TRANSACTIONS OF SECTION G. 649 
the beam), as the bending moment is increased. We may suppose it to extend 
uniformly, by Hooke’s law, till the stress upon it becomes equal to 41,000 lbs. per 
square inch as shown. If unconnected with the fibres below it would then elongate 
by about 3 inch. But the shearing resistance of the fibre below will oppose this 
elongation. In other words, the equation of equilibrium for this fibre, when 
equilibrium is re-established, will be T, =T +85, where T, is the stress due to the 
bending moment, T the tensile resistance, S the shearing stress along the line of 
division between the fibre at L and the fibre next below, say at the 40,000 line. 
det us now turn to this second layer, next below. The shearing stress, 8, will 
produce an extension in it, which must be added to the extension due to the ex- 
ternal stress, T, ; and when equilibrium is restored, this double stress, S + T,, will be 
balanced (1) by an increase in the elastic tensile reaction due to this extension in 
length ; (2) by an increased shearing stress, acting between this second fibre and 
the next below. And the same will hold of the third fibre ; that is to say, its length 
will be increased, producing an increase of the elastic reaction, and at the same 
time of the shearing stress between it and the fourth layer; and so on down to the 
neutral axis. 
We thus see that the effect of the shearing resistance at L, when the strain 
approaches the breaking point, will be to increase the elastic tensile resistance T, 
for every point of the section from L to P, where P is a point at the neutral axis. 
But it is the sum of the moments of these successive tensile resistances which 
balances the external bending moment. 
Hence the effect of this shearing resistance will be that an increased proportion 
of this bending moment will be balanced by the elastic reactions of the material 
in the parts near the neutral axis, and this will leave a smaller part to be balanced 
by the elastic reactions of the parts near the outer fibre. In other words, the 
effect is to throw a greater duty upon the parts of the beam near the neutral axis, 
and to relieve those at a distance from it, and so to increase the effective strength 
of the beam. 
This investigation seems fully to account for the fact that the transverse 
strength of a beam is always found to be much greater in practice than when it is 
calculated by the ordinary theory of elasticity ; ¢.e. when the stresses on the different 
fibres are assumed to be still proportional to their distance from the neutral axis, 
and the outside fibre is assumed to be strained by its breaking load. 
This investigation has also a very important effect on the question of employing 
solid or open beams, solid or hollow shafts. The ordinary theory of elasticity 
shows that if we wish to carry the greatest load with a given depth and weight 
of beam, we should dispose the material in two flanges or ribs, as far apart as 
possible, and only connected by cross-bracing or a thin web, such as will enable 
them to work together. Alliron and steel girders, &c., are constructed on this 
theory. Now, in such structures, the maximum load can usually be calculated 
beforehand with tolerable accuracy, and the girder is always so designed that the 
greatest stress this load can impose is well below the limit of elasticity. Hence, 
in such cases the ordinary theory (which is not at all affected by this investigation) 
may be used with safety. But the case will be quite different for any structure 
which, by accident or otherwise, is liable to be strained much beyond its limit 
of elasticity. 
For in the same figure suppose the metal from P to L to be absent, and only 
that beyond Lto remain. Then when a stress = 41,000 lbs. per square inch comes 
on the fibre L, there is no shearing resistance below to take up any part of it: the 
fibre will extend the full distance accordingly ; and the relief to the outer part 
of the beam, which we have seen to be given by the increased strain thrown upon 
the inner parts, cannot occur. 
This applies especially to shafts, such as the axles of railway vehicles, or the 
crank shafts of steamers. Both these are liable to be broken, and are not unfrequently 
broken, by special strains, induced under peculiar circumstances. It has been 
attempted to render these shafts stronger (for the same weight of metal) by 
making them hollow. In the case of railway axles the attempt was soon 
abandoned: but in the case of marine shafts it has been largely carried into effect 
