Mi; I, v sorTHWKLL ON THE GENERAL THEORY OF ELASTIC STABILITY. 



From an inspection of these curves it is easily seen that as the axial wave-length 

 increases the type of distortion which involves the least value for the collapsing 

 thrust (and which the tube therefore tends naturally to assume) changes. For very 

 short lengths we shall expect the circular type (k = 0) ; then, as the length increases, 

 lobed forms of distortion, in which the value of k becomes less as the length increases. 

 The limit is reached when k = 1 ; hence, the tendency of very long tubes is always to 

 collapse in the manner discussed by EULER. 



It is also to be noticed that those parts of the different curves which lie to the 

 right of their lowest points have no practical significance. The actual curve, which 

 shows the effect of length upon the value of the collapsing thrust, will approximate 

 to the form shown in thick lines, since the wave-length (which varies as q~ l ) will 

 naturally not increase beyond that value which involves the least collapsing thrust. 



Comparison with Existing Formula. 



The formulje of equations (90) and (104) may be compared with the results of other 

 discussions of this problem. Equation (90) has been obtained by LORENZ,* and 



LILLY* has given the same result, except that the factor A/ - m - is omitted. t 



The only existing solution for lobed forms of distortion is due to LORENZ,* and this 

 is not in agreement with equation (104). In support of the latter formula it may be 

 urged that LORENZ' formula does not agree with EULER'S result when k = 1. 



It may also be remarked that the foregoing results for the tubular strut problem 

 contradict BRYAN'S theorem, that a closed shell cannot fail by instability, because 

 distortion would involve extension of the middle surface; for although the first 

 terms in equations (102) and (104) are due solely to extension of the middle 

 surface, yet the compressive stress at collapse, as given by (90) or (105), may be 

 insufficient to produce elastic breakdown in the position of equilibrium, if the ratio 



- has a sufficiently low value. 



Stability of Tubes under Combined End and Surface Pressure. 



We shall not treat this case in any detail, but it requires notice in connection with 

 the " localization of collapse " which is observed in experiments conducted upon long 

 tubes tested under hydrostatic pressure, the permanent distortion being generally 

 confined to a portion only of the length of the tube. This result is not predicted by 

 the theoretical formula (83), which suggests a steady fall in the value of the collapsing 

 pWMure as the wave-length increases ; and a partial explanation may possibly be 

 found in the fact that the method of test has generally left a wholly or partially 



* Cf. footnote, p. 209. 



t For a similar omission in a solution of the boiler-flue problem cf. p. 225. 



