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Problem 3.1111
Given:
A bar of length L
T
carries linearly varying
axial load. Consider FE models having one, two,
and three elements.
Find: a) Consistant Nodal Loads
b) Solve for displacements and compute
element stresses.
Solution:
a)
r
e
0
L
x
N
T
q
⋅
⌠
⎮
⌡
d
=
qN
1
q
1
⋅
N
2
q
2
⋅
+
=
r
e
0
L
x
Lx
−
L
x
L
⎛
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎠
−
L
x
L
⎛
⎝
⎞
⎠
⋅
⌠
⎮
⎮
⎮
⎮
⎮
⌡
d
q
1
q
2
⎛
⎜
⎝
⎞
⎟
⎠
⋅
=
0
L
x
−
()
2
L
2
−
L
2
x
⋅
−
L
2
x
⋅
x
2
L
2
⎡
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎦
⌠
⎮
⎮
⎮
⎮
⎮
⎮
⎮
⌡
d
q
1
q
2
⎛
⎜
⎝
⎞
⎟
⎠
⋅
=
L
6
2
1
1
2
⎛
⎝
⎞
⎠
⋅
q
1
q
2
⎛
⎜
⎝
⎞
⎟
⎠
⋅
=
where:
0
L
x
−
2
L
2
⌠
⎮
⎮
⎮
⌡
d
L
3
→
0
L
x
−
L
2
x
⋅
⌠
⎮
⎮
⎮
⌡
d
L
6
→
0
L
x
x
2
L
2
⌠
⎮
⎮
⎮
⌡
d
L
3
→
1
2
cL
T
2
/3
cL
T
2
/6
For one element, L = L
T
q
1
cL
T
⋅
=
q
2
0
=
r
e1
L
T
6
2
1
1
2
⎛
⎝
⎞
⎠
⋅
T
⋅
0
⎛
⎜
⎝
⎞
⎟
⎠
⋅
=
c
L
T
2
6
⋅
2
1
⎛
⎝
⎞
⎠
=
1
2
5cL
2
/6
cL
2
/6
3
4cL
2
/6
2cL
2
/6
For two elements, L = L
T
/2
q
1
T
⋅
=
q
2
T
⋅
2
=
q
3
0
=
r
e1
L
T
2
6
2
1
1
2
⎛
⎝
⎞
⎠
⋅
T
⋅
T
⋅
2
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
⋅
=
c
L
T
2
24
⋅
5
4
⎛
⎝
⎞
⎠
=
c
L
2
6
⋅
5
4
⎛
⎝
⎞
⎠
⋅
=
r
e2
L
T
2
6
2
1
1
2
⎛
⎝
⎞
⎠
⋅
T
⋅
2
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
⋅
=
c
L
T
2
12
⋅
1
1
2
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
=
c
L
2
6
⋅
2
1
⎛
⎝
⎞
⎠
⋅
=
R
2
⋅
6
5
6
1
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
=
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2
8cL
2
/6
cL
2
/6
4
7cL
2
/6
5cL
2
/6
3
4cL
2
/6
2cL
2
/6
For three elements, L = L
T
/3
q
1
cL
T
⋅
=
q
2
2c
⋅
L
T
⋅
3
=
q
3
T
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 Fall '10
 Folkman
 Stress

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