138 LECTURE 



same, the actual deflection increases with the length, and so it comes 

 in again, giving us I 3 . 



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 d 3 . 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 with the distance of the weight from that point, such a beam 

 will break at the wall, and if it is strong enough there, it is unneces- 

 sarily strong at all other points of its length, and we may econom- 

 ically taper it off to the end in the forms shown in Figs. 12 and 13, 



_©©©©_ 





Fig. 12. Fig. 13. 



where Fig. 12 is a beam loaded with a weight uniformly distributed, 

 and Fig. 13 one loaded at the end, the under side in this case having 

 the form of a parabola. 



In engineering structures, such beams supported only at one end do 

 not frequently occur, and we must, therefore, consider how our expres- 

 sions already deduced, must be changed to apply to beams supported at 

 both ends and loaded in the middle. Such a beam may be considered 

 as fastened in the middle and acted upon by two forces, acting upwards 

 at its two ends. 



