The power by which the particles in any body resist the action of a force tending to separate those particles in the direction of the length of the body may be considered as constituting the direct or absolute strength; and it is evident that, if the body were of a homogeneous texture, that strength would be proportional to the number of particles in a tranverao section; that is, to the area of such section, while the strain is proportional to the weight applied. Therefore, if F designate the cohesive power in a unit of such area, as a square inch, a square foot, &c.; also, if a represent the area and w the weight applied, including that of the body itself, we should hare F A = w when the strength and strain are in equilibria. This formula for the absolute strength may be considered as nearly correct with respect to most of the bodies in nature; and hence (r being determined by experiment) the strength by which a rod of any material resists this kind of strain may be found when the dimensions of the rod are given.
For complete details concerning the experimental values of r, tho reader must be referred to the extensive tables which have been pub lished by Barlow (` Essay on the Strength of Timber '), Rennie (` Phil. Trans.,' 1818), Trcdgold (` Principles of Carpentry '), Ifodgkinson (` Experimental Researches into the Strength of Cast Iron,' &c.), and Morin (‘ Lecons do bi6canique Pratiqu6; voL iv.), our limits permitting us to introduce only the few determinations which follow.
The area of a transverse section of each rod is one square inch, and the values of r are expressed by the breaking weights in pounds avoirdupois.
Notwithstanding the irregularities in the column containing the values of F for ropes, it may be concluded that ropes of a given diameter have less strength on each square inch of their transverse section than those of less diameter ; and this is owing, no doubt, to their threads being less twisted together. It may be observed, also, that those woods whose fibres are nearly straight bear much greater weights suspended from them than those whose fibres have considerable curvature.
According to the experiments of Mr. Barlow, it appears that a bar of malleable iron is extended one ten-thousandth part of its length by a direct strain equal to one ton for each square inch in the area of the transverse section : when stretched with ten tons per inch its elasticity was injured, or the bar did not return to its original state.
If the fibres in any material body were exactly rectilinear, so that, a rod being placed on ono end in a vertical position,no one of the particles were opposite to the intervals between any two in a transverse section below it, it might be conceived that no force compressing the rod in the direction of its length would produce any other effect than that of diminishing its length. But as we find that all bodies when so com
pressed may be bent, and finally broken, such a disposition of the particles is destitute of probability. In fact, when a pillar is com pressed by a gnat weight above it, either the fibres, already curved, have their curvature increased, so that the whole pillar bends; or the particles in some of the transverse sections are forced outwards by lateral pressures arising from the particles above and below their intervals being thrust between them, and then the pillar swells on its whole periphery. The consequence in either case is, that the cohesion of the longitudinal fibres is impaired or destroyed, and the pillar is at length broken or crushed.
The strength of a pillar when so compressed must evidently depend upon the number of particles in a transverse section—that is, upon the area of such section ; but since, besides the displacement of those particles from the longitudinal pressure, their lateral cohesion must be overcome before they can be thrust outwards, it is evident that the strength is not proportional to the area, simply, but to some function of that area. No law on which any dependence can be placed has yet been discovered for the strength of a pillar in such circumstances. Euler, from analytical considerations, concluded that it varies as the square of the area ; but late engineers have supposed that the square root of the third power of the area more correctly represents the law of the strength.
If a bar or pillar, resting on one end in a vertical position, and con sidered as a perfectly elastic body, be compressed by a weight acting vertically above it, the purely mathematical theory gives the following equation for the value of the compressing weight when the pillar begins to bend cur w= w2 8 (Poisson,' Dlecanique,' tom. i., No. 313) ; where w= the compressing weight ; l=the length of the pillar • a= the area of the transverse section ; d= the thickness perpendicularly to the bending surface ; 8= the element of deflection ; and w = It follows that, when in two bars of like material a and d are respectively equal, the weights which those bars will sustain without bending are inversely proportional to the squares of the lengths.