122 APPLIED MBCHANICS 
130. Beam of Uniform Scction, Fixed at one End, Supported at 
the Other, and Loaded at the Centre.—AB (Fig. 163) is a beam fixed 
at B, resting on a support at A, and carrying a load W at the centre C. 
The first step is to determine the reaction P of the support on the 
beam at A. Suppose the support at A removed, as shown in the upper 
4 \w 1 
oy 
i 
ion ‘ 
1 i 1 
Dre th - 
re ~ PL — nl 
fC ee 
aE 
ll 
ir 
------- +e 
\ 
Gen 6 oe ee 
Fig. 162. Fig. 163. 
part of Fig. 162, then the load W will produce a deflection in AB at 
W(3L)?_ WL? 
C=4,= T° 94T' The bending moment diagram due to W on 
- 8EL 
BC will be a triangle BCD, and the tangent to the bent cantilever B,C, 
at C, will meet the horizontal through B, at O, which is vertically under 
G, the centre of gravity of the triangle BCD. Hence oc =3 . oa 
If a is the inclination of OC, to the horizontal, then tan a=u, +2 <9) 
The portion C,A, of the deflected cantilever will remain straight, but will 
be inclined to the horizontal at an angle a. Hence the deflection A, A’ 
3 
of the cantilever at Ay =d=u,+4L tan o=$u,= aes 
Next suppose that the load W is removed, and. the reaction P at the 
support to act as shown in the lower part of Fig. 162. An upward deflec- 
PLS 
3EL 
Now if P and W act together, the deflection at A due to W will be 
neutralised by the deflection due to P. Hence d=u,, that is, 
tion will be produced at the free end of the cantilever =u, = 
BWL? PLS 
Mad; 
ASE > SEY therefore P=3,W. 
