﻿72 Mr. R. V. Southwell on the 



be explained ; but what is needed in practice is a simple and 

 comprehensive formula, invoicing constants which can be 

 determined for any given material from one or two of the 

 ordinary tests, without the employment of elaborate tube-testing 

 apparatus ; and I do not know of any formula hitherto 

 published which satisfies these conditions. 



The theoretical formula for long tubes has been given in 

 equation (8) of this paper. In form it has been found satis- 

 factory as a representation of experimental results for thin 

 tubes : the experimental constant given by Carman for steel 

 tubes is some 25 per cent, less than the theoretical value *, 

 but this reduction may be explained as due in part to un- 

 avoidable inaccuracies in his experimental tubes and in part 

 to his employment of rather too large a range of experiments 

 (his formula is known to give excessive values for the col- 

 lapsing pressures of his thicker tubes) in the determination 

 of the constant. 



In endeavouring to explain the complete failure of the theo- 

 retical formula (8) to give the collapsing pressures of fairly 

 thick tubes, I have emphasized t the important part which 

 elastic breakdown plays in accelerating collapse. It would 

 seem, indeed, that w T e must not expect any long tube to 

 withstand a pressure which is more than sufficient to impair 

 its elastic properties. Thus, if y c is the stress corresponding 

 to the yield-point of a material in compression, we ought to 

 base our design upon the hypothesis that a tube of this 

 material will certainly collapse under a pressure given by J 



P = 2^« (11) 



For tubes of less than a certain limiting thickness the 

 formula (8) gives a smaller value of the collapsing pressure 

 than this, collapse being possible, owing to the occurrence 

 of elastic instability, under a pressure which is not sufficient 

 to impair the elasticity of the tube, so long as it remains 

 circular. What we want, then, is an expression for P which 

 is practically equivalent to (8) in the case of very thin tubes 

 and which in no case exceeds the value given by (11). 



* Cook (loc. cit. p. 52) estimates the reduction as 30 per cent., but I 

 think this figure is somewhat excessive ; my estimate is based on the 

 figures E = 30,500,000, l/m=0-3. 



t Phil. Mag. September 1913. 



% Equation (11) tacitly assumes that the compressive stress is uni- 

 formly distributed over an axial section of the tube-wall. Though in- 

 accurate in the case of very thick or short tubes, this assumption is 

 substantially correct for all tubes of practical dimensions. 



