Closed and Open 'lubes. 541 



finding Q x by our approximate theory. Then, to our order 

 of approximation, 



Q=^r ™ 



It is usual to make box sections by rivetting plates 

 together. Such rivetted girders may be treated as if they 

 were solid for the purpose of torsion provided the rivets are 

 near enough and strong enough. There may be, of course, 

 considerable shear on the rivets. 



The formulae obtained in this paper are suitable only for 

 long thin sections in which the shear lines are nearly parallel 

 curves over practically the whole of their length. A rod 

 whose section is a complete circle could not reasonably come 

 under either the rule for a closed tube or the rule for a long- 

 thin unclosed section. Nevertheless we shall, just for the 

 sake of noting to what extent the formulae fail, treat the 

 complete circle 



(1) as a closed tube with no central hole, 



(2) as a long thin section whose central line is a 



diameter. 



In the first case the central line is a circle of half the 

 radius of the section. If r denotes the radius of the section 

 then equation (35) applied to this section gives 



l. v > = =±7rnTi A . . . . (by) 



ITT 



Again treating the section by the second method, equation 

 57) gives 



Q=4wT(i7rr 4 ) = ™Tr 4 . .... (70) 



The correct result for this section is 



Q=i7m7T 4 (71) 



• 



Of the two results in (69) and (70) one is half the correct 

 result and the other is twice the correct result. It is not 

 intended that the formulae shall be used for such extreme 

 cases, and they are given here only to indicate that fair 

 accuracy can be expected from the formulae for all reasonable 

 oases. 



