through, and
when the operation is performed, the resilience of the apparatus in assuming normal conditions is accomplished with a
jerk.
From this it will naturally be inferred, that the process
of shearing and punching requires a greater immediate
mechanical force to accomplish the object, in that way, than
would be required to cut the portion with an incisive instrument by a more gradual cutting or perforation. There is,
therefore, from such an action, a constant repetition of
sudden strains coming upon the tools, which, although lasting only for a second, yet when united in effect, tend to
rupture the apparatus, unless it is made exceedingly strong,
and this accounts for the ultimate breakage which generally
overtakes all such machines, and which is fully explained by
the results, shown in the Tables, relating to the elasticity of
cast iron.
The resistance that is offered to shearing is a fraction
under that of punching, and both are found to be a little less
than the tensile strength of a piece of the material of equal
area; and from some experiments made on shearing and
punching wrought iron, it appeared that, with fair uniform
bearing instruments, to punch a hole one inch diameter in
an inch plate, or to shear through a bar three inches broad
by one inch in thickness, required nearly 8o tons of stress,
the stress being as the metallic surface disturbed or broken
through, which in these two cases would be nearly the same.
With a half-inch plate or a bar half an inch thick, the stress




Resistance to Torsion and Shearing.      I45
would be about the half of 80 tons, and so on in proportion.
This amount of stress is considerably higher than that
which appears to have been obtained, in some other published experiments, which go to show that the force required to shear or punch is about 80 per cent. of the tensile
strain. But in making such experiments, however carefully, the result will much depend upon the precise condition of the acting surfaces of the punch or shears; for unless
these surfaces have a perfectly flat and fair bearing, at the
commencement of contact of,the instruments, the work of
the shearing or punching will be performed in detail, and
thus the apparent strain will be reduced by being spread
over a longer time. Hence arises the great practical advantage which is found to result from making the face of such
punches so as not to have a fair bearing at the commencement, but slightly out of parallel with the bolster, and likewise the arrangement of the shears so as to act like a pair
of common scissors, which give a gradual shear.
When it is said that the resistance to a shearing action is
nearly the same as the resistance to tensile strain, it has to:
be borne in mind that the effect will depend on various
conditions, for there are many forms of construction which
may and do call the shearing action into activity, unintentionally, from some modification of the mechanical
arrangements; to take, for example, the case of long links,
in which the connection is formed by means of an eye or
hole at the ends, into which a pin or bolt is inserted.
It might be inferred that the,bolt should have the same
sectional area as the body of the link; but it is found that
when so made, the round pin acts as a blunt wedge within
the hole, and thus tends to tear it open. To give uniformity
of strength, it is found necessary to provide a larger bearing
surface, in order to counteract this detrusive tendency, and
it is for this reason that some engineers have the bolts or
pins of such structures, apparently out of proportion, geneL




146         On t he Strenzgth/ of llaterzais.
rally to the extent of half a diameter, but this is true
economy, notwithstanding.
The practical question is frequently raised, in regard to
the comparative merits of punching and drilling, in perforating the rivet holes in boiler plates and for similar
purposes. Punching is the cheaper process, and has the
advantage of finding out and breaking an exceptionally
bad or brittle plate, but, on the whole, it is not so efficient as drilling. The drill, being a cutting instrument,
deals more gently with the material, and the plate, so treated,
is more likely to maintain its full strength after the operation. As the line of rivets is the weakest part of the
structure, it seems throwing away an important advantage
not to drill the holes, even if it is a little more expensive to
do so. It is probable, however, that if the drilling system
were established, by proper organisation of plant and welldirected arrangements, it would not be a more costly process
than punching. Already some of the larger engineering
houses are drilling their boiler plates, and are able to make
boilers almost as economically, on that system, as by the
older plan of punching.
From some valuable experiments made in 1858, which
are recorded in the Proceedings of the Institute of Mechanical Engineers, and were published in the' Artizan' of
that year, it was found that the stress required to punch a
hole one inch in diameter, through a wrought-iron plate 3
an inch in thickness, was equal to 36 tons, and to force
the same punch through a plate of double the thickness
required 69 tons, which is not quite the double of the
former result. The mean of the two experiments gives
22'5 tons, per square inch of detruded sectional area.
A punch 2 inches in diameter was pushed through three
plates in succession, the respective thicknesses being l an
inch, I inch, and Ix inch, and the varying pressures required
were 65, I32, and I86 tons. This gives a lower mean than
the former-namely, I9'4 tons per square inch of detruded




Resistance to Torsion and Szearzing.           I47
sectional area-thus showing that, by increasing the size,
the proportionate stress diminishes, as in the above examples, to the extent of I4 per cent.
The following Table contains the results of another set of
experiments, made with actual weight on the end of a lever,
the weight being gradually applied, until the iron was detruded:TABIE XXXV.
Diameter of  Thickness of  Sectional area  Total weight   Stress per
pc  wrought-iron   of detrusion.    bearing  square inch
plate.                  on punch.    of area.
Inch.      Inch.     Square inches.    Tons.     Tons.
0'250      0'437        0'344         8'384       24'4
0'500      o-625       0o982         26'678       27'2
0'750     o0625         1 472        34'768       23 6
o'875     o0875         2'405        55'000       23'I
Io000     Ioaoo -        3'1416     77' 70        24 6
In the several experiments, the average pressure required
for punching seems to be greater for thin plates than for
thick plates, which is probably due to the greater relative
proportion of hard exterior surface in the thinner plates.
From  a comparison of experiments, in punching  and
shearing, the stress required to detrude, per unit of sectional area, appears to be nearly the same in both cases,
with moderately small holes; but with large holes there is -a
perceptible difference, nearly 5 per cent. less pressure being
required to shear than to punch a given sectional area, which
is probably owing to the effect of friction at the entrance oi
the punch.
For comparison, the following Table, taken from the
Proceedings of the Institute of Mechanical Engineers, will
show the stress required for shearing:L 2




148             On the Strength of Materials.
TABLE XXXVI.
Sectional area of surface    Pressure on the detruding
detruded.              instrument.
Direction                                       Stress per   Remarks.
Thickness and    Area in              square inch
breadth in    square    Totales-  of area deinches.     inches.   sure in tons.  truded in
tons.
Mean stress.
flat   0'50 x 3'00       I'50       33'4        22'3       
edge   0'50 x 3'00       I-50        34'6        23.I    f
flat   I'00 x 3 00       3'00        69-2       23'I 
edge   I00oo   x 300     3'00        68-i        22'7
flat    I00oo   x 3'02  302          59'7       19'8      2I'5
edge   Ioo00 x 3-02      3.02        621-        20'6
edge   I 80 x 5'00    I0'20         2I0o6        20'6
The foregoing  experiments were made with the shearing
instruments parallel, when the work of detrusion was done
after the first pinch. It was found that by spreading the
operation over a longer period by means of inclined instruments, the stress required was considerably reduced, as will
be seen by the following Table:TABLE XXXVII.
Stress with bar   Stress with bar    Percentage of
Size of bar in inches.  laid flat upon its  laid on edge, in  less stress required
side, in tons per   tons per square  when detruded on
square inch.       inch.         flat surface.
3  x I              18-2            2081              10
42 x I8I            143              17'9             20
3  x I              15'7            21I1              26
5I X I              1i6'7           22-6              26
6  x Is             I5'o            I184              18
The above experiments were made with shears, the blades
of which were at an inclination of i in 8. The less stress
required when the same bar of iron is cut on the side, is




Uniformzity of Sectional A rca.           149
most probably due to the circumstance that the work to be
done is spread over a longer period.
By comparing the stress per square inch in Table XXXVII.
with that in Table XXXVI., we obtain the averages given
in the following summary of results:Mean force required to   Mean force required to
Description of  shear bars on flat, in tons,  shear bars on edge, in
sher square inch.                     tons per square inch.
Parallel..          2I 7                 21'7
Inclined..          6             20
Thus teaching  four important lessons.  First, that with
parallel shears, the shearing force required is equal in either
position of the bar. Second, that with inclined shears, the
shearing force is 20 per cent. less for the bars on the flat,
than it is for the same bars on edge. Third, that when the
bars are cut when placed on edge, only 8 per cent. is saved
by using inclined  shears.  Fourth, that when bars are
sheared on the flat, a saving of 26 per cent. is effected by
the inclined, as compared with the parallel shears.
CHAPTER XI.
ON THE IMPORTANCE OF UNIFORMITY OF SECTIONAL
AREA.
IN designing machines or structures, not only is there a
correct form, but it is of the greatest importance, that that
form  should not be departed from  unnecessarily.  The
correct form is that, in which every part is proportioned to
the straining action to which it is subjected, so that the stress
reaches the same maximum limit on every section. If, instead of being thus proportioned, a structure has a surplus




150         On the Strength of Materials.
of material in some parts, that surplus may not only not
strengthen it, but may positively weaken it, and may render
parts, otherwise of sufficient strength, incapable of sustaining
-the load for which they have been calculated. Thus, for
example, a chain with one link smaller than the others, a
shaft with one journal of insufficient diameter, or a tie-rod
nicked at any point, is reduced in strength, not only in proportion to the reduction of section at the weak point, but in
a still greater degree. In consequence of the surplus resistance in the other parts, most of the elastic work will be
concentrated on the weak link, the weak journal, or the
nicked section of the tie, as the case may be, and those
parts will gradually be more weakened, and at last overcome, before the other parts of the structure are affected.
Even before the principle just indicated was fully understood, it was more or less anticipated by experienced
engineers; and it is interesting to recall the intuitive skill in
construction, often displayed in the forms adopted for the
details of structures. For example, in parts of early locomotive engines, where cotter holes or keyways were formed,
an increased depth or thickness was generally given at that
point, so as to make up for the material cut away. By this
means the whole structure was brought approximately to
equality of resistance, and the concentrationi of the elastic
work on a particular section was prevented.
Although, strictly speaking, we cannot increase the
strength of a structure by securing equality of sectional area,
because the strength still remains dependent on the weakest
section, yet virtually we increase the strength, by preventing
the gradual deterioration of the resistance at one point, and
thus diminishing the risk of fracture.
As a simple example, let us consider an ordinary screw
bolt. A screw bolt may be formed, either with the screw
thread in relief, the shank being of the same diameter as the
screwed part at the bottom of the threads, or the screw may
be cut into a bar, and the shrink is then of the diameter of




Uniformity of Scctional Area.          i51
the screwed part, measured to the outside of the threads.
Two bolts may be thus formed, of the same section at the
weakest part, but the former bolt will yield equally throughout when loaded, while the latter will yield chiefly at the
point of junction of the screw thread with the shank. Hence,
when subjected to stress, the stretching will be distributed
in the former, but in the latter the screw threads will be
drawn out of pitch, and will no longer match the nut. If
these two bolts are treated alike, say by using a screw key
with a long handle, so as to twist off the nut by tension, or
by combined tension and torsion, then the former bolt,
although lighter in weight, will yet require more turns of the
screw key before it will give way than the latter. Practically
it is the stronger, and it is important that this should be
understood, because the same principle is applicable in
many other cases.
It is not essential that the screw thread should be in relief,
with a small shank, the uniformity of sectional area may be
maintained in other ways. For many purposes it is convenient that the shank of the bolt should fit the bolt hole,
and the bolt hole must be large enough to admit the screwed
part. Hence, in these cases, the shank of the bolt requires
to be maintained equal in diameter to the outside diameter
of the screw thread. To secure uniform section in the bolt,
various plans have been adopted. In some bolts, the shank
is made hollow from the head up to the neighbourhood of the
screw thread. These bolts have been used in fixing armour
plates. In other bolts, the shank has been made of cruciform section, or of a square or triangular section with
rounded corners. All these plans give satisfactory results,
so far as depends on the uniformity of the section.
The condition of uniformity of section, in practical operations, is liable to be modified, either for the better or the
worse, in many ways. For example, the effect produced in
the forming of the screw thread of bolts such as those before referred fo, may be seriously injured unintentionally,




I52         On the Strengthl of Materials.
from the mode of making, cutting, or forming the thread, as
depending on the sharpness or bluntness of the screwing
dies.  With good instruments, which do really cut out the
thread, and do not distress the internal structure of the
material, the bolts are more reliable than those that are
formed with blunt dies, which act more as abrasive, or even
detrusive instruments, than as genuine cutters. The effect
of such violence of treatment is to harden or loosen the iron,
and to render the bolt more brittle. With good instruments
and ordinary care in the manufacture, and when the form of
bolt section has been arranged on correct principles, the
strength of bolts is about equal to that of the iron out of
which they are made, ranging between 20o and 25 tons per
square inch of sectional area at the weakest point, according
to quality.
Similar remarks apply to the strength of chains. The disastrous effect of a weak link is seen, more especially, in those
chains that are used for cranes and in slings for lifting heavy
weights. As a rule, such chains are proved up to about the
half of the ultimate tensile strength, or about the double of
the intended ultimate working load, nevertheless it is frequently found that, after working for a short time, some
weak link begins to stretch, and if this is not observed in
time, it will break unexpectedly. Such a link must have been
weak originally, and, during its working career, it must have
had more than its fair share of the elastic duty to perform;
hence if it is not cut out of the chain in time, it will ultimately
break with a considerably less load than that which it was
intended to support, or less than the quarter of its original
ultimate strength in its early days; on the other hand, when
a chain is so well made that all its links are of uniform
strength, or nearly so, the endurance is much greater, and it
will go on for an indefinite period without breaking. At
the same time, a change for the worse overtakes all chains
in course of time, that is, if they are in constant use; and, in
consequence, it is a rule in the War Departnment, that all




Uniformity of Sectional A. rea.             153
chains of cranes or slings are to be drawn through the fire,
and thereby annealed, periodically, which has the effect of
restoring the quality, by putting the material of the links
into a condition of equilibrium. By this course, their life is
protracted indefinitely. After the chains are annealed, they
are then subjected to a proof strain, which is the half of the
ultimate strength, and double the usual working load, according to size, which is as under:TABLE XXXVIII.
Table of Proof Strains.
Size of iron in          Size of iron in
chain,    Proof strain,  chain,     Proof strain,
in inches.    in tons.   in inches.   in tons.
3            5
8 158                   I8            28
7-          21          1_          Io0
16           4           1(i
2           3             I          12
Yit                      II          I5~
51            51 3                  225
31                       1
44           2          27
Chains are of two sorts first, the open link chain, which is
commonly used for cranes and like purposes; and second,
the stayed link chain, which is used for cables and some other
purposes. It is to be noted that, whatever the explanation
may be, the stayed link, when made of the same iron as the
open link, is stronger than the other, nearly in the proportion
of 9 to 6. To take, for example, an inch chain, that is to
say, a chain made with iron one inch in diameter, the ultimate strength of the open link is about 24 tons, while that
of the stayed link may reach 30 tons. The office of the
stay is to prevent the collapse of the link, and thereby to
intercept the shearing action due to the wedge action of




154         On the Strength of Materials.
the one link within the other. The great practical objection to the stayed link chain is its weight, and its extreme
roughness in working over pulleys.
A good approximate rule, for mentally estimating the
strength of chains, is to remember the strength of one particular size-say of one-inch chain-the safe working load
of which is 6 tons for the open link, and 9 tons for the
stayed link: the comparative strength of other chains is to
that of the inch chain, as the squares of the diameters of
the iron out of which they are severally made. This does
not hold strictly correct with very large, nor with extremely
small sizes, which both lose strength to a small extent, the
former from the iron not being quite so good, and the latter
from the welding not being quite so perfect.
Another simple rule, for the approximate safe working
load of chains of different sizes, is to square the number of
eighths of an inch, in the diameter of the iron out of which
the link is made, and to strike off the last figure as a
decimal.  For example, in the inch-chain there are 8
eighths of an inch.
8 x 8 = 64,
or, striking off the last figure, 6'4 tons, the greatest safe load.
Again, for a 3 inch chain, 3 X 3- 9. Striking off the last
figure we get o0g tons for the greatest load consistent with
safety.
Uniform and proportional sectional area adds to the
strength of beams, when subjected to transverse strains.
Any notch made on the under side of a beam for the insertion of bolts, nuts, or washers, should be studiously
avoided.
In the construction of heavy timber roofs, where diagonal
bolts or tie-rods have to be introduced, at an angle, through
the principal beam, some persons are so unskilful as to make
a notch on the under side of the beam, for the nut and washer.
Such an arrangement is most objectionable, because the damage done to the strength is permanent. The proper course,




Uniformity of Sectional A rea.        155
under such circumstances, is to employ a broad iron washerplate, with its flat surface bearing upon the under side of the
beam, the washer-plate having a hole cored out, at the proper
angle for the reception of the bolt, and with a corresponding
boss, having the bearing surface for a nut and supplementary
washer, at right angles to the axis of the tie-rod.
Another objectionable practice sometimes seen, in connection with rough structures, is the formation of slits or
cotter-ways in bolts, and in having that part of the bolt made
with the same exterior diameter as the solid portion of the
main body. Such a bolt would be really stronger, for all
practical purposes, if the main body was reduced to a smaller
diameter, equal in sectional area to the part in which the slit
has been made. The proper course is, to upset the bolt at
the part which has to be punched, and then to form the
cotter-way by taper punching, in order to leave the metal
entire, and thus obtain uniformity of sectional area.
The Strength of Ropes.
The strength of hempen ropes is found to depend
primarily on the quality of the hemp fibre with which they
are made. The fibres vary from 3 to 31- feet in length, and
a number of them are twisted together to form a yarn, this
twisting introduces the element of friction, which effectually
prevents the fibre from being drawn asunder. When a rope
is twisted it necessarily becomes shorter than the strands
of which it is made, and the amount of twist given to ropes
is usually expressed by the extent to which they are shortened; thus if shortened one quarter of their length, the twist
is then said to be one quarter. The nominal size of ropes
refers to the circumference, thus a six-inch rope is a rope
of six inches circumference.
As a rule, new white ropes are stronger and more pliable
than ropes made with tar. The tarred rope, however, retains
its original strength for a longer period, especially when ex



156        On the Strength of Materials.
posed to wet.  Tarred ropes spun hot are stronger than
tarred ropes spun cold, and are more impervious to water.
The ultimate strength of ropes is usually considered to be
about 6400 lbs. per square inch of sectional area. From
some recent experiments made at Woolwich, the strength
was found to range from 9874 lbs. in a 9-inch to Io,783 lbs.
in a 2-inch rope. The same series of experiments showed
that Italian hemp ropes are stronger than those of Russian
hemp.
The Woolwich experiments teach also three important lessons to the student: -st. That there is much variation in
the strength of pieces of rope even when cut from the same
coil. Thus, the 4-inch ropes range in tenacity from 55 to
7;3- tons. The 6-inch ropes from I4- to I7 tons. The
9-inch ropes from 25 to 291- tons. The minimum strength
observed in the experiments is that which should be relied
on in practical calculations. 2nd. That there is considerable loss of strength from the tear, wear, and exposure to
the weather during a few months working. Thus the
strength of an apparently good 6-inch Italian hemp rope,
after working, was only io3 tons, as compared with I43 tons,
the strength of a new rope. The old 6-inch Russian hemp
rope broke with 5,, while the new one required II-1 tons.
This fact suggests that we ought to allow a large margin between the working safe load and the ultimate strength of
new ropes. Thus, we may make the safe load 5th of the
ultimate strength. 3rd. That a double rope is in certain
cases weaker than would be expected from experiments with
single ropes. All the double rope slings, suspended from
an ordinary crane hook, broke with less than double the
strength of a single rope. On the other hand, when thimbles
or pulleys were used, then the double ropes nearly always
required double the maximum load which would break a
single rope.




PART II.
ON THE STRENGTHI OF STRUCTURES.
CHAPTER  XII.
BEAMS AND GIRDERS.
HAVING described the physical properties of the various
materials employed by the engineer, we have next to consider those structures, either simple or complex, the strength
of which depends on the form, arrangement and construction, of the parts, as well as on the material of which they
are composed.
In studying the laws of the resistance of structures, a
knowledge of elementary mathematics is necessary. In the
following chapters, the subject is treated in the most simple
manner, in the hope that they may thus be rendered useful
to all classes of readers, and especially to those whose
mathematical knowledge is limited, by enabling them to
calculate the strains in the more simple structures required
in practical operations.
Every practical engineer should be able to ascertain,
approximately, the strains which are likely to be brought
upon the component parts of any structure, both by the
weight of the structure itself, and the load which it is
intended to support. He should likewise know the amount
of stress which may, with perfect safety, be put upon each
square inch of section of the material of which the structure
is composed. It will then become a matter of simple pro



I53        On the Strength of Structures.
portion, to ascertain the number of square inches of any
given material, that should be introduced into each part of
the structure, in order to enable it to resist with safety the
strain which it will be required to sustain.
Of late years, unfortunately, too much reliance has been
frequently placed upon the formulae which are to be found
in different books; these are drawn from a variety of sources,
and are generally given without the data from which they were
originally deduced. This reliance upon mere formulae is a
great mistake, and often leads to expensive errors. In some
cases, formulae, originally intended for one kind of structure,
have been applied in designing a different kind of structure,
and the incorrect result arrived at has only been discovered,
when too late to avoid the consequences of the mistake.
In the following chapters, the natural laws of mechanics
are brought into prominence, so that the student may be
led to use his judgment in the application of rules. At the
same time, a general knowledge of mathematics will be
found of the greatest value, giving a more comprehensive
grasp of the whole subject, and fitting the student to use
formulae with intelligence.
From a very early period, it was known that the rectangular beam, placed on its edge, would carry safely a heavier
vertical load than the same beam placed upon its side. In
the same way, it was understood that a given quantity of
material, arranged in a hollow form, was stronger than when
in the solid shape. The reasons why these things are so,
are now also well understood, and the practical engineer
should make himself acquainted with them, if he would
carry out works with safety and economy.
The present chapter will relate chiefly to beams and
girders, and the principles applicable to these are mostly
derived from the investigations and experiments of Hodgkinson, Fairbairn, and others. An effort has been made to
explain these principles in as simple a manner as possible;
but the full treatment of the subject would involve a higher




Beams and Girders.                 I59
degree of mathematical knowledge than is here assumed.
The student who wishes to pursue the subject further is
recommended to consult the treatise by Professor Twisden,
in the same series as the present volume.
Beams or Girders.
The term' beam' is generally applied to any large piece
of material, which carries a load at a distance from the
point or points of' support upon which it rests, and which is
thereby subjected to transverse strain.  But this term, conjoined to another word, is often used for certain parts of
structures, which in reality are subjected to longitudinal
strains only, such as, for instance,' tie-beam' or'strainingbeam,' both of which are subjected to tensile strains only.
The term'girder' is now almost universally adopted by
engineers, as the'name for beams, which are supported at
both ends and subjected to transverse strain, and the term' cantilever' is generally used for beams, which are supported
at one end only and subjected to transverse strain.
Girders are made of many forms, depending on the
material employed. Those in most common use are of the
rectangular, flanged, or tubular forms of section; the flanged
girder is indiscriminately made of cast, or wrought iron,
and the term includes all forms in which a flange or flanges
are connected to a single vertical web, whether the web is
a solid plate or an open lattice, or whether the flanges are
parallel or otherwise.
In tubular girders, which are mostly built up of wrought
iron, the two flanges are connected by two webs, and in the
case of the celebrated tubular bridges of Fairbairn, the
tubular girders are of such large dimensions, as to admit of
the load being carried inside the tube, instead of on the
outside in the usual manner.
In considering the strength of beams or girders, when
subjected to transverse or cross strains, three chief points
present themselves for consideration, namely:



I6o         On t1he Strngoth of Structures.
First, the mechanical effect which any given load produces upon the section of fracture, of whatever form, under
varying conditions of support.
Second, the nature of the strains that are brought to bear
upon the girder, and the manner in which its resistance to
those strains is affected by the form of the section of the
girder.
Third, the actual strength of the material composing the
girder, which is to be thus strained, and which can only be
determined by direct experiment, or by placing reliance on
the numerous published experiments, which have been so
carefully carried out, by Barlow, Fairbairn, Hodgkinson,
Kircaldy, and many others.
The relative strength of a girder, so far as it depends on
the manner in which it is loaded and supported, can be expressed very simply. Taking a given girder, of a given span,
the load may vary in the ratio of I to 8, according to the
different ways in which the girder is supported and the load
distributed.
Position of Support and Load.
Relative
strength.
When supported at one end and loaded at the other..I
When supported at one end and load distributed.. 
When supported at both ends and loaded at the centre  4
And when supported at both ends and the load equally
distributed..                                 8
When a uniform beam is securely fixed at both ends, by
encastrement or otherwise, instead of merely resting on supports, it is then under an entirely different set of conditions.
Theoretically it would break, simultaneously, at three different sections, instead of giving way in the middle only, but
practically this would never occur, unless the fixing of the
ends were exceedingly firm, as, for instance, when the beam
and the abutments were part of the same casting. The increase of strength, due to secure fixing of the ends, has been
variously stated; according to some, the relative strength of




Beams and Girders.               1 6
the beam above referred to would become equal to 12, and,
according to others, even to I6  but it is obvious that the
value will depend on so many conditions, that it is impossible
to say precisely what it is, beforehand; it must, therefore, be
considered as doubtful, unless all the conditions are known
definitely: at the same time the importance of secure fixing,
wherever it can be employed should be noted.
The reason why the same girder possesses the above relative strengths, under the given varying conditions, may be
very clearly shown, by supposing the girder in question to
act as a bent lever, one arm of the said lever being equal in
length to the distance from the point of support to the weight,
and the other arm of the lever to be the depth of the girder.
In considering the strength of such a girder, in which the
amount of deflection due to the load is inconsiderable, we
may neglect the small curvature of the beam when loaded,
and consider that the straining forces act perpendicularly to
the longitudinal axis of the beam. They are at right angles
to the fibres of which it is composed, if it is a timber beam,
and at right angles to the fibres, of which we may imagine it
to consist, for theoretical purposes, if it is a metal beam. But
the longitudinal tensions and compressions in the beam act
parallel to the axis, or in the direction of the fibres of which
it consists or may be imagined to consist. The loading forces
act at right angles, therefore, to these tensions and thrusts.
Let figs. I6, I7, I8, Ig represent four similar flangedgirders supported and loaded under the above-stated conditions. We know, by experience, that a weight acting at
the end of a cantilever, as shown in fig. I6, will tend to bend
the free end of the girder in a downward direction, as shown
by the dotted lines a, b, c, which represent a bent lever;
we also know that, in order that such a lever may be in
equilibrium, the product of the weight at the end of one
arm, multiplied by the length of the arm, must be equal to
the product of the length of the other arm, multiplied by the
weight acting upon it.
M




162         On the Strength of Structures.
Supposing, as an example, that a weight of I ton is required at the free end of the girder to break it, and that the
4yC~lr4 -tons.'~
girder is 4 feet long and I foot deep, then the weight or
force, required at the other end of the lever, will be the product of the weight of i ton, multiplied by 4 feet, the length
of the long arm of the lever, and divided by i foot the
length of the other arm, and will be equal to 4 tons.
The same girder (fig. I7), supported in a similar manner,
but with a distributed load, will break with a weight of two
FIG. I7.
2ft.
2 tons.
tons, or twice the weight required to break the first girder,
because the mean distance at which the weight acts upon
the beam, or, in other words, the centre of gravity of the
weight will be only I the distance from the support that it is
in the former example, and hence it will have a leverage of
2 feet instead of 4 feet.
The breaking strain, or the force required at the end of




Bcams and Girders.                I63
the short arm of the lever, is of course the same as in the
first example-namely, 4 tons-and the weight at the end of
the long arm, necessary to produce the strain of 4 tons, will
be 4 multiplied by I and divided by 2, that is 2 tons,
or twice the weight required to produce the same strain
when placed at the end of the cantilever.
A similar girder (fig. I8), supported at both ends and
loaded at the centre, will require 4 tons to break it, or 4
FIG. I8.
4ton"
_    n 4 tonm.  4-tons.
times the load required to break the first eam. The weight
upon a girder, supported at both ends, is transmitted with
undiminished force to its supports, and each support will react upon the girder with a force equal to a certain portion of
the total load, according to the position of the weight relatively to the supports.
The amount of each reaction is determined by the principle of the lever; the reaction of either support bears
exactly the same proportion to the total weight, that the distance from the weight to the opposite support bears to the
total distance between the supports; for example, if we suppose a weight of 4 tons to act upon this girder, at a distance
of I foot from one support, the distance to the opposite support will be 3 of the whole distance between the supports,
and the weight upon the near support will be 3 of the total
weight, or 3 tons.
When the weight is at the centre of the girder, as shown in
fig. I8, the distance from the weight to either support is I
M2




I64        On the Streungth of Structures.
the total distance between the supports, and hence. the upward reaction of either support is equal to I the total weight
upon the girder.
The leverage of each of these reactions is equal to I the
span of the girder, or 2 feet, and the force required to break
the girder, when acting at the short end of the short arm of
the same, is 4 tons as before, and therefore the reaction of
each support must be 4 multiplied by I and divided by 2,
which equals 2 tons; and the total weight required upon
the girder to produce the reaction of 2 tons upon each support is 2 X 2 =  4 tons, and hence, the girder will not
break, until it is loaded with 4 times the weight which would
break the same girder, when supported and loaded as shown
in fig. I6.
The same girder, loaded with an uniformly distributed
weight, and supported at each end, as in fig.  9, would break
FIG. I 9.
with eight times the weight required to break it, if supported
and loaded as in fig. I6, or twice the weight which would
break it if supported and loaded as in fig. I8.
With an uniformly distributed load, each particle of the
load acts with a gradually increasing leverage from the support, where it is nil, to the centre, at which point it is equal
to I the length of the girder, and the mean leverage of all
the particles composing the weight upon each half of the




Ecants aznd Girders.              I65
girder is at a point midway between the centre of the girder
and the support. Its effect upon the material at the centre
of the girder is precisely the same, as if half the weight upon
each half of the girder was placed at a point twice the distance from the support, which would be at the centre of the
girder; or, to state the result shortly, an uniformly distributed
load has the same effect at the centre of the girder as half
the same load acting at the centre. An uniformly loaded
girder will not break until it is loaded with twice the weight
which would break it, when placed at the centre of its length.
The above has reference only to the theoretical conditions
of beams, and does not take into account the weight of the
beam. In the application and use of beams, practically,
the weight of the beam itself must be considered. In small
beams, of short span, the weight of the beam itself is small
compared with the load it will carry. But in large beams,
of long span, the weight of the beam itself becomes a very
important element in the calculation. In long cast-iron
beams, the weight of the beam itself may be equal to half
the load it will carry, exclusive of its own weight.
The strength of a beam is likewise affected, by the manner
in which the materials of which it consists are disposed in
its construction.  Many theories have been propounded
by eminent mathematicians, but scarcely any of these
theories fully explains the law which gives the results found
by experiment. The knowledge derived from experiment
is, as a rule, quite sufficient for the greater number of constructions in ordinary practice, and on this the following
observations depend.
We have already seen that the girder tends to bend under
its load, and hence the top layer of fibres must necessarily
be compressed, and the bottom  layer extended, and this
compression and extension is not confined to the outer
layers of fibres, but affects those in the interior of the girder
to a gradually decreasing extent, as their distance from the
top and bottom of the girder increases, until, when a certain




I66         On the Strength of Structures.
point is reached, the fibres are neither compressed nor
extended, and hence retain their original length.
This plane or surface of unaltered length, at or near the
centre of the depth of the beam (when the material is nearly
equal in tensile and compressive strength) is called the'neutral surface,' and the line in which this surface cuts
any transverse section of the girder is called the'neutral
axis of the section.'
The neutral axis of any section of a girder is, therefore,
the line of demarcation between the forces of tension and
compression exerted by the fibres crossing that section. In
addition to these strains of tension and compression, there
is a shearing strain, caused by the tendency of the weight
to separate the part of the beam upon which it immediately
rests from the adjoining part; this tendency is resisted by
the material, and the weight is thus transmitted to the adjoining part, and by it to the next adjoining part, and so
on to the supports. In parallel flanged girders, this strain is
chiefly resisted by the web.
That there is no tensile strain upon the upper half of
such a girder may be shown by an experiment.  If we
take a girder of the class here referred to, either of wood or
metal, and cut it across its upper surface, and then insert
a thin piece of wood or metal into the saw-cut, simply to
prevent the opening from closing again when the weight is
put on, the beam will be found to be as strong as before;
but, on the other hand, the slightest notch on the under
surface will weaken it considerably, and this will become
apparent if the beam is tested.
It has been previously stated, that the strain upon the
fibres of the beam gradually decreases from the top and
bottom to the neutral surface of the beam, and hence the
nearer the fibres are to that point the less they contribute
towards supporting the load, and the farther they are removed from it, the greater is the power which they have to
assist in its support; therefore, the neutral line is the part




Beams and Girders.                 167
where holes, for fixing other parts connected with the girder,
may be made with impunity.
In correctly proportioned flanged girders, the position of
the neutral surface is considered to pass through the centres
of gravity of the transverse cross sections, and the intensity of stress on each flange is directly proportional to the
distance of the flange from the neutral axis. The section
of each flange should be so proportioned, that the intensity
of the stress exactly corresponds with the resistance of the
material to tension or compression.  It must, however,
be borne in mind that the material of the compression
flange is subject to distortion, and the real resisting power
of this flange is its power to resist distortion or crippling,
and not absolute compression.
The position of the neutral surface of a rectangular beam
depends entirely upon the nature of the material; if its
tenacity and power of resisting compression are equal, then
the neutral surface will be in the middle of the depth, but
not otherwise. Cast iron, for example, has about six times
the power of resisting compression that it has of resisting
extension; therefore the depth of material above the neutral
surface will be to that beneath it as  / i is to  /6, or as I is
to 2'449, or, in round numbers, as 2 is to 5; and it would
only bear the saw-cut 2 of its depth, without loss of
strength.
Let fig. 20 represent an exaggerated view of a cantilever
loaded at its outer end, and a b c d a portion of the beam
before deflection; let the upper edge, after deflection, be
extended from the length b a to the length b e, and the lower
edge compressed from d c to d f, then the lines across the
two triangles represent the alteration of length of the intermediate fibres, the neutral surface g h dividing the depth of
the beam into two equal parts.
The total stress upon the fibres composing each half of
the beam is the product of the area of half the beam multiplied by the mean strain upon the fibres, and if the strain




i68         On the Strength of Structures.
upon the extreme fibres is 4 tons, the mean strain upon all
the fibres will be one-half 4 tons = 2 tons, and the total
strain upon either half of the beam will equal 2 tons multiplied by the breadth of the beam, and by half its depth.
FIG. 20.
Further, as the lines across the triangles are proportional
to the strains upon the several fibres, the centre of the tensions and compressions (that is to say, their resultants) will
coincide with the centre of gravity of the two shaded triangles, which will be. of their height from the neutral axis,
and hence the distance between the two centres of strain
will be 2 of the depth of the girder; and, supposing one of
them to act as a fulcrum, then the moments of the forces
acting about that point will be, on one side, the product of
the weight multiplied by its distance from the section of the
girder in question, and, on the other side, the product of the
tensile or compressive strain multiplied by 2 of the depth of
the beam.
If we knew in all cases the position of the neutral axis




Beams and Girders.               I69
and the tensile strength of the material, we could then compute the strength of rectangular beams without any additional data; but without such precise knowledge, the best, or
at least the easiest, method of determining the resistance of
rectangular beams is to make a series of comparative experiments upon model beams, and determine their strength
and deflection. From these experimental results a constant
number may be deduced for each description of material.
Such experiments have been made, and the table at page
176 contains the constant numbers for some of the materials
which are most commonly used in works of construction.
The strength of flanged girders, of similar section, varies
inversely as their lengths, and directly as their depths, and
as the sectional area of their flanges;'for if we double the
length of either of the beams in the foregoing examples,
then only one-half the weight upon the long arm of the lever
would be required to break the beam; or if, on the other
hand, we double the depth, we thereby double the strength,
because the short arm is twice the length, and hence the
flange, from its position, would be able to resist twice the
weight acting at the end of the long arm of the lever; or if
we double or increase the area of either flange, we also
double or increase the strength in the same proportion.
It may be stated, as a general rule, that the strength of
solid rectangular beams varies inversely as the length,
directly as the breadth, and directly as the square of the
depth, but up to a certain extent only-namely, so long as
the said beam can be kept from twisting. For if we double
the length of the beam, the resisting power of the section at
any particular part will still remain the same, so that the
actual weight required to overcome it will thus be reduced to
one-half, because it acts now with twice the leverage; or, on
the other hand, if we double the breadth of the beam, we
thereby double the number of the resisting fibres, but do not
alter the leverage, and consequently it has twice the strength;
but if we double the depth, we then have twice the number




I70         On the Strength of Structures.
of fibres and also twice the leverage, and therefore we have
four times the resisting power. Hence, the strength is as the
square of the depth.
The strength of square beams varies inversely as the
length, and directly as the cube of the side of the square,
because, if we double the side of a square beam, we then
have four times the number of fibres or molecules, acting at
twice the leverage, which gives a resistance equal to eight
times that of the original beam; in other words, the strength
is as the cube of their sides.
The strength of cylindrical beams varies inversely as their
length, and directly as the cube of their diameter, for the
same reason as that stated for the square beam.
The strongest solid rectangular beam, which can be cut
out of a round log of timber, has a cross section, the square
of the breadth of which is equal to one-third of the square of
its diagonal. In other words, the proportion of breadth to
depth is as 5 to 7 nearly. Hence, we have the following
construction for cutting the strongest beam out of a round
log.
FIG. 2r.
/C
Describe a circle equal to the size of the log, and draw a
diameter a d; divide it into three equal parts a b, b c, and




Beams and Girders.                171
cd; erect perpendiculars at b and c upon opposite sides
of the diameter intersecting the circle, and then join the
points in which tile diameter and perpendiculars intersect
the circle, and the rectangle so formed is the section of the
strongest beam that can be cut from the log.
Rectangular beams are often made much deeper, in
proportion to the breadth, than the beam given above,
but they have to be kept from twisting by resorting to cross
bracing or otherwise; still, for independent beams, the above
proportion should always be employed where practicable.
In some instances, the strength of the web of a cast-iron
girder, considered as an independent rectangular beam, is
added to the strength of the flanges, and a much greater
apparent strength is thereby assumed than is justified either
by experiment or in the practical constructions of the workshop.
The strength of a square tube when used as a girder, with
its sides placed in a vertical position, is to the strength of a
round tube of equal thickness and span, and having a diameter equal to the side of the square tube, as I 7 is to Io.
The strength of a similar round tube, having a diameter
equal to the diagonal of the square tube, is to that of the
square tube as Ioo is to 85.
With square and round tubes of equal thickness and weight,
their peripheries will be equal, and, when the sides of the
square tube are vertical, their relative strengths are as  o5 is
to  oo. The square tube, when placed with its sides in a vertical position, has thus a slight advantage over the iound tube;
but in such instances as crane-posts, which are subjected to
transverse strains in all directions, by the load being slewed
upon them all round the circle, the round form is preferable,
because it is much stronger than a square tube of equal
weight, when the load acts upon it in the direction of its
diagonals, there being then so much more of the material
in the vicinity of the neutral axis.




172        On the Strength of Structures.
Beanis of Uniform Strenglth.
For most arrangements of the load on a beam, the bending moment at various points of the length varies. It is not
generally necessary, therefore, that the section of the beam
should be uniform from end to end, and material may be
economised by reducing the section where the bending
action is less. When a beam is so proportioned that the
moment of resistance of the section, at each point of its
length, is proportional to the bending moment at that point,
the beam is called a beam of uniform strength.
It will only be necessary to consider two kinds of loading-that in which the load is concentrated at one point
of the beam, and that in which the load is uniformly distributed over the beam. In the former case, the bending
moment is greatest at the place where the load is applied,
and in the latter case, the bending moment is greatest at
the centre of the beam.  In both cases there is no bending
moment over the supports.  There are also two ways in
which the section of the beam may be varied so as to fulfil
the conditions of uniform strength. The breadth of the beam
may be maintained constant, and the depth varied, or the
depth may be maintained constant, and the breadth varied.
If the beam is a flanged beam, the thickness of the flanges
is supposed to be constant throughout the beam in either
case. For a beam of uniform strength and uniform depth,
carrying a single fixed load, the plan of the flanges should
be two triangles, with their bases united in a line, passing
through the point at which the weight is applied and their
apices at the supports. But if the flanges are of uniform
breadth, then the side-view of the girder should be a triangle,
the base of which is the top flange, and the apex of which is
directly under the load. For a beam of uniform strength and
uniform depth, carrying a uniformly distributed load, the
plan of the flanges should be formed by the overlap of two
parabolas, whose vertices are at the ends of a line drawn




Beams and Girders.               173
across the centre of the girder, equal in length to the breadth
of the flange. But, if the breadth of the flanges is uniform,
then the depth must be variable, and one flange may be
straight and the other curved in the side-view. The curved
flange of the girder should be a parabola whose axis is vertical, and its vertex at the centre of the girder. This form
should also be adopted for a girder which is to carry a single
moving load, such as the gantry to carry the crab of a
travelling crane, and for the girders of bridges to carry
weights upon a single line of rails.
Flanged cantilevers (if intended to be of uniform strength)
should be of exactly the same form as the half of one of the
above girders, according to the manner in which they are
loaded. There are often, however, practical difficulties in
the way of making beams of the above forms, and they are
frequently made half as deep at the ends as in the middle, so
as to obtain an abutment for fixing, and they are curved so
as to include a parabola drawn through the three points thus
determined.
In order that the student may readily ascertain the
strength and deflection of any rectangular or cylindrical beam,
it is necessary, as already stated, owing to the defective state
of our knowledge of this part of the subject, to use certain
numbers called constants, which have been determined at
various times by different persons.
Practically speaking, there is a great objection to the use
of such numbers, but until the position of the neutral axis of
such beams as are employed in practice is accurately determined, it is the only course open; and if an engineer has to
make a structure of some new sort of timber or other material, upon which no previous experiments have been made,
he will have to make experiments upon model beams of the
material, to determine the constant number, before he can
proceed with any proper feeling of certainty that, in his
structure, he has obtained the requisite strength with the
minimum quantity of material.




174        On the Strength of Structures.
The following table contains, in the column marked'Value of Strength,' the constants required for ascertaining the ultimate strength of a rectangular beam, of a given
size, of most of the materials commonly used in structures;
and the strength or breaking weight in lbs. of any such beam,
when supported at each end and loaded at the middle, may
be found by multiplying the breadth in inches by the depth
in inches squared, and by the constant number in the column'Strength,' and dividing the product by the length of the
beam in feet.
The value of the constants for strength may be found, by
taking a bar one inch square, twelve inches between the
supports, and observing how many pounds' weight applied
in the middle is required to break it; the number of pounds
so ascertained is the constant. But the experimental beams
need not be of the above dimensions; any convenient size
of beam may be used for the experiment, the larger the
better, and the breaking weight of the experimental beam is
then reduced to the weight required to break a beam of the
above dimensions.
For example, the constant for the strength of teak, contained in the table, was determined from three experiments
made by Barlow upon beams of that material, seven feet
long by two inches square, which broke with a mean weight
of 938 lbs.
It has been already stated, that the breaking weight of a
rectangular beam varies as the breadth and the square of
the depth, and inversely as the length, or thus:
breadth in inches x depth squared in
Breaking weight in lbs.       inches x constant
length in feet;
and all these particulars are furnished by the experiment,
except the constant, which may be ascertained by the inveisionl of the above statement; thus:
Constant -=    length in feet x weight in lbs.
breadth in inches x depth in inches squared.




Beams and Girders.                  175
Then, by filling in the known quantities and working out the
sum, we obtain the constant; thus:
7 x 938
Constant= -2 x — = 820, which is the number placed opposite teak
in the table.
To find the breaking weight of a beam of any of the materials in table:
Breadth in inches x depth
in inches squared x constant  breaking weight in lbs.
length in feet
For example, find the breaking weight at the centre of a
beam of Memel deal, I4 inches deep, io inches wide, and
20 feet between the supports:
Io x 14 x 14 X 545 - 53410 lbs.
20
To find the breadth of beam required, so that it will just
break with a given weight, when the depth and length between supports are given:
Length in feet x breaking weight in lbs.
= breadth required in inches.
depth in inches squared x constant
To find the depth of beam required to support a given
weight, when the breadth and length between supports are
given:
Length in feet Y breaking weight in lbs. = depth in inches squared.
breadth in inches x constant
And the square root of the result will be the depth required
in inches.
For example, find the depth of a beam of English oak
required to support 2- tons at the centre with safety, to be
io inches broad and 25 feet between the supports. In this
case we must first determine how many times the breaking
weight of the beam is to exceed the working load; in permanent structures it should be ten times, but in temporary
constructions it may be reduced to six times. In this ex



176          On the Strength of Structures.
ample we will take the former, and then the breaking strength
becomes2 x IO -- 25 tons, and 25    25 x 2240 = 25I3;
IO  557
and the square root of 25I-3=1-58, the depth, in inches, of
beam required to meet the above condition.
Constants of Strength and D9eflection.
The latter will be explained hereafter.
TABLE XXXIX.
Tralue of   Value of
Nature of Material.    Strength.  Deflection.  Authorities.
Strength.   Deflection.
Teak...                820       5588
English Oak                    557       3359
Canadian Oak.       ~        588       4964
Dantzic Oak...        485       2757
Adriatic Oak                   461       2255
Ash..                         675       3807
Beech                          518       3133      P. Barlow.
Elm....  337  1620
Pitch Pine..                  544       2837
Red Pine...      447       4259
New England Fir..        367       5072
Riga Fir..                    369       3079
Mar Forest Fir...       381       20I3
Larch             ~            284       2437
Norway Spar                    49I       3374
Mahogany, Spanish..       425       2906,,    Honduras.            637       357I      Tredgold.
Memel Deal  545                          4500
Christiana Deal...      686       4176
Cast Iron                     2548                 Banks.
Do.                         2532   f.41740       Banks.
Wrought Iron, Swedish         3473      64221      Kircaldy.
Hammered Steel                6403      78822
The relative strength of beams of rectangular or other
sections, supported and loaded in any other manner, will be




Beams and Girders.               177
to that found by the foregoing rules, in the various proportions given at page i6o.
With the flanged girder, the case is widely different, for
in its construction we have the means of placing the material (whether of wrought or cast iron) in the most advantageous position, to resist the strain, and, hence, it would be
a deliberate waste of material, to make a beam of either of
those substances of the form employed for a timber beam.
Having this power, we place the bulk of the material at the
greatest distance from the neutral axis.
In designing flanged beams, the nature and amount of the
strain on each part of the flanges should first be ascertained,
and then, if the proper amount of material to resist the
strain is introduced at each point, the greatest economy of
material will be attained.
It has been previously explained that the top flange will
be compressed, and the bottom flange extended, or, in other
words, that the material in the top flange will be subjected
to a compressive, and that in the bottom flange to a tensile
stress. The amount of these strains can be found by considering each half of the beam A B c, in fig. 22, as a bent lever, the
long arm being equal to the distance between the weight and
the support-namely, in this case, ten feet-and the short arm
to the depth of the beam, or in this case two feet. The force
acting at the end of the long arm will be the upward resistance of the support, in this case equal to half the weight, or
six tons; and by the principle of the lever, if we multiply the
weight acting at the end of the long arm by its length, and
divide the product by the length of the short arm, we obtain
the weight that will be required at the end of the short arm
of the lever to balance the upward resistance of the support. Thus, in the case shown, io multiplied by 6, and the
product divided by 2 = 30 tons, the total strain on one
flange.
The total amount of strain on the top flange is the same
as that on the bottom flange, as indicated in fig. 23.
N




I78           On the Strength of Structures.
I 
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0
1~~~~~~~0
-                            -~~ r
iL
1~~~~~~~~~~~~~~~~~~~~~~~~~~~j\
But the strain on the top flange is compressive, and that. on
the bottom flange is tensile. Supposing that the greatest safe!lI                                      I    rl!I      I                       I 
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Bu hesrin ntetpfag  scmrsie  n   hto
th  otmfag   stnie upsn ht   the grats s af




Beams and Girders.              I79
intensity of stress in tension, for the material of the beam,
to be Ix ton per square inch, the bottom flange must have
30 -- I' = 20 square inches of sectional area. It may be
Io inches wide and 2 inches thick, for instance, or of any
other dimensions giving the same area.
Professor Hodgkinson found, by a number of experiments,
that the top flange of a cast-iron girder only required to be
th of the area of the bottom, or, or, in other words, that cast iron
might be subjected to a compressive stress of nine tons per
square inch, but yet we find in practice that the top flange is
seldom made less than 4th of the area of the bottom flange.
The reason is that, if the top flange is made only -th of the
area of the bottom flange, and of sufficient width to give
the required lateral stiffness to the upper part of the beam,
it must be very much thinner than any other part of the
girder, which would induce an initial strain upon some part
of the girder by unequal contraction, due to the different rate
of cooling after being cast.
Take, for example, the top flange of the girder in question.
If it were made 4th of the area of the bottom flange, it would
only contain 20 *6=3 square inches; and if it were made
4 inches wide, then it would only be 3 + 4 -=  ths of an
inch thick, whereas the bottom flange would be 2 inches and
the web I4 inch thick; but by making the top flange 4th
of the area of the bottom its section would be 20 - 4 = 5
square inches, and if made 4 inches wide it would be 5 -4= II
inch thick; that is, equal to the thickness of the web, a
much better proportion. Some experience is necessary, to
determine the best form and dimensions, so as to meet not
only the theoretical requirements of strength, but the conditions imposed by practical experience in the foundry.
By assuming a compressive stress of 6 tons per square
inch of section, an area of 30 - 6 -= 5 square inches will
be required to produce the same result, and the flange may
be made 4 inches wide and iI inch thick, or 5 inches wide
and I inch thick; but the former would be preferable, because
N2




I80         On the Strength of Structures.
the cooling in the mould would be more uniform, and therefore the contraction of the top flange would be more nearly
equal to that of the other parts of the girder.
If the girder is to be of wrought iron, a similar method is
adopted, but, with that material, the strain per square inch
which is generally allowed is 4 tons for compression and 5
tons for tension, and, according to the Board of Trade regulations, it is 5 tons per square inch both for tension and compression, which simplifies the calculation, is easily remembered, and is sufficiently near for ordinary purposes. At the
same time, it would not be applicable to structures exposed
to contingencies, such as the surging of slings by falling
weights.
Deflection of Beams.
When a beam is supported at each end, the distance to
which the middle of the beam is forced down below its
original position, by the load, is termed its deflection, and in
parallel girders, with the flanges of uniform strength, the
deflection curve is found to be circular.
The deflection of solid rectangular beams varies directly
as the load and the cube of the length, and inversely as the
breadth and the cube of the depth.
In flanged girders, the amount of deflection varies directly
as the load, the sum of the areas of both flanges, and the
cube of the length, and inversely as the product of the areas
of the two flanges multiplied together and the square of the
depth of the web.
It appears from the reasoning and experiments of Professor Barlow, that the deflection of a rectangular beam, fixed
at one end and loaded at the other, is equal to that of a
beam of twice the length, supported at both ends and loaded
at the centre with double the weight. The deflection varies
as the cube of the length, and if we reduce the girder until it
is equal in length to the semi-girder, the ratio of the deflections
will then be as the cubes of the lengths; namely, as I is to 8.




Beams and Girders.                   8 I
In other words, the deflection of a semi-girder will be eight
times that of a girder, equal to it in all respects, when the
latter carries double the load. If we now reduce the weight
upon this shortened girder —say l —so as to make it equal to
that carried by the semi-girder, then we reduce its deflection I,
and hence the amount of deflection of the semi-girder will be
sixteen times that of the girder when they are equally loaded.
It has already been showil that the strength of the girder is
four times that of the semi-girder, whereas from the above
reasoning the stiffness of the girder is shown to be sixteen
times as great as that of the semi-girder.
If we require two beams of the same breadth, but of
different lengths, to be equal in stiffness, then their respective
depths must be in proportion to their lengths, because deflection, or want of stiffness, varies directly as the cube of the
length, and inversely as the cube of the depth, or as the
cubes of both these dimensions.
For example, let these lengths be 24 and 12 feet; then if
the latter is 12 inches deep, the former will have to be 24
inches deep to be equally rigid, whereas it would be equally
strong if made 17 inches deep.
If the beams are equal in depth, but of different lengths,
and are required to be equal in stiffness, then their breadths
must be as the cubes of the lengths. Taking the same
lengths as before-24 and 12 feet-the breadths would have
to be in the ratio of 24-cubed to I2-cubed; that is, as 13824
is to 1728, or as 8 is to i. In other words, the long beam
would liave to be eight times as broad as the shorter one to
be equally rigid, whereas it only requires to be twice as broad
to be equally strong.
We have already seen that the strength of a rectangular
beam varies as the square of the depth, multiplied by the
breadth, and divided by the length, but the stiffness varies as
the cube of the depth multiplied by the breadth and divided
by the cube of the length.
The stiffness of cylindrical beams varies as the fourth




182          On the Strength of Structures.
power of the diameter, and inversely as the cube of the
length.
The deflection of similar girders also varies with the
manner of loading, and if the load is uniformly distributed
over a beam supported at both ends, the deflection will only
be 5ths of that of the same beam, loaded with the same weight
collected at the centre of its length; and the deflection of a
semi-girder uniformly loaded is only Aths of the deflection
caused by the same weight acting at the end.
The foregoing investigations and results of experiments
upon the deflection of beams are true, provided the visible
limit of elasticity of the material is not exceeded, but they
are not true of the ultimate deflection, because the law of
deflection is very uncertain after the elasticity of the material has ceased to be sensibly perfect, and this condition is
reached long before rupture takes place.
The constant numbers which are used for ascertaining the
deflection of rectangular beams are contained in Table
XXXIX. page 176, in the column marked'Value of Deflection,' and these constant numbers are deduced from the
experiments in the following manner:
The deflection, in inches, of rectangular beams supported
at both ends and loaded at the centre is equal tothe cube of the length in feet x weight on beam
Deflection  =  in lbs.
in inches f   breadth in inches x the cube of the depth in inches
x constant.
And all these elements are determined by the size of the
beam, the weight applied, and the deflection which took
place during the experiment, excepting the constant, which
is found by the inversion of the above statement; thusthe cube of the length in feet x weight on beam
in lbs.
breadth in inches x the cube of depth in inches x
deflection in inches.
Take for example the constant for deflection of teak,




Beams and Girders.                 I83
which has been determined from the same experiments as
those from which the constant for strength was obtained,
but with this difference, that the weight upon the beam is
not the breaking weight, but the greatest weight the beam
bore, while its elasticity remained visibly perfect, and the
deflection is that which was caused by that weight. These
two quantities are stated by Barlow to have been 300 lbs.,
and I'15I inch; then, filling in the quantities, we haveConstant      773 x 300     343   300     5588
2 x 23 X II5    2 x 8 x I-'5I
which is the number opposite teak, in the column headed'Value of Deflection,' and all the constants D in Table
XXXIX. page 176, are calculated in this way, from experiments carried out by different individuals.
The deflection of timber beams should not exceed in
practice T~th of their length, and it appears, from Tredgold's
experiments on cast iron, that if the deflection of bars of that
material exceeds M-th of their length, a permanent set is
caused; from Kircaldy's experiments on bars of wrought
iron and steel, that a deflection exceeding -ao0th of the length
causes a permanent set, upon bars of those materials.
To find the size of beam, supported at both ends and
loaded at the centre, capable of supporting a given weight
with a given amount of deflection:
The cube of the length in feet x
weight in lbs.             f breadth in inches x cube of
deflection x constant  depth in inches.
For example, find the size of a beam, of English oak, supported at each end and loaded at the centre with a weight
of 2- tons, the distance between the supports being 25 feet,
and the deflection not to exceed -with of the length; the
deflection in inches will be25 x 12 X       300=    _ 4_  5 _'625 of an inch;.48o  480   8
then 25' x 2'5 x'2240 =          the breadth x the cube
~625 x 3359             \ 774 of the depth:




184          On the Strength of Structures.
and, assuming io inches for the breadth, we have 4I774 = 41774 =
the cube of the depth, and the cube root of 4I77'4 = I6-I = the
depth of the beam required.
If the beam is to be square, the fourth root of the quotient
will be the side of the square; thusThe fourth root of 41774 = I4'3 nearly, for the side of the square.
If the beam is to be cylindrical, first multiply the quotient
by I 7, and then extract the fourth root, which will be the
diameter of the beam required, because the deflection of a
cylindrical beam is I 7-ths that of a square beam, all other
circumstances being the same, and hence it requires I'7-ths
the material to render it equally rigid.  The diameter
of a circular beam will therefore be the fourth root of
41774 x I'7 = I6'3 inches = the diameter of the beam.
If the beam is to be rectangular, either the breadth or
depth must be fixed, and the other dimension will be found
thus.
To find the depth of beam required to carry a given
weight, with a given amount of deflection, when the length
and breadth of the beam are given:
The cube of the length in feet x weight
in lbs.          _    -          = the cube of the depth;
breadth in inches x deflection x constant
then the cube root of the result is equal to the depth in
inches required.
To find the breadth of beam required when the weight to
be carried, the amount of deflection, and the length and
depth are given:
The cube of the length in feet x weight
in lbs.                              the breadth  in inches
the cube of the depth in inches x   -  required.
deflection x constant.




Beams and Girders.                    185
To find the weight which will cause a given amount of
deflection upon a beam:
Breadth in inches x the cube of the depth in inches
x deflection x constant.
= weight in lbs.
the cube of the length in feet
To find the deflection of a beam supported at both ends
with a load uniformly distributed over its entire length, take
tths of the result, given by the rule to find the deflection of a
beam loaded at the centre.
To find the deflection of a semi-beam, supported at one
end and loaded at the other, multiply by I6 the deflection
of the same beam supported at both ends and loaded at the
centre.
And, lastly, to find the deflection of a semi-beam, supported at one end, with the load uniformly distributed over
its entire length, multiply by 6 the deflection of the same
beam supported at both ends and loaded at the centre.
The relative deflection of similar beams, each supporting
the same weight, with the supports and load in various
positions is, as follows:
Position of SzJplort and Load.
Relative
deflection.
When supported at both ends and the loads evenly l5
distributed, the deflection is..    J
When supported at both ends and loaded at the    8
centre, the deflection is...    J
When supported at one end and load distributed,  48
the deflection is.....
And when supported at one end and loaded at the  I28
other, the deflection is..... 
The above table teaches a very instructive lesson to engineers, and shows how wrong in principle it is, to have
wheels, pinions, or pulleys, which have hard work to perform,
overhanging the bearing upon which they are supported, and
in the case of machines, where such an arrangement is




186        On the Strengtzh    of Structures.
rendered necessary, the shaft should then be made of proportionately increased diameter in the bearing, and tapered
off, being not abruptly, but gradually diminished to the
diameter of the remaining portion of the shaft.
These relative deflections of the beam and semi-beam,
loaded in different ways, apply equally to beams of any section, so that if the deflection of the simple girder, supported
at each end and loaded at the centre, be found, that of the
others can be ascertained by simple multiplication, as above
stated.
If the student requires to find the deflection of a beam of
any other form of section than the rectangular or circular,
he must first deduce the constant for that particular form of
section, from an experiment upon a similar beam, and then
proceed in a similar manner, or else he may make an experiment on his own account, which will be useful to himself
in other respects.
Resilience of Beams.
The resistance of beams to transverse impact, or, in other
words, to a suddenly applied load, is termed their'resilience,'
and it follows a very different law from that of their strength,
for it is simply proportional to the mass or weight of the
beam, irrespective of the length, or the proportion between
the depth and breadth.
It appears from the published experiments and statements
of the Railway Commissioners, that a beam I2 feet long will
only support 2 of the steady load that a beam 6 feet long
of the same breadth and depth will support, but that it will
bear double the weight, suddenly applied, as in the case of
a weight falling upon it; or if the same weights are used, the
longer beam will not break by the weight falling upon it,
unless it falls through twice the distance required to fracture
the shorter beam.
This law was apparently proved by a large number of




On the Strength of Gearing.          187
experiments carried out for the Railway Commissioners, in
which the beams appear to resist the sudden application of
the load, by gradually absorbing the work accumulated in
the falling weight, and to bend through a certain distance,
until the work done in bending the beam through that
distance is equal to the work accumulated in the weight, due
to the distance which it has fallen. Hence a very strong
beam may be broken, if it is not sufficiently elastic to bend
through the required distance, and thus absorb the vis
viva of the falling weight.
CHAPTER XIII.
ON THE STRENGTH OF GEARING.
IN the chapter on torsion, reference is made to the strength
of spindles and shafts, employed in conveying power from
one point to another, to give motion to machinery. It is
equally important for the engineer to know, how to proportion the other parts of the transmissive machinery-the spurwheels and bevil-wheels, for instance-which are the agents
by which power is transferred from one shaft or spindle to
another. More especially is it important, to be able to calculate the strength of the teeth of wheels, these being the
agents through which the driving forces are directly transmitted.
When properly made, the teeth of the two wheels act
against each other, as the wheels revolve, with comparatively
little thrust, noise, or friction. To secure proper mutual
action of the teeth, their form must be determined on the
principles which are explained in Professor Goodeve's'Treatise on Mechanism.' But, besides determining the
best form for the teeth, the engineer requires, also, to ascertain their size for any given case, and this is usually simply
a question of strength.




i88         On the Strength of Structures.
In estimating the strength of the teeth of mill-gearing,
we have to attend, first, to the strength of the material of
which the gearing is made; second, to the forces which act
on the teeth, due to the power transmitted; and, third, to
the way in which the teeth resist fracture under the action
of the ascertained forces.  We shall best explain the method
of proceeding, by taking an example. Suppose it is required
to determine the dimensions of the teeth of a wheel on the
main axle of a thirty-ton crane. Let the barrel be four feet
diameter, and the wheel six feet diameter, measured to the
pitch-line. Further, let it be assumed that the tension on the
chain which is coiled on the barrel is reduced from 30 tons to
72 tons, as the result of the mechanical advantage of the
chain tackle. The total load on the teeth of the wheel will
be 72 tons, multiplied by the radius of the barrel, and divided
by the radius of the wheel, that is, 72 x 2. 3 = 5 tons,
or I I,200 lbs. We may presume, that this load is distributed
over two teeth, there being always at least two teeth in full
bearing at the same time. The load on one tooth is therefore 11,200 + 2 = 5,600 lbs., which acts on the tooth like
a load on the end of a projecting cantilever, tending to break
it by transverse fracture at the root.
Let it be next assumed that the wheel is of cast iron, and
that, for safety, the strain on the tooth is not to exceed I-th
of that which would fracture it. A cast-iron bar of good
quality, I inch long and I inch square, loaded at the end,
would break with about 6,ooo lbs.  The wheel tooth is to be
considered, as in similar conditions to such a bar, or, in other
words, it is in the condition of a beam loaded at one end
and fixed at the other, and its resistance to fracture is proportional to the square of its depth (that is, the thickness of
the tooth) to the breadth (that is, the width of the face
of the wheel) and inversely as the length (that is, the
projection of the tooth measured from the root to the
point).
As the length and thickness of the teeth of wheels are




On the Strength of Gearing.             89
usually in some definite proportion to the pitch, it will now
be necessary for the student to assume, to the best of his
judgment, a suitable pitch of tooth for the required purpose,
which will give the length and thickness.  For example,
assuming the pitch to be 2'5 inches, the length will be I-875
inch, and the thickness at the root I-55 inch, the breadth
across the wheel being three times the pitch, or 7'5 inches.
Squaring the depth of the tooth, multiplying by the breadth
and the strength of the iron, and then dividing the product
by the length of the tooth, we get-'552 x 7'5 x 6ooo = 57660 lbs.,
1'875
the ultimate strength of one tooth; and the double of that,
namely, 115,320 lbs. for the two teeth, that are supposed to
be in gear. The ratio of this to the strain upon the teeth
will therefore be II5320, or I0'3, the ratio of the strength
11200
to the working load. Should the assumed size be found
either too strong or too weak, the assumed pitch must
be decreased or increased until a suitable tooth is discovered
In practice, the strength will much depend upon the
accuracy of the adjustment of the gear, for if the teeth bear
only at one end-a condition frequently seen-then the great
advantage to be derived from a broad wheel is necessarily
lost, and might as well not have been provided. In addition to proper adjustment, a great increase of strength is
gained by flanging the teeth of both wheel and pinion up to
the pitch-lines; this arrangement is now generally adopted
for wheels of importance.
In determining the substance to be given to the teeth of
wheels to afford a given amount of strength, it has to be
kept in view that such wheels, and the pinions especially, are
subjected to rapid wear when in daily use, thus reducing the
thickness of the teeth; in other words, the depth of the




Igo         On the Strength of Strtuctures.
beam and its strength, which is as the square of its depth.
Hence it is necessary to make some allowance for the
future wear in the strength of the original construction.
From the circumstance that the pinion, in a given period,
revolves so many times oftener than the wheel, the usual
custom of making the teeth of both with the same allowance for wear may seem to be incorrect, and so far that is
the case; but still, as a rule, there are practical objections
in the way of the more correct arrangement, which prevent
its general application, although for special purposes it
is sometimes otherwise.  Thus, for gunpowder machinery,
where great pains are taken, and where gun-metal and hornbeam work together, it is usual to reduce the metal and
add a corresponding extra thickness to the wood.
In the foregoing remarks, cast iron is chiefly referred to,
because it is used more than any other metal for wheel
purposes. But that material is not selected on account of its
special fitness or superiority in any respect, so far as the
duty to be performed is concerned; it is chosen more on
account of its cheapness, and because it may be readily cast
into any form of wheel.  Such considerations have great influence in the settlement of such points, and particularly so
in this case, for, owing to the peculiar form given to spur
and bevil wheels, it is much easier to cast them than to
forge them; indeed, to make large toothed wheels by forging is scarcely practicable, on account of the cost. For
smaller wheels, termed pinions, and where a great number
are required of a definite size, it is perfectly easy to make
them  by forging.  Simple dies are prepared, in which a
roughly shaped, viscous mass of white-hot wrought iron is
placed, and then subjected to the blows of a steam-hammer.
This, together with the aid derived from a taper punch,
driven through the centre, forces the soft iron into every
crevice of the die. By such means, wrought iron is used for
wheels to some extent, and the dies may be so formed that
the teeth will be flanged up to the pitch-lines, and pinions




On the Strength of Gearinzg.          I9I
so made are at least three times stronger than the same
forms when made of cast iron.
In circumstances where only one pinion or a few pinions,
made of wrought iron, are required, it is more convenient (in
order to avoid the cost of the dies) to forge them into solid
blocks, and then cut out the teeth afterwards. This arrangement, however, scarcely admits of flanges, on account of the
cost of manufacture, and the want of flanges may be said to
reduce the strength by one-third; consequently, such wheels
have only double the strength of cast-iron pinions or wheels,
made with flanges.
Of late years, the material called malleable cast iron has
been employed for many such purposes; that is to say, the
wheels are made of cast iron in the ordinary manner, and
then subjected to a course of annealing, while embedded in
some substance rich in oxygen, which combines with a portion of the carbon in the iron, and leaves the metal malleable.  This process is the reverse of steel-making by
cementation. Wheels made by this method may have
flanges, and have double the strength of cast iron, and are
used extensively where great accuracy is not essential, but
it is found difficult to maintain the truth of the pitch and
form of the teeth, during the annealing process. During this
process the casting is kept for a long time at a red heat,
and this frequently causes it to twist or warp.
All difficulty is, however, now removed by the use of cast
steel, which may be cast into any form in an earthen mould,
in the liquid state, in the same manner as cast iron, and, when
properly made, such cast-steel wheels have four times the
strength of cast-iron wheels, the only objection being the cost,
cast steel in this form is nearly as expensive as bronze.
The amount of mechanical power that may be transmitted
by a pair of wheels, depends entirely upon the speed at
which they are driven; this opens up another and entirely
different question, from that of the strength of the teeth of a
wheel for a crane. In ordinary mill-gearing a wheel that




I92         On the Strength of Structures.
might be capable of transmitting Io horses' power at 60
revolutions per minute, will be able to transmit 20 horses'
power, if it is driven at 20o revolutions; the same remarks
apply to all the other parts, such as pulleys, and every kind
of moving mechanism. But, practically, this is only true
within certain limits, for, on reaching high velocities, other
conditions of impact, vibration, and centrifugal force due to
velocity step in, to keep the mechanical application within
bounds, as depending on the material which is employed
for the construction.
Strength of Screws.
A little consideration will show  the student that the
strength of the thread of a screw-bolt, or of its nut, or that
of the tooth of a tangent wheel, depends on principles
similar to those applicable to ordinary gearing. Looking at
the longitudinal section of a portion of a screw, the thread
will be found under the same conditions as a beam fixed at
one end and with the load distributed uniformly, or precisely similar to that of the tooth of a spur-wheel; if, for
example, the screw thread is made of a square form, it will
have less breadth of root, and consequently less strength,
than the same pitched screw, when made in the usual conical
form, or even when made with the top and bottom rounded
off, inasmuch as both of the latter shapes have a broader
base to be broken, or even detruded, thus affording a better
resistance.
Although the common angular thread, in one respect, may
be considered as the strongest form, still it is found that, in
another respect, it is not so strong as the square threadnamely, in resisting the influence of wear, as arising from
the inclined plane action of the angle upon a corresponding
angular surface in the nuts-which is remedied in the flat
bearing of the square thread. Consequently the square thread
is found, in practice, to be not nearly so likely to override
the nut, by any excessive wear or by any inordinate straining.




On the Strength of Gearing.         I93
Screw threads with a round top and round bottom are intended to combine both advantages-to possess the broad
base of the angular thread, and at the same time to have
some portion of the flat-bearing surface of the square thread.
By this compromise they have, upon the whole, a decided
advantage, which gives them the preference for many purposes.
As a rule, screws act in one direction only; such is the
case with screw-bolts and nuts; but there are many exceptions, such as the leading screw of a lathe or planing
machine, or wherever the screw is employed as an agent to
impart motion; screws for the latter class of purposes are
correctly formed when both sides of the thread are alike,
because both sides have to perform the same duty. It is
different, however, with the majority of screws, and especially so in the example given-namely, the bolt and nut.
There is no mechanical or manufacturing reason, why such
screws should not be made at right angles to the axis, upon
one side of the thread, so as to have the bearing surface
similarly placed to that of a square thread, and with the
other side inclined, as in the ordinary triangular thread. The
threads would then be, in section, similar in form to the
tooth of a ratchet-wheel, which acts only in one direction.
Such a form has a self-evident advantage, in the case of the
ratchet-wheel, but it would have a still greater advantage
in the screw, because it would not only give the same
breadth of base as the common angular thread, but that
advantage would be combined with the complete flat bearing surface of the square thread, or, in other words, it is a
form possessing the full advantage of both, and without
any of their disadvantages.
This form of screw thread was adopted by Sir William
Armstrong, for the breech-screws of his celebrated guns,
and its superiority in strength over the square threaded
screw, formerly used, was so conspicuous as to leave no
doubt in regard to its comparative advantage; the form
0o




I94        On the Strength of Structures.
is correct, in principle, for screws that act only in one
direction, and, as the arts advance, mutst become general.
Strength of Cutting Instruments.
Similar principles might be advantageously applied in
many other cases, but they are frequently lost sight of, and,
although a great change for the better has taken place, of
late years, more especially in regard to the forms given to
cutting instruments of every description, so as to combine
great strength with incisive penetration, we are yet far from
having the minds of our workmen so imbued with the
natural principles involved, as to keep them right, by an intuitive perception of what that right is, due to the natural
reason which guides us in so many other things, and which
is independent of men's inventions, and is the effect of time
and training.
To select a cutting instrument for the lathe or planing
machine, as an example; when such an article is formed on
correct principles, with the side under the cutting edge
nearly perpendicular to the work upon which it has to bear,
in order to support the cutting edge, and to give great
strength, and with the other side bevilled off from the cutting edge, like the saw-tooth, so as to give-the knife-action
of penetration; such an instrument will be immensely more
efficient, and many times more economical, than when the
form given to the same instrument has been devised without
knowledge and with reference to penetration only. Forms
which are good in the estimation of the workman, have frequently neither strength nor penetration, the metal of the
instrument being ground away where it should be left, and
being left at the part where it should be ground away.
Such a lack of proper conditions arises, entirely, from not
knowing the true principles, as there is no difference in cost
or trouble. The result is a want of strength, which incapacitates the instrument from grappling with the work, and




On Columnzs.                  I95
tiny shavings are peeled off by it, in a broken condition and
in miserable quantity. An iron shaving of one continuous
curl was shown in the I862 Exhibition, II40 feet in length,
the length of the curl being 462 feet; and some of the Royal
Gun Factory iron shavings are I of an inch in thickness by
42 inches in breadth, and are curled up of any length, as if
they were wooden spills. All this efficiency is due to the
form which is given to the cutting instrument, which of
course has to be supported by a machine of corresponding
strength and power, but the perfection of cutting is entirely
owing to the application of the true principles of strength and
penetration, in shaping the small piece of cast steel which is
used as an instrument. The mass of iron in the shaving indicates the strength of the tool, and the unbroken condition
of the curl is an evidence of its fitness for the removal of
the superfluous material, and at the same time shows that it
is accomplished, without that useless expenditure of mechanical power, which is the case with more detrusive instruments.
With badly formed tools, the shaving is broken up into small
chips, to no useful purpose. Thus it is shown that' knowledge is power.'
CHAPTER XIV.
ON THE STRENGTH OF LONG COLUMNS.
THE resistance of a long column, to loads acting in the
direction of its axis, depends mainly on three conditions:
first, on the proportion which its length bears to its least
transverse dimension; second, on the form of the ends
of the column; and, third, on the direction in which the
load acts, with reference to the axis of the column.
In the case of short columns-that is, columns whose
0 2




I96        On the Strength of Structures.
length is only slightly in excess of their transverse dimen5ions-the material is ruptured by simple crushing alone;
but when the height exceeds from three to eight times the
least transverse dimension, according to the nature of the
material, the rupture is caused partly by bending and partly
by crushing; and, when the length exceeds from twenty-five
to thirty times the transverse dimension, then the column
will fail by bending, and the material will be subjected to
strains, similar to those of a beam, when under a transverse
load; one side will be crushed and the other will be
extended.
There is, at the present time, no completely satisfactory
theory of the ultimate resistance of long columns. Euler
investigated the law of resistance, on the assumption that the
elasticity of the material remained perfect up to the point at
which rupture was imminent. On this assumption, he found
that the resistance of long cylindrical columns would be
proportional to the fourth power of the diameter and inversely as the square of the length.
Hodgkinson found, however, by his experiments, that
the ultimate resistance was proportional to a power of the
diameter rather less than the fourth, and decreased in a
much less ratio than the square of the length.
Table XL. p. I97, gives the mean results of a number
of experiments, made by Hodgkinson, and shows clearly
that the strength of pillars depends greatly on the secure
fixing of their ends. With both ends rounded, the strength
is only ~ of that afforded by similar pillars, having both
ends flat, and abutting on flat surfaces. Pillars, with one
end round and the other flat, have 2 of the strength
of those with both ends flat. These facts show the advantage derived from correct bearing surfaces in structures
exposed to compression. The table also shows that pillars
of timber, when the length exceeds seventeen times the
diameter, are destroyed by bending, and that even those of
seventeen diameters are partly bent as well as crushed.




Results of Experiments made by Hodgkinson to ascertain the Ultimate Strength of Pillars
of Timber.
TABLE XL.
Name of |  Description of  \ Length  Sectional    Ratio of  Mean weight
Nf in   dimensions  length to  in lbs. under       Remarks.
timber.        pillar.                                     which the
inches.  in inches.  least side.  pillars sank.
pillars sank.
r Uniform with
both    ends   605                  34'5 to I     3, I97   Bent and broke.                    C
rounded.       
Uniform   with                                                                                  D
one end round
a  nd the other   60.5              34'5 to I     6,1o09   Round end crushed.
Dantzic    flat.                    t square                    
oak.                     r 6o'5                34'5 to I     9,625   Sank by bending.
297 1I              7  to I    I3,083   Crushed at the ends.
3025     I7'3 to I    14,305    Wrinkled at the ends, bent
and sank.
Uniform   with 1 48    9          L 27'4 to I      9,229   Bent and sank.
both    ends i 46       I square    46  to I      I,754   iBent without crushing at the
C  flat.            46     I'5 square  37  to I      7,888      ends.
58    2 square    29  to I    I2,385    One end crushed-remaining
lRed                       58    2 square    29  to I    I, 6oi        two-thirds of column crippled.
deal.                        8    2-9 x I 4    43  to I      7,68I    Sank by bending in the direc58    3'47 x II15  50  to I     4,349     tion of smaller side.,
_~~~~~~~~~~~~~~~~~~ _I5




198        On the Strength of Structures.
It was found in these experiments that a deflection was
visible, in one case, with a little under ~ of the destroying
load, and that, generally, there was a considerable deflection
with between ~ and } of the destroying load.
The visible result of the crushing strain upon short
columns varies with the nature of the material, and rupture
is caused either by splitting, shearing, or bulging; the
former is characteristic of the hardest cast iron, the hardest
description of stones, and also of timber. Crushing by
shearing is exhibited by cast iron, when the height of the
column is about one and a half time the transverse dimension, whilst crushing by bulging takes place with short
specimens of wrought iron, mild steel, gun-metal, lead, and
other ductile metals.
There are two other forms of crushing: first, crushing by
wrinkling up, commonly called' crippling' or'buckling;'
and, second, by cross-breaking. The former is sometimes
seen in columns of wrought iron, which are too long to be
crushed by bulging, and too short to be bent by flexure;
and the latter occurs in cast-iron columns, when the length
exceeds thirty times their diameter.
The manner in which the column gives way depends
upon the form of the ends of the column, in so far that, if
the ends are rounded, the column is as liable to flexure as
one of the same diameter and twice the length, with both
ends flat and firmly fixed. Hence, if we break two similar
columns, the one with the ends rounded, and the other
with the ends flat, the length being about twenty times the
diameter, the material in the one with rounded ends will
be ruptured entirely by cross-breaking or transverse strain,
while that of the other will be ruptured partly by crushing
and partly by cross-breaking.
The way in which the column yields also depends upon
the direction of the load, which has heretofore been supposed
to pass along the axis of the column. If the direction of
the action of the load forms only a very small angle with the




Oiz Columnns.                 199
axis of the column, it induces a strain upon the column
at right angles to the direction of its axis, and it is then in
a precisely similar condition to a beam supported at each
end and loaded at the centre.
In short columns, possessing sufficient rigidity to resist
the bending strain caused by this indirect action of the
load, the material is liable to be ruptured in detail, from the
fact that the whole of it has not the opportunity of taking
its full share of the work in resisting the load, and therefore
cannot give due support to the smaller portion, upon which
the load acts.
There are a few practical deductions to be drawn from
these three considerations: First, that a column should be
made as short as possible, in proportion to diameter; in
other words, it should be capable of maintaining itself
vertical by its stiffness. Second, that the ends of cast-iron
pillars should be cast with broad bracketed flanges, considerably larger than the transverse dimensions of the
columns at the centre, for although the flanges may not add
to the strength, in a direct manner, yet they certainly add to
the stability, and will prevent the column from bending;
that is, if the ends are made sufficiently stiff and strong to
resist the thrust. Third, that the ends should be at right
angles to the axis of the column, and placed so that the
load may act upon the whole surface of the capital and
base, and its resultant may pass directly along the axis of
the column.
As a matter of economy, all long columns should be of the
form recommended for cylindrical beams-namely, that of a
parabolic spindle-but this is a form which does not please
the eye so well as the graceful outline of a taper column,
and hence, no doubt, strict economy is often sacrificed to
appearance.
In considering the strength of columns, the probability
of crushing by splitting or by shearing may be entirely
neglected, because, practically, columns are never made so




200         On the Strength of Structures.
short that these forms of rupture are called into play, and,
so far as timber and cast-iron columns are concerned,
rupture will always take place by cross-breaking, and wroughtiron plate columns should either be properly stayed, to
prevent buckling or bulging, or else sufficient material should
be put into the section to resist the stress tending to cause
local distortion or wrinkling up of the metal, until the crossbreaking or bending strain can come into play.
The most reliable experiments upon the strength of
columns are those of Hodgkinson, who carried out an
elaborate series of experiments to determine the laws which
govern the strength of cast-iron columns, and who also
made a considerable number of experiments for the Railway
Commissioners, in order to determine the best form of section of a wrought-iron tube, to resist compression, when
the load is applied in the direction of its length.
With respect to cast-iron columns, he found, first, that the
strength of solid cylindrical columns with both ends rounded,
and the length of which exceeded fifteen times the diameter,
varied as the 3'76th power of the diameter; second, that
the strength of solid columns with both ends perfectly flat,
when the length exceeded thirty times the diameter, varied
as the 3'6th power of the diameter nearly; third, that when
the diameters remained the same, the' strength varied inversely as the I'7th power of the length; fourth, that the
strength of hollow cast-iron columns, with rounded ends when
the length exceeded fifteen times the diameter, varied as the
difference of the 3'76th power of the external diameter and
the 3'76th power of the internal diameter, divided by the
I -7th power of the length; fifth, that the strength of hollow
cast-iron columns with both ends perfectly flat, when the
length exceeded thirty times the diameter, varied as the
difference of the 3'6th power of the external diameter and the
3'6th power of the internal diameter, divided by the I'7th
power of the length; sixth, that the strength of hollow castiron columns of the same diameter varied inversely as the
I 7th power of the length, as in the case of solid columns.




On Columns.                  201
From the same experiments he deduced the following
rules, to assist us in ascertaining the strength of such
columns: First, for solid cast-iron columns with both ends
flat, when the length exceeds thirty times the diameter, the
breaking weight, in tons, equals the product of the 3.76th
power of the diameter, in inches, multiplied by 44'J6 (a coefficient deduced from the experiments), and divided by the
I'7th power of the length, in feet. Second, for hollow castiron columns with both ends flat, when the length exceeds
thirty times the diameter, the breaking weight equals the product of the 3'6th power of the external diameter, in inches,
less the 3'6th power of the internal diameter, also in inches,
multiplied by 44'34 (a coefficient deduced from the experiments) divided by the I' 7th power of the length, in feet.
For columns with both ends rounded, or fixed by pins
passing through the ends, Brd of the above strength only
should be reckoned upon, and for columns with one end flat
and the other end round, Irds of the result by the above rules.
The iron, from which the columns used in the experiments
were made, possessed a crushing strength of 49 tons per
square inch, and the coefficients should be increased or
decreased, in the ratio of that strength, compared with the
strength of the iron of which any other columns are made.
It was also found by these experiments that, if columns
with flat ends are less than 30 times their diameter in
length, or if the ends be rounded and the length is less
than I5 times the diameter, that they will be partly crushed
as well as bent; and their strength is ascertained as follows:
First ascertain their strength as columns with the ends
flat, by the rule for columns exceeding 30 times the diameter
in length, or, if the ends are rounded, 15 times the diameter;
then their real breaking strength will equal the product of
the strength found as above, multiplied by the crushing
strength of the material of which the columns are to be
made, as ascertained by experiment, divided by the strength
found as above plus 3 of the crushing strength of the
material.




202            On the Strength of Structures.
From the experiments carried out with solid wroughtiron columns, it appears that the strength of square pillars,
which are long enough to be bent before the material is
much crushed, varies nearly as the 3'6th power of the side
of the square, the lengths being equal.
The results of the experiments upon rectangular tubes of
wrought iron are shown in the following Table, from which
it appears that, in tubes of equal thickness, the strength per
square inch of section of the smaller tubes is greater than
that of the larger, the 4-inch square tube, *o6 of an inch
thick, giving 8'6 tons per square inch; the rectangular tube,
8 inches x 4 inches, giving 6'79 tons per square inch; and the
8-inch square tube giving 5'9 tons per square inch of section.
Result of Experiments made to ascertain the Resistance to
Compression of Rectangular Tubes of Wrought Iron.
TABLE XLI.
Form of section.
Rectangular 8 inches   Square 8 inches  Square 4 inches
Length of tubes   x 4 inches.         side.             side.
in feet and
inches.
Strength          Strength          Strength
Thickness per square Thickness per square- Thickness per square
of plates  inch of  of plates  inch of  of plates  inch of
in inches. section in in inches. section in in inches. section in
tons.            tons.              tons.
Feet. Inches.
IO     0       -        -        -        -       03        4'9
2     6                _                          03       5'5.
Io     o      o6       6'79'o6         5 9'o6          8'6
2     4       o06     7.1                    -             -
I0o    o       -.                     o8-3     I I'24
2     6       -                 -        -.083    12'24
0I            -        -         1'I39   9'I'134      9'63
7     6   I                           I'I34    IO35
Io     0'219    I 1'84             -
Io     0'264    I2-O1I              - _
All these tubes failed by buckling or wrinkling up of the
plates.




On Columns.                       203
It will be seen also from Table XLI., that, to obtain twice
the strength, four times the thickness of plate had to be
used, whereas in some previous experiments on the crushing
strength of wrought-iron plates, similar to a side of these
tubes, the strength varies as the cube of the thickness.
Result of Experiments made to ascertain the Resistance to
Compression of Cylindrical Tubes of Wrouglht Iron.
TABLE XLII.
Length.'i             m )            Remarks.
feet.             CS /d o~; I  -"~= I aoIoI    oI
10                                     6
I0                        601          9'035  All  sank  by
5            10964  4I 6I04   30   -9   1486   flexure.
25Ii65
10I   30     *5         4             I 3'29  Sank byflexure.
I75    249   07            *     328045   2, 8 15667
/2~5~~~ /  I      I        I6'29
10  I                          1       9-6  B oth  sank by
2I 5  2'35 /242'o605'             14378   flexure.
10                        40          12-36
75  30o I    1/51  1349 I            I33'34
2-5                                  1 I   1  i6'7  Slightly bent.
10,                                   12034 }Crushed.
7/5  4'05'o4'.7       22o4       I48
2    1     6'02   Failed    by
36'5            2'5449   i8'6               crippling.
[ 7'5 /3'o |'~5~|t'349/ ~  [ ~15




204         On the Strength of Structures.
It also appears that, in square tubes compressed to such a
high degree, the length has little effect upon the strength,
for in these experiments the Io-feet tubes stood nearly as
much as the 2 feet 9 inch tubes.
It is obvious, from careful study of these tables, that
wrought iron is unsuitable for columns, for in Table XLII.
we see that columns only I2, I3, and I5 times the diameter
sank by flexure, and in one case, where the length was only
Io times the diameter, the column was bent, and in Table
XLI. we find a 4-inch tube 2 feet 6 inches long, or only
73 times the side of the square, failed by the buckling of
the plates; and, in one case, a very short tube (I foot
7- inches only), with a section of 8 inches x 4 inches and
o06 of an inch thick, failed in the same manner, although
the length was only 5 times the width of the least side.
Still, columns are often made of wrought iron, and notably
in the case of poles for sheer-legs, sometimes to over
Ioo feet long, to lift I20 tons and upwards, and by using
these tables of strength, and allowing such a margin that
the breaking strength shall be Io times the working load,
and stiffening by internal T iron ribs or L irons, the structure will be perfectly safe.
To find the strength of a wrought-iron column, multiply
its sectional area in square inches, by the strength per
square inch given in these tables, for the column bearing the
nearest relative proportion of diameter or side of square to
length, and of thickness to diameter or side of square; this
will approximately give the strength of the column required.
Steel is used in the present day, to some extent, for the
poles of sheer-legs, and will no doubt in time supersede
wrought iron for such purposes; for it is equally safe, and
combines with its safety a compressive strength superior to
either wrought or cast iron, and is not nearly so liable to
flexure as the former, nor to fracture as the latter.
Some portable steel sheer-poles were recently constructed
for the War Department to carry I8 tons on each pole, or




On Columns.                 205
36 tons altogether, both when in a vertical position and,
also, when fixed at such an angle that the top end is 15 feet
from the vertical through the bottom end. These poles are
40 feet long, 15 inches in diameter at the centre, and 8 inches
at the ends, the thickness of plate being only'I85 of an
inch, the ratio of diameter to length being  2 at the centre
and 1  at the ends, and the ratio of the thickness to
diameter IT at the centre, and Is at the ends.
The lower ends of these poles are fitted with ball and
socket joints, in order to allow the necessary amount of
lateral movement, and the blocks, which carry the weight to
be lifted, are suspended from a pin, passing through the
upper ends of the poles. Consequently these poles must
be treated as columns with rounded ends, and they are
only capable of supporting ~ of the weight which similar
poles, with ends flat and fixed, would support.
This pair of sheer-poles was tested with a weight of
36 tons, when in the inclined position of 15 feet out of the
perpendicular, in which position a load of 36 tons is equal
in effect to a load of 40 tons with the poles in the vertical
position, and it produces a stress of 2'3 tons per square
inch of section at the centre of the poles. This load they
withstood, without the slightest perceptible deflection.
It will be seen by reference to Table XLII. page 203, that
a cylindrical column of wrought iron with both ends flat,
and a diameter equal to I of the length, and a thickness
equal to - of the diameter, was crushed with I2'34 tons
per square inch of section.
The crushing strength of a similar wrought-iron column,
with both ends rounded, would be ~ of I2'34 tons, or
4'II tons per square inch of section; and, allowing the safe
load of such a column to be - of the crushing load, the
working strength would be about i ton per square inch of
section, or less than - the stress to which the above steel
poles were subjected.
This single example will serve to show that steel may, in




206        On the Strength of Structures.
time, be extensively applied with great advantage, in connection with such structures, and more particularly will
this be the case, when they have to be transported from
one place to another, and frequently dismounted and reerected in an entirely different situation, because the whole
of the gear, both for their transport and re-erection, would
be proportionately lighter.
CHAPTER XV.
ON THE STRENGTH OF CRANES ANlD ROOF TRUSSES AS
EXAMPLES OF COMPLEX STRUCTURES.
WHEN the student has mastered the preceding chapters
upon beams and pillars, comparatively little difficulty will
be experienced in understanding the present chapter, which
treats of the strength of more complex structures. Most of
the examples here given are chosen from actual works, which
have recently been carried out in the War Department, and
although some of them may at first appear difficult to
understand, they are really simple, if the principles applicable
to them are carefully studied.
Before commencing to ascertain the strength of a structure, the whole question raised should first be broadly and
closely considered, in order to determine the direction and
nature of the forces to which the structure may be exposed
in its various parts. It will then be found that by far the
larger proportion of these forces, in modern structures, are
in the direction of the length of the parts of the structure,
and even the majority of the transverse forces are so supported, that they give rise to simple tensile and compressive
strains, which act in the direction of the length of the component parts of the structure. It is comparatively rare to
find parts exposed to forces which tend to rupture them by
cross-breaking, as in the case of a beam.




On Roof Trusses and Cranes.             207
This conversion of the direction of the strains is mostly
effected, by skilfully forming the structure of one or more
triangles or systems of triangulation, or, in more familiar
terms, by bracing the component parts of the structure.
The triangular form of structure is now used, in preference
to any other, simply because a triangle is the only figure
whose shape cannot be altered, while the length of its sides
remains constant.
One of the simplest and most familiar forms of a braced
structure is that of a common roof truss of small span, as
shown in Fig. 24. It consists of two oblique rafters and a
tie-beam; the weight upon the entire truss is transmitted to
the walls by the rafters, and the office of the tie-beam is to
prevent the lower ends of the rafters from spreading outwards, and thereby overturning the walls or columns supFIG. 24.
porting the truss. Each truss has to support one bay of
the roof, and consequently each of the two rafters has to
support one-half of the weight of that part of the roof
which lies between two adjacent trusses, and as this weight
is uniformly distributed, it may be represented as if it really
acted at one point; namely, in the middle.  The whole
structure is balanced or kept in equilibrium, on each side, by
three forces: first, the reaction of the wall, which is equal
to one-half the weight upon the entire truss; second,




208         On the Strength of Structures.
the oblique thrust of the rafter; and, third, the horizontal
tension of the tie-beam.
It is one of the fundamental principles of mechanics, that
if three forces acting upon the same point are in equilibrium,
then three lines, drawn parallel to the directions in which the
forces act, will form a triangle, the lengths of whose sides
are exactly proportional to the magnitudes of the forces.
Hence, if three forces act at any point of a braced structure,
the directions of which are known, we can determine their
relative magnitudes by drawing a triangle. Further, if' the
magnitude of one of the forces is known, and the side of
the triangle corresponding to that force is made equal to it
in magnitude, on any scale of equal parts, then the other
sides of the triangle will be equal to the other forces, on the
same scale.
Referring to Fig. 24, the triangle a b c is formed by
drawing lines parallel to the directions of the acting forces,
one of which passes along the oblique rafter, the other
along the tie-beam, the third being the upward vertical reaction of the wall. As the weight upon the wall is the only
known quantity, the student will first draw a vertical line
a b to represent, on any convenient scale, the upward reaction; that is to say, if the weight or upward reaction is 4tons, then the length of the line drawn must be made 42inches, or 4- half-inches, or }4 units of any other convenient
dimension. From the ends of the line so obtained, let two
other lines b c and a c be drawn, parallel to the other two
forces, so that the three lines may form a triangle; then the
length of the sides of the triangle, when measured by the scale
used for the vertical force, will represent the exact amount of
the thrust upon the rafter and the tension on the tie-beam.
The strains that come upon the diagonals of lattice girders
and the more complicated forms of roofs may also be found
in a similar manner, but the student is referred to the literature specially devoted to those structures for further information. He is, also, advised to make himself master of




On Roof Trusses and Cranes.            2c9
the foregoing simple illustration of the truss, before going
farther, because, when it is thoroughly understood, the remainder of the chapter will be found comparatively easy, and
if it is not understood, to go farther will be useless.
Fig. 25 is a skeleton diagram of the form of trussed beam
used for overhead travelling cranes, for road and railway
bridges of short span, and sometimes for the transverse
girders of a bridge carrying three or four lines of rails. To
FIG. 25.
6tons
ascertain the strain upon the top beam of such a structure,
which is generally of timber, find the weight acting at the
centre; multiply that weight by half the span of the truss,
and divide by the depth of the truss, as in the case of the
flanges of an ordinary beam. The stress upon the strut and
tie-bar may be found, by drawing the triangle a b c, as explained in the previous example. Let the beam be 2o feet
span and required to carry 6 tons at the centre; the depth
of the truss is usually made -th of the span, and in this
example is 2 feet 6 inches.
The strain along the top bar will be io x 6.2 2-24 tons.
The compressive strength of timber may be taken at 6,ooo
lbs. per square inch, and the safe working stress at LIth of
that amount-namely, 600o lbs.-and the sectional area of
material required will be the strain in lbs. divided by the
working stress per sq. inch to be put upon the material;
thus, 24 X 2240 -+- 600oo - 89 square inches.
The strains upon the strut and tie-bar are shown by the
diagram to be 8'54 tons on the former, and 3 tons upon the
latter; the weight upon each strut is equal to the total weight,
p




210         On the Strength of Structures.
divided by the number of points of attachment of the struts.
If the load moves from one end to the other of the beam,
or acts, as shown in Fig. 26, sometimes at one end and
sometimes at the other, the truss must be counterbraced,
as shown by Fig. 26; because, when the weight is over the
strut A, the tie-rod a b c tends to straighten itself, and forces
the other strut B upwards, and with it the beam; this force
is generally counteracted, in light crane beams, by putting
sufficient material in the timber beam to make it stiff enough
FIG. 26.
6tons           t\    i
A                  B
a                 b
to bear the stress without deflection; but when used for
heavy cranes, small bridges, or the transverse beams of large
bridges, the truss should be counterbraced, because the
continual moving load would, otherwise, cause an inconvenient amount of deflection in the top beam, and at the
same time would tend to rupture it transversely.
The strains upon the tie and counterbraces are shown by
Fig. 26, where the weight is directly over one of the struts;
the downward pressure is resisted and balanced by the
tension upon the tie-bar and'counterbrace, and the strains
are represented by the triangle a d e, whose sides are lines
drawn parallel to the directions of the forces, and as the
line a e is made equal to the weight by scale, the other lines
represent the magnitude of the other strains.
The hydraulic wharf crane (Fig. 27) is another familiar
example of the employment of a braced structure, the
framing by which the top cap and slew drum are carried,
forming one side of the triangle, and the tension rods and
jib the other two sides. At the jib head three forces meet




On Roof Trusses and Cranes.             211
and are in equilibrium-namely, the downward pull of the
weight, the resistance of the tension rods, and the oblique
thrust of the jib-and they may be represented by the sides
of the triangle a b c, Fig. 28, which are drawn parallel to
them. If the side a b (which is drawn parallel to the downward pull of the weight) be made by scale to represent the
weight, the lengths of the other sides a c and a b will repreFIG. 27.                      FIG. 28.
sent.the tension uponthetie-rda
sent the tension upon the tie-rod and the compression upon
the jib. In this case, with a weight of 6 tons, the tension is
I8 tons, and the compression is 2,-i tons, by scale.
The crane post is subjected to a transverse strain caused
by the oblique pull of the tension rods at its upper end;
the amount of this pull in the horizontal direction can be
ascertained, by resolving the oblique pull into its two component parts, one horizontal or at right angles to the post,
and the other vertical.  This latter may be disregarded for
the present, as it causes no transverse strain upon the post.
This resolution may be made as shown on the upper part of
F 2




2I2         On the Strength of Structures.
Fig. 28, in which a c represents the pull of the tension rods
in direction and magnitude, d c the horizontal component of
the pull of the tension rods, and da the vertical component.
The post of this crane may be considered as similarly circumstanced to a cylindrical or other beam, projecting from a
wall and loaded with a weight at its outer extremity, equal
in amount to the horizontal component of the oblique pull
of the tension rods, as found by the diagram.  It is also
subjected to a compressive stress in the direction of its length.
The compressive stress in the direction of the length of
the crane post involves a more recondite calculation; it will
be found equal in amount to the weight of the crane
structure, plus the effect of the weight lifted; which latter
will be equal to the algebraical sum of the vertical components
of the oblique pull of the tension rods and the thrust of the
jib; that is to say, the vertical component of the former will
act upwards upon the post, but that of the latter downwards, and the resulting effect of these two opposite forces
will be the remaining portion of the greater stress after the
lesser stress has been subtracted from it, because the
greater neutralises the lesser, and the remainder only acts
upon the post. This force is comparatively so small that
the calculation may be disregarded, because if the crane
post is strong enough to resist the transverse strain, then
the compressive strain will not be called into such exercise
as will affect the stability of the structure.
On the Strength of Sheer-legrs.
The sheer-legs, or poles, shown by skeleton diagram in
Fig. 29, are an example of a temporary braced structure.
In this figure B c represents the position of the sheer-legs,
and A C that of the tension rod. The tension rod in this case
is termed a'guy,' and may be either a hempen or wire rope,
or chain. The holdfast for this guy is furnished by strong
pickets driven into the ground, and the thrust upon the
poles is supported by a small foot plate, laid upon a platform
of timbers at B.




On Roof Trusses  and Cranes.           2'1 3
The diagram (Fig. 30) shows the method of ascertaining
the strains upon the poles and the guy, by the triangle
i~~~~~~~~~~~~~~~~~~
~~  3Otomi~~
0~~~~~~~~~~
-  ~~~~~~~~~~~~~~  6~~~~~~~~




214        On the Strength of Structures.
of forces. With a 3o-tons load, hanging i2 feet beyond
the perpendicular, the thrust on the sheer-legs will be
found to be 34'5 tons, and the tension on the guy so'6 tons.
The amount of the strain upon the guy may, however, be
found by the principle of the lever, or, as it is sometimes
called, the principle of moments, as shown by Fig. 29.
The pole B c may be regarded as a lever turning upon B as
a centre; then the weight multiplied by the perpendicular
distance at which it acts from B-in this case I2 feet-and
divided by the perpendicular distance of the guy from i;,
will give the tensile stress upon the guy. The best method
to find the length of the perpendicular from the fulcrum B,
to the guy is to make a skeleton diagram similar to Fig. 29,
but to a scale of not less than 5 feet to I inch.
The sheer-poles are 40 feet in length, the distance from
the holdfast to the bottom of the poles is 120 feet, and the
top of the poles is I2 feet beyond the perpendicular line
raised from the centre of the feet of the poles; with these
data, the length of the perpendicular B D will be found
to be 34 feet. Then 30 tons x 12 feet ~ 34 feet =  o'59
tons, the strain on the guy rope A c. Although there are
two guy ropes used with these sheer-poles, there is of course
no strain upon the fore guy when the weight is in the position shown on the diagrams.
The two guys are provided to enable the sheers to pick
up a weight at the point shown on the diagram, and transfer
it to the position shown by dotted lines, the strain upon the
one guy rope gradually decreasing till the poles reach the
perpendicular position, at which point there is no strain
upon either guy; but immediately that point is passed the
other guy is subjected to a strain gradually increasing, until,
when a point is reached I 2 feet upon the other side of the
perpendicular, it is of course strained to the same amount
as the other guy was in the first position of the load.
To permit this movement and to render it a perfectly
safe operation, and also to enable a light hoisting crab,




On Roof Tresses and Cranes.            215
with its small complement of men, to perform the work,
a portion of the guy is composed of an ordinary tackle,
consisting of two blocks with three pulleys in each, and the
necessary length of rope to allow the guy to be lengthened
the required amount; the running end of the tackle is wound
direct upon the crab barrel. Then as the strain is distributed
to six reduplications of the rope, the strain upon each of
them and on the rope leading to the barrel will be only I-th
of the total strain upon the guly, or io-6 + 6 = I-76 ton.
To enable the weight to pass between the poles, they are
set 20 feet apart at the bottom; and if the hold fast ends of
the guys are fixed on a line bisecting at right angles the
line joining the feet of the poles, there will be no lateral
strain upon the structure, except such as may be due to
an external circumstance, such as a side wind, and which
the spread of the feet of the poles enables them to resist
effectually.
On (he Strength of a 30-Ton Steam Crane.
Fig. 3I shows an arrangement of steam crane, for lifting
weights of 30 tons, and as there are in the various parts of
this structure examples of transverse, tensile, compressive,
and shearing strains, it is selected as an illustration, to show
the student a method of ascertaining the amount of the
stresses upon such a structure, and also the quantity of
material required to resist them.
This crane works upon a centre pivot at B, and is prevented from rising by nuts, which act as a collar, and are
fixed to the pivot in such a manner as to prevent unscrewing.
The jib of the crane is upon one side, having two rollers
which travel upon a circular rail at A, and is partly counterbalanced by the weight of the steam boiler, steam engines,
and platform, but still leaving a weight of about 2 tons, so
that the weight of the jib may be disregarded when the
crane is unloaded. The steam power is used for lifting or




2I6         On the Strengtht of Structures.
lowering, and is also employed for turning the crane round
upon its pivot, by giving a slow motion to one of the rollers,
and the apparatus is so arranged that the motion for raising
ed \
— a —- --




On Roof Trusses and Cranes.           2I 7
or lowering, and that for turning in either direction, may
both be in gear at the same time.
The whole crane structure is kept in equilibrium by three
forces-namely, the downward pull of the weight, the resistance to the pressure upon the guide rail A, and the
resistance to the upward pull upon the centre pin or pivot B.
It may be treated as a large bent lever loaded at the end of
one arm with a weight of 30 tons.
The direction of these three pressures are shown in
Fig. 33. The weight, multiplied by its perpendicular distance from the guide rail or fulcrum of the lever, and
[1oI'              FIG. 33. -
A;O
I'.ll tto]
233-  tolls
FIG. 34.
divided by the length of the other arm of the lever, gives
the amount of force that is acting at the short end of the
lever, to balance the weight of 30 tons; thus, 30 x 25 -
9I7 = 82 tons nearly, and this force of 82 tons, resolved
(as shown by the diagram, Fig. 34) into its horizontal and
vertical components, gives 39'2 tons for the former and
72 tons for the latter. To find the pressure upon the
guide rail, multiply the weight by its distance from the
centre pivot, and divide by the distance from the other end
of the lever to the guide rail; thus, 30 x 33 - 9'I7
To8 tons.




218         On the StreMgth of Structlurs.
Having found the strain upon the centre pin and guide
rail, and agreed upon the nature of the foundation best
adapted for the situation, it becomes a simple matter to
determine the amount of material necessary to resist those
strains.
The' strains upon the jib and tension rods of the crane
may be found by means of a diagram, in a similar manner
to that described for finding the strains upon the same parts
of the hydraulic wharf crane, and by reference to Fig. 32
it will be seen that they amount to 77 tons compression on
the jib, and 56 tons upon the tension rods, while lifting a
weight of 30 tons, and the quantity of material required to
resist the strains upon the tension rods, when the iron is
subjected to a stress of 2 tons per square inch, will be 562 = 28 square inches, and as there are two rods, there will
be I4 square inches required in the section of each, and if
the rods are round they will be 4~ inches in diameter.
The jib of these cranes is a most important part of the
structure, and the great compressive strength of cast iron
would seem to indicate that it would be the best material of
which to construct it; but there are many difficulties in
making a cast-iron jib of such proportions, capable of withstanding with safety the shocks to which all crane structures
are more or less subjected, in working, and hence wrought
iron is used as it is more reliable.
Long pillars of wrought iron have to be carefully braced
to resist the tendency to flexure, and more generally is this
the case when they are not perpendicular to the line of
thrust, or when they are (as it is sometimes termed)'bowed.'
Fig. 36 shows an arrangement of wrought-iron pillar very
commonly used for the jib of large cranes, and also for the
piers of viaducts and other similar structures; it consists of
two side plates of sufficient area to resist the compression
to which they are subjected, when well braced with lattice
bars, dividing the long unsupported column into a number
of short pillars, and keeping the side plates in a direct line;




On Roof Trusses and Crancs.              2 I9
FIG. 35.            FIG. 36.          FIG. 37.:'                 fre of
~~ ~~~ ii~~~




220         On the Strength of Structures.
two angle irons, extending the whole length of these plates,
are riveted to their inner side, for the attachment of the
lattice bars.
The side plates of this jib are subjected to a transverse
strain, along their whole length, tending to produce flexure,
the value of which may be found by resolving half the
direct pressure on the jib (because there are two side plates)
into its two component parts, as shown on the diagram
(Fig. 35), the one, acting along the line of the side plate,
equal to 38'3 tons, and the other, at right angles to it, which
is equal to a compressive stress of 3'6 tons. This latter is
the force which tends to produce flexure, and it is to resist
this strain that the lattice bars are inserted; these bars are, as
shown in Fig. 37, of a T section, 6 inches x 3 inches x a an
inch, and have an effective area to resist tensile strains of 3square inches, after deducting the rivet holes at their end.
The tensile strain upon these bars is also shown on the
diagram (Fig. 36), and is found by resolving the transverse
stress of 3'6 tons upon the side plates into two component
forces acting in the direction of the lattice bars, and equal
to 2'5 tons on one and I'75 ton on the other. A simple bar,
32- inches x I an inch, would have been sufficient to resist
this strain, and therefore there must be some other reason
for making them stronger and of a T section, and it is
to enable them to prevent the deflection of one bar without the other, and thus to form the whole into a structure
capable of resisting the compressive and transverse strains
above described, and also the. tendency to deflection, or'sagging,' as it is sometimes called, engendered by its own
weight.
The jib and tension rods are connected at the jib head
by a strong turned pin, which also carries the two chain
pulleys; this pin is subjected to a shearing strain at each
end, equal to the pull upon one tension rod, and it should
at least be of the same sectional area as the tension rod,
and hence of the same diameter; the lower end of the jib




On Roof Trusses and Cranes.           221
is connected to the crane framing by two strong turned pins,
fitting into the framing and the side plates of the jib, which
are made thicker at this point, and also at the point where
the pin at the jib head passes through them, and bored out
to the exact size of the pin. These pins are very important parts of the structure, and should be well fitted, so that
they cannot bend in the holes through which they pass.
Each pin at the lower end of the jib has to resist a pressure of 38'5 tons-that is, one-half the total thrust-and, in
order that the material may not be subjected to a greater
stress than 2 tons per square inch, must have a shearing
section of 38'5  - 2 =  I9'25 square inches.  But as the
pin would have to be cut across at each side of the side
plate, it need only have an actual sectional area of I9'25
2 = 9'625, which is equivalent to a diameter of 3'- inches";
but if the pin is not supported at each side, it must then
have an area of I9'25 square inches, as above stated, or a
diameter of about 5 inches.
Each tension rod is also connected by a turned pin to
the top of the side frame of the crane, similarly to the
lower end of the jib; these pins are of course subjected to
the same stress as the pin at the jib head, but, as they are
supported at each side of the eye of the rod, they only
require to have half the sectional area; that is, 14 -- 2 = 7
square inches, or a diameter of 3 inches.
The centre pin, or pivot, upon which the crane revolves, is
fitted at the top end with a strong nut, shown on Fig. 33,which
is screwed down to its bearing upon the girder in which the
pivot works, and may be secured by a steel pin, passing
through the pivot and nut, or otherwise, so that the latter
cannot work back; and the girder is bored truly and the
pivot is turned to fit nicely, so that there may be no play.
This pivot and nut have to resist the whole of the upward
pull due to the 3o-tons weight; the magnitude and direction
of this force and its vertical and horizontal components are
shown on Fig. 34. The latter of these strains acts with a




222         On the StrengZth of StrtZcturs.
leverage equal to the whole length of the pivot, tending to
break it across in a sfmilar manner to that described in the
example of the wharf-crane post; and if the pivot is made
strong enough to resist this stress, it will be of ample sectional area to resist the vertical stress, which causes only a
tensile strain upon the material.
The formul,  generally given, to find the diameter of
round bars of iron required to carry a weight in this manner,
are founded upon two facts: first, that a bar of wrought
iron I inch in diameter and I2 inches long, supported at
each end and loaded at the centre, will break with a weight
of about 2,000 lbs.; secondly, that the strength of such bars
varies directly as the cube of their diameter and inversely
as their length.
We know that a bar or beam, similar in every respect to
the above, but supported at one end and loaded at the
other, will only support 1 th of that weight, or 500 lbs.; and if
a I-inch bar, I2 inches long, breaks with 500 lbs., a simlilar
bar, 32 inches long, (the length of the centre pin) will
only bear 500 x I2 + 32 = I87? lbs.  Therefore, the
centre pin would require to be equal in diameter to the
cube root of its breaking weight, divided by i87-, the
breaking weight of a i-inch bar, or, which is the same thing
put in another way, the cube of the diameter of the centre
pin, must be equal to the breaking weight in lbs. which it
will have to support, divided by I872-.
The breaking weight should be 392 tons-that is, ten
times the working stress of 39'2 tons, and 392 x 2240-+I872
- 4683-and the cube root of 4683 = I6'73-say, i64
inches-the diameter of pivot required, if it did not fit the
hole in the girder.
By making the pivot to fit the hole in the girder exactly,
the load upon the pivot would be distributed over its entire
length, and its strength would thereby be doubled, and
hence the cube of the diameter of the pivot required would
be only one-half of that required under the former circumstances.




On Roof Th'vzsses and Cranes.         22 3
The cube of the diameter in the former case is 4683,
and in the latter case would be 4683 divided by 2, which
is equal to 234I'5, and the cube root of 234I'5 is I3'26Say, 13} inches-the diameter of pivot required, provided it
is made to fit in the girder exactly.
The sectional area of pivot required to resist the vertical
component of the load only, supposing the material to be
subjected to a stress of 2 tons per square inch, would be
72 tons - 2 = 36 square inches, or an equivalent diameter
of 6'8 inches nearly; so that if sufficient material is provided
to resist the lateral or transverse stress, there will be ample
to resist the vertical or tensile stress.
The cast-iron girder A, Fig. 33, in which the pivot is
fitted, is 8 feet long between the supports, and 2 feet
6 inches deep, and has to resist the same strains as the
pivot-that is, a pressure at the centre of 72 tons, the vertical component of the upward pull of the crane, and, as
previously explained, in the case of the common cast-iron
girder at p. I75, the bottom flange will, by the principle of
the lever, be subjected to a stress caused by half the weight
upon the girder, multiplied by half the span, and divided by
the depth of the girder-thus, 36 x 4 -. 2 -= 56'4 tons,
and allowing that cast iron in a crane structure should not
be subjected to a tensile stress of more than I ton per
square inch of section, the flange must contain 56'4 square
inches of iron in its section: each flange has to resist the
same amount of lateral strain, and it would therefore be
advisable, in this case, to make them of the same sectional
area at the centre, and reduce them each to -2rds the area at
the ends.
The girder above the guide rail is cast hollow, so that the
upper half of the guide wheels may work within it; the transverse section is shown in Fig. 39, and the side elevation in
Fig. 4i.  This girder has to resist an upward force equal
to the pressure upon the guide rail, shown by the diagram
(Fig. 33) to be equal to io8 tons; the top flange of this
girder win be subjected to a tensile strain, because the load




224         On the Strengtht of StrZuctures.
upon it acts vertically upwards, instead of downwards, as in
the case of any ordinary weight.
This girder is also 8 feet span and 2 feet deep, and it is
loaded at two points; namely, directly above each guide
wheel at 2 feet from the ends, with a pressure at each point
of half the total load, equal to 54 tons, as shown in Fig. 40.
Each of these forces is equal in its effect upon the girder
to 27 tons, acting at the centre of its length, because,
although the weight is reduced one-half, its distance from
the support, and consequently its leverage, is doubled, and
a total force of twice 27 = 54 tons. acting at the centre of
the girder, would be precisely equivalent to the two forces of
54 tons acting as shown in the diagram. By a similar process to that pursued in finding the strain upon the flange of
the girder A, we find, that this weight of 54 tons would cause
a tensile strain of 54 tons upon the top flange of this girderthat is to say, the weight upon one support, due to the 54
tons acting at the centre of the girder-nainely, 27 tons,
multiplied by half the length of the girder, 4 feet, and
divided by the depth, 2 feet, = 54 tons —and therefore the
sectional area required in the flange to resist it, when the
iron is subjected to a stress of I ton per square inch = 54
square inches. As there is no lateral strain upon this girder,
the bottom flange may be reduced to ~th the area of the
top, or I32 square inches; but as one-half of this flange will
be placed on each side of the guide wheel, the webs should
be connected by a cast-iron rib across the middle of the
girder, as shown by the dotted lines in Fig. 40.
The side frames of a crane are necessarily made of such a
form as may be best adapted to give room for the various bearings and supports for the connection of the jib, tension bars,
and connecting girders. The principal stress, upon the side
frames of these cranes, is that which is caused by the pull of the
tension rods at the top, and these frames may be considered
to be subjected to a similar strain to a wall crane, when lifting a weight at a distance beyond its proper radius, as shown




FIG. 4T.
-7O~~~~~~~~~~~~~~~FO
to                                                                                                  11o
FIG- A38                         FIG. 39-             FIG. 40



226         On the Strength of Structures.
in Fig. 41, the horizontal members of these side frames being
in a similar position to the wall upon which the crane is
fixed, and the other members fulfilling the same office as
the tension rod and strut of the wall crane, the pull of the
tension rods of the large crane representing the weight.
The total pull exerted by the tension rods is 56 tons;
that is, 28 tons at the top of each of the side frames. This
force is balanced by the resistance to tension supplied by
the vertical member, and the compression of the oblique
member, and if, as before explained, lines are drawn parallel
to the direction of the three forces, and the one representing
the pull of the tension rod be made equal by scale to 28 tons,
the length of the other lines, measured by the same scale,
will represent the strains on the vertical and oblique members, as in Fig. 4i. By the diagram, these forces are found
to be 44 tons on the vertical, and 37 tons upon the oblique
member of the frame, and hence an effective area, when all
holes are deducted, of 44 square inches, will be required to
meet this strain alone in thle vertical member, and additional
material to resist the load caused by the gearing shafts.
The oblique member has, in addition to the compressive
strain caused by the pull of the tension rods, to withstand
the pull, at the centre of its length, of the hoisting chain
upon the barrel, which amounts to 7} tons, as shown in
Fig. 38.
The shafts which carry the gearing of a heavy crane
are of great importance, and the method of finding the
diameter of shaft, required to resist the strains, can be best
shown by a few examples; we may take, for instance, the barrel
shaft, and the shaft of one of the guide wheels, which is fitted
with gear and driven to slew the crane round. The principal
strain to which these shafts are subjected is that of torsion,
and the torsional strength of cylindrical shafts varies as the
cube of their diameters; this knowledge, coupled with the
fact that a bar of wrought iron I inch in diameter is twisted
asunder by a weight of 8oo lbs., acting at the end of a




Crane Gearhing.                227
I2-inch lever fixed upon one end, enables us to determine
the size of shaft required to resist any torsional strain.
The barrel shaft of this crane has to resist a twisting force
of 71 tons, acting upon a barrel 4 feet diameter, the length
of the lever being one-half the diameter or 2 feet, and the
breaking strength of the shaft should be equal to ten times
that amount, or 75 tons acting with a 2-feet leverage. The
breaking strength of a I-inch bar with a 2-feet lever will
be 800 *- 2= 400 lbs., and hence by dividing 75 tons
reduced to pounds by 400, we obtain the cube of the
diameter of shaft required.  Thus 75 x 2,240 -- 400 =
420, and the cube root of 420 = 7'488, say 71 inches, the
required size of shaft.
The load upon the guide-wheel shaft is determined by
the adhesion due to the weight upon the wheel, which under
favourable circumstances amounts to 600 lbs. per ton of the
load, the weight upon the wheel is 54 tons, and 54 x 6oo
32,400 lbs. as the adhesion of the wheel upon the guiderail. The diameter of the wheel is 3 feet, and therefore the
length of lever i8 inches; the weight required to break a
I-inch bar with I8-inch lever = 800 x 12 -- 8 = 533 lbs.
The breaking strength of the shaft should be ten times the
adhesion acting at the end of an i8-inch lever, that is
32,400 X 10 = 324,000 lbs., and 324,000. 533 = 608, the
cube of the diameter of shaft, and the cube root of 6o8 =
8'47 inches, the diameter of shaft required.
The size of the remaining shafts is determined in the
same manner, that is, by first finding the twisting load, then
the strength of a I-inch bar under the same conditions,
dividing the former by the latter and extracting the cube
root of the quotient.
Referring to the remarks made at page I85, on the comparative strength of shafts with overhung gear, and to
gear having an outer bearing, let us apply them in such a
crane as that now under consideration.
Let us suppose the pinion which gears into the wheel
Q 2




228         On the Strength of Structures.
on the barrel shaft, namely, the main driving pinion of such
a crane, to be keyed upon the outer end of a shaft, say
4 inches in diameter, and that the pinion is i2 inches wide
and has to drive the large spur wheel, with a load of 7'- tons
at the pitch line. We have already said, that a bar of
wrought iron i inch in diameter and I foot long, supported
at both ends, will be crippled with a weight of 2,000 lbs.
acting at the centre. Now a similar bar, supported at one
end only, and having the load distributed as in the case of
the above shaft, would only support one-half of 2,000 lbs.,
or I,ooo lbs., and as the strength of cylindrical beams varies
as the cube of their diameters, all other conditions being
the same, the above-mentioned shaft would be crippled
with a weight of I,ooo lbs. multiplied by the cube of its
diameter, or I,ooo x 64 = 64,000 lbs. The actual load
on the shaft is 7-3 tons, or I6,8oo lbs., and hence the ratio
of the working to the breaking stress is, as I6,8oo is to
64,000, or as I is to 3'8.
Such a margin of safety, although sufficient for many
purposes, is not so in the case of a crane. The breaking
strength (as has been several times stated) should be at
least ten times the working load, to ensure perfect safety.
This margin of safety is more especially required when the
load is being lowered, under the control of the friction break,
and then suddenly checked; the effect of the load, in consequence of the stoppage of the motion, is increased to an extent which may imperil the whole fabric.
This margin would be more than obtained, by simply
increasing the length of the 4-inch shaft, and putting a
bearing outside the pinion instead of having it overhung,
as the breaking stress of a I-inch bar, under these altered
circumstances-that is to say, when supported at both ends
and with the load distributed-would be 2,000 X 2 =
4,000 lbs., and hence the strength of the given shaft would
be 4,000 x 64 - 256,000 lbs., and the ratio of the breaking
strength to the working-load would be as 256,00o is to
I6,8oo, or as I5 is to I.




Crane Gearing.               229
Although, when the pinion is overhung, the arrangement
may not necessarily cause the breaking of the shaft directly,
still it will permit it to deflect and thereby cause the load
to be thrown upon one end of the tooth, namely, upon
that which is next to the bearing, instead of upon the
whole length as it ought to be, and will thus endanger the
safety of the gearing; and even if this latter should be of
ample strength, the deflection of the shaft will continue,
until in time the shaft will take a permanent set, and ultimately break off, close to the bearing.
The method of finding the strength of the teeth required
in the respective wheels has been explained in Chapter XIII.,
p. I87.
Having thus far shown how to find the quantity of
material required in the various parts of the crane, we have
now only to deal with the chain for carrying the weight,
and by general consent the Admiralty formula given at p. 153
is adopted.
In these cranes the weight or load upon the chain barrel
is diminished, by multiplying the number of chains to carry
the required weight. The total weight of 30 tons is supported by four chains, thus: The chain is anchored to the
underside of the jib-head, and passes down under one of
the pulleys in the movable block, up to and over a pulley
at the jib-head, and again descends, passes under the
second pulley in the movable block, and finally ascends
and passes over a second pulley at the jib-head and thence
to the barrel,
The load on each of the lengths of chain above the
movable block will, therefore, be 30 tons -4 = 7- tons.
Another way of finding the strain on the chain, which
may be used when there is any exceptional arrangement of
the blocks or tackle, is to see how much chain is taken up
in lifting the weight i foot; then if, as in this example, 4 feet
of chain are taken up on the barrel, the load upon the chain
will be 4th of the total weight lifted; if 6 feet be taken up,




230         On the Strength of Structures.
the load will be ~-th, and so on; for, by a well-known principle, a small weight acting through a long distance is precisely equal to a large weight acting through a proportionately shorter distance.
On Crane Foundations.
In order that such cranes may be reliable, it is necessary
to have a sound and unyielding foundation. Where the
ground is firm and solid, there is comparatively little difficulty in obtaining the required stability, but, as a rule, in
the places in which cranes have to be erected, it is otherwise. To mreet the requirements fully, the foundation should
be such, that the pressure on the guide wheels does not cause
the guide rail to yield perceptibly, as it passes round the
circle; a yielding rail adds considerably to the motive power
required to turn the crane round.
In Figs. 42 to 49 are shown three descriptions of foundation, which have been constructed for 3o-ton cranes. The
foundation illustrated in Figs. 42 and 43 is adapted for a
crane situated upon an ordinary solid wharf; that in Figs.
44 and 45 is for a crane which had to be erected at some
distance from a wharf, in order to obtain a sufficient depth
of water to allow vessels of the larger class to come under the
crane. From the isolated position of this crane, sufficient mass
had to be placed in the foundation to render it perfectly
stable and capable of resisting any strain tending to overturn
it, without throwing any strain upon the light jetty, erected
to carry the Loads to and from the crane. The foundation
shown in Figs. 46, 47, 48, and 49 is for a crane erected at
the head of a long pier.
In each of these three descriptions of foundation, sufficient
weight or resistance is provided in order to counterbalance
the upward pull of the centre pin, on which the crane turns,
and also for any increased strain that might be caused by
the sudden stoppage of the descending weight, which is the
chief cause of accidents. Hence, although the actual strain




Crane Foundations.              231
Plan
FIG. 42.
Section
FIG. 43.




232        On the Strength of Structures.
is 82 tons, the weight or resistance of material which would
have to be lifted, before the crane could overturn, is 328 tons,
or four times the amount of the ordinary working strain.
Concrete is used in two of the foundations (Figs. 43 and
45) to furnish the necessary weight, and the cast-iron block,
in which the centre post is fixed, is secured to the mass by
means of four holding-down bolts a a, each bolt being fixed
to a strong cast-iron plate 6 feet square, which is embedded
in the bottom of the concrete mass, and drawn up tight by
strong nuts fitted upon the upper end of the bolts.
In each case, these holding-down bolts have to resist an
ordinary working load of 82 tons (Fig. 33), and as they may,
by the carelessness of the breaksman, have to resist a much
greater strain, they should not in ordinary working be called
upon to bear more than 2 tons per square inch of their sectional area, when each bolt will have to resist a vertical
strain of 72 tons -- 4 = i8 tons, as will be seen by Fig. 34.
This vertical strain, in consequence of the oblique position
of the bolt, produces on it a tensile strain of I9 tons, and
therefore they must each have a sectional area of 9 -- 2 =
9l- square inches (the stress per square inch to which the
iron is to be subjected being 2 tons).  To obtain 9- square
inches of section, the diameter must be 3- inches nearly; the
actual diameter of the bolts used was 4 inches.
The second class of foundation (Figs. 44 and 45) depends,
entirely, upon its mass for stability. It consists of a cast-iron
cylinder 20 feet in diameter, and 36 feet high, filled up to
within 5 feet of the top with concrete, the remaining space
being occupied by masonry to form a bed for the centre
block and guide rail; the total weight of the cylinder, its
contents, and the crane, is 690 tons, and a force equal to a
weight of 300 tons, hanging from the crane-hook, would be
required to balance it; for, by taking moments about the
bottom edge of the cylinder, we have on the one side (go
tons multiplied by Io feet, and upon the other 300 tons
multiplied by 23 feet, each of which equals 6,90o. Hence




Crane FoucndLations.            233
Half PSlan
FIG- 44~
1~                      ~~~~ —,~'~:~. ~~                 f
ijf fi1..fi../fi./,/w, X/,/Wi,..S >, Q
Section:~~? \~~_~\-~I~)
FIG.~s~ 45.




234         On  the Sf Structurcs.
the whole structure is just balanced; any excess of weight
above the 300 tons, upon the crane, would overturn the
whole foundation.
Before erecting such a mass, the bottom upon which it is
to be built should be carefully examined, and if it is not
sufficiently hard to resist the weight, it must be piled, as
shown in the wharf foundation, Fig. 43; in the case of the
cast-iron cylinder foundation, the bottom was solid rock,
and consequently quite able to resist the pressure, which
amounted to 2'3 tons per square foot of surface.
In the construction of a large cast-iron cylinder, much
judgment is required. A comparatively weak cylinder would
suffice, after it is erected and filled in with concrete, but the
dangerous period in its history is during the time of its erection, when it is empty and subject to the pressure of a rising
tide, tending to crush it inwardly by a collapse of the fabric.
To meet this, the joints of the plates ought to be broad and
planed, so as to have a uniform bearing, and to fit all over
the surface, in order that the arch, formed by the cylinder,
may be rigid. The reason of this will be more apparent to
the student after reading the next chapter, where reference
is made to the collapse of steam-boiler flues, and the laws
by which they are governed.
In the third class of foundation, shown in Figs, 46, 47,
48, and 49, which has been erected at a pier-head, the conditions are more complicated, and will require some attention
to enable them to be understood.
The centre support consists of a cast-iron cylinder, seven
feet in diameter. This cylinder is carried down to the solid
gravel, and is filled with concrete. The top of this cylinder
is made to receive the lower ends of the four holding-down
bolts, a a. It will be evident that this centre pillar would
be insufficient of itself to furnish the necessary stability, not
only in consequence of its want of breadth of base, but
likewise on account of the great intensity of the pressure at
the base. It will be seen, by referring to Fig. 47, that a




Crane Foundations.
Fi. 46.
FIG. 47.




236           On the Strength of Structures.
FIG. 48.-Plan on Lower-tier Girders.
FG. 49Plan on Upper-tier Girders.
FIG. 49. —Plan on Upper-tier Girders.




Crane Foundations.                  237
portion of the weight of the foundation of the pier crane
structure is thrown upon the eight surrounding piles, each
of which is fitted with a screw blade 3 — feet in diameter.
These eight piles have therefore a bearing surface of 8o
square feet, and together with the 7-feet cylinder represent
bearing surface of I i83 square feet.
The wrought-iron guide rail, upon which the crane revolves, is laid upon a strong cast-iron ring or girder, and the
load is transmitted from it to the central cylinder and the
surrounding piles, by twelve cast-iron struts; eight of these
struts rest upon the central cylinder, and the remainder upon
the lower tier of girders and against the four corner piles of the
surrounding square. The strains upon these struts are shown
in Figs. 50 and 5I. The weight upon the guide rail is Io8
tons, distributed by the two guide wheels or rollers, and it
is supported by at least two of the central struts in any
position of the crane; and when the crane is directly over
one of these struts, one half of the load is carried by the,
two adjacent struts, and the other half by the strut directly
under the centre of the crane; hence the greatest load upon
either of the struts is equal to Io8    2 =  54 tons of vertical pressure.
These struts are in a similar position to the jib of a crane,
and the pressure upon them may be found by a triangle of
pressures a b c (Fig. 50), as explained in the example of the
wharf crane. The amount of this strain is shown by the
diagram to equal 56 tons, and as the length of the strut is
only about eighteen times its least breadth, it may be subjected to a stress of I- ton per square inch of its sectional
area, at the ce;ltre of its length, with safety; and the sectional area required at the centre will be 56 *.I = 371
square inches. The actual form of section is shown in the
figure, and the actual area is equal to 3943 square inches.
When the crane is directly over the diagonal of the square
formed by the surrounding piles, the vertical pressure of 54
tons is supported by one central and one diagonal strut,




238          On the Strongth of Structures.
FIG. 50.
Section
FIG. 51.




Crane Foundations.                239
and the strains upon each are shown, by the diagram a b c,
(Fig. 5 ) to equal 33 tons on the former and 24 tons on the
latter.
The weight of the crane and its foundation, including the
cast-iron cylinder filled with concrete, is about  90 tons; this,
with the weight lifted, is equal to a total of 220 tons, a portion of which is transmitted to the surrounding piles by the
eight diagonal struts, already referred to, abutting upon the
underside of the lower tier girders, and fixed at their lower
extremity to the piles, by a joint pin 3 inches in diameter,
These eight struts have each to support a vertical pressure
of 220 —. 8 = 271 tons; and as they rest upon a joint pin
at one end, they are only capable of carrying ~rds the load
of a similar strut with both ends fixed, and therefore must
not be subjected to a greater stress than I ton per square
inch of sectional area at the centre. By diagram, as shown
in Fig. 53, the longest struts, namely those fixed to the
corner piles, are subjected to a compressive stress of 3343
tons by the action of the vertical pressure of 27' tons upon
the upper end, and hence the sectional area required is 33',
square inches. The actual form of section is shown in
Fig. 52, and it has an area of 36 square inches.
If the student has been able to understand this chapter, he
will be in a position to apply the knowledge acquired in other
directions. The elementary principles here explained are
not by any means confined to the examples that have been
selected for illustration; they are equally applicable to a
variety of other structures. The remarks that have been
made, upon the direction and amount of stress which is
brought to bear upon the members of roofs and cranes, relate
also to any other structure in similar circumstances, and it
will be useful for the student to apply the knowledge gained
in other cases with which he has to deal, so as to fix the
principles in his mind. The same may be said of the observations on form and proportions, and on the quantity of
material to be introduced so as to afford the requisite




240         On the Strength of Structures.
strength, that the structure may not only be able to balance
the stress, but to have in addition the necessary margin of
safety. The margin of safety necessary in different cases is
a subject which admits of great difference of opinion; in this
FIG. 52.
0                            I
FIG. 53.
volume a leaning to extra caution has been chosen with a
purpose, so that if the student errs at the outset of life, he
may err on the safe side; and as he gains experience, the
margin of safety can be modified if it is found advisable.




241
CHAPTER XVI.
STRENGTH OF RIVETED STRUCTURES-STEAM  BOILERS, ETC.
BUILT-UP structures, such as steam boilers, which are made
of thin plates, riveted together by clenched iron rivets,
depend for their strength upon a number of conditions.
As a rule, the boiler plates have not the same ultimate
tensile strength, per square inch, in the direction of the fibre,
as the same iron when made into the form of thick bars.
The difference amounts even to several tons, and still more
unsatisfactory is the reduction of strength in the other, or
transverse direction. This limited tenacity in boiler plates
is no doubt due to the rolling of the plates to such thinness,
but the weakness must be recognised in estimating the conditions of strength in a steam boiler.
In the chapter on wrought iron, it will be seen that from
23 to 25 tons, per square inch, is the usual tensile strength of
such a quality of wrought iron as is required for this purpose; but when this quality of iron is made into thin
boiler plates, the tenacity is frequently reduced to 20 or
21 tons, and is seldom over 22 tons; and in the other, or
transverse direction, it is rarely over I9 tons per square inch
of sectional area.
An important practical lesson to be drawn from this circumstance is, that the boiler-maker, in arranging the order
of the plates for the construction of a boiler, should have
their strongest direction put in the line of the greatest stress.'rake, for example, the shell of an ordinary cylindrical
boiler; the least strain is in the direction of the length,
hence the strongest direction of the plates should be put
circumferentially. This is also the better arrangement, on
account of the greater extension or wire-drawing that the
iron will submit to, when it is drawn or stretched in the direcR




242          On the Strenzgth of Structures.
tion of the fibre. For reasons to be shown hereafter, boilers
are seldom found to give out in the direction of the length,
from constructive weakness; that is, if they are sufficiently
strong in the other direction, namely, circumferentially.
When these boiler plates are joined together by the usual
butt or lap joints and riveting, the joint becomes still
weaker than the plates across the fibres.  In thus joining
the plates together, by a butt or lap over and rivets, the
strength will obviously depend on the resistance both to
shearing and to tearing. The rivets may be considered as
pins which may be shorn, the force which is required to
shear a rivet being the shearing strength of the iron, per
square inch, multiplied by the sectional area of the rivet.
The tearing of the plates may occur in two ways, either
by the plate tearing along the line of rivet holes, or from
the pieces between the hole and the edge of the plate being
forced outwards, by the simple detrusion of the stronger
rivets, the force required being as the shearing strength per
square inch, multiplied by the area of the pieces thus
pushed out of place.
The shearing strength of rivets depends on the quality of
the iron, and ranges from nearly 23 tons, in the very best
descriptions, down to I8- tons; but 22 tons per square inch
is usually considered to be about the average strength of
the best Yorkshire iron suitable for rivets. The iron of the
best plates is about the same in quality as that of the rivets,
and, in the direction of the fibre, may be considered to have
a tenacity of 22 tons per square inch.
In order to have the rivets equal in strength to the plates,
it is necessary to observe a definite proportion between the
diameter and pitch of the rivets and the thickness of the
plates. The usual rule is to make the diameter of the rivet
equal to two thicknesses of the plate, which, as regards
strength, is a small fraction under the correct equivalent,
but is sufficiently near for all practical purposes.
The plates at the lap or butt being double, will be about
equal, in strength, to the rivet, if the space between the hole




Strenztth of Riveted Structures-Steam Boilers, &c. 243
and the edge is equal to the diameter of the rivet; hence a
single riveted joint has a breadth equal to three diameters
of the rivet, and the pitch, or distance from centre to
centre of rivets, is three diameters. From this it will be
seen that one-third of the metal has to be cut out of the
plate, and its strength is thereby reduced one-third. This
diminution of strength may also be aggravated by punching
and rough usage in putting the plates together for riveting;
the tenacity of plates, originally 22 tons, may thus be brought
down to I8 tons, but this may be avoided by piercing the
hcles with a drill.
The following Table is taken from the Proceedings of the
Mechanical Engineers, for 1872; it forms part of a paper which
was read by Mr. Walter R. Browne, bearing on the subject of
riveted structures, containing the results of experiments made
by Mr. Kirkaldy.
TABLE XLIII.
Riveled 7oihats.
Proportions.      Ratio of
strength
Description of   Riveting.   Rivet                        of joint
joint.    Riveting. holes.  Diameter  Lap or   Pitch  to that
to thick-  cover to  to dia- of plate,
ness.  diameter.  meter. per cent.
Lap.
Lap           Single  1 Punched    2         3       3       55
~Lap    Single   Drilled      2        3        21
Chain. Zig.
Lap           Double  Punched       2    5    6      41     69
Drilled     2    5    5       4      75
cover.
Butt,   cover  Single   Punched    2   6        3      55
Butt, I cover (Singe IDrilled      2        6        2I      62
Butt, I cover  Double  Punched    2    11   12    4         69
Drilled     2    10   11    4        75
Butt, 2 covers Single   Punched    It       6        3      57
S Punched   I      I   13   5~j    72
Butt, 2 covers  Double  Drilled      I   10   12    4       79
t 2




244         On the Strength of Structures.
The resistance to shearing is increased by the frictional
force, arising from the grip, due to the contraction of the
hot rivet in cooling, and to the action of the hammer or
riveting machine. This frictional adhesion is very considerable, but when added to the resistance to shearing it
does not make the joint equal in strength to the plate, even
in its weakest direction, which should be noted.
From a number of valuable experiments, made by Sir W.
Fairbairn, it appeared that, assuming the strength of the
plate to be Ioo, the strength of an equal length of the single
riveted lap joint would be 56; but by having a double row
of rivets applied, either to form zigzag or chain riveting,
the strength was increased to 70. Putting it in another
form, and assuming the strength of the iron to be 50,000 lbs.
per square inch, then the strength of various kinds of riveted
joints would be as under —lbs.
Iron....   50,000  per square inch of the
Double riveted joint.    35,co0  per square inch of the
Single riveted joint.   28,000 )  section of the plate.
The reduction of strength due to riveting greatly affects
the construction of boilers. The joint is evidently the
weak link of an important chain; consequently the strength
of the boiler is only equal to that of the joints; the greater
strength of the plates is so much useless metal, which it is
the object of the engineer to eliminate, so far as he may be
able to do so, by the practical devices which are at his
disposal.  This may be done either by increasing the
strength of the joints, so as to bring them up to that of the
plates, or by reducing the substance of the remainder of
the plates, or otherwise, the desideratum being uniformity
of strength throughout the entire boiler structure.
Since the above facts were made known, a considerable
amount of attention has been directed to the subject, in
order to devise some arrangement, whereby the strength of




Strenth of Riveted Strizctzires-Steawi Boilers, &c. 245
the joint might be brought up to that of the plate. With
this object in view, it has recently been suggested, to
strengthen the joints of boilers and other riveted structures
by modifying the shape of the rivet. Rivets are usually
round, and about 3-inch in diameter; if, however, the same
quantity of iron was made into an oval form, with the thickness of the narrow part 2 inch, the remainder of the metal
going to increase the length in the other direction, the
strength of the joint would be altered for the better, because
a larger area would be left between the rivet holes.
The resistance to shearing of an oval rivet will be proportional to its area, and will be nearly alike in either
direction, as will be seen by referring to the chapter on
shearing, p. I43. It is there shown that greater stress was
required to shear a bar when placed on its edge than when
laid upon its side, but it is more than probable that the
increase of resistance was due to the shorter period occupied
in the performance of the operation, in the former case,
with shears in which the blades were not perfectly parallel.
This conclusion is supported by the general result of the
experiments made on shearing and punching, which go to
show that the stress required is as the area of surface which
has to be detruded.
With the boiler plates the case is different, because the
weakening of the plates is in proportion to the area punched
out, in the line of fracture through the rivet holes; therefore,
if a rivet, with the same strength as a 3inch cylindrical rivet,
can be put into a hole which is only half an inch wide,
measured along the joint, it is evident that, so far as this
part of the joint is concerned, the plates would be made
stronger by one-sixth, all other conditions remaining the
same. At the same time the rivet can be strengthened to
the same extent, by increasing its substance in the longer
dimension, so as to make the plates and rivets equally
strong at the point of junction, so far as the resistance to
shearing depends on the sectional area. But here another




246         On the Strcn'gtkh of Structzires.
condition must be taken into account; a very narrow rivet
acts injuriously on the plate, consequently the limit of this
narrowing of the rivet is soon reached, because when it
becomes too thin, its wedge action comes into play with
greater force, from want of the necessary breadth of bearing
surface upon the plate, on the same principle that the
(liameter of the bolt or pin of a tie rod requires to be of
larger sectional area than the rod itself, or that the double
rope sling over a crane hook does not afford the double
strength of two single ropes, which it always does when
passed over a pulley of proper diameter, thereby avoiding
the wedge action, due to the concentration of force upon a
small bearing surface.
Should such an arrangement of rivets be found to give
satisfaction, there will be no practical difficulty in carrying
out the system; an elongated hole can be punched as
easily as a round hole; nay more, an oval hole can be
drilled as easily and as cheaply as a round hole, when
proper appliances are provided. This latter statement may
be doubted by those who are only familiar with the piercing
of holes of the round shape, but such is the case, nevertheless, and as more refined systems of manufacture are introduced, and the forms given to boilers are simplified, the
whole row of oval slits may be made at the same time, by
a corresponding number of revolving chisels, the plate
meanwhile having the requisite amount of transverse motion
to give the elongation.
The arrangement of joint-construction is also being modified for the better in other directions. By looking at the
section of a joint of the best form, when the ends of
one plate or of two plates butt, and are riveted to one
cover-plate, it is evident that the rivet is not in the best
condition to do its full duty, that it would be stronger
if there were two cover-plates, one situated as at present,
only a little thinner in section, and having a corresponding
one placed on the opposite side of the joint, thus supporting




Strcngti ofRiveted Structurcs —Steaz Boilrs, &c. 247
the rivet at both ends, like the links of a pitch-chain, and
thereby adding to its efficiency.
By referring to page 244, it will be seen that the strength
of single riveting is to that of double riveting as 28 is to 35;
hence, in the construction of the better class of boilers, the
joints that have to resist the greater stress-that is, the
circumferential stress-are double riveted, while those that
have to resist the longitudinal or lesser strain are made with
single riveting; yet even the double rivets are relatively the
weaker for their work, because the circumferential stress is
doubly as great as the longitudinal stress. And when all
has been done by these modifications, the joints are weaker
than the original plates, to the extent of one-fifth of the
strength of the plates.
With the view, therefore, of raising the strength of the
joints to that of the plates, Sir W. Fairbairn introduced the
system of rolling the plates with a thicker substance at the
edges, where the holes have to be pierced, so as approximately to bring up the sectional area between the rivet
holes to that of the general section of the plate, and
thereby to obtain uniformity of strength throughout.
Another most ingenious device which has recently been
proposed is, to arrange the boiler-plate joints diagonally,
instead of longitudinally and transversely, according to the
usual method; by the new system the shell of the boiler is
formed by winding the plates in a diagonal or spiral
direction, whereby the difference of strength in the two
directions is considerably modified for the better. With this
arrangement the joints are at an angle of 450 with the axis
of the boiler, and the strength of the joints in the weakest,
that is, the circumferential direction, is thereby considerably increased.
Mr. W. R. Browne says that,'taking the angle of the
joints at 450, and considering any square portion of the
boiler surface of which the joint forms a diagonal, it will be
seen that, instead of the two pairs of equal and opposite




248         On the Strength of Structures.
tensions, one pair double in amount of the other pair,
which would act on the sides of the square in ordinary
square jointing, there are, with the diagonal jointing, two
equal and opposite resultant tensions acting across the
diagonal joint. The resultant tension, per inch run of the
joint, is found on calculation to be about four-fifths of the
greater tension, and it acts not exactly'at right angles to
the joint, but at an angle of about 72o with it. The latter
circumstance, however, does not materially affect the result,
and the tension on the diagonal joint may, therefore, be
taken at four-fifths of that on the longitudinal joint; consequently the effective strength of the joint and of the
boiler is increased in the ratio of four to five. Thus, in a
punched lap  joint, if single riveted, the proportionate
strength of joint as compared with that of the entire plate
is increased from 55 to 69 per cent., and if double riveted
from 69 to 86 per cent. The diagonal joints have, in most
cases, to be replaced by transverse seams at the ends
of the boiler; but as the tension on a transverse joint is
only one-half of that on a longitudinal one, this does not
form any objection.'
The great change which has taken place in the manufacture of plates by the rolling process, whereby they are produced of great length, breadth, and thickness, is leading
rapidly to a simplification of the forms given to boilers, and
to an increase of strength at the same time. The circular form
of boiler is now employed for marine purposes, where
rectangular boilers, with flat stayed surfaces, were formerly
used, and are even yet employed in many cases. This
modification, combined with elongated rivets, diagonal
riveting, thickened edges, and double cover-plates, will
increase the strength of the structure at its weakest points,
and render the strength more nearly uniform.
The strength of a boiler is also much affected by the
mode of putting the plates together, previous to riveting.
When the holes are punched by careless workmen, it is not




Strength of Riveted Structures- Steam B'oilers, &c. 249
uncommon to find them not opposite to each other when
the plates are brought together, and in order to be able
to introduce the rivet, a taper-drift or punch is driven in,
with violence, by means of a heavy hammer, thus damaging
the line of holes to an incalculable extent, by sheer thoughtlessness. In the construction of the best class of boilers,
the specification provides that tne holes are to be drilled,
and if any error is found to exist in the correct meeting of
the holes, no drift is to be employed, any metal that has
to be removed being drilled or rymered out with a cutting
instrument, thus avoiding injury to the metal.
The strength of boilers depends upon the nature of the
material employed for their construction: wrought iron is
that which is most generally adopted, but steel is now
rapidly coming into use, and in past times copper was not
uncommon. Stated in even numbers, the strengths of the
three kinds of material stand to each other about as
follows:
Steel                             go,ooo lbs.
Iron....             50,000,,
Copper.                           34,000,,
From  this comparison, it may be correctly inferred that
steel will be very generally used for boilers in the future.
The strength of iron boilers is not much affected by
the working temperatures, up to considerably over 400~,
nor by low temperatures down to the freezing-point. But
when the temperature of the plates, through the absence
of water or any other cause, rises much above 500~, then
a change commences. Above 750~ the tenacity diminishes
very rapidly, and when the plates become red hot, they have
lost fully the half of their usual strength.
In those parts of steam boilers where the pressure acts
upon the exterior surface of tubes, as in the internal flues of
the cylindrical boiler before referred to, new conditions are
established. Here the tendency is to crush, and to cause




250         On the StrezgtAz of Structures.
a failure by collapse of the tube. To resist external pressure
the circular is obviously the strongest iorm, and, consequently, it is that form which is now always preferred
wherever it is admissible. In such constructions, lap-joints
are avoided, and welded-joints are generally substituted,
although sometimes the ring is made in one or two pieces
with butt-joints and cover-plates, to which the ends of the
rings are riveted, either with single or double riveting.
The greatest tendency to weakness, in such flues, arises
from the difference of expansion and contraction, due to the
changes of temperature that are likely to arise in ordinary
working, thus causing a difference of length between these
flues and the exterior shell of the boiler: hence many plans
have been resorted to, in order to give the different parts
some freedom  to act, according to their several requirements; one of the best is that shown in Fig. 54.
This diagram represents part of the internal flue of a
cylindrical boiler, such as is shown in Fig. 57; the tubular portions marked a a a a in the diagram are made by bending
a plate into an overlapping circle, and then the ends are
welded into a perfectly cylindrical, but short tube, of the
breadth of the boiler plate. These short tubes are then
joined by means of rivets, in the usual manner, to the corrugated rings or hoops, b b; the left-hand part of the diagram
shows the exterior of the flue, while the other part sho.vs it
in section, from which it will be perceived, that the curved
form of the ring gives the flue considerable liberty to
expand and contract, as may be necessary, during the
lengthening or shortening that takes place in the ordinary
working of the boiler, but which would not be the case
if the ring was made of a jL-iron or other solid form, or
even if the flue were of one piece. This hollow form which
is given to the ring likewise secures another important
advantage, inasmuch as it allows the entire substance of the
flue to be of nearly uniform thickness, or rather thinness,
because so far as the transmission of the heat and the




Strength of Riveted Structlures- Steam Bo]ilers, &c. 25 I
endurance of the material are concerned, the thinner the
flue, the better it is. Another advantage arises from the corrugated shape, in that it becomes a sort of circular beam
to prevent the collapse of the flue.
b                                       W
__o I  I_ -          0_W I                      _
0_. _ = _  _     _l l0_           _      _
FIG. 54.
This arrangement of flue construction, therefore, not only
affords liberty of motion in the direction of the length, blut.a'c the same time it adds greatly to the strength of the flues
in resisting collapse, at the edge of each plate. Front
certain experiments, made by Sir WNT. Fairbairn, the marked
superiority of circular to elliptical or other forms of tube, in
resisting collapse, was most decided, and where  strength
combined with thinness are of such importance, the circular
form should not be willingly departed from.
In order to make sure of the strength of steam boilers,
when in ordinary use, the best practical course is to subject
them  to a hydraulic test, periodically, say after every 500
hours of actual work. The pressure applied should be the
double of that to which the safety valve is ordinarily loaded,
or one-tatird of the ultsae time it adds greate strength of the flutructure. There
or one-third of the ultimate strength of the structure. There




252           On the Strength of Structures.
is no risk of danger arising from this test, and if only a
moderate degree of intelligence is brought to bear upon the
boiler, during the process, in order to watch its behaviour, all
risk of an explosion from weakness is avoided, and the
operation is accomplished at a cost so trifling as not to be
worth regarding, when valuable life and property are at
stake.  Belonging to the War Department, there are about
two hundred steam boilers, which are all treated as here
recommended, and hitherto with the most satisfactory results.
From these general remarks, it will be seen that the
strength of steam boilers depends on many conditions, but
if the exact conditions are precisely known, then there will
be no difficulty in estimating the strength, as due to form,
dimensions, quality of material, and workmanship.
To enable the student to understand the nature of the
strains in steam boilers, a few simple examples are here
selected, for his guidance in the investigation of these and
other similar structures.
Previous to estimating the strength of steam boilers, it is first
necessary to examine and consider their shape, and the direction of the strains to which they are exposed. Let us select,
as a simple illustration, the case of a plain cylindrical boiler,
with hemispherical ends, as shown in Figs. 55 and 56. This
boiler is exposed to rupture in two directions, as shown by
the arrows, first from the ends being pushed asunder, which
is resisted by the iron contained in the transverse section of the shell; second, by the bursting the cylindrical
shell in the direction b b, which is resisted by the iron contained in the substance of the shell, when considered as a
ring. This latter strain is similar to that in a water-pipe,
and consequently the stress is as the diameter of the pipe or
boiler, that is to say, a boiler of double diameter would
necessarily require to have double the thickness of iron to
give the same strength, whereas in the other direction,
although the surface exposed is as the square of the diame



Strength of Riveted Structures —S/cam Boilers, &c. 253
ter, yet for certain reasons, which will be presently stated,
it is still the stronger. Let us suppose this boiler to be
20 feet long, by 54 inches in diameter, and that it is constructed with plates of wrought iron of three-eighths of an
inch in thickness. The strength in the direction a a depends on two conditions: first, the greatest sectional area
which is exposed to the pressure, namely, the diameter
squared and multiplied by'7854, which is equal to an area
of 2290 square inches; second, the resistance offered by an
entire ring of the iron of the shell, taken in the transverse
direction, which in this case will be 54 x 3'1416, equal to a
circumference of I69'64 inches, the thickness of the iron
being three-eighths of an inch.
Here an important question arises, namely, What is the
ultimate strength of the shell per square inch of section?
The ultimate strength of the iron may be 50,000 lbs. per
square inch.  But the shell is weakened by the riveted
joints which connect the plates together.
Suppose that the joints are double-riveted, the rivets being
placed in a zigzag direction, so as to obtain the best result.
Then the tenacity of the joint may be taken at 35,000 lbs.
per square inch of the gross section of the plate, as has been
already mentioned. For a single-riveted joint, we should
have had to reduce the strength to 6,ooo or 7,000 lbs. per
square inch less. To allow an ample margin of safety, we
will assume the strength of the double-riveted joint to be
34,000 lbs. per square inch.
The length of the ring of the shell is I69-64 inches, the
thickness is 3ths of, an inch, therefore I69'64 x  -3ths x
34,000 will give a total resistance equal to 2,i62,400 lbs.;
then if we divide that strength, by the area of the section of
the boiler, we shall have the pressure per square inch that
would be required to burst the shell in the longitudinal
direction; 2,162,400     4 lbs. ultimate bursting pressure
area 2290
per square inch.




254          On the Strengthl of Structures.
A                                     IAi
in 
t                     I vt




Strength  of Riveted Struzctres-Steams Boilers, &c. 255
As regards the strength in the other direction b b, let us
suppose a ring in the middle of the shell, one inch in length,
this ring will be exposed to a certain pressure per inch, on a
surface of 54 square inches, with an amount of iron to resist
that force equal to twice 3 x I, or "ths of a square inch.
Then, taking the strength of the weakest point, as before,
34,000 lbs. x 3 = 25,500 lbs., the resistance of the ring, and
this divided by the area of 54 square inches will give a total
resistance of 472 lbs., which is exactly the half of the
strength in the other direction. The lesson is here taught,
that the shell is twice as strong in the direction a a, as it is
in the direction b b, for a given uniform internal pressure.
The ultimate strength at the weakest point is thus equal
to a pressure of 472 lbs., and the boiler would be perfectly
safe if worked at 60 lbs. steam pressure, so that 672= 7'86,
showing that it would have a margin of safety fully 7- times
the working pressure, which, however, is not by any means
too much for a new boiler, because corrosion and other
causes will soon reduce the original strength.
To select another familiar example, that of the ordinary
flat-ended cylindrical boiler, with two internal flues, passing
from one end of the boiler to the other end, and both riveted
to the flat ends, as shown in Figs. 57 and 58.  Here the
conditions are widely different to what they were in the
former example. In considering the extent of surface which
is exposed to pressure at the two ends of this boiler, it will
evidently be necessary to deduct the area of the two flues,
and, still farther, in reckoning up the quantity of iron which
resists pressure in that direction, it will be equally necessary
to take the tensile strength of the iron, composing both the
shell and the two flues, which together will place the strength
of the ends in a still higher ratio to that of the shell, in a
longitudinal direction, than in the case previously considered.
There is one weak point, however, in these flat endsnamely, at the point marked a in Fig. 58; this arises from




256          On7 t/e Strenrgt  of StrzctlLres.
the flatness of the form and the thinness of the plate. If
the ends were hemispherical, which is the strongest form,
then it would be different. Still, for convenience, they are
flat and hence weak at the point a, and therefore have to
be supported, either by tie-rods passing from end to end
of the boiler, or by what are termed' gusset-plates,' b b, as
shown. These are plates combined with angle irons, which
are riveted both to the shell and to the upper part of the
flat ends.
The course to pursue, in finding the strength of this shell,
would be exactly the same as in the former example, because
the shell is not assisted by the flues. Suppose, then, the
boiler to be 7 feet in diameter, by 28 feet long, with two
flues, each 30 inches in diameter, and for convenience let
us suppose the iron to be - an inch thick throughout,
then 34000 _ 404 lbs. as the bursting pressure per square
84x I
inch of the shell. The strength of the ends will depend
upon the area exposed to pressure.
84 x 84 x.7854 = 5542 = whole area.
30 x 30 x'7854 x 2= I4I3 =flue area to be deducted.
or 4I29 inches of surface exposed to pressure.
The circumference of shell equals..  264 inches.
Do.      do.   of the two flues..  I88
452,,
of 2-inch iron, which is equal to  226  square inches,
in strength, 34,000 lbs. per inch; then 226 X 34000 =
4I29
I86o lbs. as the ultimate pressure per square inch, which
would be required to rupture the boiler transversely, thus
showing that this boiler is fully four-and-a-half times as
strong to resist the longitudinal pressure, as it is to resist
the pressure which tends to rupture the shell in the other
direction.




Strength ofRiveted Structures-Steam Boilers, &c. 257
But here it has to be remarked, that such boilers, or,
indeed, boilers of any other shape, may be weakened to
any extent by improper arrangements, as, for instance, by
cutting out holes for the steam-dome, or manholes, without
introducing material to counteract the loss of strength thus
occasioned. The judicious boiler-maker will always introduce strengthening rings, or otherwise stiffen the part
affected, in order to fully make up for the weakening
of the general structure, due to the cutting away of the
material.
There is another important practical point, which affects
to a considerable extent the endurance of steam boilers,
namely, the arrangement of the plates that have to be
joined. This remark applies more especially to the shell
and the flues of boilers of the cylindrical shape. Formerly, it was usual to unite the shell and flues to the end
plates by means of an angle iron, but the better mode is to
flange over the iron plate, in order to shorten the distance
between the rivets and the point of exposure to pressure,
and thereby to reduce the leverage of the stress which tends
to open the joint. The same general remark applies to the
system of overlap joints, either in the shell or the flues;
there is more'work' or stress thrown upon them than is
the case when the shell is more perfectly cylindrical, with the
shell-plates butted to each other, and then united by coverplates. By this arrangement the strain is more direct, and
anything that reduces the tendency to movement contributes
to the permanence of the boiler.
When a boiler structure is so combined that certain parts
are kept in a state of restlessness, either through differences
of expansion and contraction, or by having parts of the
steam engine attached to different points, all such restlessness tends to wear out the parts so exposed, not so much
by the mere friction, as by a process of disintegration,
whereby the incipient oxide is prematurely scaled from the
surface of the plates or joints.
S




258           On the Strength of Structures.
We may select a third example of boiler, still more complicated, namely, the tubular variety, of which the boiler of
the locomotive engine may be considered as the most
familiar type. As the present purpose, however, is not to
explain boilers, but rather to consider their strength only,
let us for convenience take this boiler in its simplest form,
as consisting of a fire-box a, barrel b, and smoke-box c,
as shown in Figs. 59 and 60. -A slight examination of
its several parts will convince the student, that its three
divisions are under very different conditions as regards
strength.
To select the cylindrical barrel first, because it corresponds more nearly to those boilers which have been
already considered, the strength of the cylinder or barrel
portion will be exactly the same as that of the shells of
the two former examples, and therefore need not be further
investigated. The small tubes likewise may be disregarded,
because other conditions interfere, which demand for them
a substance that places them far beyond the danger of
collapse.
The fire-box consists of two boxes, an internal and an
external, and the whole of their parts, as well as the partitionplate of the smoke-box, are all most disadvantageously placed
as regards resistance. They are thin flat plates of iron or
copper, which, if left unsupported, would yield with a few
pounds of pressure; and when considered as a beam, with
the load uniformly distributed, the strength of these plates
to resist either breaking or deflection or collapse, from the
great pressure of steam, may be set down as approximately
nil, in consequence of the want of depth, due to the thinness of the plates.
To those who are not familiar with steam boilers, the
suggestion may occur, that this difficulty could be overcome
by making them thicker, but such a course is impracticable
on account of the plates of the fire box (which is the most
vulnerable part) having to transmit the heat of the furnace




Strength of RivetedStructures- Steam Boilers, &c. 259
to the water, as rapidly as possible, so that it becomes
necessary to resort to other means of strengthening, in order
to have the whole boiler structure of equal strength, and
without requiring or employing an inordinate quantity of
material to accomplish this condition.!ooooooooc
00000000 
00000000 _'00000000
000000000
00oooo00oo0 0
0000000000
0000000000
aOOOOO OOO
FIG. 59.
0000000000.
0000000000 a,         -            - a
0000000000 
0000000000o  o  o                   Bo o
0000000000
0000000000
0000000000
0000000000
1z5           a
FIG. 60.
Looking at Figs. 59 and 60 it will be seen that the steam
or fluid pressure is acting upon all their flat surfaces, and
tending to push the inner fire-box into the furnace, and the
plates of the outer box outwards. Both are equally loaded
S2




260        On the Strength of Structures.
with fluid pressure, on every inch of exposed surface, and
necessarily in opposite directions. This pressure, in opposite'directions, is directly counteracted, and the two thin plates
are united into one beam, by means of a series of copper or
iron stays, these stays being repeated at every few inches of
surface. The stays are inserted in such a manner as really
fits them to take hold of the plates, with a grip equal
to their own strength; the plates are evenly drilled through,
a suitable tap is employed to screw them both, at the same
time, then the stay or bolt is screwed through both, and
afterwards riveted over at each extremity, thus taking a
double hold of both plates. Hence, the question of strength
is transferred from the plates to the stays; these stays are
made of iron or copper, according to the class of boiler.
Selecting the weaker metal, copper, for an example, let us
suppose its ultimate strength to be I5 tons per square
inch, and that we are disposed to put a stress upon it equal
to 3 tons per square inch, which is as much as is suitable,
and let us assume also that a stay is introduced to every
25 square inches of surface, and that each stay has an area
eqiual to the half of one square inch, this will give us a
resistance of Ix ton to be distributed over 25 square inches,
or I34 lbs. per square inch, in addition to the modicum of
assistance derived from the plates themselves. As such
a boiler would not be worked at a pressure over I20 lbs.
per square inch, it will be seen that the margin of safety is
very considerable.
The crown of the inner fire-box is a part which is much
exposed to the fire, and stands out in a solitary position
away from. external support.  We saw, in reference to
copper, p. 79, that its strength does not improve when heated,
on the contrary, it becomes weaker; hence, the crown-plate,
taking everything into account, is badly situated. In this
strait, recourse is had to a series of thin beams on edge, or
jL-irons variously applied, which are placed on its upper
surface, and at regular distances apart, and to which the




Strenzgth of RivetedStructzures-Steam Boilers, &c. 26 
crown-plate is either stayed, or held up by ordinary rivets;
the stays or rivets being of such dimensions and number as
will give the required resistance. The united strength of
these beams, together with that of the stays attached to the.
top of the outer box, must be at least equal to six times
the entire pressure which comes upon the crown.
Here another difficulty occurs: the presence of such
beams or j-irons necessarily interferes with the passage of
the heat from the fire to the water, which is not only
damaging to the metal, but is disadvantageous in other
respects. To obviate this objection, the beam or j-iron
does not rest upon the crown all over, but is separated from
it, either by distance-washers, or by relieving the under surface of the beams, or by other arrangements, so that the
water is in contact with a surface of thin metal, which is
prevented from deflecting by stays or rivets, that derive their
support from an extraneous source, and the amount of
resistance may be calculated in the same manner as for the
copper stays of the two fire-boxes.
From several causes, the crown of this inner fire-box is
more critically situated than any other part, and is more liable to be weakened by the action of the fire; its temperature
is always greater than that of the water, which is still farther
aggravated by the rapid formation of steam-bubbles, which
form a medium between the hot metal and the body of solid
water over it, so that in determining the strength of such a
vital part, all these points must be taken into account, and
due allowance made for the tear and wear, during the period
which it may be expected to last.
The partition of the smoke-box and the corresponding
part of the inner fire-box are exposed to the same fluid
pressure as the other parts, just referred to, and as they are
thin they would have, if left to themselves in that position,
comparatively little strength. As a rule, however, they are
thicker than the other plates, not from choice but from




262         On the Strength of Structures.
necessity, inasmuch as they have to hold a series of small
tubes, through which the fiery gases pass from the furnace to
the chimney. Hence, advantage is taken of these tubes to
contribute assistance to the tube plates, by offering resistance
to the fluid pressure as an additional duty. Corresponding
holes are formed in both plates, the tubes, nicely prepared,
are securely inserted, and as additional aid, ferrules (accurately fitted) are driven into the interior of the ends,
thus converting the tubes into stays.
It so happens, however, that these stays are not to be
entirely relied upon, inasmuch as they are apt to work
loose in the course of time. Hence, other more permanent
stays of solid metal are introduced, at intervals, with nuts on
the outer surface of both plates, thus rendering the structure
more reliable.
From the foregoing remarks on boilers, the student will
perceive that the question of strength is not so difficult as it
appeared at first sight; it has only to be taken in detail, the
various strains carefully worked out, and the strength of the
material which is, or which has to be provided to meet the
stress, may be calculated.
Resistance to Coliapse.
The flues and other parts of boilers, which are exposed to
the steam pressure acting from without inwards, tend to give
way in another manner-by collapse. The flues are the
most vulnerable part of the cylindrical boiler, unless they
are properly supported, either by the corrugated rings already
referred to, or by some other arrangement. The subject of
collapse is so important that it is desirable that it should
receive some further explanation.
This most important subject was thoroughly investigated
by Sir W. Fairbairn, who carried out a long series of experiments upon wrought-iron tubes at the request of the
Royal Society and the British Association, the object being
to ascertain the laws which govern the resistance to collapse




Strengthz ofRiveted Structures-SteamBoilers, &c. 263
in such structures. The results have been freely published
in Sir W. Fairbairn's'Useful Information for Engineers.'
As these experiments were extensive, the following brief
remarks on the subject are only intended to describe the
most important of the results which were obtained, and
likewise to draw attention to some of the more important
laws deduced from the experiments.
In these experiments, the tubes were firmly secured at
the ends, so as to be in the same condition as flues in
steam boilers, and the end fastening probably contributed
in some degree to the remarkable results obtained in these
experiments.
By these experiments it was shown, first, that with tubes
of the same diameter and the same thickness of metal, the
strength to resist collapse was nearly inversely as the length;
and, second, that with tubes of the same length and thickness, but of different diameters, the strength is nearly inversely as the diameter.
In the most important series of these experiments, the tubes
were all cylindrical and of the same thickness, but varied
in length and diameter. The tubes were made from a single
plate, bent to the cylindrical form, and the joints riveted
and brazed to render them water-tight; the ends of the tubes
were riveted to rigid cast-iron discs, and the air from the
interior of the tube was allowed to pass out, when the tube
collapsed, through a pipe screwed into one of the discs.
The tubes were placed in a strong cast-iron cistern.filled
with water, and the pressure was applied by a hydraulic
pump, and its amount was registered by two gauges of the
best description. It is to be noted that the term collapse,
as here used, does not imply the crushing of the material,
but simply that the circular or elliptical arched shape of the
transverse sections changes its form, at the weakest part of
the tube, by the thin plates either bending or wrinkling.
The results of this set of experiments are shown in the
following Table:



264              On the Strength of Structures.
Results of experiments made to ascertain the resistance
of cylindrical tubes of wrought iron to collapse when the
ends are firmly fixed, and the tubes are subjected to an external pressure.
TABLE XLIV.
I         2         3         4         5         6         7
Size of the tubes.               ComparaMean pretive num____   Mean p   hers showiNumber o                               sure in lbs.  ing the!the experi                             per square collapsin  Mean of
mthepenl          Tck asinch at                comparative
mental    iameter  Length    ickness whichthe  pressure   numbers.
t:be    Diaeter Iength  iMnes s wic tue
tube.   in inches.   n feet.ofin decimals  tub    reduced to
in inches.  in feet.                    unity of
of an inch. collapsed. length and
diameter.
1'2'6               I  12              149         930
~3       3 13'               65        866
~4            " 1o                65        823
4        -) 6'043                          891'5
45              5                   43         860     895
27                  5                  47         940
29                421                  93         930
IO' II   |           2                   58'5       877
6         1'043       32              o 105
9                 41-,                 32        944
T3                  2 2                39         780
14.   8         32'043       32         832     812-6
i'5                3                   31         826
6                 41                  19         791
Io'043                           808
I~~~~7        2        ~~         33           5
I        I.
1 8         12 2   4a                  1 I        654
920    }9  12'043     z125        750     688
20  2 1        22        66o
22       I8 75    51'2        25       420      40031
2 4    3J  g        3.-   }' 31 4     262       7270
24 } 9                        14        378    _         _1
24  312378    I 10489




Strength of RivetedStructu res-Steam Boilers,&c. 265
It has to be noted that the experimental tube No. 6,
although it was really 5 feet long, yet had two rigid castiron rings soldered upon it, dividing it into three equal parts,.
I9 inches in length between the rings, and hence this tube
is placed in the table with the tubes whose real length was
I9 inches. The result shows that, although the tube was
really the same length as Nos. 5 and 27, it yet possessed
three times the strength, to resist collapse, a result which
was due solely to the tube being kept in the true circular
form by the cast-iron rings, and thus made virtually into
three short tubes instead of one long tube.
There are two great lessons taught by this set of experiments. Ist. The resistance of tubes of uniform thickness
varies inversely as the length, or nearly so. This law is
clearly shown by the experiments, and by comparing the
4-inch tubes Nos. 5 and 27 with No. 29. It will be seen
that the former, which are twice the length of the latter,
collapsed with about half the pressure, and Nos. I and 2,
which were rather less than one-third the length of Nos. 5
and 27, bore rather more than three times the pressure.
The 6-inch tubes also follow the same law; Nos. Io and ii
collapsed at 58'5 lbs. pressure, and No. 9 should have borne
a proportionate pressure, as due to its length, which may be
calculated thus;-59 inches: 30 inches:: 58'5 lbs.: 29"74
lbs. But the actual collapsing pressure of the latter tube was
by experiment 32 lbs., so that the difference between the
calculated and actual pressure was only 21 lbs. per square
inch, which is very slight.
The resistance of the 8-, Io-, and I2-inch tubes also bears
the same relation to the length, and hence it may be concluded that the strength of such tubes, to resist an external
pressure, varies inversely as the length, at least between the
limits of I  and Io feet.
The furnace or flue tubes of steam boilers, at the time of
these experiments, were made without any support between
the extremities,. but an alteration speedily followed, so soon




266         On the Strength of Structures.
as the knowledge gained by these experiments was made
known to the world. At the present time, the tubes of all
well-constructed steam boilers are strengthened by rings
of a T section, or of a corrugated section, these rings
being riveted to the tube at intervals of about 3- feet, and
this has added greatly to the security of boilers.
The second lesson taught by these experiments is, that
the resistance of such tubes to collapse, when the length
and thickness are the same, varies inversely as the diameter.
As an example, take No. 29, a 4-inch tube 28 feet long,
with a collapsing pressure of 93 lbs. per square inch, as a
standard. Then, by calculation, the 6-, 8-, Io-, and I2-inch
tubes of the same length and thickness should collapse with
pressures of 62, 46'5, 37'2, and 3I lbs. per square inch
respectively. By experiment, the pressures were found to be
58'5, 39, 33, and 22 lbs. per square inch.
The column No. 6 is composed of numbers obtained, by
multiplying the collapsing pressure of the tubes by the
length and diameter of the tube, to show how nearly the
resistance of tubes of the same thickness is inversely proportional to their diameter and length; and, had the tubes
followed this law exactly, the numbers in that column would
have been equal. The greatest discrepancy occurs with the
I2-inch tubes.
The differences in these numbers, and likewise between
the calculated and experimental collapsing pressures, may
be accounted for, by the varying strength of the material and
differences of workmanship, and, in the case of the I2-inch
tube, by the difficulty of making so large a tube of such thin
material, so that it may be perfectly cylindrical in all parts.
The experiments Nos. 22, 23, and 24 in this Table were
made with three tubes having a greater thickness in proportion to their diameters. Two of these tubes were precisely similar in all respects, but that the one was made
with a'lap' joint (Fig. 6i), and the other with a'butt
joint and external cover plate (Fig. 62).




Strength of Riveted Structures —Steam Boilers, &c. 267
These experiments were made to ascertain, first, to what
extent the strength of the tube was reduced by the slight
departure from the true circular form, which is unavoidable
when a'lap'joint is used; and secondly, to ascertain, if
possible, the different powers of resistance of thick tubes of
different diameters. The result of the experiments shows
that a loss of rather more than one-third of the strength
was caused, by the slight deviation of a quarter of an inch
from the true circular form in the'lap'-jointed tube;
this should be carefully noted.
In well-constructed boilers, any deviation from the true
form is avoided by making the tubes in short lengths, and
FIG. 6i.                   FIG. 62.'lap-welding' the joint, care being taken that the thickness
of the tube is maintained, but not exceeded, at the weld.
The cylindrical form is further secured by the strengthening rings, which are now rolled out of the solid'bloom,'
and are therefore weldless and perfectly cylindrical, so
that any departure from the proper cylindrical form is
inexcusable.
The resistance of the above three tubes was found to be
much greater than that of those having a less thickness in
proportion to their diameter, and Sir W. Fairbairn concludes
that the strength of tubes of the same length and diameter,




268          Onz the Strength of Structures.
but of different thicknesses, varies as the 2Ig9th power of the
thickness. For all practical purposes and to facilitate calculation, the square of the thickness may be used instead of
the more correct 2'I9th power, so that with two tubes
precisely similar in every respect, but that the one is
twice the thickness of the other, the strength of the former
to resist collapse will be very nearly four times that of the
latter.
In all the foregoing experiments, the ends of the tube
were securely fixed, and further experiments were made to
ascertain whether similar tubes follow the same law, when
the ends are left perfectly free. Two tubes were used for
these experiments, they were 8 inches diameter and 3o and
60 inches long respectively, the result is shown in the following Table:Result of experiments made to ascertain the resistance of
cylindrical tubes of wrought iron to collapse, when the ends
are perfectly free to approach each other, and the tubes are
subjected to an external pressure.
TABLE XLV.
Size of tubes.
Pressure in lbs.
Number of experi-                              per square inch at
llental tube.                    Thickness in  which the tubes
Diameter in  Length in  decimals of    collapsed.
inches.    feet.
an inch.
25        8            5         4322
26      f2                    f 04336
It appears from  this Table that the strength is not inversely as the length, as was the case when the ends of
the tubes were fixed, or the shorter tube should not have
collapsed until the pressure reached 44 instead of 36 lbs.
per square inch.
Comparing these experiments with No. 13 in Table XlIV.,
it will be seen that a similar tube to No. 26, but with the




Strength ofRivetedStructueres-Steanz Boilers, &c. 269
ends fixed, collapsed with 39 lbs. pressure, so that it appears
that the freedom of the ends to approach each other does
not decrease the strength to any considerable extent, provided the circular form is still retained at the ends of the tube.
The next experiments were made with tubes of an elliptical
form of section; the result is shown in the following Table:Result of experiments made to ascertain the resistance of
elliptical tubes of wrought iron to collapse, compared with
that of cylindrical tubes, when subjected to an external
pressure.
TABLE XLVI.
Size of tube.
Pressure in lbs.
Number of the ex-       -                      per square inch at
perimental tube.                   Thickness in  which the tjbe
Diameter in  Length in  decimals of    collapsed.
inches.    feet.    an inch.
34       I4 x  IO      5'043         6'5
35        o40 X 151    512'25         I27'5
The tube No. 34 was of the same thickness, and had the
same quantity of material in its section, as the cylindrical
tube No. 19 in Table XLIV.; the latter collapsed with a
pressure of I2'5'lbs. per square inch, and the former with a
pressure of 6'5 lbs. per square inch, thus showing that a loss of
strength equivalent to a pressure of 6 lbs. per square inch, or
48 per cent., was caused by flattening the tube, so that its least
diameter was reduced IRE inch, although  the quantity of
material in both tubes was exactly the same.
The tube No. 35 was of the same thickness as No. 22 in
Table XLIV., and had only 3ths of a square inch less
material in its section, but it collapsed with a pressure of
I27'5 lbs. per square inch or 272'5 lbs. less than the cylindrical tube, showing a loss of strength of 64'8, or nearly 65
per cent., by flattening the tube until its least diameter was
-th less than that of the true cylinder.




270         On the Strength of Structures.
Tubes exposed to Internal Pressure.
A few experiments were carried out at the same time to
ascertain whether the length of tubes similar to some of
those previously enumerated, affected their resistance to an
internal pressure. The results were, however, far from satisfactory, only two of the five tubes experimented upon being
sound at the joint. These two were 6 inches diameter and
12 inches and 48 inches long, respectively; the former burst
with a pressure of 475 lbs., and the latter with 375 lbs. per
square inch, apparently showing that the resistance varied in
some degree with the length; but when practically considered,
the result would appear to show that the length of the tube
does not affect its strength under such circumstances, and
both tubes would have burst with about the same pressure,
had not the shorter one, owing to its extreme shortness and
the consequent proximity of the point of fracture to the end
supports, derived a great amount of its resistance from the
fixing of the ends.
Some further experiments were then carried out with
leaden tubes of 3 inches diameter, and I4i inches and 3I
inches in length, respectively. These tubes were of the same
thickness of metal and burst with a pressure of 374 and 364
lbs. per square inch respectively, from which it may be
inferred that the shortness of a tube does not contribute to
its resistance to an internal pressure, unless it is extremely
short, say less than two or three times its diameter in length,
when its apparent strength would be partly due to the
support given by the ends of the tube.
In the Perkins system of steam boilers, the boiler is
composed of small tubes, which sustain with safety a pressure
of 5,000 lbs. per square inch. To obtain greater economy,
steam pressure increases from year to year; it is therefore
probable that the steam boilers of the future will be
gradually modified, by the more general employment of
small tubes.




Strength of RivetedStzructzrcs-Steamn Boilers, &c. 27 I
CHAPTER XVII.
ON STRUCTURES SUBJECT TO INTERNAL PRESSURE.
THE present chapter will treat of the strength of cast-iron
pipes and water tanks, and also of the relative advantages
of hooped and solid cylindrical structures exposed to internal pressure, as, for instance, in the case of guns and
hydraulic press cylinders.
Cast-iron Pipzes.
Cast-iron pipes are now so extensively used for various
purposes, that it is of importance for the student to examine
their power of resistance.
In the majority of purposes for which cast-iron pipes are
employed, as for the conveyance of gas or of water for the
supply of towns, or for steam pipes or steam cylinders, or
even for high-pressure hydraulic pipes, the question of
strength is not so important as might be inferred from the
trouble experienced in keeping them sound and water-tight.
They are usually cast of such a thickness that their strength
is apparently in excess of that required to resist the pressure
acting on them. This superabundance of strength is given,
because other practical considerations step in, which, as a
rule, render it absolutely necessary to make such articles
considerably stronger than is actually required for the work
which they have to do. The practical considerations here
referred to are the limits to thinness and soundness attainable by the founder, and the comparatively fragile nature of
cast-iron pipes, in bearing the rough handling to which they
are subject in transit, and more especially the straining due
to the subsidence of the earth from under them when laid




272         On zthe Strength of Structures.
underground, or to its compression during the passage of
heavy waggons over them.
Reckoning the tenacity of cast iron of the quality which
is frequently used for common pipes at i6,ooo lbs. per
square inch, it will be found by calculation, that very thin
pipes of such iron would resist a great water pressure. Let
us take a ring cut from a pipe, with a bore of Io inches in
diameter, and suppose the said ring to be I inch in length
and' an inch in thickness, as in Fig. 63; this will give a
FIG.  63. l- l —-ll. -  lll ii
FIG. 63-.
substance of iron equal to one square inch of section to
resist the water pressure. But one square inch will have an
ultimate strength of i6,ooo lbs., and may in such a case be
safely strained to 4,000 lbs. To produce that stress, the
pressure in the pipe must be 4,000 -- Io, or 400 lbs., per
square inch, or above 26 atmospheres. Hence, it will appear
that, but for the reasons already stated, gas and ordinary
water pipes might be much thinner than they are usually
made.
In the Belgian Annexe of the Paris Exhibition of I867, a
cast-iron pipe was shown, 20 feet long and 28 inches in diameter, and varying from ~th to 3ths of an inch in thickness.
This pipe was proved with a pressure equal to 5 atmospheres.
In this case the stress put upon the iron at the thinnest part
of a ring, one inch in length, would be as follows:-The
pressure of 5 atmospheres would be 75 lbs. per square inch,
the number of inches being 28. The total substance of the
two sides of the iron ring would be equal to - an inch.
Then 75 x 28 will give 2,Ioo lbs. as the total pressure of
the fluid which had to be resisted by the a inch of iron, or
equal to a stress of 4,200 lbs. per square inch, thus only




Structures subject to Internal Pressure.    273
straining the iron up to the quarter of its ultimate tenacity;
such a pipe, however, would be easily broken, unless great
care was exercised in handling it.
The water pressure pipes employed in connection with
hydraulic crane works, are seldom used under a pressure
of less than 700 lbs. per square inch, and the pressure varies
from 46 to 68 atmospheres in different cases. The pipes
conveying the water, when 3 inches in diameter, are made
with -ths of an inch of thickness, and are proved with
2,500 lbs. internal pressure per square inch. Taking a
ring of this pipe I inch in length, the water pressure tending
to burst it will be 2,500 x 3 = 7,500 lbs., which has to be
resisted by - x 2 =- I square inch of iron, which, for
such a purpose, would be of rather a better quality than
that for common pipes, probably having a tenacity of
I8,ooo lbs. per square inch. This would give an ultimate
resistance of 22,500 lbs., or three times the stress to which
the ring is exposed, under proof, and the proof stress is
fully three times greater than the stress during ordinary
working. It might be inferred that such pipes were unnecessarily strong, but such is not the case; owing to the
numerous contingencies to which they are exposed, and
to the effect of continued rusting while buried in the earth,
they require an excess of strength, and experience fully confirms the wisdom of allowing a large margin at the outset.
Cast-iron Tanks.
In the construction of round, square, or rectangular tanks,
built up of cast-iron plates, which are united with wroughtiron bolts by means of flanges or ribs, which are cast upon
the edge of the plate at right angles to it, different conditions exist to those met with in pipes which are of
comparatively small diameter.  Let us select a round
water-tank as an example, which may be compared to an
immense tube placed upon its end. The enlargement of the
pipe into a water tank of say Ioo feet in diameter by 25 feet
T




274         Onz the Strenzgth of Struct'i-es.
in depth, brings the question of strength into greater prominence. It is a very common error to suppose that the
strength of the tank is not affected by the diameter, but, as
we shall see, the diameter has a most important influence
on the resisting powers of a tank.
In the first place, a pressure or weight of 25 cubic feet of
water, equal to 25,000 ounces, rests upon every square foot
of surface of the bottom plates, but as these are here supposed to be lying upon a solid foundation, the strain on
them may be disregarded.
The pressure per square foot that comes upon the sides
of the tank is only equal to the half of that which rests on
the bottom, because at the surface the pressure is nil, while
at the bottom it is, as before stated, equal to 25,000 ounces;
consequently the fluid pressure due to I21 feet, or I2,500
ounces, is the average pressure that the side or circumference of the tank has to sustain, per square foot of surface.
The above remarks apply to every kind of tank, but it
would be a great waste of iron to make the upper tier of
plates, in a tank, sufficiently thick to withstand the pressure
of water at I24 feet in depth, and it would be still worse
to make the lower tier of plates only equal to the pressure
of I21 feet, seeing that that they have to withstand 25 feet. This
renders it obvious that, in order to obtain the requisite
strength with thre minimum of iron, the bottom tier must be
equal to the strain that comes upon them, and the thickness
upwards must be gradually diminished to the top plates,
which need only be of such a substance as will meet the
various contingencies referred to, in connection with the
casting and conveyance of pipes.
If we select one foot of the lowest ring of iron, and
assume the average pressure upon it to be 25 feet of water,
and then consider it as similar to an inch ring of a pipe,
we shall then see that the question of strength is important.
The iron used for such a purpose would probably have a
tenacity of 1 6,ooo lbs. per square inch; then we may inquire




Structures subject to Internal Pressure.    275
how much of such iron will be necessary to withstand the
pressure, leaving out of consideration, for the present, any
assistance which it derives from being fixed to the edge of
the bottom of the tank.
The tank is Ioo feet in diameter, which, multiplied by
25,000 ounces, the water pressure on every foot, is equal to
a total pressure tending to burst the lower ring, equal to
2,500,000 ounces. Suppose we resolve not to strain the iron
above -rd of its ultimate tenacity; Ird of its ultimate tenacity; 6,ooo lbs., when
reduced to ounces for convenience, is equal to 85,333-say
85,000 ounces. Then, dividing the total water pressure on
the ring, tending to tear it open, by 85,000, will give
29'4 as the number of square inches of iron required (say
30 inches); that is to say, the two sides of the ring should
have between them 30 inches, and, each being I2 inches in
length, the thickness would consequently have to be  14
inch.  If the tank is considered merely as a pipe, then
I~ inch of thickness would be required, but if we take
into account the assistance derived from its connection to
the bottom plates, as well as the support due to the projecting flanges, which act as ribs, then it will be seen that
the I, inch may be reduced to I, inch, with perfect
safety. In the same way will have to be calculated the
number and size of the wrought-iron bolts that are required
to hold the plates together, and which ought to be at least
equal in strength to the 30 inches of cast iron.
In order still farther to reduce the substance of the plates
and the cost of the tank, it is now customary not to rely
entirely upon the cast iron, but to reduce the substance of
the lowest plate still farther-say to I inch in thicknessand to make up the difference by wrought-iron bands or
hoops, which are put on under tension, and which, by their
greater tenacity and reliability, afford the necessary security.
By such combinations, the practical engineer is in some
measure enabled to attain the maximum of strength and
minimum of cost.
T2




276         On the Strength of Structures.
In the foregoing remarks on the strength of pipes which
are comparatively thin, it has been assumed that, when they
are exposed to internal pressure, the metal composing the
pipe is all performing duty in its resistance, in the same
manner as a bar of iron when pulled asunder by tensile force.
This is not, strictly speaking, the case, although in large
structures, such as the water-tank, it is very nearly accurate
to assume the metal to be uniformly strained; and, even in
small water pipes, no error of practical importance is introduced by that assumption. In thick pipes, or in cylinders
for hydraulic presses, or in gun structures, it is, however,
widely different; then it becomes imperative to treat the
question in another manner.
On the Strength of footed as compared with Solid
Structures.
It was pointed out by Professor Barlow, many years ago,
that the strength of a pipe, hoop, or cylinder to resist internal pressure is not in proportion to the mass or thickness
of material of which it is composed, and that by adding to
the thickness of a tube or a gun, or to the substance of the
cylinder of a hydraulic press, or by increasing the thickness
of an iron pipe, we do not thereby increase the strength, in
proportion to the quantity of metal which is thus added.
When a hollow cylindrical vessel is exposed to internal
pressure (amounting in hydraulic presses to 3 tons per
square inch of surface, and in guns to a still greater intensity) the pressure does not affect the whole mass of metal
opposed to it in the same degree. The metal composing
the inner surface of the bore is first affected, and as that
part is extended, so the stress gradually reaches to the next
lamina, but in a decreasing ratio, and this transmits it to
the next, and so on, until at length the resistance of all are
in some degree brought into active exercise.
The important point to observe is, that the work which is




Structures slubjct to Internal Presszure.    277
performed by the successive concentric laminae, in resisting
the internal pressure, is in exact proportion to the amount of
stretching to which they are severally subjected. It is found
by experiment that, when a ring is stretched by mechanical
means, the outside does not expand so much as the interior;
this will appear to be self-evident, because if it did so,
then the volume of the material composing the ring would
actually be increased thereby, which is impossible; but
besides and independently of any experiment, as the outer
circumference of the ring is necessarily longer than the inner
surface, even if they did stretch an equal amount (if that
were possible by a narrowing of the ring), the metal composing the two surfaces would not even then be strained
equally, on account of the stretch on the outside being
distributed over the circumference of a larger ring than
that on the inside. And upon the above conclusion some
most important practical deductions rest.
In attempting to design a gun or cylinder, which shall be
theoretically perfect, so that the whole of the molecules
which compose the cylindrical part of the structure shall
stretch alike, then the metal should be so disposed that, at
the moment of pressure or explosion, every part thereof
shall be equally ready to take its'full share of the duty at
once, and in proportion to its ability, and should be constructed with the tension previously put on the alert, so as
to be always in readiness, and without the outer portion
having to wait for the stretching of the interior to give it
employment.
Up to the present time, no practical system of constructing either guns or hydraulic cylinders comes nearly up to
these theoretical conditions; the nearest approach is probably attained by building up the cylinder with fine wire,
which is wound round an interior barrel, the wire being
put on under definite tension; but such an arrangement,
although fulfilling one set of conditions, would evidently be
weak in the longitudinal direction of the gun, unless all the




278        On the Strength of Structures.
wires were soldered together into a homogeneous mass, and
even then the result would be doubtful.
Another approximation to the theoretical conditions is
secured, in the American system of casting iron guns, by
cooling the mass of hot metal from the interior of the bore,
instead of by the usual method of allowing the mass to
remain in the foundry mould, until the heat has passed away
by conduction through the exterior. In the latter case, it
may be inferred that the outside of the mass of metal is
necessarily colder than the centre of the solid block, and
consequently will have contracted more on that account,
and so have become stretched upon the warmer and more
expanded interior, until at length the whole of the heat has
passed away, and the inside consequently has become
equally cooled, which will then likewise have contracted
and taken up its normal dimensions, and will thus find
itself at a disadvantage on account of the previous stretching and final setting of the outer portion. It will then
necessarily become less dense, and, hence, to some extent,
it will lose the full grip and support of the exterior metal
composing the mass.
By the American plan, on the other hand, the gun is cast
hollow on a mandril, with what is termed a water-corenamely, a tube in which a constantly circulating stream of
water is kept up, to carry off the heat rapidly from the
centre, thus entirely reversing the conditions, and so causing
each lamina in succession to grip hard upon that which it
encloses.
In carrying out this cooling operation, shortly after casting,
the stream of water is first directed through the core-barrel,
entering by a pipe down the centre and then rising through
the annular space between the pipe and the core-barrel, and
escaping by passing away over the top of the dead head.
In this process of cooling a heavy gun by a stream of water,
the procedure may have to be continued for a couple of
days, and when the gun has partly cooled, the core-barrel




Structures subject to Internal Pressure.    279
is extracted, and the water is made to flow  into the
bore by a pipe, in a similar manner as before, but the
water now escapes in contact with the actual bore of the
gun; meanwhile a fire is kept up all round the exterior
mould of the gun, in order to protract the high temperature
of the outside, and prevent the heat from escaping in that
direction.
Independently of the foregoing consideration, the question
of strength in a solid cast-iron cylinder or gun is likewise
affected by the exterior form, the presence of the gun
trunnions, or, indeed, of any other massive projections or
irregularities of outline, and still more by the shape given
to the breech.  All these points determine strength or
weakness, inasmuch as these forms or shapes create new
conditions, which greatly affect the direction in which the
waves of heat pass out, from the interior of the mass,
to the nearest point of exit into space. It is found that all
such irregularity introduces elements of discordance, into
that which would otherwise be the harmonious and natural
order of crystallisation, any departure from which is always
accompanied by corresponding weakness.
A good illustration of this kind of weakness was afforded
by the accident which occurred in raising the Britannia
Tubular Bridge. The hydraulic cylinders were originally
made with a flat bottom, like that of a drinking-glass
(as shown in Fig. 64); the cylindrical part of the casting
had the crystals radial from the inside, but in the bottom
part the crystals were perpendicular to the flat end, and
at the points where the two different arrangements of
crystals come together, at an angle-namely, at a line
drawn from  the inner to the outer corners-there were
the lines of weakness. Hence it was that, although considerably thicker, the cylinder failed at those points.
The second or substitute cylinder was made with a hemispherical end (as in Fig. 65), and in it the radiating crystals
were all arranged in lines more nearly parallel, although of




280         On the Strength of Structures.
course not truly so. As thus made, it was found to be
amply sufficient in strength, even with the same amount of
metal in the mass.  The foregoing accident was the means
of drawing public attention to the subject. Many engineers
found it to aoree with their former experience, and many
FIG. 64.                       FIG. 65.
had previously been modifying the forms of structures without knowing the natural law.
The subject is well illustrated by a singular phenomenon
that is observed in chilled shells, in which, from the effect
of the sudden deprivation of heat, the lines of crystallisation are very clearly marked, and, when broken, the order
of crystallisation is exhibited most convincingly. In such
shells there occurs an apparent inconsistency with this reading
of the law, namely, at the point of the shell, where in
every case the crystals take a curved direction rather- than




Structures subject to Internal Pressure.    281 I
straight out into space (as shown at Fig. 66). This is
due to the following cause:-The
shell is cast on end, with the point
downwards. When the liquid metal              /
is poured into the mould, it is, as
a matter of course, in contact all /
over; and during the period that
it remains a liquid, it naturally follows the gradually expanding mould
or vessel in which it is contained;
but as it begins, from the effect of
the chill, to form an exterior crust,
the time arrives when the body of the
casting is not in actual contact with
the mould, and consequently the
rate of conduction of heat is thereby
lessened, but not so with the point Pi     i_
of the shell, which is kept at the _/   _!    _
bottom of the mould by the effect;_
due to the gravity of the mass.
Hence, the point of contact becomes     FIG 66
a new line of direction, competing with the sides of the
mould, and the effect of the two determines the curved form
of the crystals, as shown in every instance.
In past times, it was assumed that cast iron must necessarily be homogeneous, instead of which it is otherwise,
any sudden divergence of the escaping lines of heat causing
a change in the direction of the crystalline formation, and
rendering that part weaker than the general mass; and as
with a chain so in a gun or cylinder, the strength is only
equal to that of the weakest point. The knowledge and
right application of this law will in time affect the form of
many structures, and lead to the use of cast iron for purposes where wrought iron is now employed.
Already most of our engineers are constructing their
hydraulic cylinders on the plan shown in Fig. 65, and




282        O0i thie StreZgtlZ of Strzuctures.
the Americans are constructing their cast-iron guns in form
not unlike a soda-water bottle, which is nearly in strict accordance with this law of crystallisation, the exterior surface
being arranged so as to invite the heat outwards in a nearly
uniform current in all directions.
Buillt-uz Guzns.
We have already indicated that strength is derived from
putting the interior mass of a cast-iron gun or cylinder
under compression, by the initial tension of the metal
nearer to the outside, due to cooling from the interior,
and thus approximating to the condition of the gun made
with fine wire. But it has further to be observed that the
same conditions would be obtained if the structure were
composed of a great number of thin hoops, put on one
over the other, but under the same tension as the wire, if
this were practicable, which it is not commercially, on
account of the expense in producing such accuracy as
would ensure the specified tension. But although it may
not be convenient to construct such articles with a great
number of thin hoops, still, by a slight departure from the
theoretical conditions, namely, by using a smaller number of
thick hoops, an approach may be made to great strength, and
by thoroughly practicable means. In this waywe can fabricate
large guns or cylinders, by taking the several parts in detail,
and combining them into a comparatively perfect whole,
at a moderate expense, and with such an approximation to
the theoretical conditions as affords great satisfaction; the
great condition aimed at being to equalise the stress
throughout the mass,,when under tension from internal
pressure, and this is the principle of the system now pursued
both at Woolwich and Elswick in the construction of what
are termed'built-up guns.'
The wrought-iron hoops are made of any diameter or
length, by first making a bar of the required section, and of




Structures subject to Internal Pressure.    283
sufficient length to contain the necessary quantity of iron;
this long bar is then heated in a furnace, and by mechanical
means it is wound round a mandril into the' condition of a
coiled bar. This coil is removed from the mandril and then
put into a furnace, made welding hot, and then put under
a steam hammer, and the loose coil is then welded into a
close cylinder or hoop of wrought iron. Or, the above
process may be varied, in the formation of exceptionally
thick hoops, by the winding of one coil over another, before
the welding operation is performed. WVhen the hoop is
forged, it is put into the lathe or other machine, for boring
and turning it to the required dimensions.
On the first consideration of this subject, the mind is
rather unwilling to believe that in such articles as are here
referred to, when composed of fine wire or a number of thin
rings or hoops, or even of a smaller number of thick
hoops, as used in the construction of modern built-up
guns or cylinders, that such a mode of construction can
have the same solidity and strength to resist internal
pressure, as a similarly formed homogeneous mass in one
forging or casting; still more so, when that mass is made
of the best material, and in practice shows great endurance, as is the case with the fine cast-steel guns made
by M. Krupp of Essen. Besides, it might further appear
that in the built-up structure, when consisting of a hooped
fabric, in which the mass composing it is discordant, being
neither homogeneous nor working in harmony-that each
hoop is under different conditions, that the barrel and
the inner hoops are existing under compression, while
the exterior hoops are under great tension. Nevertheless,
the positive results show that no sensible practical disadvantage is found to arise from the want of homogeneity,
while the homogeneous or solid guns are not more reliable,
if indeed they are equally reliable.  By the building-up
system, we are enabled to have the finest steel for the
interior barrel, and a cheaper material, wrought iron, for




284         On the Strength of Structures.
the remainder; and although wrought iron is a cheaper and,
in one sense, an inferior material, yet for this special purpose,
from its great toughness, it may, as a general rule, be considered fully equal, if not superior, to steel of greater ultimate tenacity.
From these remarks, it will be seen that the subject admits
of a difference of opinion, and such a difference does exist
amongst those well qualified to judge, and who have given
the subject great attention; but still the fact remains that
the strain which comes upon the interior of a gun first acts
on the inside of the bore, and as that part of the metal becomes stretched, so the stress gradually reaches the outside
in a diminishing degree, and hence the outer metal cannot
contribute its full share of duty, unless it has an initial
tension.
This principle of initial tension is employed, in the
modern manufacturing system of building up wrought-iron
and steel guns and hydraulic cylinders, by the various modes
of either shrinking or pressing one hoop, under tension, over
another hoop.
The shrinking system, which was introduced by Sir William
Armstrong, for the construction of artillery, has been extensively applied, and at the present time it appears likely to
supersede all other systems, from the circumstance that, by
this simple arrangement, the best practical results as to
quality and cost have been obtained.
After taking all things into consideration, greater practical
advantages have probably been attained by this system than
by any other, and as the same principle of construction is
equally applicable to hydraulic cylinders, the student should
endeavour to understand its general bearing.
To take the gun as an example, the principal part is
the bore, formed within the inner barrel, which constitutes
the foundation or core upon which the exterior hoop structure is to be built up. The chief object of the hoops is to
render all the support of which they are severally capable,




Structures subject to Internal Pressure.    285
by the principle of having the portion of work that they can
do, partly put upon them in the original making of the gun.
The inner barrel is, then, the most important part; it is
made of cast steel of a mild quality, and carefully tempered
in oil, so as to bring out its strength and elasticity.  It is
then covered with a thick hoop, or a series of thick hoops in
succession, the one over the other, and each under regulated tension, as determined bxy calculation of the relative
dimensions of the parts that have to grip and the part which
has to be gripped, and they are so combined into a whole
that each hoop shall take its full duty at the critical moment,
and that the total tension put upon the several hoops shall
be at least equal to or exceed the effect of the explosion.
The possibility of attaining such conditions is due to the
knowledge of the fact that iron stretches about the R- Ith
part of an inch, per inch of length, by a ton of stress. It
therefore becomes a simple matter of calculation, to determine the difference of dimensions between the outside of
the inner surface and the interior of the outer that will give
the required tension, and consequently the stretch and
specified grip in tons of positive support.
When the two surfaces are made to the required diameters,
then the outer hoop is heated to redness, which causes it to
expand about ToIth part of its linear dimensions; it is then
carefully slipped over the inner part, upon which it gradually
cools and contracts, with the force, in tons per square inch,
upon the metal, intended to be applied.
As the greatest stress or pressure from the explosion will
be brought to bear, first, upon the interior of the inner barrel,
and from that will be passed on to the hoops, the object is
to concentrate the grip of all the hoops upon the inner
barrel.  The effect of the united grip of a series of hoops,
due to shrinking one hoop over another, is, that the grip of
an outer hoop serves partly to undo a portion of the grip of
the hoop or hoops which are under it, thus putting the
entire structure into a state of lively activity; and the




286          On the Strength of Strzzctzl-s.
several hoops are always standing at attention, to perform
the required duty; but in determining the precise dimensions which will assign the proper tension and duty to each
hoop, all the alteration or disturbance of dimension, and
consequently of tension, thus caused by the successive grip.
of hoop over hoop must be taken into account, in establishing the difference of dimension between the inner
surface of the gripping hoop and the outer surface of the
under hoops, as also the relative grip to be put upon
the hoop next to the barrel, and the intermediate and outer
hoop.
In the attempt, so far as it may be practicable, to place
the whole structure in such a condition of lively tension as
that, at the moment of explosion, each hoop in succession
will be already under the assigned load, and all of them take
the same duty in tons per square inch, it is necessary that
the amount of tension put upon the several hoops should
not be equal, but should be inversely as the stress that
would reach them severally, if the structure were a solid
mass of metal.
Until recently it was generally considered, on theoretical
grounds, that the force or resistance exerted by the different parts of a solid cylinder, was inversely as the square
of the distance of the parts from the centre, but recent
experiments, made by stretching rings within rings, would
seem to prove that the stress is nearer to the inverse ratio,
or at least nearer to that than to the former. The cause
for any such uncertainty is owing to the physical nature of
materials, and arises from a complication of various reasons
due to the properties of the metal-its elasticity, ductility,
and compressibility, which all step in to interfere, complicate, and introduce uncertainty, and all come into active
play and thereby affect the apparent result. But all experiments point in the one direction-namely, that the strain
comes upon the interior in all its force, and gradually
decreases inversely as the distance from the centre, or




StrYucmcurcs subzjct to Internal Pressure.    287'nearly so. Therefore when a gun is completed, the tension
of the structure should be greatest at the outside, and
gradually decreasing to the inner barrel, which should be
under compression, so that at the moment of greatest stress
no part should be strained beyond the apparent limit of
elasticity.
To those who are not familiar with the practical working
out of this system of building up concentric structures, it
might appear that some difficulty would be experienced, in
obtaining such accuracy as would ensure the required degree
of tension, with uniformity; but such is not the case, from
the circumstance that the system of manufacturing has been
so organised as to reduce the results of measurement to a
certainty. The hoop is first bored to the intended dimehsion;
it is then carefully measured by instruments, that by the aid
of the vernier or micrometer read off the dimensions to
the -0sIth part of an inch.  As the interior surface may
not be perfectly parallel, the correct size is noted at different points and written down on paper. To these dimensions are added the amount of difference required (namely,
r-Ir-th of an inch, per inch, for each ton of tension or
grip).  The total dimension gives the size of the part on
which it has to be shrunk. These new dimensions are
recorded, and with the corresponding measuring instruments duly marked in ink, and the same dimension written
upon a sketch; the paper and guages pass on to a turninglathe, where the respective parts of the structure are reduced
to the required diameter, an operation necessarily requiring
skill and care on the part of the workman, but no difficulty
is experienced.
Should the operation be performed erroneously, by making
the grip too small, then it is only necessary to reserve'the
hoop for the next gun, and select a smaller hoop for the
one in hand, these differences being only a few thousandths
of an inch.  If, on the other hand, too much metal has
been left on the exterior diameter, thus creating a greater




288         On the Strength of Structures.
tension upon the metal composing the hoop than was
intended, the error will be detected by the system of
check, when it reaches the official viewer at the next
stage, and is then either passed or returned by him for
correction by the turner.
From some observations which were made on the above
point at the early stage of the manufacture, before much
experience had been gained in regard to the shrinking of
exterior hoops, it seemed more than probable, from certain
results, that the strength of the hoop was not much, if at
all, imperilled by too much tension being put upon it; that
the iron hoop, from being hot, probably allowed itself to
be stretched or permanently wire-drawn to a certain extent
without injury, and thereby was enabled to accommodate
itself to a wrong diameter. With a hoop under different conditions, as regards temperature, the result would be disadvantageous, and might end in fracture, owing to the want
of sufficient ductility in cold iron.
Lonigitudinal Strain.
In considering the strength of guns, and the causes which
determine their failure, it has to be clearly understood that
besides the force tending to burst them in the lateral
direction-that is to say, outwards circumferentially-they
are also subjected to another force which tends to destroy
them in the direction of their length; the same force, which
sends the shot forward, has an equal reaction on the breech
of the gun, thus tending to separate it from the barrel.
The student, on considering these two distinct strains
or forces, will be led to perceive that the breech of a gun
is in some respects like a beam or girder; it must have a
neutral surface; there must be some point where these two
forces are not favourably situated for harmonious action.
If we suppose the gun to be composed of some substance
resembling highly elastic india-rubber, it will be readily




Structures subjcct to Internal Pressure.    289
conceived that, under such imaginary circumstances, at the
moment of explosion, the whole of the rear part of the gun
would be extended in every direction, like a bladder, or as
we see in the operation of glass-blowing.  It is quite
different, however, in the case of an iron or steel structure;
there is not the same freedom for distension in every
direction; and hence, when the metal is overcome, the
usual result of the explosion is the blowing out of the breech,
away from the barrel, and the bursting of the cylindrical
part next to the breech, generally leaving the fore part of the
gun entire.
The longitudinal strain, then, which comes upon the
breech, is derived from the same force as that which sends
the shot forward, but with this difference, however, that the
gun not being so light as the shot, it recoils a proportionately less distance. The safety of a gun will depend
upon the mass of solid metal composing the breech, even
when the metal is merely considered to resemble an
ordinary steam-hammer anvil-block, which is the instrument
for receiving the force of the blow of the falling hammer;
and every observing forgeman knows how efficient is the
blow with a heavy anvil-block, and how well it stands up
to its work, as compared with a light one. Upon the same
principle, the heavier or more massive the breech of. the
gun, so the less is the tendency for it to recoil and break
away from the cylindrical portion, by the effect due to the
blow of the explosion.
For the foregoing reasons, all designers of guns who
know their work introduce as much solid metal as possible,
or as much as may be admissible, into the breech of a gun,
merely to act the part of an anvil-block, and thus prevent
the neutral surface, where the two forces unite, from being
unnecessarily disturbed, or to such an extent as would
endanger the safety of the gun structure.
u




290         On the Strength of Structuries.
Strengthening of Cast-lron Guns.
The gradual dawning of the foregoing principles, on the
minds of many individuals, has led to several attempts at
strengthening ordinary cast-iron guns, either by exterior
hooping with wrought iron, or by lining the interior, either
with wrought iron or with oil-tempered steel. The former
system has hitherto been unsuccessful, no doubt owing -to
the weak interior of cast iron giving way, before the assistance of the hoops came into exercise, and such guns have
seldom done more work with the hoops, than might have
been expected without their presence; hence, for the present,
that system has been abandoned.
Cast-iron guns, when strengthened on the latter system
-namely, by the introduction of an inner lining of some
stronger material-have afforded most satisfactory results,
and many guns have, therefore, been treated on this principle. Various plans have been proposed, both by Major
Palliser and Mr. Parsons, but these do not differ much in
regard to their general principle; the chief peculiarities of
the different plans are in the details, which may here be
disregarded.
The general principle upon which all such guns are
strengthened is founded on the natural law, discovered by
Barlow, and by him  demonstrated mathematically, and
which is generally assented to by all who have followed
in his footsteps, and which has already been referred to in
connection with the subject of built-up structures of wrought
iron.
On the assumption, therefore, that when a cast-iron gun
is exposed to internal pressure, the metal composing it is
thereby strained unequally, if the solid gun cylinder were
supposed to be divided into a number of thin concentric
cylinders, the extension of each under strain would be, according to Barlow, inversely as the square of the diameter;




Structures subject to Interncal Pressure.    29I
and as the strain is in exact proportion to the extension,
then the strain will vary in the same ratio.
The interior of the gun is therefore subjected to the
greatest strain, and the exterior to the least strain; consequently the interior has to stretch more than the outside.
If, then, the interior is made of a stronger material and a
more extensible metal than the exterior, by inserting into
the cast-iron gun a tube of wrought iron or steel, the metals so
arranged will be placed in accordance with the true theoretical
conditions, and each portion will take its appropriate share
of the work. When a discharge of the gun takes place, the
inner wrought-iron or tempered steel tube will offer the first
resistance and take a large portion of the strain; but, in
doing so, it will stretch three times as much as it would
have done had it been made of cast iron, and yet without
exceeding its apparent elastic limit. But its exterior surface will necessarily stretch less than its interior, and therefore the cast iron into which it is fitted will only be stretched
to the same extent, a point of importance. If the steel tube
is properly fitted into the cast-iron bore, the fit being what
may be termed easy, or so proportioned as to give such an
amount of extension to the steel tube as will be just sufficient to stretch the cast iron up to a point a little under its
elastic limit, at the moment of explosion, then the cast iron
will not be injured. But these conditions imply a degree of
nicety which may seem  difficult to attain, by those who
are not initiated, but which in reality is easily accomplished,
practically, by the aid of the vernier.
A remarkable feature in connection with this part of the
subject, and which the student should note, is the paradoxical fact, which can be satisfactorily proved, that by
boring out a cast-iron gun to receive a steel tube, even
to the extent of half its thickness-that is to say, by
cutting away half its substance-the gun is thereby rendered
stronger than it was before-not much stronger, but still the
difference is for the better. This, however, will only hold
u 2




292         On the Strengt/z of Structrles.
good when the internal strain is applied on the same diameter of bore as it was originally. This is simply due to a
greater harmony existing among the lamine, because when
the difference between the diameter of the enlarged bore
and the external diameter is smaller than it was before,
there is now less difference in the relative extensions, and
conseq-uently the various laminae act together more uniformly, and therefore more advantageously.
Likewise the longitudinal strength of the gun is not injured, for the areas of circles being as their diameters
squared, half the thickness of the gun may be bored out,
and it will still retain more than half of its longitudinal
strength; and as the longitudinal strain on a gun is only
about one-fourth of the tangential strain, the gun will still
possess an ample excess of longitudinal strength. These
considerations have, therefore, led to the system of strengthening cast-iron guns by the insertion of a steel or wroughtiron lining, which has given high results, although not
equal to those obtained with built-up guns of steel and
wrought iron.
The student should not overlook the foregoing principles,
because his pursuits may lie in some other direction.  The
same principles apply in numerous cases, wherever hollow
cylinders of cast iron or other metal are exposed to internal
pressure, of sufficient force to call their full strength into
requisition.
If the study of this unpretending little volume has produced the desired effect on the mind of the earnest student,
he will, doubtless, have perceived how much his daily
routine duty in the workshop is associated with natural
science and practical art, and how very little even of art
belongs entirely to man or to man's doings; that all the
materials to which reference has been made, together with
all their properties, exist irrespective of man; and that the
principles which determine strength in structures of any
kind-beams, gearing, pillars, cranes, boilers, and even guns




Structures subject to Internal Pressure.   293
— are all natural principles which have simply to be complied
with or taken advantage of, in order to make the best of
things as they are found in the world. There is a sort of
conventional notion, especially in the workshop, that to
men belongs the credit of inventing principles; those involved in the forms given to beams, for example, in order
to obtain the greatest strength with a given quantity of
material. But it is otherwise; men have only gradually
perceived the natural principles on which strength depends,.nd then endeavoured approximately to comply with their
iequirements. Generally, natural principles can only be
approximately complied with, because numerous practical
difficulties present themselves in consequence of our imperfect knowledge, and bar the way to the true form in
carrying out practical work. So it is in everything; the
natural law, for example, upon which the mechanical device
called a lever is founded, is the same principle which
governs all the other so-called mechanical powers, however
variously devised, and explains all our mechanical expedients for the conversion of force or motion. By no
agency can mechanical work be lost or gained, however
ingenious the contrivance; it can only be wasted or misapplied for the intended object, by the unnecessary friction
or by the imperfect arrangements of man in accomplishing
his purpose. The law itself is perfectly simple, and when
the full knowledge of the law is possessed by all workmen
engaged in its application, in the wide domain of applied
mechanics, we are fully warranted in the anticipation that
our apparatus will be gradually simplified, that the various
members of complicated structures will be so proportioned
to the stress which comes upon them, that a greater uniformity of strength and saving of material will thereby be
achieved. So it is likewise with the most common operations of the workshop; all of them depend on some natural
principle; the tempering of a chisel, the angle of a cutting
tool, the shape given to a hammer, so as to give the required




294         On the Strength of Strzuctres.
quality of blow, the speed suitable for the cutting of different materials, and every other kind of operation or
process, all are founded upon unalterable laws of nature;
our province is merely to apply materials, wherewith to direct
and control the natural laws, in order to effect some definite end of our own.
From day to day, men are seeing new ways of turning
these natural principles to account; it has been the same
for thousands of years; these new applications are sometimes called discoveries or inventions, but all these new
contrivances of the present, existed at the beginning of
applied mechanics, only with this difference, that men did
not perceive them. So it is now; we are groping in the
dark for new appliances, but all the appliances of the future
are already existing in nature, but our ignorance prevents
us from seeing them at the present time; and every fresh
ray of light, that shows us how to apply natural law and
natural material in a new way, will never bring us nearer to
the end: every improvement will only form a clue to other
applications, so that invention will never cease, nor will
men's work be done;-the gradual result will be to enable us
to live with less expenditure of material and labour, and
thus to ameliorate the condition of the whole human family.
It is scarcely one hundred years ago since it was considered
next to impossible to turn or bore cast iron; now we can
perceive how thin the veil was which obscured that possibility-that it was only necessary to move the surface at
a certain velocity. At that period it was thought essential
to take the power from the motor at a slow velocity, thus
entailing heavy cumbrous mechanism to transmit the power;
yet men knew the law then just as well as we do now,
but they did not realise it with our vividness, so as to
warrant them in changing their system; quickening the
motion would at once have reduced the cost both of the
power required and the materials employed.
It is not many years since it was deemed impracticable




Structures subject to Internal Pressure.    293
to sharpen hard steel circular cutters, without first softening
them, yet we now see that it can be done, as easily as
sharpening a drill at the grindstone, and by an arrangement
as old as the cutting of diamonds, and upon the same
principle. The effect due to velocity has yet to play an
important part in many of our modes of treating materials;
the old experiment of firing a candle through a door, or of
firing a pistol bullet through a pane of glass, without cracking the pane, exhibited an important principle, but the last
generation could not read the lesson; by understanding
this principle and applying it in other directions, refractory
granite or other hard substances can be treated with ease
and facility.
Again, take the steam-engine, for example. The little
steam-engine which worked in the courtyard of Hero of
Alexandria, two thousand years ago, was no doubt considered wonderful in its time; but if the indicator diagrams
of its performance, together with the quantity of fuel consumed, per horse-power per hour, had been handed down
to us, we, with our more advanced experience, would not
be surprised to find the consumption of coal equal to ioo
lbs. -or, going back only two hundred years, it is probable
that Captain Savery's engines consumed 50 lbs.-per horsepower per hour.
Then James Watt, by fresh devices, took advantage
of nature's secret working in better ways, and thus reduced the consumption to Io lbs. of coal, and, by the
further development of the same principles, the consumption of some modern engines is within a fraction of 2
lbs. of coal, per horse-power per- hour. But are we to remain at 2 lbs.? We learn from Joule's equivalent that
the heat which is required to raise a pound of water Ix
would raise 772 lbs. i foot, if entirely convert6d into mechanical work; if so, and there is no reason to doubt,
then our work with coal and steam is scarcely begun; our
best engines do not perform one-tenth of that duty. Those




296        On the Strength of Structures.
who are now engaged in the contest are prone to think,
that with 2 lbs. of coal, per horse-power per hour, we are
near the limit of improvement, but let not the student think
so; we have every reason to believe that unlimited devices
are stowed away in the archives of nature, which, one by
one, will in due time be found by the earnest seeker,
and will open up avenues of economy that are unknown
in our present philosophy. The thin curtain which hides
from us the means of obtaining a horse-power, with 1 lb.
of coal per hour, is gradually being withdrawn, and the way
of overcoming certain practical difficulties, that bar the way
at the present time, is clearing up from year to year.
When the majority of men, who are engaged in the
fabrication of materials into definite forms, in order to
prepare them severally for some mechanical combination, for
the conversion of force or motion, are all well imbued with
the knowledge of natural principles, the effect must be
great. There are now many thinking minds directed to
the study of such questions, by the teaching of art and
science, whose influence will soon make itself felt in every
direction, and through all our operations.
The object of these concluding remarks is to draw the
attention of the student more to the physical and experimental basis of the study of applied mechanics, force,
motion, structures, and material than is usually given.
The cultivation of the habit of looking for the natural
laws which underlie all mechanical operations will prepare
the mind for the higher and, at present, unknown practical
questions that are laid up in the future.




INDEX.
ALU                          CAS                             COL
A LUM INIU M                  Brass-cont.                    Caston-cont.
BRONZE, 97              - specific gravity of, 88     - fluidity of, 35
Ash timber, io8               - tenacity of, 89             - for castings, 35
Bridles, testing, I7          - girders, 35, I67
Bronze aluminium, 97          - guns, 32
ABBITT'S  METAL,    - casting of, 97                        - in dry mould, 35
98                         - composition of, 97             testing machine, 32
Beams and girders, I57-I59    - cost of, 97                 -limit of stress of, 33
- cylindrical, stiffness of,    - experiments with, 97      - liquid, 3I
I8I                       - and gun-metal, 82           - mixing of, 32, 36
- strength of, I7o            - compression of, 87          — Nova Scotia, 38
- flexible, made rigid, 8     - experiments  with,  at    -obscure points of, 30
- of uniform strength, I72        Woolwich, 86              - ore refractory, 30
- rectangular,  deflection    - exposure of, in furnace,    - permanent elongation
of, I8o                       84                            of, 37
- -solid, I69                 - furnace, treatment of, 86    - - set of, 33, 38, 39
-     stiffness of, I8I       - fusibility of, 84           - proportion of carbon in,
- - strongest form of, I70    - Muschenbroeck's   ex-           32
-rigid, 8                         periments with, 85        - remelting of, 44
-secure fixing at ends of,    - Rennie's  experiments    -- rigidity of, 35
I60                           with, 85                  - rough testing of, 35
-square, strength of, I70     - smelting, 84                - specific gravity of, 27
- -stiffness of, I8I          -- specific gravity of, 83    - specimens, short, crush-trussed, counterbraces,    - Wade's   experiments              ing of, 44
210                           with, 83, 86              - strength of, 32, 34, 45
- -diagram of, 209            Butt joints, 267              - tensile strength of, 33
- -movable load on, 209                                     - test bars, 36
- -stress of, 2o9                                           - - beam, 37
Beech timber. Io9             C  ANTILEVERS,                - testirg   by  hydraulic
Bell-metal, 98,           FLANGED, 273                press, 36
-preservation of, 98          Carbon in cast iron, 3I       - - io-feet bars, 37
Boilers, arrangement of    -  - wrought iron, per-    - tests of, 36
plates in, 241                centage of, 48            - under impact, 45
- copper for, 79              Cast-iron, 30                 Chains, proof strains of, I53
- strength of, 253, 256       - carbon in, 3I               - sorts of, I53
- testing of, 25I             - closeness of structure, 35    - strength of, rules for, I54
- tubes, small, 270           - columns, 200                -- wrought-iron, 52
- tubular, 257, 258           - - with flat ends, 20o       Collapse, 250
- -fire-box, 260              - - -rounded ends, 20I        - experiments by  Fair- -smoke-box, 26i             - compared with wrought           bairn, 263
- - stays, 260                    iron, 39, 54              - - with buttjoints, 672
- -strength of, 258           - compression of, 40          - experiments with over-weakened by improper    - crude metal, 30                      lap do., 267
arrangement, 257          - depending on tempera-    - of circular tubes, 267
-wear through restless-           ture, 44                  - - elliptical tubes, 269
ness of, 257              - effect of repeated de-    -- resistance to, 262
Brass, 87                         flection, I28             Columns, 298
- composition of, 88          - elasticity of, 8            - cast-iron, 200
- effect of lead in, 88       - experiments with, 38, 39    --   with flat ends, 2ot
-fusibility of, 88            - fluxes of, 30               -     — rounded ends, or




298                                     Index.
COL                             FOU                           GIR
Columns-cont.            -         EFLECTION    and           Foundation —ont.
- crushing of, i98             DJ  strength, constants    - corner piles, 238, 239,
- cylindrical,   wrought-          of, i76                        240
iron, 203                 - of columns, i98              — diagrams of, 23I, 233,
- deflection of, I98           Double riveting, 247               234, 235, 237, 240
- flexure of, I98              Dry mould, cast iron in, 35    - distribution of weight,.
- long, strength of, 195                                          238
- parabolic, I99                                             - for pier head, 232
- practical deductions for,       -DGES knife,                  - solid wharf, 230
I99n._, wEffect of lead and    - holding down bolts, 23- rectangular,  wrought-           tin in brass, 88          -piles for, 238
iron, 202       -         - of repeated deflection on       pressure   on   vertical
- rigidity of, I99                 cast iron, 128                struts, 238
- short, strength of, 195        -  temperature, 249         -resistance to  counter- strength  as depending      Elastic flexibility, I3            balance weight, 230
on fixing, I96, I97       Elasticity, 3                  - struts, stress on, 236
- taper, 99                   - of ariform bodies, 5         -surrounding piles, 236
- wrought-iron, 202            - an advantage, 7             -- upper tier girders, 235
- - cylindrical, 203          - Hooke's law of, 4            Frictional  resistance   of
- -strength of, 204               imperfect, 4                   joints, 244
Compressioi of cast iron, 40   -limit Of, 9, I  12 22         Fusibility of brass, 88
- of wrought iron, 40         - of cast iron, 12, 38, 28
Constants, examples of,
Constants,  examples - - copper and tin, I2
174, I75                  - - hard steel, 12,    overhun
-of strength and deflec       ---  liquids, 5                       bearing, i85
tionpl,       6             strength of, I87
lO~l 76         -- — plastic materials, 5        strength of, 87
Ctti -us e of, I73a  expasi                                   Girder, cast-iron, 35, I67
Contraction and  expansion    - - watch-spring,              -considered  as a cantiin boilers, 250              -wrought iron, 6                lever, I6I
Copper, 77                    - perfect, 3, 4, e, II, 12     -        I --  lever, I6t
-  affected by temperature,    -  used up, 7                 - - practically, I65
79                         Elliptical tubes, collapse    - deflection  of, i8o, 182
- alloys of, 77                    of, 269                   -- examples of, 182, I83,
- and tin, 83                  Elm timber, io09                   1S
- annealing of, 78             Exmples   of   constantshen uniformly load- effective working of, 79            es,         ed, 182
-  forging of74, 175                                          - disposal of material in,
forging Of, 78       Expansion and contractional
- for steam boilers, 79           in boilers, 250                 I65
- in castings, 79              Experiments, collapse  of    - experiments  on,   by
- purity of, 77                    butt joints, 267          -      RaHodgkinso,'177
- refining of, 78             - --  overlap joints, 267      - -       deAection   of, 877
- rigidity of, 78             - Fairbairn's, 263             - -      tin of, I
- specific gravity of, 79     - of Railway Commission,    -    strains of, 77
- strength of, 77                  II, 22               X    - neutral axis of section,
- tenacity of, uncertain, 77    - with aluminium bronze,          66
-with phosphorus, 8i                                         -resilience of, I86
Corrugated ring, 251          _        97b                  -- semi, deflection  of, I8
Crane,  hydraulic (see  hy-                                 --  cast iron, 38, 39  -square and round tube,
draulic crane)                                               strength of, I7I
- d rtauli (setc crane)  Extension at weak point, 7    - stiffness of, I7
-steam (see steam crane)                                        stiffness of, I8i
- overhead travelling, 203                                   - strength of, I60
Crank shaft, 140                                             - stress on, I68
Crude metal, cast iron, 30      F LANGED cantilevers,    - tubular, I59
Crushing cast iron, short           I73 —under different condispecimens of, 44          Flexible beam made rigid, 8        tions, I62, 163, 164
Cutting   instruments,         Flexure of columns, I98       Girders and  beams, I57,
strength of, I94          Fluidity of cast iron, 35          I59
Cylinders, hydraulic, 279      Fluxes of cast iron, 30       -  cylindrical, stiffness of,
Cylindrical beams and gir-    Forces, triangle of, 208            I8I
ders, stiffness of, I8I   Formulae, 158                  -  -  strength of, 170
-       --  strength of, I70  Foundation, cast-iron cy-    -  of  uniform    strength,
— boiler with flues, 255          linder, 232                    172
- wrought-iron   columns,    - centre support of, 236        -  rectangular,  deflection
203                       - concrete, 230, 232               of, I80




PICznex.                                   299
GIR                           NAT                          SCR
Girders and beams-cont.       Iron, cast (see cast-iron)     f "AK  TIMBER, III
- rectangular, stiffness of,    - wrought isee wrought-        J   Ore of cast iron, reI8I                           iron)                          fractory, 30
-   strongest form cf, I70                                   Overhead travelling crane,
-  igid, 8                                                        209
- secure fixing at ends of,     OINTS, riveted, con60o                       J   struction of, 243
- squtare, strength of, I70   -diagonal, 247                  PARABOLIC    COstiffness of, i'i           - frictional resistance of,     [    LUMNS, I99
Guns, cast-iron, 32                244                       Permanent elongation  of
Gun construction, by build-    - lap and butt, 267               cast iron, 37
ing up, 282               - strength of, 241, 247        - set, io
- strength depending on                                      — of cast iron, 33, 38,
outward form, 279                                            39
-grip of hoops, 283                NIFE-EDGES, I7             Pine and fir, 2io
-homogeneity, 283              K                              Pinion, wear of, 290
inner barrel, 285                                          Pipes, cast-iron, 27I
- lining cast-iroi gun, 29                                    - high pressure, 273
-longitudinal strain, 28             UID CAST IRON           --         sthig rength Ofssure, 273
- Palliser and  Parsons,        IQ     D  CAST IRON,    -- strength of, 273
290    31                     - strength of, 272
290                       Lap joints, 267                Planks, behaviour of, Io6
- principles of, applicable                                   Plate and joint, relative
to  other  structures,                                       strength of, 244
shrinking,  system   of,   MIACHINERY,    air    Plates, arrangement of, in
284 srnig sytmo,                 and water, 6                 boiler, 241
284 slow cooling,         Mahogany, Ii:                  - strength of, 24i
stress on hoolps, 286         Material, 2                    - thickness at edges, 247
the breecss on hoops, 289    - constructive value of, I      Power  transmitted   by
- theory of lining, 289       - endurance of, I                   wheels, I9I
ustheory of verninger, 287   - fitness of, I                 Proof strains of chains, I53
use of vernier, 278           - for boilers, 249              Punching and drilling, I46
- wrought-iron hoops, 283    - instruments for testing,    - careless, 247
Gu n-metal (see bronze)           2                          - experiments on, I46
G- permanence of form, 2       - resistance to, I44
- reliability of, I            - stress, I45
M ARDNESS and brit-    - rigidity of, 2
HRtleness, 12                 - soft and ductile, 22
Hornbeam timber, Iio          - stretching of; 3                 EMELTING   OF
Hydraulic crane, 2ro          Metal, Babbitt's, 98            R        CAST IRON, 44
--  diagrams of stress,         bell, Q8                    Rigidity of cast iron, 35
r211                      - - preservation of, 93        Rivet, elongated, 245
-— jib of, 21-i  ductile, 27                                 Riveted structures, 241
— stress on component        flo f,, 26, 27, 28           Rivets, strength of, 242
parts of, 2I2             -specific gravity of, 27       Roof-truss, strength of, 207
-- post of, 212               - malleable, 27                Ropes, experiments with,
-clir, 279 IC- melting point of, 99                    6
-- cylinders, 279                       -56
-resistance  to  compres-       strength of, I55
sion, 00o                  Rectangular  beams  and
TMPACT, 125                 - -- tension, Ioo                 girders, strongest form
- cast iron under, 45     Muntz-o etal, 89                   Of, I70
- deflection due to, I26         for guns, 89                   --  deflectionof, I8o
-effect of, on cast iron,    - strength of, 89                  -    stiffness of, I8I
I25                                                      - wrought-iron  columns,
-experiments on, I25                                             202
-sets under, I26               MATURAL  PRINCI-    Rules  for  strength  of
- vertical, is26               N     PLES, 293                   chains, I53
Impurity, cast iron  ex-    - approximately complied
posed to, 32                  with, 293
Instrument for testing ma-    - in grinding, 295                 CREW   BOLT,
terials, 2                - - steam engine, 295          S        strength of, I50
Internal flue, 250            - - tools, 294                 Screws, square threaded,
- pressure,  experiments    -- underlying  mechanical            I93
on, 270                       operations, 296            - strength of, 192




300                                   Itdcrx.
SCR                          STR                            TIMN
Screws, strength of-cont.    Steam crane-cont.                Strength-cont.
- - angular thread, 192   — diagrams of, 2i6, 217,    - surplus, eftect of, I50
- -- round top and but-            219, 225                  - torsional, 27
tom, I93                  - for 30-tons load, 215        Stress in  proportion  to
-thread of, right-angled    - girders, strength of, 223           stretch, 277
triangle, I93             - jib, construction of, 2I8,    - limit of, II
Shafts, 134                       2i9                        - on 5o-feet bar, iI
- cast-iron, 132              - pinion overhung, 228         -- - joints, 257
- correct proportions of,    - pressure on guide rail,    - theory of, 276
135                           217                        Structures,  complex,
- crane, strength of, 139     - side frames, 224, 225            strength of, 206
- diameter of, I34            - strain on centre pivot,    - hooped and solid, 276
- elasticity of, 136               218                       - riveted, 241
-engine, expansive, 4         - - - chain, 229
- hollow, strength of, I3'    - --  guide rail, 236
-stiffness of, 143, I34,      ---   jib and  tension
- strength of, I3t, 134, I37,     rods, 218, 221             TANKS, cast-iron, 273
139                       - strength of shafts, 226          -hooped, 275
-- wrought-iron, I32          - stress  on  guide-wheel    - strength of, 274
Shearing and punching, I43        shaft, 227                 Teak timber, 112
- detrusion, i44              -- - side frames, 226          Temperature, dependence
- effect of, I44              Steel, 68                          of cast iron on, 44
- experiments on, I48         - Bessemer, 69, 76             T'enacity of brass, 89
resistance to, I44          - cementation, 68              -- cast iron, 34
- stress, I45                 - compression of, 76           Tensile strength of brass,
Sheer-poles (or legs), 212.-determination   of                89
-diagram of, 213.               strength, 73               - - - cast iron, 33
-- guy chains of, 214         - effect of tempering, 73      Test bars of cast iron, 36
- steel, 204                  - elasticity of, 7I            - beam of cast iron, 37
- strength of, 212            - for roulghl structures, 72   Tests of cast iron, 36
-- stress on guys, 2I4        - goodness of, 71              Testing boilers, 251
-- strength of, 205           — gun, 7I                      - bridles, 17
Sheer-pole tackles, 2I5       - in testing machine, 75       - by gauge, 21
Shell, chilled, 28i           - - large masses, 69           - - impact, 12
Solids, elasticity of, 6, 7   - percentage of carbon in,    - - lever, I5
Specific gravity of brass, 88     67                         --  torsion, 26
- - - cast iron, 27           - porosity of, 70              - care necessary in, i8
---  wrought iron after    -- related to cast iron, 68      - cast iron  by hydraulic
compress on, 67           - tempered in oil, 72              press, 36
Specimen bridles, 20          - tenacity of, 73              - Machine, Io, i2
- each part of, independ-    - testing of, 72                - - adjustment of, I9, 26
ent, 10                   — testing gun blocks of,    - - American, 15
- elongation of, per ton,          74                        - - balance of, 17
21                        - \Whitworth's  treatment    - - cast iron in, 32
- holder for punching, 24          of, 70                    - - compound lever, 14
---   transverse resist-    Sterro-metal, go               - -  compression by, 23
ance, 24                  - composition of, go           - - lever, i4
Specimens for compression,    - compressibility of, 91       - - simple, 29
14                        - experiments with,  go,    --   single lever, 14
- - extension, I3                  92, 93                    - -- strength of, 13
- -testing, 9                 - tenacity of, 9i              — Woolwich, I4, I6
-long and short, 9, 37         Strength  and  deflection,    - rectangular bars, 25
-prepared in lathe, I3, 28         constants of, I76         - ten-feet bars of cast iron,
-rupture of, i9               - of boilers, 253, 256              37
-sketch of, i9, 2i            - - cast iron, 32, 34, 45      - time an important ele-unequal stretching of, 9    - - chains, rules for, I54           ment,in, 10
Spiegel-Eisen, 69             --   columns depending    - weights, I8
Springs, railway carriage, 8      on fixing, I96, I97        Thrust on triangular struc-steel, 5                     --  columns, long and              ture, 208
- watch, elasticity of, 5          short, I95                Tie beam, I59
Steam crane, centre pivot,    - - plates, 24i                - rods, 8
215, 221, 222             --   square beams  and    Timber, IoI
-deviation of pressures,           girders, I70              - age of, o02
217                       - - rivets, 24I                - ash, io8




Index.                                    301
TIM                           WRO                           WRO
Timber-coant.                 T   NDER-GIRDERS,             Wrought-iron-cont.
- beams, strength of, I21     U       conditions of load,    - ductile property of, 52
-beech, Io9                        60o                      - effect of overheating, 49
- change in form of, I04      Uniformity of sectional area    - - of time on, 6I
- commercial value of, Io2        of screw bolt, 149, 150   - elasticity of, 6i
- density of, o02             -- section in chainls, I52    - elastic limit of, 47
- elm, o09                    Uniform strength of beams    - equilibrium in, 29
- hornbeam, IIo                   and girders, I72          - experiments with, 56
-mahogany, III                Use of constants, I73         ---   for compression,
- medullary rays in, I04                                        64
- oak, II                                                   --  -  so-feet  bars, 57
- pine and fir, IIo                                             to 63
- resistance to crushing,r        TATCH-SPRING,             - extension of, 6i
II4, 1x5                    V   elasticity of, 5        - fibre of, 28, 50
- seasoning of, 103           Water, elastic, 6             - for columns, 204
-shearing of, iI6             Wheels  for  transmitting    - forging of, 48
- shrinking of, 103               power, 19I                - fracture of, 52
- strength of, o02            Wheel teeth of, i87           - hard and soft, 55
-teak, II2                    - - adjustment of, I89        - in bar, 49, 56
- tenacity of, II3           - - cast-iron, I88, I90        - - cylinder, 56
- transverse strength of,    - - examples of, I89           - loss of strength of, 56
II7                      - - malleable  cast-iron,    - made with charcoal, 47
- under compression, II5          I9I                       - malleable, 46
Torsion and shearing, i30,    - - steel, i9I                - mechanical  impurities
137                      - - strength of, i88               of, 47
-experiments on, I32, I35    - - wrought-iron, I91          - modulus of elasticity, 62
- resistance to, 131         Wires, bundle of, 28           - permanent set of, 54,
Torsional elasticity, W33     Wrought-iron, 28, 46              62
-- stiffness, 133            - affected by temperature,    - piling, 47
strength, 27                   50                        - puddling of, 46
-  of cast steel, I33         - annealing, 52, 67           - purity of welding sur--  _- copper, I 33           - appearance of fracture,         face of, 49
--   - shafting, I3I              5I                        - rectangular columns, 202
---  wrought iron, 13'        - behaviour of, 54            - red-short, 53
Transverse flexure, experi-    - broken suddenly, 5I        - rigidity of, 53
ments on, 127             - butt-welding, 56            - rolling of, 47
- strength,  experiments      - bulging, 67                 - safe load for, 53
on, 123                   - carbon, percentage of,    - scarf joints, 56
-- of iron, 122                   48                        - soft, 48
- -ratio of, I25              - care required in manu-    - specific gravity of, after
Travelling crane, overhead,       facture, 47                   compression, 67
209                      - cementation of, 47           - steely, 49
Triangle of forces, 208       - chains, 52                  - strength at right angles
Trussed  beam,  counter-    - change from cast iron,            to fibre, 50
braces, 210                   46                        - - of, 47, 50, 204
- -diagram of, 209            - cold rolling, 49            - strength of thin plates,
- -movable load of, 209       - cold-short, 53              -    - - welds, 56
- -stress of, 209             - columns, 202                - testing for hardness of.
Tube boilers, small, 270      - compared with cast iron,        48
- circular, collapse of, 267      39, 54                    - ultimate  strength  of,
- elliptical, collapse of, 26)    - compression of, 40          49
Tubular boilers, 257, 258     - conditions for welding,    - under compression, 63
-    fire-box of, 260             57                        - unlike wood, 28
- -smoke-box of, 26I          - deflection of, 67           - welding of, 46
- -stays of, 260              - difference in quality of,    - when turned, 49
--  strength of, 258              47                        - wire-drawn, 54








WORKS
BY
JOHN TYNDALL, LL.D. F.R.S.
Professor of Naltural Philosophy
in the Royal Institution of Great Britain.
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WORKS BY PROFESSOR TYNDALL-continued.
NOTES OF A COURSE OF SEVEN  LECTURES ON
ELECTRICAL PHENOMENA AND THEORIES, delivered
at the Royal Institution of Great Britain, A.D. 1870. Crown 8vo.
price is. sewed; is. 6d. cloth.'Dr. TYNDALL, whose lectures   sible to those who are but imperand works are of world-wide cele-  fectly acquainted with the physical
brity, is one of the very few men   sciences.  The work on diamagwho have acquired a solid and well-  netism  and the two courses of
deserved reputation amongst both  lecture-notes on light and electrinon-scientific and scientificpersons.  cal phenomena, although of most
His works on magnetism, light,  interest to scientific men, may still
sound, heat, and electricity, have   be read with pleasure and profit
obtained as great a degree of favour   by non-scientists.   The lectureas his lectures themselves; and the   notes  especially  should  be in
simple and clear style in which   possession of every lecturer on
they are written render the infor-  physics.'
mation which they contain acces-       DUBLIN MEDICAL JOURNAL.
NOTES OF A COURSE OF NINE LECTURES ON LIGHT,
delivered at the Royal Institution of Great Britain, A.D. 1869.
Crown 8vo. price is. sewed, or is. 6d. cloth.'The contents of this little   not been deterred, when the occavolume fully justify the Author   sion required, from stepping beyond
in his prefatory remarks, and the   the physical region into the physiointelligent student or teacher will  logical. Thus we have some very
find very great benefit by a perusal  good remarks upon brightness, as
of these Notes. Every statement   well as upon the eye and its pecuis extremely clear, and the experi-  liarities with respect to light. On
ments hinted at are all extremely  the other hand, he has not pergood. Such a publication is ex-  mitted himself to enter largely on
ceedingly well adapted to a certain  the subject of dark rays, but has
class of minds, of which the latent   confined himself to those which
powers are better brought out by   affect the eye. A perusal of these
hinting at solutions than by de-   Notes will benefit all who wish to
tailed explanations. The skeleton   become acquainted with the laws
is brought before them, and they   of light, and even if they sat down
are called upon to clothe it for  to such a task, having a previous
themselves. In fact, if physical  acquaintance with every statement,
science is to be used in order to  they will rise with benefit; for a
educate and train as well as to in-  branch of knowledge, like a landform the mind, we cannot dispense   scape, is never fully understood
with a set of notes of this descrip-   until it is regarded under different
tion. The Author has dealt very   atmospheres and from  different
fully with his subject, and he has   points of view.'   NATURE.
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