﻿in a Rectangular Bar by means of Polarized Light. 11 



loading as before, y is measured vertically downwards from the 



point of concentrated loading, r' = \/ x 2 +y' 2 is the distance from 



x 

 this point, <£'=tan -1 - is the inclination of the radius 



vector from this point to the vertical, b has the same meaning 

 as before, and the H's are constants given by the following 

 equations 



Jo \sinh 2 2w-4^ 16a 2 / 



H _ C ( iP + \vr + \u-\ ue-** 3 \ 



1 Jo I sinh 2 2w-4u 2 16u 9 / 



2 "~ ' sinh 2 2w-4a 2 



< 2 "+ 3 + in 2l/+2 + iu 2vi l -W v+l e-* 



-S\du 



II 



2v+l 



j; (- 



sinh 2 2u—4:U 2 



Jdu 



(v>0) 



(. (10) 



Also there is a constant couple on the bar adjusted so that 

 there is no bending moment under the concentrated load, so 

 that the above represent only the disturbing effect of this 

 load in its immediate neighbourhood. 



If in addition we superimpose the stress system due to any 

 constant couple M , we have the following additional term's 

 in the stresses 



P=- 



3M ft y 



2b d 



, Q = 0, S = 0, 



y being measured vertically upwards from the neutral axis. 

 Hence 



P-Q= 2 ^,f'cos2<i>' 



ITT - r 



8W a fry-TT cos 2v<f>' 



9 , 2Wy' . 8W a /r\* sin 2v4>' 



wr 7r/> „ =1 \A / (2v) ! 



Now if we retain only terms of order r' in these expansions 



