Civil Engineering Reference
In-Depth Information
of the arch forms the major portion of the load the arch has to carry. In Section 5.2 we
saw that a cable under its own weight takes up the shape of a catenary. It follows that
the ideal shape for an arch of constant thickness is an inverted catenary. However, in
the analysis of the three-pinned arch we shall assume a general case in which shear
forces and bending moments, as well as axial forces, are present.
6.2 T
HE
T
HREE-PINNED
A
RCH
A three-pinned arch would be used in situations where there is a possibility of support
displacement; this, in a two-pinned arch, would induce additional stresses. In the
analysis of a three-pinned arch the first step, generally, is to determine the support
reactions.
SUPPORT REACTIONS - SUPPORTS ON SAME HORIZONTAL
LEVEL
Consider the arch shown in Fig. 6.3. It carries an inclined concentrated load,
W
,ata
given point D, a horizontal distance
a
from the support point A. The equation of the
shape of the arch will generally be known so that the position of specified points on
the arch, say D, can be obtained. We shall suppose that the third pin is positioned at
the crown, C, of the arch, although this need not necessarily be the case; the height or
rise
of the arch is
h
.
The supports at A and B are pinned but neither can be a roller support or the arch
would collapse. Therefore, in addition to the two vertical components of the reactions
at A and B, there will be horizontal components
R
A,H
and
R
B,H
. Thus, there are
four unknown components of reaction but only three equations of overall equilibrium
(Eq. (2.10)) so that an additional equation is required. This is obtained from the fact
that the third pin at C is unable to transmit bending moments although, obviously, it
is able to transmit shear forces.
C
W
a
D
h
h
D
A
B
R
A,H
R
B,H
R
A,V
R
B,V
a
F
IGURE
6.3
Three-pinned arch
L
/2
L
/2
