Page 425 - Civil Engineering Formulas

HYDRAULICS AND WATERWORKS FORMULAS 351

usually considered as gravity dams, although some parts of the loads may be
carried by arch action. Many early arch dams were built of rubble, ashlar, or
cyclopean masonry. However, practically all arch dams constructed during
recent years have been built of concrete. Arch principles have been used in
dams since about 2000 B.C.
Full Load on Arches. Formulas for analyzing circular arches of constant
thickness, under uniform radial loads, have been developed by various engineers.
The studies made by William Cain were especially noteworthy. Slightly
modified forms of Cain’s equations for thrust and moment at the crown and
abutment sections, due to uniform water loads, are as follows:

pr t 2
Thrust at crown, H 0 pr 2 w sin w (12.186)
D 12r 2
sin w
w
Moment at crown, M 0 (pr H 0 )r 1 (12.187)

Thrust at abutments, H α pr (pr H 0 ) cos w (12.188)
sin w
Moment at abutments, M α r(pr H 0 ) cos w (12.189)
w
In these formulas, r is the radius to the center line of the arch, p the normal
radial pressure at the center line, t the horizontal arch thickness, and ' the angle
between the crown and abutment radii. The center-line pressure p is the extra-
dos pressure times the ratio of the upstream radius to the center-line radius (see
Fig. 12.30).
If shear is neglected, values of D are given by the equation
t 2 2 sin 2 w 2
D 1 w w 2 sin w (12.190)
12r 2
In order to simplify the formulas for crown thrust, D has been used in lieu of the
lengthy right-hand part of Eq. 12.190, which appears in the original formulas.
When shear is included, D is replaced by D , the value of which is given by
s
t 2 2 sin 2 w 2 t 2 2 sin 2 w
D s 1 w w 2 sin w 3.00 w w
12r 2 12r 2
(12.191)
Thrusts and moments having been calculated, intrados and extrados stresses
may be found by the usual formula

H 6M
S (12.192)
t t 2