Advanced Methods of Structural Analysis

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 ordinateyDD3:5m, 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)