128 APPLIED MECHANICS 
=2-10,1L3L,=7,0,L3, and z=$1p. 
Hence A,w, 13+ 3,3 + 4L,M,+}(L, + L,)M,+ $L,M,=0, 
or, multiplying both sides by 6, 
J.Le + }w,L3 + L.M, + 2(L, + L,)M, + L;M,=90. 
For a continuous beam on 2 supports the theorem of three moments 
furnishes m — 2 equations, and the conditions of support at the two ends 
furnish another two equations. These » equations are sufficient for 
determining the bending moments over the m supports. Most commonly 
the beam is free over the end supports, and the bending moments there 
are then zero. 
134. Reactions at the Supports of a Continuous Beam. eee 
the reaction R, (Fig. 170) at the 
intermediate support of two ff ak {fe ro ff 
consecutive spans BC=L, and 
CD=L,. Let M,, M,, and M, L aes 
be the bending moments over ie > i 
the supports B, C, and D re- Fia. 170. 
spectively. Let F, and Fy be 
the shearing forces on the beam immediately to the left and right 
respectively of the support at C. Then R,=F,+F\. 
Consider the span BC, and take a about B, 
M,=F,.L,+M.-—W.Z,. Therefore F, =_ —M,.+W,Z,), where W, 
is the sum of the loads on the span BC, and Z, is the horizontal distance 
of their centre of gravity from B. 
Consider the span CD, and take moments about D. 
M,=F.L,+M,—W,Z,. Therefore F,= (My —M,+W,Z,), where Wg 
. 3 
is the sum of the loads on the span CD, and Z, is the horizontal distance 
of their centre of gravity from D. 
_M,- Mo, Mp— Mes, Wels , Wado, 
L, L, L, oye 
For uniform loading of w, per unit run on BC and w, per unit run 
o, Wyliy WL : 
i De Ses 
Hence R,=F,+Fo= 
on CD. R= 
If the beam is free over the end supports, then the reaction at either 
end is equal to the shearing force at that end. 
135. Example of Continuous Beam.—A bridge ABCD (Fig. 171) 
-consists of two continuous girders having a central span BC of 200 feet, 
and two side spans AB and CD each of 160 feet. There is a uniform 
dead load of $ ton per foot run on the whole of each girder, and on each 
girder of the span AB there is an additional load equivalent to ? ton per 
foot run. The four piers are at the same level, and the ends of the girders 
