210 MK. K. V. siMTHNVKI.L (>X THK CKXF.RAL THEORY OF ELASTIC STABILITY. 



and R. LORENZ.* W. E. LiLLYt has indicated the correct form of the result for 

 an infinitely long flue, and A. E. H. LOVE{ has discussed the strengthening effects of 

 constraints which keep the tube circular at its ends. 



A. B. BASSET has given a very clear exposition of the difficulties which are 

 encountered in an attempt to construct a theory of flue collapse by usual methods. 

 To obtain sufficient equations we must assume that the middle surface undergoes no 

 extension ; and the existence of pressure on one or both surfaces of the tube not only 

 makes this assumption very improbable, but violates an essential condition upon 

 which the theory of thin shells is based. When one surface only is subjected to 

 pressure, there is reason to believe that BKYAN'S solution is substantially correct ; 

 but no treatment can be looked upon as rigorous which neglects the cross-stresses in 

 the material. 



The experimental researches of A. P. CABMAN|| and R T. STEWART^ have revived 

 interest in this problem, since they offer the first information which has been obtained 

 as to the behaviour under practical conditions of tubes which in circularity, uniformity 

 of thickness and homogeneity are fair approximations to the ideal tube of theoretical 

 analysis.** 



We commence our discussion by considering the stability of a thin cylindrical tube, 

 subjected to the combined action of end and surface pressures. We shall thus be 

 able to derive the required solutions for the thin tubular strut, and for a boiler flue 

 without end-thrust, as particular cases, and from the general solution we may obtain 

 indications of the way in which end-thrust tends to promote the collapse of a 

 boiler flue. 



In the most general form of the boiler-flue problem, as enunciated by BASSET,tt 

 pressures are acting on both surfaces of the tube, and we shall therefore investigate 

 conditions for neutral stability in a tube subjected to the following system of 

 stresses : 



(i.) An end-thrust of total amount ,&, uniformly distributed ; 

 (ii.) An external hydrostatic pressure, of intensity $! ; and 

 (iii.) An internal hydrostatic pressure, of intensity $ 2 . 



* ' Physikalische Zeitschrift,' XII. (1911), p. 241. 

 t 'Inat. Civ. Eng. Ireland,' 1910. 

 | ' Proc. Lond. Math. Soc.,' XXIV. (1893), p. 208. 

 ' Phil. Mag.,' XXXIV. (1892), p. 221. 



"Resistance of Tubes to Collapse," 'Bulletin of the Univ. of Illinois,' No. 17, 1906. 

 " Collapsing Pressures of Bessemer Steel Lap- Welded Tubes," ' Trans. American Soc. Mech. Eng.,' 

 1906, p. 730. 



1 The experiment* of FAIRBAIRN were restricted to tubes which were constructed from sheet metal, 

 with brazed and riveted seams, 

 ft Lot. eit., p. 223. 



