CHAPTER VIII 
DEFLECTION OF BEAMS - 
(118. Bending to Circular Arc.—It was shown in Arts. 109 and 110, 
. . 103-105, that 21 hast or 
a 
he 
that a beam will bend % a circular form La Se y,/f, is constant, or when 
_1/M is constant throughout the length of the beam. For a beam of 
“uniform cross section y, and I are constant, and therefore f; and M must 
alse be constant for circular bending. If M is variable, then, for circular 
bending, I must be proportional to M. 
The equations R--2 may be used for non-circular bending if 
a 1 
ormapiad short length of the beam be considered ; R will then be 
s radius of curvature at a point in the length, M will be the bending 
n aoment at that point, and /,, y,, and I will refer to the section at the 
e point. 
. “9. Deflection due to Circular Bending.—Let a beam (Fig. 153) 
he ting on supports A and B, whose distance apart 
is J, be bent to a circular arc ACB, The point O 
s the centre of curvature, D is the middle point 
0 AB, and CD is the maximum deflection 1. 
nm the triangle OAD, OA? = AD? + OD*. But 
® =R, AD =}i, and OD =R - u. Therefore 
+R2-2u,R+u;, that is, 2u,R=4}2 +0}. 
u, is a very eal quantity compared with 
d i, the term uw} may be ca a Hence, 
F ‘R- 2 Mi? | 
ey R=20, and m= 55 But R=, SEL" 
M2 
7 . ; r ’ 
_ For a cantilever of length / it is easy to show that uw, = xT 
120. Cantilever Loaded at Free End.—The cantilever (Fig. 154) is 
ipposed to be of uniform cross section pate: Heed ate 
i tl roughout. Consider an_ indefinitely shige: Oe BEY 
all portion HK of the length at a 
ef x from the free end A. Let 
be the angle between the radii drawn 
‘om H and K to the centre of curva- 
ture of ak, and let R be the radius of 
rvature of HK. Let HC and KD be . 
fs to HK at H and K, meeting the gece 
iI through. A at C and D. Then CD is the amount of deflection of 
113 H 
R=—-!=—, These equations show 
Fria, 153. 
therefore %, = 
