[ 687 ] 



LXX. On the Collapse of lubes by External Pressure. 

 By R. V. Southwell, B.A* 



IN the issue of tills magazine for September 18921" 

 Mr. Basset has given a very clear exposition of the 

 difficulties which we encounter in attempting to construct a 

 theory of flue collapse. The conclusions to which his dis- 

 cussion leads are, that the formula obtained by Prof. Bryan ± 

 for the collapsing pressure of an infinitely long flue is 

 probably correct as a first approximation, when the inner 

 surface of the tube is free from applied pressures ; but that 

 his work, as distinguished from his result, is vitiated by the 

 employment of a faulty expression for the strain energy of 

 bending, and that although the formula may be obtained by 

 other methods, these also must be based upon assumptions 

 which are certainly not exact, and may be so far from the 

 truth as to render the result faulty, even as a first approx- 

 imation. 



1 have recently investigated the problem by a method 

 which I believe to be exact, and which does not involve the 

 Theory of Thin Shells §. My results show that Professor 

 Bryan's formula is correct as a first approximation, even 

 when both surfaces of the tube are subjected to pressure, 

 the difference of the pressures being the sole criterion of 

 collapse. In the light of this conclusion, Mr. Basset's 

 remarks seem to require further examination ; for the 

 Theory of Thin Shells is apparently more capable than 

 they would suggest of providing a satisfactory formula for 

 flue collapse. I shall attempt in the following pages to 

 indicate the alterations in method which, as I believe, enable 

 us to evade the difficulties remarked by Mr. Basset, and to 

 show how Professor Bryan's formula, thus established, may 

 be extended so as to suggest estimates for the influence of 

 " collapse rings " upon the strength of flues. 



For convenience of comparison I shall retain the notation 

 of Mr. Basset's article in dealing with his discussion of the 

 boiler-flue problem, which, in its simplest form, he states as 

 follows : — " Let the flue be regarded as indefinitely long, 

 and be cylindrical ; let a + Ji and a — h be the external and 

 internal radii of its surfaces ; and let the outer and inner 



* Communicated by Professor B. Hopldnson, F.R.S. 

 f Phil. Mag. vol. xxxiv. p. 221. 

 X Proc. Camb. Phil. Soe. vi. p. 287 (1888). 



§ In a paper read before the Royal Society of London ; January 30th, 

 1913. 



