16 



On the comparative SlJffness of Beams. 



A piece of Memel timber, eight inches in deptli, two inches in 

 lireadth, and supported at each end, the supports being 18 feet 

 apart, when a weight of 480 lbs. was applied to the middle, was 

 depressed 0'73 inches. The same wefght was suspended from 

 the middle of a piece tvvo inches square, the supports were four 

 feet asunder, and it bent nearly 2*2 inches : but cakulating from 

 Cor. I. Prop. T\), Emerson's Mechaiiics, (3d edit. 4to,) the shorter 

 piece ought to have bent only half an inch. This difference will, 

 perhaps, be a sufficient excuse for an attempt to investigate the 

 theory. 



All bodies may be extended or compressed, and the extension 

 or compression, in the same body, is as the force producing it. 



Let AR represent a 

 solid beam, supported 

 at each end, and a 

 weiglit suspended from 

 the middle : by the ac- 

 tion of the weight the 

 lower side will be ex- 

 tended, and the upper 

 side compressed; but 

 the extension and com- 

 pression, both follow the same law of variation; therefore we may 

 consider the beam subject to extension only. 



Conceive the beam to lie composed of a number of equal and 

 extensible parts, the extension of the parts being as the force 

 producing it, is as the weight: but it is known by experiment 

 that the deflexion is as the M'eight ; therefore the deflexion is as 

 the extension. 



Now the effect of a force to produce extension is inversely as 

 the breadth, and square of the depth ; and the extension pro- 

 duced will vary in the inverse ratio of the distance from the cen- 

 tre of motion, or inversely as the depth ; therefore, the deflexion 

 is inversely as the cube of the depth. Also, the extension is as 

 the strain, that is directly as the weight and length ; and as the 

 number of parts strained, that is as the length : hence, the de- 

 flexion is directly as the weight and square of the length. 



Therefore, the deflexion of a rectangular prismatic beam is 

 directly as the weight and square of the length, and inversely as 

 the breadth, and cube of the depth ; that is, if L, B, and D be 

 the length, breadth, and depth of a beam, W the weight, and 



C D the quantitv of deflexion ; then, C D is as -^55-. This 



gives 2'37 inches for the deflexion of the four-feet piece; and 

 when the weight of the beams is taken into the calculation, it 

 nearly agrees with the experimeut. 



The 



