MS MH. R V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY. 



Stability of Short Struts. 



This problem has been discussed elsewhere by the author,* and it will be noticed 

 here only at sufficient length to indicate the directions in which further research is 

 needed. We have to derive an expression for the collapsing load of a straight strut, 

 when this is more than sufficient to cause elastic break-down of the material ; and we 

 proceed as before by considering three configurations of the strut : (l) before strain ; 

 (2) in a position of neutral equilibrium under uniform end-thrust; and (3) in a 

 position of infinitesimal distortion from the second configuration. 



For a first approximation we may say that cross-sections remain plane in the third 

 configuration, so that the diagram of longitudinal compressive strain for any cross- 

 section is as shown in the upper part of fig. 12 ; the horizontal line fg shows the 



f Diagram F of G Strra K 



Strain. 



Fig. 12. Fig. 13. 



uniform strain of the second configuration. Then, if fig. 13 be the stress-strain 

 diagram for a compression test of our material, and this uniform strain corresponds to 

 a stress p which is represented by the point B, we see that to the right of the point 

 F in fig. 12 the longitudinal compressive stress in the third configuration must be 

 greater, and to the left less than p. 



Now it is a well-known property of metals that if at any point B on the stress- 

 strain diagram, beyond the elastic limit, we begin to decrease the load, the diagram 

 is not retraced, but that we obtain a line BC which is parallel to OA.t It follows 

 that the ratio decrease of stress 



decrease of strain 



is still given by E, YOUNG'S Modulus for the material. On the other hand, the 

 diagram shows that if we increase the load beyond B by an infinitesimal amount, the 



r:itl" P 



increase of stress 

 increase of strain 



is a smaller quantity E', which may be found from the slope of the diagram at B. 



* ' Engineering,' August 23, 1912. 



t A. MORLEY, 'Strength of Materials,' 42. 



