A CONTINUOUS BEAM.
419
Combining,
RAa = - (a- - a b + b-)+ ~2~>
RA = ^L^a2 + ab - b*),
od
and a similar expression may be written for Rc.
In the preceding investigation, for the sake of simplicity, it has been
assumed that the three points of support are on a level. If this is not so,
and the supports, A and C, are respectively heights of yx and y2 above B
(the heights being small), it is not difficult to establish, in the same manner,
that
MAa + M0& + 2 MB(a + 6) + ^(«3 + ^ = 6EI^ + ^). (105)
And it is also clear that, if the lengths a and b be subjected to different
loads, as w± and w2 per foot run respectively, the equation will then
become
w,aa w2b3 avl (y^ ,
Ma« + M0ô + 2MB(« + ô) + ^ + ^ = 6EI
It would take too long, and it is unnecessary, to elaborate the formulæ
for these cases in detail. The preceding method may be followed, and it
will be found that where a level girder, without overhangs, is subjected to
different intensities of load upon its two sections, the reactions are given by
Ra« (a + 6) = Wj a-(^ + 2) “ W2 g• • ■ 006)
Rc b (a. + b) = w2 b"^3&r + g) -“hg^ ' '007)
RB = ^i a + rø2 6 — (Ra + Ro)-• - (108)
We now corne to the question of counterpoise. No notice has hitherto
been taken of the effect exercised by the ballast at the tail end of the biidge,
because it is much more convenient to consider this question separately from
that of the uniform load of the structure generally, and afterwards to com-
bine the results obtained in the two investigations.
To arrive at the stresses due to a sectional load, we must first considei
those due to a concentrated load. As before, let ABC (fig. 402) be a
Fig. 402.
girder continuous over three points of support, A, B, and C, and let Wj and
\y( be concentrated loads at distances, d^ and d^ from the central suppoit.