but wide flanges, but united by a narrow rib. That which constitutes the strength of the beam being the resistance of its material to compression on the one side of its neutral axis and its resistance to extension on the other side, it is

Fig.3.

evidently a second condition of the strongest form of any given section, that when the beam is about to break across that section by extension on the one side, it may be about to break by compression on the other. So long, therefore, as the distribution of the material is not such as that the compressed and extended sides would yield together, the strongest form of section is not attained. Hence it is apparent that the strongest form of the section collects the greater quantity of the material on the compressed or the extended side of the beam, according as the resistance of the material to compression or to extension is the less. Where the material of the beam is cast iron, whose resistance to extension is greatly less than its resistance to compression, it is evident that the greater portion of the material must be collected on the external side.

"Thus it follows, from the preceding condition and this, that the strongest form of section in a cast-iron beam is that by which the material is collected into two unequal flanges joined by a rib, the greater flange being on the extended side, and the proportion of this inequality of the flanges being just such as to make up for the inequality of the resistances of the material to rupture by extension and compression respectively. Mr. Hodgkinson, to whom this suggestion is due, has directed a series of experiments to the determination of that proportion of the flanges by which the strongest form of section is obtained..

"The details of these experiments are found in the following table:

Number of ex- Ratio of the section of Area of whole section in Strength per square inch periment.

1

2

5

6

the flanges.

1-1

1--2

1-4

1-4.5

1-5.5

1—6.1

square inches.

2.82

2.87

3.02

3.37

5.0

6.4

of section.

2368

2567

2737

3183

3346

4075

"In the first five experiments each beam broke by tearing asunder of the lower flange. The distribution by which both were about to yield togetherthat is, the strongest distribution-was not, therefore, up to that period reached. At length, however, in the last experiment, the beam yielded by the compression

of the upper flange. In this experiment,
therefore, the upper flange was the weakest ;
in the one before it the lower flange was the
weakest. For a form between the two, there-
fore, the flanges were of equal strength, to
resist extension and compression respectively,
and this was the strongest form of section.
In this strongest form the lower flange had six
times the material of the upper.
It is repre-
sented in the accompanying figure. In the best
form of cast-iron beam or girder used before
these experiments, there was never attained,
a strength of more than 2,885 pounds per
square inch of section. There was, there-
pounds per square inch of the section, or of
(See Moseley's "Engineering and Archi-

The Rev. Canon Moseley further observes on this point, "It is only in castiron beams that it is customary to seek an economy of the material in the strength of the section of the beam; the same principle of economy is surely, however, applicable to beams of wood."

This victory over the material foe is entirely Mr. Hodgkinson's own; and, using the language of the president of the British Association at Manchester, 1861, there is no one to divide the honor of this useful achievement with him. The Hodgkinson beam is really what its name would imply, as he originated the conception and pursued it with judgment and industry until the best form of beam was fully determined. This beam has been the pole star for engineers and builders during the last 20 years, a period in which construction of all kinds has been in great demand, and in which the ingenuity and skill of the constructor has been confronted with many and formidable difficulties. Railways, shipbuilding, and public works of various kinds have opened out new channels for the application of cast and wrought iron; and when this material is placed in new and untried positions, it is no little point which is gained when its tensile and crushing strength is determined, and the best form investigated by which the safety of large structures is secured. This was the life-work of Professor Hodgkinson.

It is a great thing, which no man of science lightly appreciates in these days. of mental activity, for a man to point to a useful discovery and claim it as his own without a rival, to say to himself, (his own precious reward,) "I drew it forth from the dark chaos in which it had been entombed for ages to the light of day, and now I leave it as a legacy to my countrymen, trusting that the chance of calamities such as that which happened at Hartley Colliery, where 200 men lost their lives by the breaking of a cast-iron beam, may be diminished, if not entirely obviated." In this paper Mr. Hodgkinson acknowledges his deep obligations to the liberality of his friend Mr. Fairbairn, in procuring for him the bcams whereon to experiment.

The contributions of Professor Hodgkinson to the "Reports" and "Sections" of the British Association were numerous and important. In proof of this it is only necessary to refer to the opening address of the president, Professor Sedgwick, at the meeting at Edinburgh in 1834: "The association may claim some credit for having brought into general notice the ingenious investigations of Mr. Hodgkinson of Manchester."

In the Report of 1833 there are two papers by Mr. Hodgkinson:

1. "On the Effect of Impact of Beams.'

2. "On the Direct Strength of Cast Iron."

In the Report of the British Association of 1834, we find an extended inquiry into the collision of imperfectly elastic bodies. After alluding to Newton's labors, as recorded in the "Principia," Mr. Hodgkinson proceeds to describe the methods by which his experiments were made, and derives from them the following nonclusions:

1. All rigid bodies are possessed of some degree of elasticity, and among bodies of the same nature the hardest are generally the most elastic.

2. There are no perfectly hard inelastic bodies, as assumed by the early and some of the modern writers on mechanics.

3. The elasticity, as measured by the velocity of recoil divided by the velocity of impact, is a ratio which (though it decreases as the velocity increases) is nearly constant when the same rigid bodies are struck together with considerably different velocities.

4. The elasticity, as defined in 3, is the same whether the impinging bodies be great or small.

5. The elasticity is the same, whatever be the relative weights of the impinging bodies.

6. On impacts between bodies differing very much in hardness, the elasticity with which they separate is nearly that of the softer body.

7. In impacts between bodies whose hardness differs in any degree, the resulting elasticity is made up of the elasticities of both, each contributing a part of its own elasticity in proportion to its relative softness or compressibility.

The following rule, given by Mr. Hodgkinson, agrees remarkably well with the results of experiments:

Let e=the elasticity of A} as determined by A striking against A, &c.

1}as

m=modulus of elasticity of A as determined by extending the material in the ordinary way.

m':

=

66

B

This paper concludes with a table of elasticities of 60 various substances used in the construction of buildings, &c.

The Fifth Report of the British Association contains a paper on the "Impact of Beams."

The author has deduced from the experiments the following laws:

1. If different bodies of equal weight, but differing considerably in hardness and elastic force, be made to strike horizontally with the same velocity against the middle of a heavy beam supported at its ends, all the bodies will recoil with velocities equal to one another.

2. If, as before, a beam be struck horizontally by bodies of the same weight, but different in hardness and elastic force, the deflection of the beam will be the same, whichever body be used."

3. The quantity of recoil in a body, after striking against a beam as above, is nearly equal to what would arise from the full varying pressure of a perfectly elastic beam as it recovered its form after deflection.

4. The effects of bodies of different natures striking against a hard, flexible beam seem to be independent of the elasticities of the bodies, and may be calculated, with trifling error, on a supposition that they are inelastic.

5. The power of a uniform beam to resist a blow given horizontally is the same in whatever part it is struck.

6. The power of a heavy uniform beam to resist a horizontal impact is to the power of a very light one as half the weight of the beam, added to the weight of the striking body, is to the weight of the striking body alone.

7. The power of a uniform beam to resist fracture from a light body falling upon it (the strength and flexibility of the beam being the same) is greater as its weight increases, and greatest when the weight of half the beam, added to that of the striking body, is nearly equal to one-third of the weight which would break the beam by pressure.

There can be but one opinion as to the importance of these deductions, direct from the voice of nature, made, as they were, at a time when such an appeal was by no means common.

There are several interesting problems on impact, of a high mathematical character, solved in this paper. In these inquiries Mr. Hodgkinson is very particular in acknowledging his many obligations to his friend Mr. Fairbairn, engineer, of Manchester, to whose labors and liberality practical science is deeply indebted.

We now pass on to notice his contributions to the transactions of the Royal Society.

In the Philosophical Transactions for 1840 there is an extensive inquiry by Mr. Hodgkinson, "On the Strength of Pillars of Cast Iron and other Materials."

The object of this inquiry is to supply a desideratum in practical mechanics, which had been pointed out by Dr. Robison and Professor Barlow. In order to accomplish this it was necessary to institute a series of expensive experiments more varied and extensive than any which had hitherto been made public. The

subject was mentioned to Mr. Fairbairn, who at once, with his characteris ac liberality, supplied his friend with ample means for investigating experimentally the strength of cast-iron pillars. For this paper the council for the Roya Society awarded Mr. Hodgkinson the royal medal as a mark of their appreciation of his labors, the value and importance of which are confirmed by every ngineer's pocket-book in Europe during a period of 20 years.

The inquiry is naturally divided into two parts, viz., long pillars and short pillars.

LONG PILLARS.

The first object was to supply the deficiencies of Euler's theory of the strength of pillars, if it should appear capable of being rendered practically useful, and if not, to endeavor to adapt the experiments so as to lead to useful results. For this purpose solid cast-iron pillars were broken, of various dimensions, from five feet to one inch in length, and from half an inch to three inches in diameter. In hollow pillars the length was increased to seven feet six inches, and the diameter to three inches and a half.

With pillars of cast-iron, wrought iron, steel, and timber, whose length is upwards of 30 times their diameter, the strength of those with flat ends is three times as great as those with rounded ends.

Experiments were next made upon pillars with one end flat and the other end rounded, and the result is summed up in the following interesting and impor

tant law:

With pillars of the same diameter and length, both ends rounded, one end rounded and the other flat, and both ends flat, their strengths are as 1, 2, 3, respectively.

When the pillars were uniform, and the same shape at both ends, the fracture took place in the middle. This was not the case when one end was flat and the other rounded, as the fracture then took place at about one-third of the length from the rounded end. Hence, in these pillars, the metal may be economized by increasing the thickness in the point of fracture.

It follows, from Euler's theory, that the strength of pillars to bear incipient flexure is directly as the fourth power of the diameter, and inversely as the square of the length.

This incipient flexure was sought for by Mr. Hodgkinson without success, and he states his conviction that flexure commences with very small weights, such as could be of little use to load pillars with in practice. Although Mr. Hodgkinson was unable to find the point to which Euler's computations refer, still he has shown that Euler's formula is not widely from the truth when applied to the breaking point of the pillar. From a great number of experiments Mr. Hodgkinson deduced the following formula for pillars with rounded ends:

The above rule applies to pillars the length of which is 15 times the diameter and upwards. Perhaps not quite so low as 15 times the diameter in large pillars, as there is a reduction of the strength of such pillars, owing to the softness of the metal in large castings. This remark is significant, and gave rise to many interesting experiments at Portsmouth dockyard by the royal commissioners, conducted by Colonel Sir Henry James.

When the pillars are flat at the ends, the formula becomes

This rule applies to pillars whose lengths vary from 30 to 121 times the diameter,

SHORT PILLARS.

In order to estimate the breaking-strength of short pillars, Mr. Hodgkinson considered the strength of the pillar to be made up of two functions.

1. To support the weight.

2. To resist flexure.

When the breaking-weight is small, as in long pillars with small diameters, then the strength of the pillar will be employed in resisting flexure. When the breaking-weight is one-half the pressure required to crush the pillar, one-half of the strength may be considered available to resist flexure, and the other half to resist crushing. And when the breaking-weight is so great as in the case of short pillars, it may be considered that no part of the strength of the pillar is applied to resist flexure. These two effects may be separated in all pillars by dividing the pillar into two portions, one of which would support the weight without flexure, and the other would support the flexure without crushing, to the extent indicated by the preceding formulæ.

Let c=the force which would crush the pillar without flexure.

Let Pthe utmost pressure the pillar would bear without being weakened by crushing.

b=breaking-weight as calculated by the preceding formulæ.
y=the actual breaking-weight of short pillars.

The value of c is obtained from the formula

c=(area of section) × 109,801 pounds.

The reasoning by which the above formulæ are established is well deserving of attention, and shows that the author was a worthy successor of Euler, Lagrange, and Poisson in this important branch of practical science.

HOLLOW PILLARS OF CAST IRON.

Mr. Hodgkinson has shown that solid pillars with rounded ends and enlarged in the middle are stronger than uniform pillars of the same length and weight. This is proved to be the case in hollow pillars. The formulæ for the breakingweight of hollow pillars, as derived from experiment, are as follows:

The strength of short hollow pillars must be calculated in the same manner as the strength of short solid pillars. These formulæ, derived from experiments made with great judgment and care, embody our present knowledge and prac tice of cast-iron pillars for bearing-purposes.

"The Power of Cast-iron Pillars to Resist Long-continued Pressure."

Mr. Hodgkinson has recorded in this paper several very interesting experiments on this subject. Two beams, rounded at the ends, six feet long and one inch diameter, were cast of Low Moor iron, No. 3. The first bore a weight of