This document provides information and equations for computing the stiffness matrix of a finite element made up of 4 identical triangles. It defines the geometry and material properties of the element. It then shows the calculations to derive the elasticity matrix coefficients and the individual components of the stiffness matrix based on the geometry, properties, and equations provided. The stiffness matrix is then computed for a given example where N=5. Forces on the element are also calculated based on the pressures and geometry.

1. 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 200000
daN
cm
2
 neu  0.16 A 37.5m
2

Elasticity matrix coefficients computed using below formulae:
e11 E
(1  neu)
(1  neu)(1  2neu)
  e22 e11
e12 neuE
(1  neu)
(1  neu)(1  2neu)
  e21 e12
e33
E
2(1  neu)

e
e11
e21
0
e12
e22
0
0
0
e33






2.13 10
10

3.408 10
9

0
3.408 10
9

2.13 10
10

0
0
0
8.621 10
9









  Pa
x1  0 y1  0
x2
b
2
 y2  0
y3
h
2

x3
b
6

Paul Ionescu
Gr. 1 N=5

2. 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      a61a16
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

3. 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 10
12

6.014 10
11

2.022 10
12

3.458 10
11

3.233 10
11

2.556 10
11

6.014 10
11

1.395 10
12

4.513 10
10

5.958 10
11

6.466 10
11

7.987 10
11

2.022 10
12

4.513 10
10

2.184 10
12

3.007 10
11

1.616 10
11

2.556 10
11

3.458 10
11

5.958 10
11

3.007 10
11

9.952 10
11

6.466 10
11

3.993 10
11

3.233 10
11

6.466 10
11

1.616 10
11

6.466 10
11

4.849 10
11

0
2.556 10
11

7.987 10
11

2.556 10
11

3.993 10
11

0
1.198 10
12























 N
k
t
4A
 a
k
1.564 10
10

4.009 10
9

1.348 10
10

2.306 10
9

2.155 10
9

1.704 10
9

4.009 10
9

9.297 10
9

3.009 10
8

3.972 10
9

4.31 10
9

5.325 10
9

1.348 10
10

3.009 10
8

1.456 10
10

2.005 10
9

1.078 10
9

1.704 10
9

2.306 10
9

3.972 10
9

2.005 10
9

6.635 10
9

4.31 10
9

2.662 10
9

2.155 10
9

4.31 10
9

1.078 10
9

4.31 10
9

3.233 10
9

0
1.704 10
9

5.325 10
9

1.704 10
9

2.662 10
9

0
7.987 10
9























kg
s
2

p1
h9.81
m
s
2
 1000
kg
m
3
4
 p2
3h9.81
m
s
2
 1000
kg
m
3
4

p1 4.905 10
4
  Pa p2 1.471 10
5
  Pa
F1
p1
2
h1m
4
 1.226 10
5
   N F2
p2  p1
2
h
2
 t 9.81 10
5
   N

4. 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

5. 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

6. 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

7. 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)