411. THE STRONGEST FORM OF SECTION AT ANY GIVEN POINT IN THE LENGTH OF THE BEAM.

Since the extension and the compression of the material are the greatest at those points which are most distant from the neutral axes of the section, it is evident that the material cannot be in the state bordering upon rupture at every point of the section at the same instant (Art. 388.), unless all the material of the compressed side be collected at the same distance from the neutral axis, and likewise all the material of the extended side, or unless the material of the extended side and the material of the compressed side be respectively collected into two geometrical lines parallel to the neutral axis a distribution manifestly impossible, since it would produce an entire separation of the two sides of the beam.

The nearest practicable approach to this form of section is that represented in the accompanying figure, where the material is shown collected in two thin but wide flanges, united by a narrow rib.

I

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 evidently (Art. 388.) 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 extended side.

Thus, then, it follows, from the preceding condition and

It is only in cast iron 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, 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:

In the first five experiments each beam broke by the tearing asunder of the lower flange. The distribution by which both were about to yield together-that 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, therefore, the flanges were of equal strength to resist extension and compression respectively; and this was the strongest form of section (Art. 388.).

In this strongest form the lower flange had six times the material of the upper. It is represented 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 2885 lbs. per square inch of section. There was, therefore, by this form, a gain of 1190 lbs. per square inch of the section, or of 3ths the strength of the beam.

* Memoirs of Manchester Philosophical Society, vol. iv. p. 453. Illustra tions of Mechanics, Art. 68.

412. THE SECTION OF RUPTURE.

The conditions of rupture being determined in respect to any section of the beam by equation (637), it is evident that the particular section across which rupture will actually take place is that in respect to which equation (637) is first satisfied, as P, is continually increased; or that section in respect to which the formula

If the beam be loaded along its whole length, and a represent the distance of any section from the extremity at which the load commences, and the load on each foot of the length, then (Art. 371.) P,p, is represented by . The section of rupture in this case is therefore that section in respect to which is first made to satisfy the equation

SI

!= ; or in respect to which the formula

section of rupture is therefore evidently that which is most distant from the free extremity of the beam.

413. THE BEAM OF GREATEST STRENGTH.

The beam of greatest strength being that (Art. 388.) which presents an equal liability to rupture across every section, or in respect to which every section is brought into the state bordering upon rupture by the same deflecting pressure, is evidently that by which a given value of Pis made to satisfy equation (637) for all the possible values of I, p1, and c,, or in respect to which the formula

If the beam be uniformly loaded throughout (Art. 371.), this condition becomes

or constant, for all points in the length of the beam.

414. ONE EXTREMITY OF A BEAM IS FIRMLY IMBEDDED IN MASONRY, AND A PRESSURE IS APPLIED ΤΟ THE OTHER

EXTREMITY IN A DIRECTION PERPENDICULAR TO ITS LENGTH: TO DETERMINE THE CONDITIONS OF THE RUPTURE.

P

If a represent the distance of any section of the beam from the extremity A to which the load P is applied, and a its whole length, and if the section of the beam be everywhere the same, then the formula (638) is least at the point B, where x is greatest: at this point, therefore, the rupture of the beam will take place. Representing by P the pressure necessary to break the beam, and observing that in this case the perpendicular upon the direction of P from the section of rupture is represented by a, we have (equation 637)

If the section of the beam be a rectangle, whose breadth is b and its depth e, then I='be', c1=&c.

If the beam be a solid cylinder, whose radius is c, then (Art. 364.) I=1πc', c‚=c.

If the beam be a hollow cylinder, whose radii are r, and r2, I=1=(r,'—r,'); which expression may be put under the former(+) (see Art. 86.), r representing the mean

radius of the hollow cylinder, and c its thickness. Also c1 = r1 = r+c;

415. The strongest form of beam under the conditions supposed in the last article.

is at A.*

P

1st. Let the section of the beam be a rectangle, and let y be the depth of this rectangle at a point whose distance from its extremity A is represented by x, and let its breadth b be the same throughout. In this case I by, cy; therefore (equation 637) P= If, therefore, P be taken

SI

х

to represent the pressure which the beam is destined just to support, then the form of its section ABC is determined (Art. 413.) by the equation

If the portion DC of the beam do not rest against

D

masonry at every point, but only at its extremity D, its form should evidently be the same with that of ABC.

2d. Let the section be a circle, and let y represent its radius at distance from its extremity A, then I=y',

c=y; therefore P+S? so that the

Ꮖ

geometrical form of its longitudinal section is determined by the equation

The portion of the beam imbedded in the masonry should have the form described in Art. 417.