414
DOCK ENGINEERING.
To find the amount of the respective pressures on the rollers and the
pivot, let P (fig. 398) be the position of the pivot and C that of the rollers.
Fig- 398. Fig. 399.
Then taking moments about P—
w(g + g)2 _w(b - g)2 w1 a2 (b — z)
2 — 2 + 2b ~ ^^ " z)’
whence,
„w((b - z)2 - (a + g)2) wat
and
Rp = AV + C - RK,................(89)
where W = w(a+ b) and C = W1T-
Case III.—An Arch (fig. 399).—In the preceding investigation each
wing of the closed bridge has been treated separately. If it be desired to
take advantage of the support afforded by the mutual abutment of the two
meeting faces of the bridge, it is evident that the tail end must be lifted in
order to develop the full thrust due to the dead weight. Assuming (as
would essentially be the case) that the underside of the cantilevers constitute
a real or virtual arc of rise r and span 2 a, we have by the conditions of
equilibrium for three forces and by similar triangles,
or, approximately,
whence,
T a
W " 27’
T_ _ _^
wa 2r
m w a2
2r......................................................................
which gives us an expression for the amount of mutual thrust. The
upward force, F0 , at C, required to develop this thrust is found by taking
moments about B—
Tr = F,.b,
so that,
b being the distance, BC. This force, the value of which is identical with
that of the counterpoise, is additional to the reaction at C, due to the load
on BC.