DEFLECTION OF BEAMS 121 
- Beam of Uniform Section, Fixed at the Ends, and Loaded 
mly.—The reasoning in this case, which is illustrated by Fig. 161, 
ilar to that in the preceding 
le, and corresponding points p 
Figs 160 and 161 have the same 
attached to them. 
“let w denote the load on the 
per unit of length. The 
i ending moment diagram for the 
thole beam considered as  sup- 
rted at the ends is a parabola 
cb, the height of the middle 
ordinate being equal to 4wL*, the 
an ding moment at the centre. 
As in the preceding Article, 
' 
Soe Fated 
the area of the re cec’, and UY uf 
therefore the Lage the’ rect- : 
angle ade’a’ is equal to the Fig. 161. 
area of the semi-parabola aecd. 
But by th the well-known property of the e parabola, area aecd = 2ad a 
hence aa’ - ad = 2ad - cd, therefore aa’ = cd, and cc’ = Jed ; a 
SE OO oT 
AIA __ ! | : 
if 
4wL?, therefore cc’ = J,wL2_- Db cee 
onthe now the-middle portion EF as a beam supported at the = 
is and loaded uniformly. cc’ = ,wL? =4wL3, therefore L; =1L2, and 
L, L,=<1L J/3=0°577L. Also L, = }(L-L,)=3L(3— /3)=0°211L. 
The cantilever A,E, of length L,=}L(3— ./3) carries a load 
ath jul ,/3 at Ey and a uniform load of # per unit of length. 
tion at E, due to the first load is, by Art. 120, 
_twLh 3, _ wl! (9 ,/3 — 15) 
3EI 648EL 
e deflection at E, due to the second load is, by Art. 121, 
_wLy _wLt (7 — 4 ,/3) 
i = SEI ~ ~. 288EI * 
‘ The total deflection of the cantilever A,E, at E, is therefore 
a wLA(9 /3-15) , wh(7-4,J3)__ wh 
648EI 288EI 864EI” 
E "tory beam E,F, of length L,=3L ,/3 carries a uniform load of w per 
uni it of length, hence by Art. 123 the deflection of C, below E, 
: * Bol} 5wLt 
q ~ 3841 3456KI" 
_ The total deflection of the whole beam at the centre is therefore 
oe wht | Swlt _ wht 
864EI 3456EI 384EI° 
A ~ 
ae 
’ 
