Collapse of Tubes by External Pressure. 



693 



after distortion, and Gauss' theorem* shows that extension 

 of the middle-surface is involved. Hence it may be seen 

 that terms will be introduced into the expression for 

 (II! — II 2 ) which involve the first power of the thickness ; 

 and these have now to be determined. 



It will be convenient to change our notation for the 

 various stress quantities, and to employ the system which is 

 sufficiently explained by fig. 2. 



Fisr. 2. 



$-x> 



We begin with the consideration of an indefinitely long- 

 tube, and take x for the length measured along a generator 

 from a fixed normal section, and a(f> for the length measured 

 along the section from a fixed generator. When the dis- 

 tortion occurs, let 2* be the displacement along the generator, 

 v the displacement along the tangent to the circular section, 

 and iv the displacement along the normal to the cylinder 

 drawn outwards. 



The equations of equilibrium may be obtained at once 

 from the figure. They are 



d<r a "dcj> a ~d% L 2 \d<j> 



~dx a d</> p 



a ~&x 

 = 



[ p *SH] 



= 



~dx a d</> 



BH j 1 3G 2 



7$x a o<p 

 -dx 



p 2 



p 



+^(1-^-11,(1+ -)=o ; y (17) 



13H 

 a ~d<f> 



+ T 2 =0; 



-T 1 = 0. 



) 



* Werke, Bd. iv. p. 217. Cf, Salmon, ' Geometry of Three Dimensions,' 

 4th ed. p. 355. 



Phil Mag. S. 6. Vol. 25. No. 149, May 1913. 3 B 



