DEFLECTION OF BEAMS 123 
_ The bending moment diagram for the beam AB (Fig. 163) may now be 
nstructed. AtC the bending moment is}PL=,5,WL. At B the bend- 
: moment is °,WL-}WL= —4%,WL. Let E be the point where the 
be ding moment is zero, and let “AE=a, then yy Wa - W(# - $L)=0, 
the terefore x= ;;L. The shaded figure on the base ab is the nding 
“mo 2€ nt diagram. As there is no bending moment at E the beam may be 
— pposed to be hinged at that point, and the whole beam is equivalent to 
ie B,E, fixed at B,, and a beam E,A, supported at its ends. 
the supporting force at A, is ,5,W, it follows that the supporting 
: f orce at E, is +} W, and this latter force will also be equal to the load on 
ie ‘cantilever Ur at E). 
__ The deflection of the cantilever B,E, at E, = 
om ‘y 
teWG'yLY_ 9WLs 
~ SEI‘ 1936EI’ 
____ By Art. 127 the deflection of C, below A,E,= ~~" _. In this 
; 3(a + b)EI * 
‘ease = ysl, and b= $1, ‘therefore the deflection of ©, below A,E, 
wi 9WL! 
4 ~ 2241 But E, deflects Fos gET 
11 9SWL? _ 9WLS heh a 
- distance equal to — 16 * 1996 5816HT" , the multiplier +} being the ratio 
of AC to AE. 
The total deflection of C, below the horizontal through B, is therefore 
, and this will lower the point C, a 
= OWLS , 25WLS _7WL 
~OB16EI * 4294EL ~ 768ET 
it the bending moment diagram on the base ad be considered as a load 
W inertae, the part below ab representing a load acting upwards, the reaction 
_ at the right-hand support will be found to be equal to ;,WL?, and the 
_ point F, where the shearing force is ee is easily shown to be at a distance 
from the right-hand support equal to 5h The complete shearing force 
3 di gram ST is shown, but this need not be drawn. 
; Still considering the bending, moment diagram on the base ab as 
a load diagram, the bending moment due to this load at a point in AC 
at a distance x from A is equal to as (« ee “> and the deflection at 
~ Wi? 528 5 
‘ ‘this point is therefore equal to —— 39EI (« - an Putting «= 5h the 
deflection at F', where the deflection is greatest, is equal to 
WLS _ WL 
48 JoEI ~ 107m °°" 
_ Putting «=4L in the same expression, the deflection at C is found to 
a 
Tose * result which has already been found in another way. 
a 
- 3 ‘ 
