Advanced Methods of Structural Analysis

4.1 Preliminary Remarks 79

a b

A x

C

B

y

Tie A x

C

B

y

Elevated

tie

RA RB RA RB

Fig. 4.2 (a,b) Design diagram of three-hinged arch with tie on the level supports and elevated tie

Thus, the arch is characterized by two fundamental markers such as a curvilin-
ear axis and appearance of the thrust.Therefore the structure in Fig.4.3presents
the curvilineartrustlesssimply supported element, i.e., this is just members with
curvilinear axis, but no arch. It is obvious that, unlike the beam, in this structure
the axial compressed forces arise; however, the distribution of bending moments for
this structure and for beam of the same span and load will not differ, while the shear
forces are less in this structure than that in beam. Thus, the fundamental feature of
the arch (decreasing of the bending moments due to the appearance of the thrust)
for structure in Fig.4.3is not observed.

Fig. 4.3 Simply supported
thrustless curvilinear member

A B

P

RA RB

4.1.3 Geometric Parameters of Circular and Parabolic Arches

Distribution of internal forces in arches depends on a shape of the central line of
an arch. Equation of the central line and some necessary formulae for circular and
parabolic arches are presented below. For both cases, origin of coordinate axis is
located at pointA(Figs.4.1band4.2).

Circular Arch

Ordinateyof any point of the central line of the arch is defined by formula

yD

s

R^2 



l

2

x

 2

RCfI RD

f

2

C

l^2

8f

; (4.2)

wherexis the abscissa of the same point of the central line of the arch;Rthe radius
of curvature of the arch;fandlare the rise and span of the arch.