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work relation (2) becomes
3
]
δ
W
=
[
EI
/(
L
/
5
)
{
δw
1
[
(
7
w
1
−
4
w
2
+
w
3
)
−
(
4
/
5
)
p
0
(
L
/
5
)
]
+
δw
2
[
(
−
4
w
1
+
6
w
2
−
4
w
3
+
w
4
)
−
(
3
/
5
)
p
0
(
L
/
5
)
]
+
δw
3
[
(w
−
4
w
+
6
w
−
4
w
)
−
(
2
/
5
)
p
0
(
L
/
5
)
]
1
2
3
4
+
δw
4
[
(w
−
4
w
+
6
w
+
w
)
−
(
1
/
5
)
p
0
(
L
/
5
)
]
2
3
4
6
+
δw
6
[
(w
+
w
)
]
}=
0
(6)
4
6
V
T
In matrix notation,
δ
W
=
δ
(
KV
−
P
)
=
0
.
Then
KV
=
P
takes the form
−
w
/
7
4100
4
5
1
−
−
w
/
46
410
3
5
2
(
/
)
4
p
0
L
5
1
−
46
−
40
w
=
2
/
5
3
w
4
w
5
EI
01
461
00011
−
1
/
5
0
(7)
K
V
=
P
From the final row,
Insert this relationship in the fourth row
of (7). Observe that this system of symmetric equations reduces to Eq. (8) of Example 8.2.
Hence, the displacements of Eq. (9) of Example 8.2 are obtained again. Furthermore, this
variationally based solution gives the same moments and shear forces as the conventional
solution of Example 8.2. This means that the variational approach led to a solution that
satisfies exactly the force boundary condition,
M
5
w
4
+
w
6
=
0or
w
6
=−
w
4
.
0, although this condition was not
explicitly imposed. It is unreasonable, of course, to expect the force boundary condition
always to be satisfied exactly, but a best fit is provided automatically by the variational
approach. As in the case of the classical trial function solution of Chapter 8, use of a principle
extended by appending boundary condition terms to the fundamental global form can ease
the task of applying boundary conditions.
The above procedure is readily generalized to apply to problems with many nodes, i.e.,
with many DOF. To see that this is the case, it is useful to repeat the above solution using
matrix notation. From (5), we can express
=
δ
W
i
as
1
−
w
i
−
1
w
i
w
i
+
1
EI
h
4
δ
W
i
=
[
δw
i
−
1
δw
i
δw
i
+
1
]
2
1
[1
−
21]
v
iT
δ
v
i
1
−
21
v
iT
EI
h
4
v
i
v
iT
k
i
v
i
=
δ
−
24
−
2
=
δ
(8)
1
−
21
k
i
where
v
i
is a vector of nodal displacements, and
k
i
can be considered to be a stiffness matrix.
Note that
k
i
is symmetric.
EXAMPLE 8.5 Torsional Stresses on the Cross-Section of a Bar
Consider the torsion of the same prismatic bar as in Example 8.3. Calculate the Prandtl
stress function on the cross-section using a variationally based approach.
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