240 MB. R. V. SOUTH \\KI.I, ON THK CKNKKAL THKOKY OF ELASTIC STABILITY. 



material will never fail The " curves of distortion," if we could determine their true 

 sli:.|*', would probably be approximately of the form shown in fig. 9. The theoretical 

 methods of this paper enable us to fix the position of A, the " point of bifurcation," 

 but give no information as to the form of AB, beyond the fact that it cuts OA at 

 right angles.* The other curves of the diagram will approach more and more closely 

 the limiting form OAB as the initial value of the amplitude is decreased. 



In the second case, we have the additional complication of elastic break-down 

 under finite stress, which reduces the resistance of the material and causes the new 

 "curves of distortion," shown by thick lines in fig. 10, to begin at certain points to 

 fall away from the corresponding curves of fig. 9 (reproduced in fine lines for com- 

 parison) ; these points will lie on some line such as CD, cutting OA at a point 

 above A, and it is clear that to the right of CD the curves of distortion refer to 

 displacements which do not wholly vanish when the load is removed. Total collapse 

 of the system will obviously occur at the points of maximum load on the curves of 

 distortion, and the locus of these points, which is shown on the diagram by the 

 dot-and-dash line EF, may lie termed the " line of final collapse." 



Amplitude 

 Fig. 9 



Natural 

 Fig. 10. 



Harmonic. 



Fig. 11. 



A knowledge of the true form of EF would enable us, when we are given the initial 

 value of the amplitude, to predict the load at which the system will collapse ; and 

 these quantities could be connected by another curve AG, which would show 

 at once whether the resistance of the system to collapse is seriously reduced by 

 practical inaccuracies of form. A complete theory of any problem in elastic stability 

 must yield information on this very important point, as well as an expression for the 

 " critical load " ; but in most cases more powerful methods would be needed for its 

 derivation than are at present available. The investigation of the " critical load" is 

 therefore not without utility, for although never realized in practice, this forms a limit 

 which should be fairly closely approached when considerable accuracy is possible. 



In our third case the " critical load," as deduced by theoretical methods, is more 

 than sufficient to cause elastic break-down. We may proceed as before to draw 

 hypothetical curves of distortion. The line CT)' (fig. 11), which corresponds to the 



* It must not be assumed that AB is a horizontal straight line; in general, since the distorting 

 effect of the applied stress-system, which varies as the deflection, increases less rapidly than the 

 resistance, which varies as the curvature, AB will tend to rise from A. 



