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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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