Advanced Methods of Structural Analysis

4.5 Special Types of Arches 101

y

1 x

1 q=2kN/m

H H

D

A′ B′

12m

RA RB

E

F

HA

B

A

C

f=12m

Parabolic

arch

f 0 =2m

ak=18m

l=48m

k

Fig. 4.16 Design diagram of the arch with complex tie

4.5.2.1 Reactions and Bending Moment at Sectionk

The vertical reactions are determined from the equilibrium equations of all the
external forces acting on the arch

RA!

P

MBD 0 WRA 48 Cq 12  6 D 0 !RAD 3 kN;

RB!

P

MAD 0 W RB 48 q 12  42 D 0 !RBD 21 kN:

(a)

Horizontal reaction at supportAisHAD 0.
The thrustHin the tie (section 1-1) is determined from the following equation

H!

X

MCleftD 0 WRA

l

2

CH.ff 0 /D 0 !HD

MC^0

ff 0

D7:2kN:

(4.20)

Equilibrium equations of jointFlead to the axial forces at the members ofAFand
EFof the tie.
Internal forces at sectionkfor a reference simply supported beam are as follows:

Mk^0 DRAxkD 3  18 D 54 kN m;

QK^0 DRAD 3 kN:

(b)

Internal forces at the pointkfor three-hinged arch are determined as follows:

MkDMk^0 H.ykf 0 /D 54 7:2 .11:252/D12:6kN m;

QkDQ^0 kcos'kHsin'kD^3 0:9707:20:2425D1:164kN;

NkD



Q^0 ksin'kCHcos'k



D.30:2425C7:20:970/D7:711kN:

(4.21)