Civil Engineering Reference
In-Depth Information
W
C
D(
x
D
,
y
D
)
y
P
1
(
x
,
y
)
P
2
(
x
,
y
)
h
B
A
R
A,H
x
R
B,H
F
IGURE
6.8
Determination of
the bending moment
diagram for a
three-pinned arch
R
A,V
R
B,V
L
W
D
B
A
R
B
R
A
x
D
L
(a)
F
IGURE
6.9
Bending moment
diagram for a
simply supported
beam (tension on
undersurface of
beam)
A
D
B
ve
(b)
and the bending moment,
M
P2
, at the point P
2
,(
x
,
y
) between D and B is
M
P2
=
R
A,V
x
−
W
(
x
−
x
D
)
−
R
A,H
y
(6.10)
Now let us consider a simply supported beamAB having the same span as the arch and
carrying a load,
W
, at the same horizontal distance,
x
D
, from the left-hand support
(Fig. 6.9(a)). The vertical reactions,
R
A
and
R
B
will have the same magnitude as the
vertical components of the support reactions in the arch. Thus the bending moment
at any point between A and D and a distance
x
from A is
M
AD
=
R
A
x
=
R
A,V
x
(6.11)
Also the bending moment at any point between D and B a distance
x
from A is
M
DB
=
R
A
x
−
W
(
x
−
x
D
)
=
R
A,V
x
−
W
(
x
−
x
D
)
(6.12)
