CHAP. HI.] SUPPLEMENT TO CHAP. VH. 115 



materials, known dimensions and given weights, we may find experimen- 

 tally T. It would seem that this value thus found should equal either the 

 tenacity or crushing strength of the material, but the results of experiment 

 show that it never equals either, but is always intermediate between T and 

 O. Calling this intermediate value R, we have 



*-'-- ........ 



The formula is based upon the condition of perfect elasticity, while R is 

 determined by experiments made at the breaking point when the condition 

 of perfect elasticity is no longer fulfilled. In the following table the tabu- 

 lated values of R are correct for solid rectangular beams, and sufficiently 

 exact for those which do not depart largely from that form. If instead of 



we use the values of T .or O, whichever is the smaller, we shall always 

 be on the safe side, since R is invariably intermediate between these. 



In general we shall refer to the equation 



when we have occasion to find the breaking strength. But it must be 

 always remembered that in any practical example we should replace T by 

 R for rectangular beams, or by T or O, whichever is the smaller, for others. 

 "We give also the values of the coefficient of elasticity E. (Wood's Resist. 

 of Materials.) 



TOR E 



Cast-iron ....................... 16,000 96,000 3fr,000 17,000,000 



Wrought-iron .................. 58,200 30,000 33000 25,000,000 



English Oak .................... 17,000 9.500 10,000 1,451,200 



Ash ............................ 17,000 9,000 10,000 1,645,000 



Pine ........................... 7,800 5,400 9,000 1,700,000 



All in pounds per square inch. 



2. Beam of uniform strength. 



Suppose the cross-section or I is not constant, but varies so that at every 

 point the strain T is constant. From (11) we have 



2 T I 



M = P x = - for the outer fibre, whence 

 fi 



T = --- . For a rectangular cross-section T = ". Now suppose the 



21 On 



breadth and height at the fixed, end are Ji and hi. Then at this end T = 

 6FZ 



. But this must be equal to T at any other point ; hence 



6 Pa: 6PZ 5 h* x 

 0i> _____ 



If we suppose the height constant, we have for the varying breadth at any 

 point ft = &i . That is, the breadth must vary as the ordinates to a straight 



I 



line, and the plan of the beam is a triangle with the weight P at the apex. 



