Advanced Methods of Structural Analysis

80 4 Three-Hinged Arches

The angle'between the tangent to the center line of the arch at point.x; y/and
horizontal axis is shown in Fig.4.1b. Trigonometric functions for this angle are as
follows:

sin'D.l2x/

1

2R

I cos'D.yCRf/

1

R

: (4.3)

Parabolic Arch

Ordinateyof any point of the central line of the arch

yD4f x .lx/

1

l^2

: (4.4)

Trigonometric functions of the angle between the tangent to the center line of the
arch at point.x; y/and a horizontal axis are as follows

tan'D

dy

dx

D

4f

l^2

.l2x/Icos'D

1

p

1 Ctan^2 

Isin'Dcos'tan' (4.5)

For the left-hand half-arch, the functions sin'>0;cos'>0, and for the right-hand
half-arch the functions sin'<0and cos'>0.

4.2 Internal Forces..........................................................

Design diagram of a three-hinged symmetrical arch with intermediate hingeCat the
highest point of the arch and with supportsAandBon one elevation is presented in
Fig.4.4. The span and rise of the arch are labeled aslandf, respectively; equation
of central line of the arch isyDy.x/.

Reactions of Supports

Determination of internal forces, and especially, construction of influence lines for
internal forces of the three-hinged arch may be easily and attractively performed
using the conception of the “reference (or substitute) beam.” The reference beam is
a simply supported beam of the same span as the given arch and subjected to the
same loads, which act on the arch (Fig.4.4a).
The following reactions arise in the arch:RA;RB;HA;HB. The vertical reac-
tions of three-hinged arches carrying the vertical loads have same values as the
reactions of the reference beam

RADR^0 AIRBDR^0 B: (4.6)