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where
y
y
3
6.5 Use the stiffness matrices obtained in Problem 6.4 to calculate the responses of the
structure of Fig. P6.4a.
y
3
(
x
2
−
x
)
+
y
(
x
3
−
x
2
)
xy
3
yx
3
x
2
y
3
−
N
1
=
,
=
,
=
2
3
x
2
y
3
Hint:
The element stiffness matrices in the local coordinate systems have to be
transformed into the global
xy
coordinate system using
k
i
k
i
T
i
,
where
k
i
and
k
i
are the element stiffness matrices in the global and local coordinate systems,
respectively.
T
iT
=
cos
T
i
11
α
sin
α
T
i
T
i
22
T
i
jj
=
=
−
sin
α
cos
α
T
i
33
α
where
x
(local) coordinate
(Fig. P6.4b) for the
i
th element. The global node number and the global DOF are
related by
is the angle between the
x
(global coordinate) and the
Global
DOF No.
Global Node
No.
u
x
u
y
1
1
2
2
3
4
3
5
6
4
7
8
The incidence table is
Element
Node No.
Number
1
2
3
1
1
2
3
2
1
3
4
and the element numbers and the corresponding global DOF are related by
Element
Element Node Numbers
No.
1
2
3
1
1
2
3
1
2
3
4
5
6
2
1
3
4
1
2
5
6
7
8
Answer:
u
x
2
=
0
.
00132 mm
u
y
2
=−
0
.
00435 mm
u
x
3
=
0
.
02169 mm
u
y
3
=
0
.
00997 mm
6.6 Investigate the completeness and compatibility of the trial functions for the elements
shown in Fig. P6.6.
1.
u
=
ξ
+
u
2
u
3
η
=
ξ
+
η
+
ξη
2.
u
u
2
u
3
u
4
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