48 Extension and Flexure, of Thin Elastic Shells. [Dec. 19, 



The variational equation of motion may be written 



aw+s& = SU+aj (v) 



where 5^ is the term depending on the time variations of the dis- 

 placements, SJJ is the work done by the bodily forces, and £jt is the 

 work done upon the edges of the portion of the shell considered, by 

 the stresses arising from the action of contiguous portions of the 

 shell. 



Applying (v) to a curvilinear rectangle bounded by four lines 

 of curvature, and working out the variation in the usual way, the 

 line integral part will determine the values of the edge stresses 

 T l5 T 2 . . ., in terms of the displacements, and ought also to reproduce 

 the values of the couples which we have already obtained; and the 

 surface integral part will give the three equations of motion in terms 

 of the displacements. These results furnish a test of the correctness 

 of the work, and also of the fundamental hypothesis upon which the 

 theory is based ; for if we substitute the values of the edge stresses in 

 terms of the displacements in the first three of (i), we ought to 

 reproduce the equations of motion which we have obtained by means 

 of the variational equation ; and this is found to be the case. 



The boundary conditions can be obtained by Stokes' theorem, which 

 enables us to prove that it is possible to apply a certain distribution 

 of stress to the edge of a thin shell, without producing any alteration 

 in the potential energy due to strain. 



The general equations, owing to their exceedingly complicated 

 character, do not, except in special cases, readily lend themselves to 

 the solution of mathematical problems ; but, for the purpose of throw- 

 ing some light upon the question raised by Mr. Love, as to the 

 impossibility of satisfying the boundary conditions at a free edge, 

 when a curved shell is vibrating in such a manner that its middle 

 surface experiences no extension nor contraction throughout the 

 motion, I have considered the following statical problem : — 



A heavy cylindrical shell, whose cross section is a semi- circle, is sus- 

 pended by vertical bands attached to its straight edges, so that its axis is 

 horizontal, and is deformed by its own weight ; required the strain pro- 

 duced. 



We shall assume that the displacement at every point of the middle 

 surface lies in a plane perpendicular to the axis, and we shall suppose 

 that the necessary stresses are applied to the circular edges. 



Measuring from the lowest point and putting R for the change of 

 curvature along a circular section, we find that 



R_E^ 3a(-J 7t — sin 0— cos 0) . 



a 2 a h' 2 {^7r— cos0-f-f E (^w— 0sin0— cos0)} 



Since h is small compared with a, this equation shows that the 



