number of pounds of strength per square inch to be the strongest. Adopting this mode of comparison, the result of these experiments will appear from the following table.

All these castings were made with iron, of which the following is the description

In every case, the casting broke by the yielding of the extended or lower portion of the section, and in all, except the fourth and fifth, there was indicated a continual increase of strength, as more of the material was accumulated in the lower portion of the casting. In the fourth, it was believed that the upper flanch, or rib, had been so much diminished as to affect not its relative, but its actual resisting power, and in the fifth, it was a little increased as well as the lower'; to this circumstance is probably to be attributed the greater strength of this section than the fourth with a less inequality of the two flanches.

Since, in the last experiment, the casting broke by the separation of the extended side, it was probable that the strongest form was not yet attained. The experiments were therefore continued. But the general form or elevation of the beam was now altered.

The form before adopted, was that recommended by Mr. Tredgold as being of equal strength to sustain a load anywhere placed upon it, and therefore as being the most economical form. Since, however, it now appeared that the width of the lower flanch was a more important element in the strength than had before been imagined, since, moreover, the effect of the tensile power of this flanch would everywhere be greater as the resisting portion of the material in the upper flanch was more distant from it, it was clear that there would be an economy of the material, and, practically, a great convenience of form, in keeping the distance of the upper and lower flanch, throughout the whole length, the same, and varying the width of the lower flanch instead of the height of the beam, as had heretofore been done.

Under this new form the beam is represented by the two following diagrams, of which the first represents the plan of either flanch, and the lower the elevation of the rib which joins the two†.

* The number of the experiment corresponds to the number of the diagram in the preceding page.

+The curved form of each of the portions, ACB and A DB, of either flanch, was that of the curve called the parabola,

from the nature of which curve it follows that the widths, PQ, of the flanch at different points of its length, will be to one another as the products of the distances, AM and B м, from the two ends. Thus, for instance, the width, PQ, will be

Having determined upon this general form of the beam, as involving a great economy of the material. Mr. Hodgkinson continued his experiments upon the best form of section, with beams thus constructed; the following table contains the result of them.

The depth of the beam, and distance of the points of support, was as before.

In this last experiment, the casting broke by the compression of the upper flanch.

The above figure represents the form of the rupture. In every experiment up to this, the rupture had been by the yielding of the lower or extended portion of the beam; under this form of section, then, the material resists the forces tending to extend it, a very little more, and but a very little more, than those tending to compress it. Here then is the point at which it was the object of these experiments to arrive. The distribution of the material which gives an equal resistance to the forces of compression on the one side, and of extension on the other, is nearly that by which the lower flanch is made to contain about six times that of the upper. This, then, is the strongest form of section. As this experiment, in point of fact, gave a greater strength per

to the width P'q', as the product AM × BM to AM' x BM'. And since the flanch is everywhere of the same thickness, its power at different points to resist the forces tending to extend or compress it there, will be as its widths at those points; this power will therefore be proportional to the products spoken of above; and since the distances of the flanches are everywhere the same, it follows that the strength of a beam, thus constructed, to resist rupture, whether by extension or compression, will, at different points of its length, be

proportional to the products of the distances of those points from its extremities. Now, it is shown by all writers on the strength of materials, that the effect of the same force, applied at different points in the length of a beam to rupture it, is in this proportion of the product of the distances of the points from the extremities. The strength, then, of a beam thus constructed, is at different points in the same proportion as the effect of the force tending to rupture it. It is therefore throughout of the same strength.

inch of section than any other; it will be well to describe the section more accurately.

9

B

Its form was that of the following figure, but four times the size. The length of the upper flanch AB was 2.33 inches, and its depth 31 inches, and in the lower flanch these dimensions were 6.67 inches, and 66 inches respectively. The thickness of the vertical part, CD, connecting the two flanches was 266 inches, and the weight of the beam was seventy-one pounds. Considerably more than two-thirds of the whole material of the beam was contained in the lower flanch. Now let us compare the strength of this beam with that of a beam of the form most approved before these experiments were made.

E

D

The girders, cast at the factory of Messrs. Fairbairn and Lillie, of Manchester, and most approved by them, were of the elliptical form (see diagram, page 196), and had the section shown in the annexed diagram. One of these was cast of the same length and depth as those in the preceding experiments, at the same time as the last, and of the same metal. Its dimensions were as follows:

It broke with a weight of 9146 lbs.; and as the area of the section of fracture was 3·17 inches, the strength may be estimated at 2885lbs. per square inch, and it may be stated, that of a number of experiments with beams of this form, there was only one which gave a higher numerical strength, and that under peculiar circumstances. Now the strength per inch of section in Mr. Hodgkinson's improved form (Experiment 9), was 4075 lbs. Here is a gain then in strength of the enormous amount of 1190lbs. on every square inch of section, by distributing the material of the beam according to that gentleman's form, being an increase of ths. of the whole strength of the beam. And this, be it observed, over a beam of the best construction before known-a construction greatly superior to that on which girders were then, and are indeed now ordinarily cast. There is, however, another method of comparing the advantages of the two methods of construction, and that is by the ratio of the strength to the weight of metal.

The casting of the best construction (Experiment 9), broke with a weight of 26,084lbs., and the casting weighed 71lbs., each pound in weight gave therefore 367-38 lbs. in strength. The casting on Messrs. Fairbairn and Lillie's original construction of the same length and depth, weighed 40lbs., and it broke with 9146lbs., so that each pound in the

weight of this casting corresponded to 228-65 lbs. in strength. The strength given by each pound weight of metal in this beam was, therefore, 138-73 lbs. less than that given by each pound of the other.

Mr. Hodgkinson's experiments had hitherto been confined to castings of the same depth and length. His inquiries were now directed to the comparison of castings of different depths and lengths. Collaterally with this inquiry, he directed his attention more particularly to the amount of deflexion, produced by a given load, and to the different stages of deflexion and pressure, under which the elasticity is destroyed, or under which the metal is technically said to take a set. This inquiry may be characterized as one into the stiffness of the casting. It is a most important one; for although beams, cast on the new principle, might resist ultimate fracture more successfully than those of the old form, yet if they deflected more under a given pressure, or if under such deflections they more readily fixed themselves in those forms into which they had been deflected, partaking thus in a degree of a quality analogous to flexibility, there might result from these causes inconveniences more than counterbalancing the increase of ultimate strength. It may be mentioned in the outset, that these suppositions were in themselves improbable, and anomalous; it was to be expected that the difficulty which attended the ultimate fracture would in a degree characterize all the stages of approach to it, and such in reality was found to be the case.

The beams were now cast 7 feet 6 inches long, and the props placed 7 feet asunder. The ratio of the upper and lower flanches was, in each experiment, that of 1 to 6, which had been ascertained to belong to the best form of section, and they were all of the same size, being, indeed, cast from the same model, which was only varied by increasing in each experiment the distance of the flanches, or the depth of the rib which joined them. The following diagrams represent the sections of fracture, each as before one-fourth its real size.

In the first of these experiments (diagram 10), the depth of the beam was 4.1 inches. The beam was loaded with 2764lbs., when the deflection was 0.25 parts of an inch, and on the removal of this load it returned to its original form, showing that the elasticity had not been strained. The load was then gradually increased up to 3339lbs., and the deflexion to 0.28 inches, and still the beam recovered its form; 3454lbs. were then put upon it, and there was now a perceptible set, or permanent deflexion, but of exceeding small amount. With a load of 3914, this permanent deflexion became 0.05 inches. The load was then further increased until it became 6215 lbs., and the deflexion 0.51, and throughout the whole of this increase no further set was apparent, the beam returning, when the load was removed, always to its first permanent deflexion of 0.05. When, however, the load was made 6971 lbs., a new set became apparent, and when it was increased to 8637lbs., this set was measured to 003 more than the first, or the whole permanent deflexion was now made 0.08 with a load of 11,397lbs., this permanent deflection became 0.09, with each further increase of weight; the permanent deflexion began now rapidly to increase, until with 12,815 lbs. it became 0.14, and with 13,543 lbs. the beam broke.

Similar circumstances characterized the other experiments. In the second (see diagram 11), the depth of the beam was 5.2 inches-the first perceptible set took place under a load of 7257lbs., with 7947 lbs. it was 0.08, and the deflexion 0.35. It bore 12,087lbs. with a deflexion of 0.63, and broke by extension with a weight of 15,129lbs. In the third experiment (see diagram 12), the depth of the beam was 60 inches, and the first perceptible set took place under a weight of 13,543 lbs., and a deflexion of 0.49 inches. The beam broke with 15,129lbs. In the fourth experiment (diagram 13), the depth of the beam was 6-93 inches, and the first perceptible set took place with a load of 14,271 lbs., and a deflexion of 0.35. The beam broke with 22,185lbs. These results may be tabulated as follows:

From these experiments, the first conclusion to be drawn is, that other things being the same, the ultimate strength is, in beams of this form, nearly as the depth, but in a somewhat lower ratio.

The second, that the stiffness of the beam rapidly increases with its depth, a weight nearly twice as great being required to produce the