98 4 Three-Hinged Arches
where span for archACB^0 with support points on the same level isLD
lCl 0 D 48 m. ForxD 42 m (supportB) the ordinateyDD3:5m, so
tan ̨D
l
D
3:5
42
D0:0833!cos ̨D0:9965!sin ̨D0:08304:
8m
C
B
A
P=10kN
ZA
ZB
R′A
ac=24m bc=18m R′B
l=42m l^0 =6m
x 12m
k=6m
D=3.5m
k f h
a
f 0 • B′
L=48m
x
y
yk= 3.5m
Fig. 4.14 Design diagram of an askew three-hinged arch
Other geometrical parameters are
f 0 DaCtan ̨D 24 tan ̨D2:0m!fD 8 2 D 6 m!
hDfcos ̨D 6 0:9965D5:979m:
(a)
ForxD 6 m (sectionk), the ordinateykD3:5m.
4.5.1.1 Reactions and Bending Moment at Sectionk
Reactions of supportsIt is convenient to resolve total reaction at pointAinto two
components. One of them,R^0 A, has vertical direction and other,ZA, is directed along
lineAB. Similar resolve the reaction at the supportB. These components areR^0 B
andZB. The vertical forcesR^0 AandR^0 Bpresent apartof the total vertical reactions.
These vertical forces may be computed as for reference beam
R^0 A!
P
MBD 0 WR^0 A 42 CP 12 D 0 !R^0 AD2:857kN;
R^0 B!
P
MAD 0 W R^0 B 42 P 30 D 0 !R^0 BD7:143kN:
(b)
Since a bending moment at crownCis zero then
ZA!
X
MCleft 0 WZAhMC^0 D 0 !ZAD
MC^0
h
D
2:857 24
5:979
D11:468kN;
ZADZBDZ;
(c)
whereMC^0 is a bending moment at sectionCfor reference beam.
ThrustHpresents the horizontal component of theZ, i.e.,
HDZcos ̨D11:4680:9965D11:428kN: (4.15)