DEFLECTION OF BEAMS 125 
| 132. Beam of Uniform Section, Resting on Three Equidistant 
‘Supports and Uniformly Loaded.—AB (Fig. 166) is a beam resting 
on three supports, one at each end and the other at the centre, and 
carrying a uniform load of w per unit length. Let each of the equal 
spans be denoted by L, and let the reactions at the ends be Q, and the 
reaction at the centre P. 
_ First consider the case where the supports are at the same level. 
_ Suppose the middle support to be removed, as shown in the upper half of 
Fig. 165. The deflection at the centre will then be de 
— 384EI 24EI 
_ Now suppose that the load is removed and that a force P acts upwards at 
the centre, forces acting downwards being applied at the ends, as shown 
~ TAAL 
: i! 
AMANITA 
LIA c B 
" 
" 
" 
4 
‘ 
— 
Ip 
Fie. 165. Fia. 166. 
in the lower half of Fig. 165. The upward deflection at the centre will 
P(2L)*_ PL ane e 
be “ISEL * GE If the conditions in the upper and lower halves of 
Fig. 165 be applied simultaneously, and there is no resulting deflection 
at the centre C, then Eee we 
; , 6EI 24ET’ 
Q=%wL. The bending moment diagram may now be constructed, and 
_ will be as shown on the base ab in Fig. 166. The beam AB (Fig. 166) 
_ may evidently be considered as two cantilevers AC and BC fixed at C, 
loaded uniformly ‘and supported at their free ends. 
_ Next, suppose that the middle support is 1—nth of d below the level 
4 
of the other supports, where d =e , the downward deflection at the 
centre when the middle support is removed. The upward deflection 
4 : 5wLt/, 1\_ PL’ far ( -1) 
_ due to P is now 4EI 1 ea therefore P= wL{ 1 a) and 
Q= gu (3+2). 
Lastly, suppose that the middle support is l-nth of d above the 
at of ee oe The upward deflection due to P is now 
w L 1 5 
- -OtKI 1+ ‘) = SRI’ therefore P = fu (1 + -) and Q= jul(3 - °), 
The case discussed in this Article is the simplest case of a continuous 
_ beam ; the general case, where there are any number of supports and any 
_ combination of loads, is considered in the next Article. 
therefore P=4wL, and consequently 
