164 APPLIED MECHANICS 
strut to bending will be proportional to /, and therefore propor- 
tional to the deflection. But when the elastic limit is passed, the 
- moment of resistance will increase more rapidly (see Art. 116, p. 109); 
and with the additional load W there may be another position of 
equilibrium for the strut, but the strut will then have taken a per- 
manent set. 
156. Approximate Theory of Long Columns.—ACB (Fig. 232) 
represents a long slender column which, when unloaded, is perfectly 
straight, of uniform cross section, and uniform elasticity. 
The ends are supposed to be rounded so that the column 
is free to bend throughout its whole length. The critical A\ 
load P is supposed to act in a fixed line which coincides 
with the axis of the column when the latter is unloaded. u 
Let the loaded column be slightly bent by the applica- 
tion for an instant of a lateral force. At any point C uj; 
there is a bending moment equal to Pu, where w is the de- 
flection at C, and it follows that the figure ACBA is the 
bending moment diagram for the whole column. ‘ It is B 
evident that the curve ACB cannot be an are of a circle, 
because that would necessitate the bending moment being yyq 939 
uniform throughout the whole length of the column. If 
the curve ACB be assumed to be a parabola, then the deflection 
of the column is the same as it would be if the column became 
a beam, supported at its ends, with a transverse load uniformly dis- 
tributed over its length. In Art. 123, p, 114, it was shown that for a 
beam of length L and uniform section, supported at its ends and loaded 
5WL 
384EL" 
If M is the bending moment at the centre of the beam, then 
“F 
1 
' 
k 
k---- - -- 2 
uniformly with a total load W, the deflection w, at the centre is 
WL, _ 5ML? wf 
M= = hence w, = FREI? but for the column M=Pw,, therefore 
= _ 5Pu,L? and p= 48EI _ 9°6EI 
1 48EI ’ 5L? L? © 
By the more exact theory of Euler, discussed in the 
next Article, Pare 
157. Euler’s Theory of Long Columns.—The column 
ACB (Fig. 233) is supposed to be under exactly the same 
conditions as the column considered in the preceding 
Article. At any point C in the column, at a distance y 
from the middle point of AB, the bending moment M is 
equal to Pu. If R is the radius of curvature of the 
column at C, then by Arts. 109 and 110, pp. 103-105, 
2 
——— But ” oa (see Art. 9, p. 9), the minus sign 
being used to maké.R positive, because as % increases 
a 2 
“ decreases. Hence PY = — The general solution of this differ- 
dy El dy 
Fig, 233. 
