TORSION OF CKI.TMN |'ol;\|s or SIIAITINC. 





I Hud the roots of tins to l:e approximately given by 



= T475 when a = 2/8, 

 = '821 when a = ir/2. 



The corresponding values of S//XTC are found to ta 2'1084 and T4041 respectively. 

 Comparing these with the values of S A //HTO given on p. 33 1 , we see that the values of the 

 stresses corresponding to the maximum on the broad side are the greater, and hence 

 ur have really a case alwolutely analogous to SAI NT- VKN ANT'S section en double, 

 xpatule, with four fail-points symmetrically distributed, all of them lying on the broad 

 wides of the contour. 



16. Ctbse where a. is made very 



It is interesting to see to what limit the fail-points tend, when a is made very 

 We have 



' = - \ tan 2/8 (cosh 2f + cos 2/8) 

 M TC v j .. 



8)8 (cosh 2 + cos 2)8) S 



=o r/o 1 1\' - iro-i u 2 + iir 



I (2n + l)nr 16)H-J cosh . 



Replace now \ tan 2)8 by its equivalent 8y8 S TT 



=0 \ ^ i 



8/9 



(cosh 2 + cos 2)9) - (cosh 2 + cos 2/9) 



, 2n n 

 cosh - 





. '>n+lira. 



i nsli 



(-'l + 



- 16/9 1 



Now if f, a be great, we have cosh = (exp.) approximately. Hence, if we 

 suppose g = a where 6 is finite, so that we are dealing with points whose 

 distances from the centre l>ear a finite ratio to the dimensions of the section, we find 





 e ' - e 



+ terms negligible in comj>arison and ultimately vanishingly small when a is made 

 infinite. 



Hence we have to make 



., (2n + 1)V- 16/9 

 a maximum. 



Differentiating, the series being absolutely and uniformly convergent, we get 



2X2 



