A CONTINUOUS BEAM. 415
The pressure on the abutment B is
^3= wa^l+ ^. .. (92)
from which it is apparent that it may be considérable and that carefully
adjusted and solid bearings are essential. It is a matter of some difficulty
to secure these in the case of swing bridges, and accordingly it is not usual
for the central reaction to be much, if at all, relied upon. In bascule
bridges, on the other hand, it is comparatively easy to provide accurate
bearing surfaces.
Case IV.—A continuons beam supported at three points (fig. 400).—Let
Fig. 400.
ABC be a girder continuous over three points of support—A, B, and C all
on the same level. Take the intermediate support, B, as the origin of
co-ordinates, and let y represent the deflection of the beam at the point X
due to a uniform load, w, per unit length. Let S be the shearing stress and
M the bending moment at the same point.
By a well-known formula establishing the connection between the bending
moment ’(M) the modulus of elasticity (E) the moment of inertia (I) and
the radius of curvature (R), we have at any point X—
Now, when the curvature is very small, as is assumed to be the case in
the foregoing relationship, we may find a very close approximation for the
value ot ^
from the principles of the Calculus, viz. :—
1 ^ y
R dxv
where x and y are the co-ordinates of the deflection curve. Hence we may
write—
M = - EI
Again, let us consider the conditions of equilibrium at the point X. If
P be a point indefinitely near to B, where the shearing stress is Sj and the
bending moment M,, it is clear that for equilibrium of the portion P X, we
have—
M = - M, + Sia: + ....................(93)