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derivative of the integral with respect to
u
equal to zero. Thus
∂
∂
f
T
W f
]
dV
u
T
T
WLN
u
u
T
T
W f
[
(
LN
u
)
u
−
2
(
LN
u
)
+
=
0
(7.31)
u
V
or
∂
∂
f
T
W f
]
dV
u
T
T
WLN
u
u
T
T
W f
u
[
(
LN
u
)
u
−
2
(
LN
u
)
+
(7.32)
2
T
WLN
u
dV
T
W f
dV
=
V
(
LN
u
)
u
−
V
(
LN
u
)
(7.33)
2
(
k
u
u
−
p
u
)
=
0
These are
mp
simultaneous linear equations which can be used to find the
mp
unknown
u
i
.
Frequently, the weighting functions
W
j
are set equal to unity.
Equation (7.32) can be considered as a special case of the orthogonality integral of
Eq. (7.26) if
Ψ
i
is set equal to
(
LN
u
)
T
W
.
Also, the least squares approach can be obtained
from Eq. (7.14b) by using
W
replaced by
∂
R
∂
h
(
R
)
=
R
and
u
W
(7.34)
EXAMPLE 7.4 Beam with Linearly Varying Load
For a uniform beam,
R
i
v
−
=
EI
w
p
z
.
With
w
=
N
u
w
and
W
i
selected to be 1, the least square
relations would be
L
L
N
i
v
T
u
N
i
u
dx
N
i
v
T
u
(
)
−
=
EI
w
p
z
dx
0
0
0
(1)
k
u
w
−
p
u
=
0
or
k
u
=
p
u
(2)
For the beam with linearly varying load in Fig. 7.1, use the trial function of Eq. (8) of
Example 7.1,
w
2
3
4
Then
N
i
u
=
L
4
w
=
N
u
w
=
(
3
ξ
−
5
ξ
+
2
ξ
)w
1
.
48
/
and (1) becomes
48
EI
dx
EI
48
L
4
p
0
L
L
48
EI
L
4
48
EI
L
4
p
0
L
2
L
4
w
1
−
p
0
(
1
−
x
/
L
)
=
w
1
L
−
−
=
0
(3)
0
p
0
L
4
Then
w
1
=
/(
96
EI
)
and the approximate deflection is
p
0
L
4
96
EI
(
2
3
4
w
=
3
ξ
−
5
ξ
+
2
ξ
)
(4)
Note that this is the same as Eq. (12) of Example 7.1
7.3.9 Symmetry and Convergence of the Weighted Residual Methods
Most of the approximate methods discussed in this chapter lead to a system of linear
equations in the unknown coefficients
u
i
,
e.g.,
k
u
=
u
p
u
(7.35a)
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