Civil Engineering Reference
In-Depth Information
a
D
M
D
P
D
D
V
D
h
D
H
A
A
R
A
FIGURE 5.10b
Free body diagram of arch rib from support A to point D.
Therefore, the influence line for the vertical components of the arch reactions,
R
A
and
R
B
, will be the same as those for a simply supported beam of length,
L
, as shown
in Figure 5.10a.
If moments are taken about the arch crown pin (point C),
∗
R
A
L
2
.
H
A
(h)
=
(5.19)
Since
R
A
(L/
2
)
is the bending moment at point C in a simply supported span, the
influence line for horizontal thrust reaction,
H
A
, is proportional (by the arch rise,
h
)to
this simple span bending moment as shown in Figure 5.10a. Therefore, the criteria for
the position of Cooper's load for maximum bending moment (see Section 5.2.1.1.3)
can be used for the determination of maximum horizontal thrust.
The arch reactions may now be used to determine the internal shear force, bending
moment, and axial force influence lines for the arch rib. From Figure 5.10b, the
bending moment,
M
D
, at a location, D, is
M
D
=
R
A
(a
D
)
−
H
A
(h
D
)
.
(5.20)
Equation 5.20 indicates that the influence line for bending moment in the arch rib
at location D can be obtained by subtracting the ordinates for the influence line for
H
A
(Figure 5.10a) multiplied by the distance
h
D
from the ordinates for simple beam
∗
Itistheinclusionofthecrownpinthatenablesthisequilibriumequationtobewritten;therebyillustrating
the benefits of statically determinate design and construction.
