MR. R. V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY. 241 



\\iw CD of fig. 10, will intersect OA at a point below A, and the other curves of 

 distortion at correspondingly lower points. We have seen that the effect of local 

 elastic breakdown upon fig. 10 was to deflect the curves of distortion from the forms 

 which they would have assumed if the material had possessed indefinite strength ; 

 and it is clear that this deflection will begin at lower values of the loads in the 

 present case. We may therefore expect curves of the type shown in thick lines in 

 fig. 11, where the curves already obtained are reproduced in fine lines for comparison. 

 As before, we may draw a line A'F "of final collapse" through the points of 

 maximum load on the curves of distortion, and connect the collapsing load with the 

 initial value of the amplitude by another curve A'G'. 



It is clear that the curves of distortion must tend to a limit which is no longer 

 OAH, but some other curve OA'H', where OA', the critical load under the new 

 conditions, is more than sufficient to produce elastic breakdown, but less than OA. 

 We can see further that the curve A'G' is not likely to fall away from A' much more 

 steeply than AG from A in fig. 10. The great weakness of short struts in practice, 

 compared with EULER'S theoretical estimate, is now explained. Whereas long struts 

 come within the conditions of fig. 10, the failure of short struts will be repre- 

 sented by fig. 11, and occurs at comparatively low stresses, not because practical 

 imperfections have a greater effect upon the strength, but because OA', the true 

 value of the critical load, is less than OA, the value which EULER'S theory would 

 dictate.* 



It is the rule, rather than the exception, that the critical load, as found by the 

 ordinary theory of elastic stability, is more than sufficient in practice to produce 

 elastic break-down. This may be readily seen in reference to any particular example. 

 In the case of the tubular strut, fig. 10 is only applicable when the ratio of diameter to 

 thickness is greater than 560 (for an average quality of mild steel), and for thicker 

 tubes the critical load falls, apparently by a very considerable amount, t below the 

 theoretical estimate. The determination of the critical load, in cases where this is 

 more than sufficient to produce elastic break-down, is thus a problem of great 

 importance, since it forms a limit which can never, under any circumstances, be 

 exceeded. In the ordinary strut problem the determination can be effected without 

 difficulty, and an apparently new field is thus indicated for research. The distin- 

 guishing feature of its problems is the dependence of the stress-strain relations upon 

 the past history of the material, rendering absolutely necessary a method which 

 follows the actual cycle of events up to the occurrence of collapse. 



[* Added May 11. Since this paper was written, the author's attention has been drawn to a 

 dissertation by T. VON K A KM AN (' Untersuchungen uber Knickfestigkeit,' Berlin, 1909), in which the 

 forms of these " curves of distortion," for solid struts of practical dimensions, are deduced both from theory 

 and from experiments. KARMA'N also gives a relation equivalent to that of equation (112).] 



t Experiments conducted by the author upon seamless steel tubes showed failure under loads which 

 were in every case little more than sufficient to produce " permanent set." 



VOL. CCXIII. A. 2 I 



