1 finite element method fem for frames for readers of all backgrounds g. r. liu and s. s. quek...
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1
FFinite Element Methodinite Element Method
FEM FOR FRAMES
for readers of all backgroundsfor readers of all backgrounds
G. R. Liu and S. S. Quek
CHAPTER 6:
2Finite Element Method by G. R. Liu and S. S. Quek
CONTENTSCONTENTS INTRODUCTION FEM EQUATIONS FOR PLANAR FRAMES
– Equations in local coordinate system– Equations in global coordinate system
FEM EQUATIONS FOR SPATIAL FRAMES– Equations in local coordinate system– Equations in global coordinate system
REMARKS
3Finite Element Method by G. R. Liu and S. S. Quek
INTRODUCTIONINTRODUCTION
Deform axially and transversely. It is capable of carrying both axial and transverse
forces, as well as moments. Hence combination of truss and beam elements. Frame elements are applicable for the analysis of
skeletal type systems of both planar frames (2D frames) and space frames (3D frames).
Known generally as the beam element or general beam element in most commercial software.
4Finite Element Method by G. R. Liu and S. S. Quek
FEM EQUATIONS FOR PLANAR FEM EQUATIONS FOR PLANAR FRAMESFRAMES
Consider a planar frame element
Y, V
X, U
node 1 (u1, v1, z1)
x, , u y, v
z
z
l=2a
node 2 (u2, v2, z2)
1 1
2 1
3 1
4 2
5 2
6 2
diplacement components at node 1
diplacement components at node 2
ze
z
d u
d v
d
d u
d v
d
d
5Finite Element Method by G. R. Liu and S. S. Quek
Equations in local coordinate systemEquations in local coordinate system
Combination of the element matrices of truss and beam elements
2
2
2
1
1
1
z
ze
v
u
v
u
d
Truss
Beam
From the truss element,
1 1 4 2
1 12 2
4 22
0 0 0 0
0 0 0 0 0
0 0 0 0
0 0
. 0 0
0
AE AEa a
trusse AE
a
d u d u
d u
d u
sy
k
(Expand to 6x6)
6Finite Element Method by G. R. Liu and S. S. Quek
Equations in local coordinate systemEquations in local coordinate system
From the beam element,
3 2 3 2
2
3 2
3 1 5 2 6 22 1
3 3 3 3
2 2 2 2 2 12 3
2 3 1
3 35 22 2
26 2
( ) ( ) ( )( )
0 0 0 0 0 0
0
0
0 0 0
.
z z z z
z z z
z z
z
z z
EI EI EI EI
a a a aEI EI EI
a a a zbeame
EI EI
a aEI
za
d d v dd v
d v
d
d vsy
d
k
(Expand to 6x6)
7Finite Element Method by G. R. Liu and S. S. Quek
Equations in local coordinate systemEquations in local coordinate system
0
00.
00
0000
00000
0000
2
22
sya
AE
aAE
aAE
trussek
3 2 3 2
2
3 2
3 3 3 3
2 2 2 22 3
2
3 3
2 22
0 0 0 0 0 0
0
0
0 0 0
.
z z z z
z z z
z z
z
EI EI EI EI
a a a aEI EI EI
a a abeame
EI EI
a aEI
a
sy
k+
a
EIa
EI
a
EIa
AEa
EI
a
EI
a
EIa
EI
a
EI
a
EI
a
EIa
AEa
AE
e
z
zz
zzz
zzzz
sy2
2
3
2
32
2
322
3
2
3
2
3
2
322
23
2
2323
.
00
0
0
0000
k
8Finite Element Method by G. R. Liu and S. S. Quek
Equations in local coordinate systemEquations in local coordinate system
Similarly so for the mass matrix and we get
2
22
8
2278.
0070
61308
132702278
00350070
105
a
asy
aaa
aa
Aae
m
And for the force vector,
13
2
2
13
1
1
2
2
s
af
syy
sxx
s
af
syy
sxx
e
m
faf
faf
m
faf
faf
y
y
f
9Finite Element Method by G. R. Liu and S. S. Quek
Equations in global coordinate systemEquations in global coordinate system
Coordinate transformation
ee TDd
where
j
j
j
i
i
i
e
D
D
D
D
D
D
3
13
23
3
13
23
D ,
100000
0000
0000
000100
0000
0000
yy
xx
yy
xx
ml
ml
ml
ml
T
D3i -2
D3i-1
D3j -2
D2j
2a
x
u1
u2
fs1
global node j local node 2
global node i local node 1
0
X
Y
o
x
y
v1
v2
D3j
D3j - 1
z
2
z1
D3i
10Finite Element Method by G. R. Liu and S. S. Quek
Equations in global coordinate systemEquations in global coordinate system
D3i -2
D3i-1
D3j -2
D2j
2a
x
u1
u2
fs1
global node j local node 2
global node i local node 1
0
X
Y
o
x
y
v1
v2
D3j
D3j - 1
z
2
z1
D3i
cos( , ) cos
cos( , ) sin
j ix
e
j ix
e
X Xl x X
l
Y Ym x Y
l
cos( , ) cos(90 ) sin
cos( , ) cos
j iy
e
j iy
e
Y Yl y X
l
X Xm y Y
l
2 2( ) ( )e j i j il X X Y Y
Direction cosines in T:
(Length of element)
11Finite Element Method by G. R. Liu and S. S. Quek
Equations in global coordinate systemEquations in global coordinate system
Therefore,
TkTK eT
e
TmTM eT
e
eT
e fTF
12Finite Element Method by G. R. Liu and S. S. Quek
FEM EQUATIONS FOR FEM EQUATIONS FOR SPATIAL FRAMESSPATIAL FRAMES
Consider a spatial frame element
v1
u1 w1
x
y
z
1
2
z2
w2
x2
y2
z1
x1
y1
u2
v2
1 1
2 1
3 1
4 1
5 1
6 1
27
28
29
210
211
212
x
y
ze
x
y
z
d u
d v
d w
d
d
d
ud
vd
wd
d
d
d
d
Displacement components at node 1
Displacement components at node 2
13Finite Element Method by G. R. Liu and S. S. Quek
Equations in local coordinate systemEquations in local coordinate system Combination of the element matrices of truss and
beam elements
3 2 3 2
3 2 3 2
1 21 21 1 1 1 2 2 2 2
2 23 3 3 3
2 2 2 2
3 3 3 3
2 2 2 2
2
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
z z z z
y y y y
y yx xz z
AE AEa a
EI EI EI EI
a a a a
EI EI EI EI
a a a a
GJa
e
u v w u v w
k
2
2
3 2
3 2
2
2 3
22 3
2
23 3
2 2
3 3
2 2
2
2
2
.
0 0
0 0 0 0 0
0 0 0 0
0 0 0 0 0
0 0 0
0 0
0 0
0
y y y
z z z
z z
y y
y
z
GJa
EI EI EI
a a aEI EI EI
a a a
AEa
EI EI
a a
EI EI
a a
GJa
EI
aEI
a
sy
v1
u1 w1
x
y
z
1
2
z2
w2
x2
y2
z1
x1
y1
u2
v2
14Finite Element Method by G. R. Liu and S. S. Quek
Equations in local coordinate systemEquations in local coordinate system
2
2
2
22
22
22
8080070.0220782200078000007060001308060130008003500000700130270002207813000270220007800000350000070
105
aa
rsya
a
aaaaaa
rraa
aa
Aa
x
xx
e
m
whereA
Ir x
x 2
15Finite Element Method by G. R. Liu and S. S. Quek
Equations in global coordinate systemEquations in global coordinate system
Y
X
Z
y x
z
D6i-1
D6i-2
D6i-3
D6i-4
D6i-5
D6j-2
D6j-1
D6j-3
D6j-4
D6j
D6i
d6
d5
d4
d3
d2 d1
d12
d11
d10
d9
d8
d7
D6j-5
y x
z
1
2 3
16Finite Element Method by G. R. Liu and S. S. Quek
Equations in global coordinate systemEquations in global coordinate system
Coordinate transformation
ee TDd where
j
j
j
j
j
j
i
i
i
i
i
i
e
D
D
D
D
D
D
D
D
D
D
D
D
6
16
26
36
46
56
6
16
26
36
46
56
D ,
3
3
3
3
T000
0T00
00T0
000T
T
zzz
yyy
xxx
nml
nml
nml
3T
17Finite Element Method by G. R. Liu and S. S. Quek
Equations in global coordinate systemEquations in global coordinate system
cos( , ), cos( , ), cos( , )
cos( , ), cos( , ), cos( , )
cos( , ), cos( , ), cos( , )
x x x
y y y
z z z
l x X m x Y n x Z
l y X m y Y n y Z
l z X m z Y n z Z
Direction cosines in T3
18Finite Element Method by G. R. Liu and S. S. Quek
Equations in global coordinate systemEquations in global coordinate system Vectors for defining location and orientation of
frame element in space
Y
X
Z
x
x
1
2 3
12 VV
13 VV
y
y
z
y
)()( 1312 VVVV
2V
1
1V 1V
3V
1
1V
ZZYYXXV
1111
ZZYYXXV
2222
ZZYYXXV
3333
lkkl
lkkl
lkkl
ZZZ
YYY
XXX
k, l = 1, 2, 3
221
221
221122 ZYXVVal
ZZYYXXVV
21212112
ZZYYXXVV
31313113
19Finite Element Method by G. R. Liu and S. S. Quek
Equations in global coordinate systemEquations in global coordinate system Vectors for defining location and orientation of
frame element in space (cont’d)
Y
X
Z
x
x
1
2 3
12 VV
13 VV
y
y
z
y
)()( 1312 VVVV
2V
1
1V 1V
3V
1
1V
a
ZZxZxn
a
YYxYxm
a
XXxXxl
x
x
x
2),cos(
2),cos(
2),cos(
21
21
21
)()(
)()(
1312
1312
VVVV
VVVVz
Za
ZY
a
YX
a
X
VV
VVx
222
)( 212121
12
12
})()(){(2
1213131212131312121313121
123
ZYXYXYXZXZXZYZYA
z
221313121
221313121
221313121123 )()()( YXYXXZXZZYZYA
20Finite Element Method by G. R. Liu and S. S. Quek
Equations in global coordinate systemEquations in global coordinate system Vectors for defining location and orientation of
frame element in space (cont’d)
Y
X
Z
x
x
1
2 3
12 VV
13 VV
y
y
z
y
)()( 1312 VVVV
2V
1
1V 1V
3V
1
1V )(
2
1
)(2
1
)(2
1
21313121123
21313121123
21313121123
YXYXA
Zzn
XZXZA
Yzm
ZYZYA
Xzl
z
z
z
xzy
xzxzy
xzxzy
xzxzy
lmmln
nllnm
mnnml
21Finite Element Method by G. R. Liu and S. S. Quek
Equations in global coordinate systemEquations in global coordinate system
Therefore,
TkTK eT
e
TmTM eT
e
eT
e fTF
22Finite Element Method by G. R. Liu and S. S. Quek
REMARKSREMARKS
In practical structures, it is very rare to have beam structure subjected only to transversal loading.
Most skeletal structures are either trusses or frames that carry both axial and transversal loads.
A beam element is actually a very special case of a frame element.
The frame element is often conveniently called the beam element.
23Finite Element Method by G. R. Liu and S. S. Quek
CASE STUDYCASE STUDY
Finite element analysis of bicycle frame
24Finite Element Method by G. R. Liu and S. S. Quek
CASE STUDYCASE STUDYYoung’s modulus,
E GPaPoisson’s ratio,
69.0 0.33
74 elements (71 nodes)
Ensure connectivity
25Finite Element Method by G. R. Liu and S. S. Quek
CASE STUDYCASE STUDY
Constraints in all directions
Horizontal load
26Finite Element Method by G. R. Liu and S. S. Quek
CASE STUDYCASE STUDY
M = 20X
27Finite Element Method by G. R. Liu and S. S. Quek
CASE STUDYCASE STUDY
-9.68 x 105 Pa
-1.214 x 106 Pa
-6.34 x 105 Pa
-6.657 x 105 Pa
9.354 x 105 Pa
-5.665 x 105 Pa
-6.264 x 105 Pa
Axial stress