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FIGURE 13.13
Loading with sinusoidal y distribution.
The loading p z can also be expanded in a similar sine series
sin m
π
y
p z (
x, y
) =
p m
(
x
)
(13.62)
L y
m
=
1
An expression for calculating p m
(
x
)
is found by multiplying both sides of Eq. (13.62) by
sin
(
n
π
y
/
L y
)
and integrating from y
=
0to y
=
L y
.
Use the orthogonality conditions for
trigonometric functions such as
L y
0
=
m
n
sin m
π
y
sin n
π
y
dy
=
L y
L y
L y
2
m
=
n
0
to liberate p m
(
x
)
.
L y
2
L y
sin m
π
y
p m (
x
) =
p z (
x, y
)
dy
(13.63)
L y
0
Since p z (
, the transformed loading function coefficient, can always be
determined from this equation.
x, y
)
is given, p m (
x
)
EXAMPLE 13.2 Transformed Loading Functions
Find the transformed loading function coefficients p m if the applied loading is sinusoidal
in the y direction as shown in Fig. 13.13.
The loading can be expressed as
sin π
y
L y
p z (
x, y
) =
p 0
(
x
)
(1)
From Eq. (13.63),
p 0
L y
=
p 0 sin π
π
if m
1
2
L y
y
L y
sin m
y
(
) =
=
p m
x
dy
(2)
L y
0if m
>
1
0
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