MI!. R. V. sorrmVELL ON THE GENERAL TIIKOKY OF KLASTIC STABILITY. 



sensibly the same for all lengths. The main interest both of CABMAN and of 

 - i \\ \KT* was confined to tubes in excess of this limit, experiments on which may 

 fairly lx- compared with the theoretical formula (80); their results showed that this 

 formula gives a satisfactory estimate of the strength of very thin brass and steel 

 tulx-s. but must not be taken as a basis for design throughout the whole range of 

 dimensions employed in practice. 



The experiments of FAIRBAIRN,! on the other hand, were restricted to tubes of 

 such relatively small length that he failed to realize the existence of a definite 

 minimum below which the strength of a tube, however long, will not fall. He also 

 neglected the possibility of discontinuities in the curve of collapsing pressure at points 

 where there is a change in the form of the distorted cross-section. In the light of 

 these facts, figs. 3 and 4 help to explain his well-known formula, by which the 

 collapsing pressure is given as inversely proportional to the length of the flue ; for a 

 curve of hyperbolic form will represent as well as any other single curve the scattered 

 points of fig. 4, and trial shows that the hyperbola 



(86) 



is very closely an envelope of the discontinuous curve CBAE in fig. 3, in each case 

 doivn to the point of least collapsing pressure. 



Validity of Investigation ly the Theory of Thin Shells. 



One important result of our investigation, which is apparently new, is shown by 

 equation (83). It may be seen that collapse is practically dependent upon the pressure- 

 difference alone, and that the absolute values of the pressures are immaterial. In 

 view of this result, the objections raised by BASSET against BRYAN'S treatment of the 

 problem { require further consideration. 



These objections are : first, that the ordinary expressions for the stress-couples in a 

 plate or shell, in terms of the curvature of its middle surface, are not valid when the 

 surfaces are subject to pressure ; and secondly, that it is not legitimate to assume, as 

 we must if sufficient equations are to be obtained, that the middle surface is 

 unextended in a configuration of slight distortion. Hence the theory of thin shells 

 is not applicable to this problem. 



The above difficulties may be almost entirely overcome by a change in the method 

 of investigation which is employed. It is customary to derive equations for the 

 equilibrium of the distorted shell directly, and without reference to the position of 

 equilibrium. Such procedure renders it necessary to make BRYAN'S assumptions, that 

 the middle surface is unextended, and that the usual expressions for the stress-couples 



* Of. footnote, p. 210. 

 t Cf. footnote, p. 209. 

 t See footnote, p. 210. 



