Civil Engineering Reference
In-Depth Information
Lh
D
/(4
h
)
a
D
(
L
-
a
D
)/
L
B
Influence line for
M
D
A
E
C
D
FIGURE 5.11
Influence line for bending moments at location D in three-hinged arch rib.
bending at location, D, described by
R
A
(a
D
)
. The construction of this influence line
is shown in Figure 5.11. The ordinates (shaded areas) may be plotted on a horizontal
line for ease of use in design.
V
D
=
R
A
cos
φ
D
−
H
A
sin
φ
D
.
(5.21)
Equation 5.21 indicates that the influence line for shear force in the arch rib at location
D can be obtained by subtracting the ordinates for the influence line for
H
A
multiplied
by sin
φ
D
. The
constructionofthisinfluencelineisshownin
Figure5.12.
Again,theordinates(shaded
areas) may be plotted on a horizontal line for ease of use in design. Location E is the
position of the moving load that creates no shear force or bending moment in the arch
at location D (Figure 5.10a).
From Figures 5.10a and b, the axial force,
F
D
, at a location, D, is
φ
D
from the ordinates for simple beam shear at D multiplied by cos
F
D
=−
R
A
sin
φ
D
−
H
A
cos
φ
D
.
(5.22)
Equation 5.22 indicates that the influence line for axial force at location D in the arch
rib can be obtained by adding the ordinates for the influence line for
H
A
multiplied
by cos
φ
D
. The
constructionofthisinfluencelineisshownin
Figure5.13.
Again,theordinates(shaded
areas) may be plotted on a horizontal line for ease of use in design.
φ
D
to the ordinates for simple beam shear at D multiplied by sin
D
E
L
/(4
h
)sin ϕ
D
[(
L
-
a
D
)/
L
]cos ϕ
D
B
Influence line for
V
D
[
a
D
/
L
]cos ϕ
D
C
A
FIGURE 5.12
Influence line for shear forces at location D in three-hinged arch rib.
