COLUMNS AND STRUTS 165 
ential equation gives w= A sin ny +B cos ny, where n? = Lm and A and B 
are constants. . 
Differentiating, ie = An cos ny — Bn sin ny. 
¢ 
When 7 =0, a 0, also sin ny =0; therefore A=0, and u=B cos ny. 
But when y=0, w=, and cos ny=1, therefore u, = B. 
When y=/, w=0, therefore B cos m/=0, hence either B=0 or 
cosnJ=0. But B=u,, therefore cos nl =0, hence nl = 3 and / vl RS =", 
2 
wEI 7°EI 
Therefore P ip i 
158. Influence of End Connections on Strength of Ideal Columns. 
—If the ends of the ideal column ACB are fixed as shown at (a), Fig. 
234, then when the column deflects the directions of 
the tangents to the curve ACB at A, C, and B must 
be vertical, and the line of action of the resultant load 
on the column will no longer pass through the centres 7 
of its ends, but must lie between the point C and the | 
straight line AB, and will cut the curve ACB at points 
Hand K. At the points Hand K there is no bending , 
moment, and these must therefore be points of con- | 
trary flexure. i 
Consider the parts HA and HC of the bent column. L 
i 
! 
{ 
L 
At points in HA and HC where the deflections, mea- 
sured from the vertical line through H, are equal, the 
bending moments are equal, and therefore the radii of (a) P 
curvature at these points must be equal, the column (b) 
being of uniform cross section. Also the curves have 
the same slope at H, and also the same slope at A and P 
_ ©. Hence it is evident that the curves HA and HC Fic, 234 
are similar and equal, and that the points H, C, and K 
_ divide the column into four equal parts. Hence the part HCK has a 
length equal to half the length of the whole column. Now the part 
_ HCK is in the condition of a column with rounded or hinged ends carry- 
ing the load P, as shown at (0), Fig. 234. Therefore, 
2 PoP 
from the preceding Article, oe tect - Hence 
Ps 2 3 A FTA 
@ column fixed at the ends is four times as strong as the L 
same column with hinged or rounded ends. , 
The formula for the strength of an ideal column fixed c— 
at one end and loaded at the other is easily deduced 4 YY 
from that for the column with rounded ends. A column , (a) (b) 
ACB with rounded ends is shown at (a), Fig. 235. At 3 
C, the middle point of its length, the tangent to the B 
curve is vertical, and if the column be held in a clamp at 
C, so as to preserve the direction of the tangent to the —Fyq, 235, 
curve at that point, the lower part of the column might 
be removed without affecting the upper part. The upper part will then 
