CIRCULAR CYLINDERS UNDER CERTAIN PRACTICAL SYSTEMS OF LOAD. IGI 
they do leave a total resultant, the effect of this resultant is practically independent 
of the manner in which it is applied. This is the celebrated “ principle of the 
equivalence of statically equipollent loads,” which was first enunciated liy de Saint- 
Venant on general physical principles, and has been considerably confirmed l>y 
Boussinesq’s researches on the effect of small local surface actions. 
It is to be borne in mind, of course, that the solution obtained in § 5, although 
making zz = 0 over the flat ends, does not at the same time ensure 7'z = 0. In other 
words, we have a determinate system of radial shears over the flat ends, but from 
symmetry this system must be self-equilibratiog. The disturbances due to it will 
therefore, by the above principle, be purely local, and, jirovlded we remove tlie cuds 
sufficiently far from the parts of the lieam wliich we desire to stud}% no trouble need 
arise on account of all the conditions not being strictly satisfied. 
§7. Nwnerical Problem. Expressions for Stra.ins anrl Stresses. 
Let us now return to the exact expressions and apply them to the case of a 
comparatively short cylinder. 
Suppose that rr = 0 all over the curved surface and that in some way, as descrUjed 
in §1, a shear rz, which we shall take uniform and equal to S, is made to act 
along two rings iqion the curved surface, so that 
= 0 when — b e < z <. h ~ e 
We have then 
z < —b — e, z> b e 
{rz),.^„ = S wlien b — e <C z <C b e 
( = — S when — b — c <*:<“-/> + e. 
= 
a., = 
«'n = 0, 
0 , 
pb,, 
88 . '2n + I've’ . 2)1 + 
sm--sm-- 
(2?^ + I)7r 2 c 2c 
The expressions for the constants, stresses, and displacements then reduce to 
A: = 
48 . 2n lire . 2n + IttI) va-I, -t- (1 — 7 ) aU 
-sm sm- - - - -^^ 
{ 2)1 + 1) ir/ji 2c 2c - (1 + Id 
. 48 . 2n + lire . 2n 4- lirh (1 + 7 )aln — (^ + 7 ““)Ii 
\ — —- oiiq _Clio _ - _- i -1 —L 
- (2n+l)7rri' 2c 2c juHf - {1 + 
0 _ 48 . 2n + ]7rc . 2n + XttT) 7 «To — 7 I 1 
k ~ {2n + Dtt/I 2c ¥c^ - (1 + 
VOL, CXCVTII,—A. 
Y 
(47). 
