aoa 



APPLIED MECHANICS. 



[TRANSVERSE STRAIN. 



intervals of a foot by weight* each 10 cwt. (Fig. 69), 

 by the principle of the lever, the offoot of each of 



<-l -*-!- I I l-K-l-X-i-> 



cot 



V T v V ^ 



10 U It 10 



i 



the weights, to break the beam at A, may bo reckoned 

 as follow* : 



B=IO cwt. acting at 1 ft horn A hu the effect of 10 cwt at 1 ft. Icrprmge. 

 '=! ,. J M 1 



n=io ., a so i 



E = IO ., 4 .. .. 1 



F=10 .. S M 1 



Total SO cwt. dUtributtd equally. liO cwt. at I It. leverage. 



Or, a* 150 cwt. at 1 foot leverage are equivalent to 60 

 cwt. at 3 feet, we find that the total strain is the game 

 a* if the total weight were collected at D, the middle 

 point. Were we to assume a greater number of weights 

 at smaller intervals, we should still find the same result ; 

 and the more numerous the weights, and smaller the 

 intervals we assume, the more nearly do we approach to 

 the case of a beam uniformly loaded over its whole 

 length ; whence we conclude that the effect of the dis- 

 tributed load is the same as if it were collected at the 

 middle of the beam, and therefore just half of what it 

 would be if hung at the end. Or, conversely, if the 

 beam bear a certain load at its extreme end, it will bear 

 double that weight distributed over its whole length. 

 The same law applies in the case of a beam supported at 

 both ends, and loaded uniformly throughout its length ; 

 the strain of the load is reduced to half what it would be 

 if collected at the centre ; or the beam will bear twice 

 as much distributed weight, as it can bear at its middle 

 point. 



When a beam is not merely supported at each end, 

 but fixed firmly there, its strength is increased by one- 

 half. It would appear at first sight, that by fixing the 

 ends of a beam we should double its strength, for the 

 following reasons : when the beam is merely supported, 

 an extreme load in the middle has only to effect one 

 fracture at A (Fig. 70), the two ends B and C being free 

 rig. 70. 



only to break the beam at the middle point D, but also 

 at E and F ; and to do this it must be double what it 

 would be if the ends wore free, as may be very simply 

 computed thus. Suppose, in the first case, it r&i 

 12 cwt to effect the single fracture at A, then in the 

 second it would require likewise 12 cwt. in the middle to 

 effect the fracture at D, and 6 cwt in the middle to effect 

 each of the fractures at E and F ; making a total of 

 24 cwt. in the case of the beam with fixed ends. 



Many writers have taken this view, neglecting a cir- 

 cumstance which must in such cases occur, and which 

 greatly modifies the distribution of strain on the middle 

 and ends. A single glance at the figure shows that the 

 amount of tension and compression on the fibres at the 

 middle fracture must be double that at either of the end 

 fractures ; and hence each half of the weight required to 

 produce either of the end fractures, or the total weight 

 due to the middle fracture, must be 4 times either of 

 those due to the end fractures. If then we supposed the 

 total breaking-weight divided into 6 equal parts, 4 of 

 those parts would act to break the beam in t lie middle, 

 and 2 to break it at the ends. But the 4 parts required 

 to effect the middle fracture must make up the breaking- 

 weight due to a beam merely supported without being 

 fixed at the ends ; and the other 2 parts that is, half as 

 much more make up the additional weight required 

 when the ends are fixed. Takingthe numerical example 

 as before, if it required 12 cwt. to effect the single frac- 

 ture at A, it would require as much to break the beam at 

 D, and one-fourth, or 3 cwt. , to break it at each of the 

 points E and F, making a total of 18 cwt., which is the 

 sum of 12 and 6, the ordinary breaking-weight increased 

 by its half. 



As this principle of computation accords better with 

 experiment than the former, it is important that its 

 demonstration should be clear, inasmuch as numerous 

 theorists have adhered to the principle, that by fixing the 

 ends of a beam its strength is doubled. According to 

 them, the circumstances correspond with those of a beam 

 (Fig. 71) resting on supports A and B, and projecting 

 each way one half of its length beyond. A load of 1 at 

 each end, balanced by two loads, each 1, in the middle, 

 throws a breaking strain of 2 upon the beam at each 

 prop ; and an additional load of 2 in the middle will 

 measure the breaking strain there, so that the total 

 middle load is 4, or double the ordinary breaking strain. 

 This is no doubt true, because if we suppose the fracture 

 effected, the amount of compression and extension of the 

 fibres at each of the points A, C, and B is the same ; and 



n*>n. 







rUe. But when the ends are fixed, the load has not 



therefore each fracture requires the same load to effect it. 

 But in the case of the beam with ends not balanced but 

 fixed, as we have already explained, the end fractures 

 demand only half the amount of extension and compres- 

 sion due to the middle fracture. 



When a load acts on a beam not perpendicularly or 

 square to its length, but at some other angle, it throws less 

 strain upon it, because the actual leverage of the weight 

 is not the length of the beam as measured from A to B 

 (Fig. 72), but the length of a line measured from A to 

 C perpendicularly to the vertical line in which the weight 

 acts. So when a beam is supported at both ends, but 

 lies obliquely, the transverse strength is to be reckoned 

 as that due to a beam of the length indicated by tliu 



