DEFLECTION OP BEAMS.] 



APPLIED MECHANICS. 



807 



horizontal line D E (Fig. 73) measured between the 

 supports. 



Fig. 71 



This law, however, has not by any means been found to 

 be true m practice. More accurate investigators have 

 Kg. 73 



DEFLECTION. Hitherto, for the sake of simplicity, 

 we have discussed the question of transverse strain as 

 applied to materials perfectly inflexible, such as break, 

 but cannot bend. We have, however, no practical expe- 

 rience of materials of this character, although stones, 

 slates, or even cast-iron, approach very nearly to it. 

 Timber beams, and those made of wrought-iron, bend 

 considerably before they become fractured by transverse 

 itrain ; and as, in such beams, deflection from their 

 straight or horizontal condition may be inconvenient and 

 unsuitable, it becomes important to estimate the amount 

 of deflection which they will exhibit under certain loads, 

 so that they be made slightly curved in the opposite 

 direction before the load is placed upon them. If it 

 were found, for instance, that a beam loaded with a cer- 

 tain weight deflected so far from the straight line that 

 its middle point R (Fig. 74) sunk a certain distance say 

 6 inches below the horizontal line ; then if the beam, 

 instead of being made straight, were made somewhat 

 arched, or cambered, as it is technically called, the load 

 being placed upon it would still deflect it, and thus bring 

 its surface to a horizontal line, if the camber or amount 

 of arching C D were properly estimated. 



The complete investigation of the question of deflection 

 would involve us in mathematical reasoning of rather a 

 complex character, which would scarcely be in place here ; 

 and, indeed, practical results as to deflection present so 

 many irregularities, and so many deviations from any 

 apparent law, that it is questionable whether theory 

 would prove a very safe guide. Some writers on this 

 subject have determined theoretically, that the amount 

 of deflection of a beam of certain length, increases in the 

 same proportion as the load, and that the deflection 

 under a certain weight varies as the square of the length. 

 If this law were true, a beam 20 feet long, with a certain 



Fig. 74. 



weight on it, would be deflected four times as far as one 



feet long, because the one has twice the length of the 



other, and the square of 2, or 2 multiplied into itself, is 4. 



furnished a law 

 which, while it ap- 

 pears to be theore- 

 tically correct, pre- 

 sents results very 

 nearly according with 

 those of experiment. 

 This law is, that the 

 deflection of a beam 

 (having a rectangu- 

 lar section) varies di- 

 rectly as the weight 

 and as the cube of 

 the length (or the 

 length multiplied 3 

 times into itself), 

 and inversely as the breadth and the cube of the 

 depth. If, then, we knew the deflection of a certain 

 beam, we might estimate that of another of the same 

 material, but varying in all its dimensions. Sup- 

 pose, for example, that a fir batten 1 inch broad 

 and 2 inches deep, with, a load of 1 cwt., deflected 

 ,Vth of an inch in a length of 3 feet, and we de- 

 sired to know the deflection of a fir beam 3 inches 

 broad, 10 inches deep, and 15 feet long, under a weight 

 of 1 ton, or 20 cwt. In the case of the batten, which is 

 2 inches deep, since 2 x 2 X 2= 8, the deflection is only 

 Jth of what it would be were the depth 1 inch, because it 

 is inversely as the cube of the depth. The deflection, 

 then, of a batten 1 inch deep, under a load of 1 cwt., 

 would be 8 x i 1 * = inch, the length being 3 feet. 

 Further, the cube of 3, or 3 x 3 X 3, is 27 ; and were 

 the batten only 1 foot long instead of 3 feet, the deflec- 

 tion would be Vrth ; the deflectiou then of a batten 1 inch 

 broad, 1 inch deep, and 1 foot long, would be Jj-th of \ an 

 inch, or j"rth of an inch, with a load of 1 cwt. Having 

 thus got an estimate for a beam with all the dimensions 

 reduced to unity, we can apply it to any other beam, ac- 

 cording to the law we have stated. This law is arith- 

 metically applied by multiplying the deflection found 

 above, Ath of an inch, by the weight 20 cwt., by the 

 cube of the length 15 feet (or 3 times by 15 feet), and 

 dividing the product by the breadth 3 inches and the 

 cube of the depth 10 inches ; or the deflection is 



in. x 20 X 15 x 15 X 15 

 3 X 10 x 10 X 10 



Aths of mch ' 



or nearly i an inch. 



As to the deflection of beams of various forms and 

 materials, and subjected to strains under various con- 

 ditions, although numerous experiments have been made, 

 yet they have been conducted with too little reference 

 to each other for us to develop any law of general appli- 

 cation. In Barlow's Treatise <m the Streiigth of Materials, 

 the question of the deflection of wrought-iron, as applied 

 in the construction of rails, is treated at considerable 

 length ; but the conditions of strength in malleable iron 

 must differ very considerably from those in other 

 materials, because their comparative tensive and com- 

 pressive strengths differ very widely. Mr. Fairbairn, of 

 Manchester, has lately conducted some extended experi- 

 ments on this and allied subjects, which are well worthy 

 of the attention of practical men. To show how cautious 

 we ought tu be in applying to any particular material the 

 results deduced from experiments on other materials, we 

 may instance the peculiar difference between cast and 

 wrought-iron in respect of tensive and compressive 

 strength. The direct cohesive strength of wrought-iron 

 s about 3 times that of cast-iron ; or, to bear a certain 

 ;ensive strain, the area of section in cast, should be about 

 3 times that of wrought-iron. As to compressive strain, 

 on the other hand, although it may be true that wrought- 

 ron will sustain much greater compression than cast- 

 ron before it becomes entirely crushed and disintegrated, 

 et the greater softness of the wrought-iron permits it 

 o yield under pressure, and to become compressed to a 

 considerable extent ; while cast-iron scarcely yields per- 

 ceptibly until it entirely gives way. 



