Fig. 2 represents an ingenious contrivance for strengthening the wooden beams sup portino. a bridge. An iron rod fixed to the beam AB at the ends, is kept at a distance by struts c, c'. The beam cannot now be bent downward without stretching the rod; which thus has to bear the tensive strain while the beam itself sustains only the compressive strain.
Another way of removing part of the strain from a girder, is to fix a king-post and two oblique pieces on its upper side. The whole is now one composite girder; and any weight bears upon it, the whole of the compressive strain is thrown upon the oblique pieces, and only the tensive strain is left for the beam to sustain.
When a beam AB is fixed at one end, and loaded at the other, the strain is greatest at B, and is less at other points c, c', in proportion as Ac, Ac', the levers at which it acts. are less than AB. The beam may therefore be made to taper off toward the end, and we may determine the exact form the beam should have, in order to be equally strong at every point. For supposing the breadth uniform, the strength increases as the squares of the depths c'd' , cd, while the strain increases as the levers Ac', Ac; and thus, if Ac: Ac' : : cd' to the strengths are equal at those points.
This proportion will always hold good, if the curve of the beam is that of a parabola; and, accordingly, this is the shape given to the beams of steam-engines.
In beams supported at both 'ends, the strain is greatest in the middle; girders are therefore made strongest in the middle, and taper toward the ends.
4. Shearing Strain.—This force is called into play when a plate is cut by shears, or when a riveted or bolted joint is torn asunder, in which case the rivets are sheared across. The effect of it is to cause the particles in one plane to slide over those in another; this is resisted by their mutual coherence, and the magnitude of the resistance depends on the number of the particles, that is on the area of cross-section of the body sheared. The following laws are the result of experiment: 1. The ultimate resistance to shearini is proportional to the area of section of the bar sheared. 2. The ultimate resistance o'
any bar to a shearing strain is nearly the same as the ultimate resistance of the same bar to a direct longitudinal strain.
5. lorsion.—If one end of the axle or shaft of a wheel is immovably fixed. and a power acts at the circumference of the wheel (or at the end of a lever or winch), t!.,. power may be so increased as to twist the shaft asunder at its weakest point. If a shaft A has twice the diameter of another shaft B, there will be four times as many fibers in the section of fracture of A, to resist the twist, as in that of B. But as the separation takes place by the one end of the fracture turning round upon the axis of the shaft. making the ends of the separating fibers describe circles, those fibers that are furthest from the center will have the greatest power of resistance, tvAd the sum of their moments.
or their united effect, will be in proportion to their mean distance from the center. This mean distance in A is twice that in B; therefore, the resistance in A is 2 X 4. or 8 times the resistance in B. Generally, the strength of shafts to resist torsion, is as the cubes of their diameters. The toisive strengths of shafts 1 in. diameter, and with weights acting at 1 ft. leverage, being found by experiment for different materials; the strength of shafts of other dimensions is found from these " constants" by multiplying by the cube of the diameter, and dividing by the length of the lever. It is evident that the torsive strength of a hollow shaft will be greater than that of a solid one of the same quantity of material, ou the same principle that its transverse strength is greater. The rule used by Boulton and Watt for calculating the diameters of their wrought-iron shafts was as follows: /120 X horse-power.
Diameter of shaft in inch. = Revolu. per minute.
This is found to make the shafts rather too light; and the following variation gives safer practical results: V240 X horse-power.
Diameter of shaft in inch. = Revolu. per minute.