Lecture Notes: Introduction to Finite Element Method |
Chapter 2. Bar and Beam Elements |
Distributed Load
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q |
i |
x |
j |
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qL/2 |
qL/2 |
i |
j |
Uniformly distributed axial load q (N/mm, N/m, lb/in) can be converted to two equivalent nodal forces of magnitude qL/2. We verify this by considering the work done by the load q,
Wq = ∫L |
21 uqdx = 21 ∫1 |
u(ξ)q( Ldξ) = qL2 |
∫1 |
u(ξ)dξ |
0 |
0 |
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0 |
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1
= qL2 ∫[Ni (ξ)
0
1
= qL2 ∫[1−ξ
0
N |
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u |
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(ξ) i dξ |
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j |
] uj |
ξ]dξ uuij
= |
1 |
qL |
qL ui |
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2 |
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2 |
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2 |
uj |
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1 |
[ui |
qL / 2 |
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= |
2 |
uj ] qL / 2 |
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© 1997-2002 Yijun Liu, University of Cincinnati |
38 |
Lecture Notes: Introduction to Finite Element Method |
Chapter 2. Bar and Beam Elements |
that is,
Wq |
= |
1 |
u |
T |
fq |
with fq |
qL / 2 |
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2 |
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= |
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qL / 2 |
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Thus, from the U=W concept for the element, we have 21 uT ku = 21 uT f + 21 uT fq
which yields
ku = f +fq
The new nodal force vector is
f +fq |
fi |
+qL / 2 |
= |
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f j |
+qL / 2 |
In an assembly of bars,
q
1 |
2 |
3 |
qL/2 |
qL |
qL/2 |
1 |
2 |
3 |
(22)
(23)
(24)
(25)
© 1997-2002 Yijun Liu, University of Cincinnati |
39 |
Lecture Notes: Introduction to Finite Element Method |
Chapter 2. Bar and Beam Elements |
Bar Elements in 2-D and 3-D Space
2-D Case
y |
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j |
x |
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Y |
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ui’ |
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θ |
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i |
vi |
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ui |
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X |
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Local |
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Global |
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x, y |
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X, Y |
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u' |
, v' |
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ui , vi |
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i |
i |
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1 dof at a node |
2 dof’s at a node |
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Note: Lateral displacement vi’ does not contribute to the stretch of the bar, within the linear theory.
Transformation
ui' = ui |
cosθ + vi |
u |
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sinθ = [l m] |
i |
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vi |
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vi' = −ui sinθ + vi cosθ = [− m |
u |
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l] |
i |
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vi |
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where l = cosθ, |
m = sinθ . |
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© 1997-2002 Yijun Liu, University of Cincinnati |
40 |
Lecture Notes: Introduction to Finite Element Method |
Chapter 2. Bar and Beam Elements |
In matrix form,
u' |
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l |
m ui |
i' |
= |
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vi |
−m |
l vi |
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or,
ui' = T~ui
where the transformation matrix
~ |
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l |
m |
T = |
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−m |
l |
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is orthogonal, that is, T~−1 = T~T .
For the two nodes of the bar element, we have
ui' |
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l |
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' |
−m |
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vi |
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= |
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u |
'j |
0 |
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' |
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0 |
v j |
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or,
u' = Tu
m |
0 |
0 |
ui |
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l |
0 |
0 |
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vi |
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0 |
l |
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m u j |
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0 |
−m |
l |
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v j |
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T~ 0 with T = ~
0 T
The nodal forces are transformed in the same way, f ' = Tf
(26)
(27)
(28)
(29)
(30)
© 1997-2002 Yijun Liu, University of Cincinnati |
41 |