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VI. On the Extension and Flexure of Cylindrical and Spherical Thin Elastic Shells. 
By A. B. Basset, 31.A., F.E.S. 
Received December 9,—Read December 19, 1889. 
1. The various theories of thin elastic shells which have hitherto been projjosed have 
been discussed by Mr. Love* in a recent memoir, and it appears that most, if not all of 
them, depend upon the assumption that the three stresses which are usually denoted 
by B, S, T are zero : but, as I have'recently pointed out,t a very cursory examination 
of the subject is sufficient to show that this assumption cannot be rigorously true. It 
can, however, be proved that, when the external surfaces of a plane plate are not sub¬ 
jected to pressure or tangential stress, these stresses depend upon quantities propor¬ 
tional to the square of the thickness, and whenever this is the case they may be treated 
as zero in calculating the expression for the potential energy due to strain, because 
they give rise to terms proportional to the fifth power of the thickness, which may be 
neglected, since it is usually unnecessary to retain powers of the thickness higher 
than the cube. It will also, in the present paper, be shown by an indirect method 
that a similar proposition is true in the case of cylindrical and spherical shells, and, 
therefore, the fundamental hypothesis upon wdiicli Mr. Love has based his theory, 
although unsatisfactory as an assumption, leads to correct results. A general expression 
for the potential energy due to strain in curvilinear coordinates has also been obtained 
by Mr. Love, and the equations of motion and the boundary conditions have been 
deduced therefrom by means of the Principle of Virtual Work, and if this expression 
and the equations to which it leads were correct, it would be unnecessary to propose 
a fresh theory of thin shells ; but although those portions of Mr. Love’s results which 
depend upon the thickness of the shell are undoubtedly correct, yet, for reasons which 
will be more fully stated hereafter, I am of opinion that the terms which depend upon 
the cube of thickness are not strictly accurate, inasmuch as he has omitted to take 
into account several terms of this order, both in the expression for the potential 
energy and elsewhere. His preliminary analysis is also of an exceedingly complicated 
character. 
2. Throughout the present paper the notation of Thomson and Tait’s “ Natural 
Philosophy” will be employed for stresses and elastic constants, but, for the purpose 
* ‘ Phil. Trans.,’ A, 1888, p. 491. 
t ‘ London Math. Soc. Proc.,’ vol. 21, p. 33. 
.3 K 
MDCCCXC.—A. 
31.7.90 
