Advanced Methods of Structural Analysis

4.2 Internal Forces 81

0

l

C

Reference beam

R^0

B

RB

x 1

x 1

x 2

x 2

HB

A

C

l

f

B

y

xk

xk

HA

yk yk

jk

RA RA

RA

P 1 P 1

P 1

Pn

Pn

k

P 2 Tangent

P 2

y(x)

H

Tangent

Mk

Nk

Qk

k

x

a b P

2

jk

Fig. 4.4 Three-hinged arch.(a)Design diagram and reference beam.(b)Positive internal forces
at any sectionk

The horizontal reactions (thrust) at both supports of three-hinged arches subjected
to the vertical loads are equal in magnitude and opposite in direction

HADHBDH: (4.7)

Bending moment at the hingeCof the arch is zero. Therefore, by definition of the
bending moment

MCDRA

l

2

P 1



l

2

x 1



P 2



l

2

x 2



„ ƒ‚ ...

MC^0

HAfD0:

Underlined set of terms is the bending moment acting over sectionCof the ref-
erence beam (this section is located under the hinge of the arch). Therefore last
equation may be rewritten in the form

MC^0 HAfD0;

which allows immediately to calculate the thrust

HD

MC^0

f

: (4.8)

Thus, the thrust of the arch equals to bending moment at sectionCof the reference
beam divided by the rise of the arch.