Mil. K. V. ROITTH\VI-LL ON THE OKNT.K'AL THEORY OF ELASTIC STABILITY. 



In practice, the end constraints will also tend to maintain the cylindrical form at 

 the ends of the flue, and this effect will strengthen the tube, by an amount which is 

 not easy to determine exactly. In any case we may say that 



UJ. 



a q 



and we may illustrate the way in which the end effects die out by plotting the 

 pressure differences ($.-&,) against the quantity q~\ To do this we must take 

 some definite value of the ratio tfa, and plot different curves for the values 2,3, .... 

 Ac., of k. The result is shown by fig. 3, in which the following values have been 

 assumed for the constants : 



E = 3 x 10 7 pounds per sq. inch, 

 = 2-07 x 10" dynes per sq. cm. 



m = J ^, - 



From an inspection of the different curves we see that long tubes will always tend 

 to collapse into the two-lobed form, since the curve for k = 2 then gives the least value 

 for the collapsing pressure, but that at a length corresponding to the point A the 

 three-lobed distortion becomes natural to the tube, and for shorter lengths still, of 

 which the point B gives the upper limit, the four-lobed form requires least pressure 

 for its maintenance. Thus the true curve connecting pressure and length is the 

 discontinuous curve CBAE, shown in the diagram by a thickened line. 



Whatever lie the relation between q and the length of the flue, it is clear that 

 instability is theoretically possible in cases where the distortion involved is not even 

 approximately " inextensional." For if T is sufficiently small, the collapsing pressure, 

 as given by (83), need not involve elastic break-down in the position of equilibrium, 

 even though the first (or " extensional ") term in (83) be equal to, or even greater 

 than, the second. Of course, elastic break-down will occur by reason of the extension 

 very soon after the commencement of the distortion. Nevertheless, failure in such a 

 case must be regarded as due entirely to instability ; for if this source of weakness 

 were removed, effective resistance could be offered for an indefinite period to pressures 

 which actually result in collapse. 



Comparison with Experimental Results. 



Although, as we have just remarked, it is theoretically possible for failure to occur 

 by true elastic instability in comparatively short tubes, yet the relative dimensions of 

 the tubes must be such as it would be quite impossible to test experimentally. In 

 any practical case, instability will not occur until the properties of the material have 

 been altered by overstrain, and the value of the pressure at collapse is therefore very 

 much less than the foregoing theory would suggest. 



