198 MR. K. V. sulTIIUKl.L <>N THK UKXKIJAL THKOKY OF ELASTIC STABILITY. 



.mil in this form they may be conveniently compared with the ordinary equations of 

 elasticity.* 



The three equations of the type (15) we shall term Equations of Neutral Equilibrium. 

 The equilibrium of the stress-system X z , Y y , Z. will be neutral, provided that solutions 

 for u', r', w>' exist which satisfy certain boundary conditions. These boundary 

 conditions are peculiar to each problem, but usually express the condition that the 

 additional stresses involved by u', v', ' shall vanish on certain boundary surfaces. 

 T/iey never determine the magnitude of u', tf, w' , so that our solution gives the farm 

 only of the distortion which tends to occur in the body under consideration when its 

 equilibrium becomes unstable. It gives a definite relation between the stress-system 

 X, ... and the dimensions of the body, which must be satisfied in order that any 

 distortion may be permanent ; but if this relation be satisfied, no limits are imposed 

 by the equations upon the magnitude of the distortion which may occur, t 



Example in Jiectangidar Co-ordinates. Stability of Thin Plating under 



Edge Thrust. 



It seems advisable, before we employ a new method on problems which have not as 

 yet received satisfactory treatment, in some degree to test its validity by the result 

 to which it leads in a more familiar example. For this purpose we may consider the 

 stability of an infinite strip of flat plating under edge thrusts in its plane. The 

 accepted formula}: for the thrust necessary to produce instability, per unit length of 

 edge, is 



where 



2t = thickness of plate, 



/ = breadth of plate, 



and the opposite edges are simply supported. If the edges are built in, the thrust 

 required has four times this value. 



To investigate this problem by the new method we take axes Oa; and Oz in the 

 middle surface of the pkte, in the direction of its breadth and length respectively, and 

 Oy perpendicular to the middle surface. The initial stress-system is then given by 



X, = const. = G (say), ~ 



* LOVK, of. at., 91, equation (19). 



are> hoWeVOr> ri g rous onl y in the case ^ infinitesM displacements; ef. footnote, 



, 40 



. LOVK, op. cti., 337 (a), whence the above expression may be obtained. 



