In the case of long columns whose length is 25 or more times their diameter, if we represent the strength of a long cast-iron column of any dimensions by 1000, the strength of a wrought iron column of the same dimensions will be 1750; of cast steel, 2,500; of Danzig oak, 110; of red deal, 80.
3. Transverse or Cross a beam fixed at one end is loaded with a weight at the other, it is bent from its original form, and takes a curved shape. The fibers on the upper or convex side of the beam are extended, and those on the under or concave compressed; while at the middle of the beam, there are fibers which are neither extended nor compressed, where the compression suds and the extension begins: this surface of fibers is called the neutral surface. As long as the beam is not strained beyond the limit of its elasticity, the extensions and compressions for a given strain are nearly equal, and therefore the neutral surface passes through the center of gravity of the cross section of the beam.
If we strain the beam beyond this limit, and approach the breaking strain, the exten sions and compressions are no longer equal, and therefore the position of the neutral surface is not readily determined. For example, iu the cases of stone and cast-iron, the amount of compression is much less than that of the extension, and in the case of tim ber greater. Also the extensions and compressions are no longer proportional to the strains. From these causes the position of the neutral axis, and the amount of strain on the different parts of the cross section at the moment of rupture, cannot be deter mined by theory.
Different theories have been proposed to determine the relative strength of similar beams, while their absolute strength is left to experiment. That of Galileo consists ha supposing the beam incompressible, and that it gives way by extension turning ronn.1 the lower edge, each point of the section giving an equal resistance before rupture. That of Mariotte and Leibnitz supposes the beam in like manner to turn rotund its lower edge, but considers that the resistance given out by each point of the section is propor tional to its distance from that edge.
The theory now.generally adopted consists in supposing the extensions and compres sions to continue up to the point of rupture proportional to the strains, as is actually the case up to the limit of elasticity, an:I therefore, that the beans turns round a neutral axis, passing through the center of gravity of the cross section, the force given out by each point being proportional to its distance from the neutral axis. This last theory is
found to give the best results in the case of timber and wrought•iron, especially wrought iron arranged in the forms usual in girders. The second represents nearly the method of failure of stone, and the first that of cast-iron.
Though none of these theories give accurate results, they yet give us means of deter mining, from experiments, the strength of any other beam whatever. For example, these theories agree in giving the strength of a beam to be propor tional to the area of cross section multiplied by the depth, and inversely proportional to the length of the beam, since the strain increases directly as the length. This, when expressed mathematically, is W = bd (L) Where to = breaking weight in tons.
b = breadth of beam in inches.
d = depth of beam in inches.
/ = length of beam iu inches.
C = a constant number for beams of the same material, to be determined by experiment.
This result is borne out by experiment—that is to say, the constant 0 being determined by experiment on one beam, the strength of any other is found by multiplying its breadth by the square of its depth and by the constant C, and then dividing by its length. In the case of a beans supported at each end and loaded by a weight in the middle, the strength is also given by the formula, W = UP; 1 but c, in this case, is 4 times the value of C in the formula for a beam loaded at one end The truth of this may be seen from the consideration that the beam may be treated as_ if it were two beams, each fixed at the middle point at one cud, and pressed upward by the reaction of the supports at their other ends. This reaction is evidently equal to 2 so that the breaking weight of the whole beam, supported at both ends, resolves itself into that of a beam of length 4, acted on by the weight at one end this by formula (I.) is, W 2 ' or, W=4C therefore, c=40 or C= c.