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Finite Element Method
Homework I
Problem’s data: H = N+15,0 m B = N+10,0 m E = 200.000 Dan/cm2 for N = odd number
= 0,16
Thickness of the element
Area of the element
Because the element is meshed in 4 identical triangles only one stiffness matrix is needed
to compute displacements. So we compute for the small element 1 and we adapt the matrix for
other elements taking care of their orientation.
daN 10 N
n 5m t 1m
h n 15m b n 10m
h 20m b 15m A
h
2
b
2
2
E 200000daN
cm2
neu 0.16 A 37.5m2
Elasticity matrix coefficients computed using below formulae:
e11 E1 neu( )
1 neu( ) 1 2 neu( ) e22 e11
e12 neu E1 neu( )
1 neu( ) 1 2 neu( ) e21 e12
e33E
2 1 neu( )
e
e11
e21
0
e12
e22
0
0
0
e33
2.13 1010
3.408 109
0
3.408 109
2.13 1010
0
0
0
8.621 109
Pa
x1 0 y1 0
x2b
2 y2 0
y3h
2
x3b
6
Paul Ionescu
Gr. 1 N=5
a11 e11 y2 y3 2
e33 x3 x2 2
a12 e21 x3 x2 y2 y3 e33 x3 x2 y2 y3 a21 a12
a13 e11 y2 y3 y3 y1 e33 x1 x3 x3 x2 a31 a13
a14 e21 x1 x3 y2 y3 e33 x3 x2 y3 y1 a41 a14
a15 e11 y1 y2 y2 y3 e33 x2 x1 x3 x2 a51 a15
a16 e21 x2 x1 y2 y3 e33 x3 x2 y1 y2 a61 a16
a22 e22 x3 x2 2
e33 y2 y3 2
a23 e12 x3 x2 y3 y1 e33 x1 x3 y2 y3
a32 a23
a24 e22 x1 x3 x3 x2 e33 y2 y3 y3 y1
a42 a24
a25 e12 x3 x2 y1 y2 e33 x2 x1 y2 y3 a52 a25
a66 e22 x2 x1 2
e33 y1 y2 2
a56 e21 x2 x1 y1 y2 e33 x2 x1 y2 y1 a65 a56
a55 e11 y1 y2 2
e33 x2 x1 2
a46 e22 x1 x3 x2 x1 e33 y1 y2 y3 y1 a64. a46
a45 e12 x1 x3 y1 y2 e33 x2 x1 y3 y1 a54 a45
a44 e22 x1 x3 2
e33 y3 y1 2
a36 e12 x2 x1 y3 y1 e33 x1 x3 y1 y2 a63 a36
a35 e11 y1 y2 y3 y1 e33 x1 x3 x2 x1 a53 a35
a34 e12 x1 x3 y3 y1 e33 x1 x3 y3 y1 a43 a34
a33 e11 y3 y1 2
e33 x1 x3 2
a26 e22 x2 x1 x3 x2 e33 y1 y2 y2 y3 a62 a26
Stiffness matrix of structure
a
a11
a12
a13
a14
a15
a16
a12
a22
a23
a24
a25
a26
a13
a23
a33
a34
a35
a36
a14
a24
a34
a44
a45
a46
a15
a25
a35
a45
a55
a56
a16
a26
a36
a46
a56
a66
a
2.345 1012
6.014 1011
2.022 1012
3.458 1011
3.233 1011
2.556 1011
6.014 1011
1.395 1012
4.513 1010
5.958 1011
6.466 1011
7.987 1011
2.022 1012
4.513 1010
2.184 1012
3.007 1011
1.616 1011
2.556 1011
3.458 1011
5.958 1011
3.007 1011
9.952 1011
6.466 1011
3.993 1011
3.233 1011
6.466 1011
1.616 1011
6.466 1011
4.849 1011
0
2.556 1011
7.987 1011
2.556 1011
3.993 1011
0
1.198 1012
N
kt
4Aa
k
1.564 1010
4.009 109
1.348 1010
2.306 109
2.155 109
1.704 109
4.009 109
9.297 109
3.009 108
3.972 109
4.31 109
5.325 109
1.348 1010
3.009 108
1.456 1010
2.005 109
1.078 109
1.704 109
2.306 109
3.972 109
2.005 109
6.635 109
4.31 109
2.662 109
2.155 109
4.31 109
1.078 109
4.31 109
3.233 109
0
1.704 109
5.325 109
1.704 109
2.662 109
0
7.987 109
kg
s2
p1
h 9.81m
s2
1000kg
m3
4 p2
3 h 9.81m
s2
1000kg
m3
4
p1 4.905 104
Pa p2 1.471 105
Pa
F1
p1
2
h 1 m
4 1.226 10
5 N F2
p2 p1 2
h
2 t 9.81 10
5 N
Now having the elemental stiffness matrix we replace the terms in the global stiffness matrix.
Note: All coefficients from the following matrixes are multiplied by 109 kg/s2 !
element 1
1 1 2 2 4 4
ui vi uj vj uk vk
1 ui 15.64 4.009 -13.48 -2.306 -2.155 -1.704
1 vi 4.009 9.297 0.3009 -3.972 -4.31 -5.325
2 uj -13.48 0.3009 14.56 -2.005 -1.078 1.704
2 vj -2.306 -3.972 -2.005 6.635 4.31 -2.662
4 uk -2.155 -4.31 -1.078 4.31 3.233 0
4 vk -1.704 -5.325 1.704 -2.662 0 7.987
element 3
2 2 3 3 5 5
ui vi uj vj uk vk
2 ui 15.64 4.009 -13.48 -2.306 -2.155 -1.704
2 vi 4.009 9.297 0.3009 -3.972 -4.31 -5.325
3 uj -13.48 0.3009 14.56 -2.005 -1.078 1.704
3 vj -2.306 -3.972 -2.005 6.635 4.31 -2.662
5 uk -2.155 -4.31 -1.078 4.31 3.233 0
5 vk -1.704 -5.325 1.704 -2.662 0 7.987
element 4
4 4 5 5 6 6
ui vi uj vj uk vk
4 ui 15.64 4.009 -13.48 -2.306 -2.155 -1.704
4 vi 4.009 9.297 0.3009 -3.972 -4.31 -5.325
5 uj -13.48 0.3009 14.56 -2.005 -1.078 1.704
5 vj -2.306 -3.972 -2.005 6.635 4.31 -2.662
6 uk -2.155 -4.31 -1.078 4.31 3.233 0
6 vk -1.704 -5.325 1.704 -2.662 0 7.987
element 2
5 5 4 4 2 2
ui vi uj vj uk vk
5 ui 15.64 4.009 -13.48 -2.306 -2.155 -1.704
5 vi 4.009 9.297 0.3009 -3.972 -4.31 -5.325
4 uj -13.48 0.3009 14.56 -2.005 -1.078 1.704
4 vj -2.306 -3.972 -2.005 6.635 4.31 -2.662
2 uk -2.155 -4.31 -1.078 4.31 3.233 0
2 vk -1.704 -5.325 1.704 -2.662 0 7.987
u1 v1 u2 v2 u3 v3 u4 v4 u5 v5 u6 v6
u1
v1
u2 15.64 4.009 -13.48 -2.306 -2.155 -1.704
v2 4.009 9.297 0.3009 -3.972 -4.31 -5.325
u3 -13.48 0.3009 14.56 -2.005 -1.078 1.704
v3 -2.306 -3.972 -2.005 6.635 4.31 -2.662
u4
v4
u5 -2.155 -4.31 -1.078 4.31 3.233 0
v5 -1.704 -5.325 1.704 -2.662 0 7.987
u6
v6
u1 v1 u2 v2 u3 v3 u4 v4 u5 v5 u6 v6
u1
v1
u2
v2
u3
v3
u4 15.64 4.009 -13.48 -2.306 -2.155 -1.704
v4 4.009 9.297 0.3009 -3.972 -4.31 -5.325
u5 -13.48 0.3009 14.56 -2.005 -1.078 1.704
v5 -2.306 -3.972 -2.005 6.635 4.31 -2.662
u6 -2.155 -4.31 -1.078 4.31 3.233 0
v6 -1.704 -5.325 1.704 -2.662 0 7.987
u1 v1 u2 v2 u3 v3 u4 v4 u5 v5 u6 v6
u1 15.64 4.009 -13.48 -2.306 -2.155 -1.704
v1 4.009 9.297 0.3009 -3.972 -4.31 -5.325
u2 -13.48 0.3009 14.56 -2.005 -1.078 1.704
v2 -2.306 -3.972 -2.005 6.635 4.31 -2.662
u3
v3
u4 -2.155 -4.31 -1.078 4.31 3.233 0
v4 -1.704 -5.325 1.704 -2.662 0 7.987
u5
v5
u6
v6
M1
M3
M4
u1 v1 u2 v2 u3 v3 u4 v4 u5 v5 u6 v6
u1 k11 k12 k13 k14 k15 k16
v1 k21 k22 k23 k24 k25 k26
u2 k31 k32 k33+k11+k55 k34+k12+k56 k13 k14 k35+k53 k36+k54 k15+k51 k16+k52
v2 k41 k42 k43+k21+k65 k44+k22+k66 k23 k24 k45+k63 k46+k64 k25+k61 k26+k62
u3 k31 k32 k33 k34 k35 k36
v3 k41 k42 k43 k44 k45 k46
u4 k51 k52 k53+k35 k54+k36 k55+k11+k33 k56+k12+k34 k13+k31 k14+k32 k15 k16 u4 F2
v4 k61 k62 k63+k45 k64+k46 k65+k21+k43 k66+k22+k44 k23+k41 k24+k42 k25 k26 v4 0
u5 k51+k15 k52+k16 k53 k54 k31+k13 k32+k14 k55+k33+k11 k56+k34+k12 k35 k36
u5
0
v5 k61+k25 k62+k26 k63 k64 k41+k23 k42+k24 k65+k43+k21 k66+k44+k22 k45 k46 v5 0
u6 k51 k52 k53 k54 k55 k56 u6 F1
v6 k61 k62 k63 k64 k65 k66 v6 0
u1 v1 u2 v2 u3 v3 u4 v4 u5 v5 u6 v6
u1
v1
u2 3.233 0 -1.078 4.31 -2.155 -4.31
v2 0 7.987 1.704 -2.662 -1.704 -5.325
u3
v3
u4 -1.078 1.704 14.56 -2.005 -13.48 0.3009
v4 4.31 -2.662 -2.005 6.635 -2.306 -3.972
u5 -2.155 -1.704 -13.48 -2.306 15.64 4.009
v5 -4.31 -5.325 0.3009 -3.972 4.009 9.297
u6
v6
M2
= x
u1 v1 u2 v2 u3 v3 u4 v4 u5 v5 u6 v6
u1 15.64 4.009 -13.48 -2.306 0 0 -2.155 -1.704 0 0 0 0
v1 4.009 9.297 0.3009 -3.972 0 0 -4.31 -5.325 0 0 0 0
u2 -13.48 0.3009 33.433 2.004 -13.48 -2.306 -2.156 6.014 -4.31 -6.014 0 0
v2 -2.306 -3.972 2.004 23.919 0.3009 -3.972 6.014 -5.324 -6.014 -10.65 0 0
u3 0 0 -13.48 0.3009 14.56 -2.005 0 0 -1.078 1.704 0 0
v3 0 0 -2.306 -3.972 -2.005 6.635 0 0 4.31 -2.662 0 0
u4 -2.155 -4.31 -2.156 6.014 0 0 33.433 2.004 -26.96 -2.0051 -2.155 -1.704 u4 9.81
v4 -1.704 -5.325 6.014 -5.324 0 0 2.004 23.919 -2.0051 -7.944 -4.31 -5.325 v4 0
u5 0 0 -4.31 -6.014 -1.078 4.31 -26.96 -2.0051 33.433 2.004 -1.078 1.704
u5
0
v5 0 0 -6.014 -10.65 1.704 -2.662 -2.0051 -7.944 2.004 23.919 4.31 -2.662 v5 0
u6 0 0 0 0 0 0 -2.155 -4.31 -1.078 4.31 3.233 0 u6 1.226
v6 0 0 0 0 0 0 -1.704 -5.325 1.704 -2.662 0 7.987 v6 0
Solving the algebraic system the displacement vector is obtained:
u4
=
1.590322092
v4 0.459324661
u5 1.415161681
v5 -0.344777906
u6 2.983102525
v6 0.228693924
= x
X105N
X105N X10-9s2/kg (X 10-4m)
