. Annual report of the Board of Regents of the Smithsonian Institution. Smithsonian Institution; Smithsonian Institution. Archives; Discoveries in science. 138 LECTURE same, the actual deflection increases with the length, and so it comes in again, giving us I3. In the denominator of the fraction, the deflection with the same weight will be diminished as the breadth is increased, simply because there will be more material to resist, disposed in exactly the same position as before; but when we increase the depth we diminish the deflection, not only by adding material, (d,) but by adding it at a
. Annual report of the Board of Regents of the Smithsonian Institution. Smithsonian Institution; Smithsonian Institution. Archives; Discoveries in science. 138 LECTURE same, the actual deflection increases with the length, and so it comes in again, giving us I3. In the denominator of the fraction, the deflection with the same weight will be diminished as the breadth is increased, simply because there will be more material to resist, disposed in exactly the same position as before; but when we increase the depth we diminish the deflection, not only by adding material, (d,) but by adding it at a greater distance from the neutral axis, so that it acts with a greater moment to resist the separating action of the weight. Thirdly. The amount of separation of the particles at the surface being the same, the deflection will be less as the depth is increased, owing to the angle of deflection being smaller; therefore, the deflection will be inversely, as d3. Although we have only considered the upper surface, the same reasoning will apply to the compressed side. The strength of the beam will also depend upon its proportions, but not exactly in the same way. It may be thus expressed : Strength = c — b I W It will evidently depend directly upon the breadth or the amount of material; and if we increase the depth we not only add material, but we add it at such points, far from the neutral axis, that it will have a greater moment, and therefore give us that advantage also, whence we have d~. In the denominator, the strength will be inversely as the length, since increase of length will give the weight additional moment, and it will be less as the weight increases, obviously. The angular deflection, which gave us one I and one d, and the in- creased number of particles, which gave us another I, in the first ex- pression, do not come into this one at all, as a careful consideration of the subject will show. Again, since the tendency to break at any point with a weight, in- creases
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