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This set of equations, which would appear to be similar to stiffness equations, gives
u
1
=
0
.
013917
u
2
=
0
.
018452
u
3
=
0
.
012567
(18)
At
ξ
=
1
/
2
,
the error is 4.8%.
=
Case c (Fig. 8.5c, M
8)
For eight segments,
M
=
=
/
.
8
,h
1
8
Equations (7) and (11) lead to
14
.
583
−
21
1
u
1
u
2
u
3
u
4
u
5
u
6
u
7
18
.
519
−
21
1
22
.
727
−
21
1
10
−
3
−
21
1
=−
22
.
222
(19)
−
21
1
19
.
231
−
21
1
14
.
286
−
2
7
.
778
K
U
P
which gives, for example,
u
2
=
0
.
013333
,u
4
=
0
.
017817
,
and
u
6
=
0
.
12165
.
The error at
midspan
(ξ
=
1
/
2
)
is 1.2%.
Case d (Generalization)
Equation (19) is readily extended to
M
segments. Then
h
=
1
/
M,
and from (11),
1
M
2
(
u
i
=
u
i
−
1
−
2
u
i
+
u
i
+
1
)
(20)
Define the coordinate
ξ
=
1
/
M, i
=
1
,
2
,
...
M
.
From the differential equation (7),
i
u
i
=−
ξ
i
(
1
−
ξ
i
)/(
1
+
ξ
i
)
(21)
From (20) and (21),
=
−
M
2
(
1
i
/
M
)(
1
−
i
/
M
)
=
−
1
M
3
i
(
M
−
i
)
u
i
−
1
−
2
u
i
+
u
i
+
1
(22)
1
+
i
/
M
M
+
i
The boundary conditions are
u
0
=
u
M
=
0
.
Equation (22) provides the system of stiffness-
like equations
−
21
1
1
(
M
−
1
)/(
M
+
1
)
u
1
u
2
u
3
·
·
·
u
M
−
2
u
M
−
1
−
21
0
2
(
M
−
2
)/(
M
+
2
)
1
−
21
3
(
M
−
3
)/(
M
+
3
)
·
·
·
·
M
3
=−
(
1
/
)
·
·
(
M
−
2
)
2
/(
M
+
(
M
−
2
))
0
1
−
21
1
−
2
(
M
−
1
)
1
/(
M
+
(
M
−
1
))
K
U
P
(23)
which can be solved for
u
i
for a prescribed value of
M
.
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