Problem 3.33.
Given:
A uniform bar elemen of length L has a node at
each end and a node at the middle as shown.
Find: the element stiffness matrix that operates on the
nodal d.o.f. u
1
, u
2
, u
3
.
Solution:
x
1
0
=
x
2
L
2
=
x
3
L
=
NN
1
N
2
N
3
()
=
N
1
x
2
x
−
x
3
x
−
⋅
x
2
x
1
−
x
3
x
1
−
⋅
=
L
2
x
−
⎛
⎝
⎞
⎠
Lx
−
⋅
L
2
⎛
⎝
⎞
⎠
L
⋅
=
1
L
2
L
2
3L
⋅
x
⋅
−
2x
2
⋅
+
⋅
=
N
2
x
1
x
−
x
3
x
−
⋅
x
1
x
2
−
x
3
x
2
−
⋅
=
x
−
()Lx
−
⋅
L
−
2
⎛
⎝
⎞
⎠
L
2
⎛
⎝
⎞
⎠
⋅
=
4
L
2
xL
⋅
x
2
−
⋅
=
N
3
x
1
x
−
x
2
x
−
⋅
x
1
x
3
−
x
2
x
3
−
⋅
=
x
−
L
2
x
−
⎛
⎝
⎞
⎠
⋅
L
−
L
−
2
⎛
⎝
⎞
⎠
⋅
=
1
L
2
2
⋅
−
⋅
=
B
x
N
1
d
dx
N
2
d
N
3
d
d
⎛
⎝
⎞
⎠
=
x
1
L
2
L
2
⋅
x
⋅
−
2
⋅
+
⋅
⎡
⎢
⎣
⎤
⎥
⎦
d
d
⋅
4x
⋅
−
L
2
−
→
x
4
L
2
⋅
x
2
−
⋅
⎡
⎢
⎣
⎤
⎥
⎦
d
d
4L
⋅
8x
⋅
−
L
2
→
x
1
L
2
2
⋅
−
⋅
⎡
⎢
⎣
⎤
⎥
⎦
d
d
L4
x
⋅
−
L
2
−
→
B
1
L
2
3
−
L
⋅
⋅
+
⋅
4
L
2
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 Fall '10
 Folkman

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