THE CBITICAL EXTERNAL PRESSURE OT CYLINDRICAL TUBES. 



By R. voa Mlses. 



Zeitschrlft des Verelnes 

 Deutscher Ingenleure, 

 (Vol.58, No. 19, 

 May, 1914, P.750) 



Translated and annotated 



F. Wlndenburg.'^' 



by D. 



A circular, cylindrical boiler flue in the pressure chamber of a steam 

 boiler is loaded similarly to a vertically, centrally loaded column: as long as 

 the pressure remains under a certain limit we have stable equilibrium and a uni- 

 form contraction of the material on all sides. For greater pressures, we have a 

 buckling or bulging of the tube. (See C. Bach: "Elastizltat und Festigkeit", 

 1905 p.275). 



Observations of the critical pressure, corresponding to the great prac- 

 tical significance of the question, have already been abundantly made. The corre- 

 sponding theory, quite similar to the Euler buckling theory, was developed first 



for the tube of infinite length only. Here, the 

 instability pressure is given by 



h< 



£ € \: 



2E 



©■ 



(a) 



-^'TTT-l 



Fig. 1. Cylindrical 



tube under external 



pressure p. 



' 1 -0-' 



where "a" designates the inner radius, 2h the 

 shell thickness of the tube (Fig. 1), E the 

 modulus of elasticity and (T Poisson's ratio. 

 This equation was first developed by Bresso in 

 1829. (See Love-Tlmpe "Lehrbuch der Elastizl- 

 tat"., Leipzig 1907, p. 637). The complete ex- 

 pression for tubes of finite lengtih was treated by R. Lorenz ("Physikal. Zelt- 

 sohrift", 1911, p. 257) who in order to avoid the difficulties of computation in- 

 troduced a series of neglected terms. R.V. Southwell (Phil. Mag., Vol.25, I913, 

 p. 687) has also recently given an approximate solution. 



The following development is from the rigorous theory of thin elastic 

 shells andderives the accurate expression for the critical pressure. For practi- 

 cal computations, the results can be easily simplified according to the circum- 

 ^ Assistant Physicist, U.S. Experimental Model Basin, Washington, D.C. 



