246 



The Want of a Uniform System 



force in oak is greater when seasoned than when^'green, but 

 that the transverse strength of different kinds of pine trees 

 is greater when green than when dry. ,: iAHoiH; ~ SJ^f! 



5. Deflection as affecting the bending moment of a beam. 

 — The customary formula for the bending moment of a 

 beam, loaded in the middle and supported at each end, is 



--T-, where w is the weight applied and I the length 



of the beam between the points of support. Now this 

 formula expresses the bending moment only so long as the 

 beam retains its original form and position, for whenever 

 deflection begins two new elements are introduced, which 

 affect the result. In the first place, the three forces acting 

 on the beam — viz., the load and the reactions of the supports 

 — cease to act in parallel lines ; and in the second place the 

 beam is lengthened by moving over the points of support 

 as it deflects. 



Barlow, in his work upon the " Strength of Materials," 

 notices the effect of this new factor, and explains by it what 

 was formerly an anomaly in the behaviour of loaded beams — 

 viz., that the strain increases in a ratio higher than the direct 

 ratio of the lengths. No doubt this is the explanation, for it is 

 very evident that it will require a much less weight to produce 

 the same strain in a beam when highly deflected than when 



straight. In cases, then, of large deflection the formula — 



is misleading, and, although sufficiently accurate for ordinary 

 practical use, should certainly be replaced in all cases of 



01)7 



experiment by the formula -^ Sec. ^0, $ being the angle of 



deflection. I obtained this result by the simple application 

 of the resolution of forces, and as this method is much 

 shorter and less involved than that given by Barlow, I 

 include it in this paper: — 



Let ADB be the beam loaded 

 in the centre with a load iv. 

 Let AB = I and the angle of 

 deflection. The reactions at A 

 and B will be at right angles to 

 the inclination of the beam at 

 these points, and may be repre- 

 sented by the lines AO BO. 

 Producing the strain in the beam 

 at the point D there is : — 



