BEAM OF EECTANGULAE CEOSS-SECTION UNDEE ANY SYSTEM OF LOAD. 
77 
§ 8. Conditiovs at the Tivo Ends x = E «• 
It IS, howGVGi, iiupossiblG to satisfy fully tliG conditions ovGr tliG two Gnds 
X = E Ihcse would roquirc that P and S should havG givGii valuGS over these 
ends. If, however, a is so large that, at a long distance from the ends, the effect 
of any self-equihhrating system of stress over these same ends may he neglected, 
then we need only consider toted terminal conditions at x = ^ a. 
These conditions will involve 
(i.) The total tension T = P c/y across either end, 
-h 
rb 
(ii.) The total shear S = S c^y across either end. 
(iii.) The bending moment M = 
rb 
Pi/ dy across either end. 
-b 
I now propose to calculate the quantities T, S and M for that part of the 
solution which has been given in the last section. 
I find, after reduction, 
(T),. = (T)_„ = s"K-.„) .(49). 
CO.S ma 
m 
(50). 
(M)^„ = - (M)_„= V 
— l^n) I V ^ / I \ 
COS ma + A —'(/C;, + v,,) cos ma 
VI 
(51). 
Now we can always adjust M and T so as to he zero, for the solutions for a 
uniform tension and a uniform bending moment, viz.:— 
u 
V 
Tx 
2bE 
2bE 
+ 
olSilxy 
We 
3M 
2¥E 
mr 
(52) 
(where y = 2 V(^ + y) K is Young’s Modulus), produce no stress across the 
faces y = i 6, and therefore such solutions can always he arbitrarily superimposed. 
I hey coi'respond to stresses which are transmitted from the ends ; and we shall find 
that it is necessary, in various cases, to add such solutions in order to satisfy the 
end conditions, which are not necessarily satisfied by the series merely involving 
circular functions. 
