Civil Engineering Reference
In-Depth Information
The vertical components of the support reactions are, from symmetry,
wL
2
R
A,V
=
R
B,V
=
Also, in the absence of any horizontal loads
R
A,H
=
R
B,H
Now taking moments of forces to the left of C about C,
L
2
+
wL
2
L
4
=
R
A,H
h
−
R
A,V
0
which gives
wL
2
8
h
With the origin of axes at A, the equation of the parabolic shape of the arch may be
shown to be
R
A,H
=
4
h
L
2
(
Lx
x
2
)
y
=
−
The bending moment at any point
P
(
x
,
y
) in the arch is given by
wx
2
2
or, substituting for
R
A,V
and
R
A,H
and for
y
in terms of
x
,
M
P
=
R
A,V
x
−
R
A,H
y
−
wL
2
8
h
wx
2
2
wL
2
4
h
L
2
(
Lx
x
2
)
M
P
=
x
−
−
−
Simplifying this expression
wx
2
2
−
wx
2
2
=
wL
2
wL
2
M
P
=
x
−
x
+
0
as expected.
The shear force may also be shown to be zero at all sections of the arch.
6.4 B
ENDING
M
OMENT
D
IAGRAM FOR A
T
HREE-PINNED
A
RCH
Consider the arch shown in Fig. 6.8; we shall suppose that the equation of the arch
referred to the
xy
axes is known. The load
W
is applied at a given point D(
x
D
,
y
D
)
and the support reactions may be calculated by the methods previously described. The
bending moment,
M
P1
, at any point P
1
(
x
,
y
) between A and D is given by
M
P1
=
R
A,V
x
−
R
A,H
y
(6.9)
