-
ASSIGNMENT 6-1
Find the shape function for the following element using zero
method.
1 =23467
2(1)3(1)4(1)6(1)7(1)
(1) { = 27
. = 9
1 =(9 + )(9 )(27 )()(9 )
(9 27)(9 + 27)(27 + 27)(9)(9 + 9)
1 =(81 2)(27 )()(9 )
5668704
1,4,6,9 =1
5668704. (81
2)(27 + )(9 + )
2 =13467
1(2)3(2)4(2)6(2)7(2)
() { = 9
. = 9
2 =(27 + )(9 )(27 )()(9 )
(27 9)(9 + 9)(27 + 9)(9)(9 + 9)
2 = (729 2)(9 )()(9 )
1889568
2,8,3,7 =1
1889568. (729
2)(9 + )(9 + )
Z1 = 27 + , Z2 = 9 + , Z3 = 9 Z4 = 27
3
= +9
2
= 9
12 11 10
6 7 8 9
5
4 1
7
6
5
4 3 2 1
7 = 9
6 =
= 9+
= 27 = +27
-
=123 7
1( )2( )3( ) ( )7( )
() { = +27
. = 0
=(27 + )(9 + )(9 )(9 + )(9 )
(27 + 27)(9 + 27)(9 27)(9)(9)
= (27 + )(81 2 )()(9 )
2834352
,10 = 1
2834352. (81 2)(27 + )(9 + )
remains constant in node 5 and 10 Therefore the negative sign
remains as well in the
generalized form.
11 =134 7
1(11)3(11)4(11) (11)7(11)
(11) { = 9
. = 0
11 =(27 + )(9 )(27 )(9 + )(9 )
(27 9)(9 + 9)(27 + 9)(9)(9)
11 =(9 )(729 2)(81 2 )
944784
11,12 =1
944784. (9 + )(729
2)(81 2 )
-
ASSIGNMENT 6-3
Find the shape function for the following element:
1 =23 6
2(1)3(1) (1)6(1)
(1) { = 0.75
. = 2
1 =(
10.75
)()(2 )
(0.75)(1
0.75 + 0.75 )(2)(2 + 2)
1 =(
10.75
)()(2 )
4
1, =1
4.
2
0.75
1, =1
4.
2 +
0.75
2 =13 6
1(2)3(2) (2)6(2)
() { = 0.
= 2
2 =(0.75 + ) (
10.75
)()(2 )
(0.75)(1
0.75 + 0 )(2)(2 + 2)
2 = 1
8. ()(18 )
(0.75 + )
(0.75 )
2,6 =1
8. (18 + )
(0.75 + )
(0.75 )
remains with the same value t in node 2 and 6 Therefore there is
no need to generalize solution for value.
Z1 = 0.75 + , Z2 = , Z3 =1
0.75
3
6 7
2
= 0, = 0
= +2 5
= 0
= 2
8 4
1
6
5
4
3 2 1
= 0.75 = 0 =
6 = 9
=
4 = 9+
-
4 =2346
2(4)3(4) (4)6(4)
() { = 0.75
. = 0
4 = (
10.75
)(2 + )(2 )
(0.75)(1
0.75 + 0.75 )(2)(2)
4 = (
10.75
)(4 2 )
2
4 = 1
2. (
0.75 )(18 2 )
8 =1346
2(4)3(4) (4)6(4)
(8) { = 0.
= 0
8 =(0.75 + ) (
10.75
)(2 + )(2 )
(0.75)(1
0.75 + 0 )(2)(2)
8 =(0.75 + ) (
10.75
)(4 2 )
4
8 =1
4.(0.75 + )
(0.75 )(4 2)
-
ASSIGNMENT 6-4
Find the shape function for the following element using Zero
method
1 =23 67
2(1)3(1) (1)6(1)
(1) { = 9
.
= 9
1 =
()(1
9 )(3 + )(3 )(9 )
(9)(1
9 + 9 )(3 9)(3 + 9)(9 + 9)
1 =
()(9 )(9 )
(9 2 )
648
1, =1
648. (9 )
(9 )(9 2 )
1,6 =1
648. (9 2 )
(9 + )
(9 )
2 =13 67
1(2)3(2) (2)6(2)
() { = 0
.
= 9
2 =
(9 + ) (1
9 )(3 + )(3 )(9 )
(9)(1
9 + 0 )(3 9)(3 + 9)(9 + 9)
2 =
(9 + )(9 )
(9 2 )(9 )
1296
2,7 = 1
1296. (9 + )(9
2 ) (9 + )
(9 )
Z1 = 9 + , Z2 = , Z3 =1
9
= +9
7 = 9
5
= +3
6
= 3
= 9
4
8 7
3 2 1
6
5
4
3 2 1
= 9 = 0 =
6 = 3
= 3+
4 = 9+
7
-
4 =23467
2(4)3(4)4(4)6(4)7(4)
() { = 9
.
= 3
4 =
(1
9 )(9 + )(3 )(9 )
(9)(1
9 + 9 )(9 3)(3 + 3)(9 + 3)
4 =
(1
9 )(3 )(81 2 )
216
4 = 1
216. (81 2 )
(3 )
(9 )
4,5 = 1
216. (81 2 )
(3 + )
(9 )
-
ASSIGNMENT 6.5
Find the shape function for the following element using zero
method?
1 = 18 + 4 = 9 + 7 = 9 2 = = 8 =
3 =1
18 6 = 9
9 = 9
= +9
= 0
= 9
= 0
= + 9 6
17
11
19
13
7
15 9
5
16
10
14
8
18
12
1
4
2 3 = -18 = 0 =
= - 9
3 2 1
6
5
4
-
1 =23 689
2(1)3(1) (1)6(1)8(1)9(1)
(1) { = 18 = 9 = +9
1 =()(
118
)()(9 )()(9 + )
(18)(1
18+ 18 )(9)(18)(9)(9 + 9)
1 =(
18
)()(9 )()(9 + )
13122
1, =1
13122. (
18 )()(9 )(9 + )
1,6,14,18 = 1
13122. (
18 )()()(9 + )(9 + )
2 =13 689
1(2)3(2) (2)6(2)8(2)9(2) (2) {
= 0
= 9
= +9
1 =
(18 + )(1
18 )()(9 )()(9 + )
(18)(118 )(9)(18)(9)(9 + 9)
2 =(18 + )(
118
)()(9 )()(9 + )
26244
1 = 1
26244(18 + )(
1
18 )()(9 )()(9 + )
2,7,1 ,17 =1
26244.(18 + )
(18 ) ()()(9 + )(9 + )
4 =234689
2(4)3(4)4(4)6(4)8(4)9(4)
(4) {
= 18
= 0
= +9
4 =
(1
18 )(9 + )(9 )()(9 + )
(18)(1
18 + 18 )(9)(9)(9)(9 + 9)
4 =
(1
18 )(9 + )(9 )()(9 + )
6561
4 = 1
6561. (
18 )(81
2 )()(9 + )
4,16 = 1
6561. (
18 )(81 2 )()(9 + )
=134689
1( )3( )4( )6( )8( )9( )
(5) {
= 0
= 0
= +9
5 =
(18 + )(1
18 )(9 + )(9 )()(9 + )
(18)(118 )(9)(9)(9)(9 + 9)
5 =
(18 + )(1
18 )(9 + )(9 )()(9 + )
13122
5 =1
13122. (
18 +
18 )(81
2 )()(9 + )
,17 =1
13122. (18 +
18 )(81 2 )()(9 + )
-
10 =234679
2(10)3(10)4(10)6(10)7(10)9(10)
(10) {
= 18
= 0
= 0
10 =
(1
18 )(9 + )(9 )(9 )(9 + )
(18)(136 )(9)(9)(9)(9)
10 =
(
18 )(81 2 )(81 2 )
3280.5
10 = 2
6561. (
18 )(81 2 )(81 2 )
11 =134679
1(11)3(11)4(11)6(11)7(11)9(11)
(11) {
= 0
= 0
= 0
11 =
(18 + )(1
18 )(9 + )(9 )(9 )(9 + )
(18)(118 )(9)(9)(9)(9)
11 =
(18 + )(18 )
(81 2 )(81 2 )
6561
11 =1
6561. (18 + )
(18 ) (81 2 )(81 2 )
8 =234679
2(8)3(8)4(8)6(8)7(8)9(8)
(8) {
= 18
= 9
= 0
8 =
()(1
18 )( )(9 )(9 )(9 + )
(18)(1
18 + 18 )(9)(18)(9)(9)
8 =
(
18 )( )(9 )(81 2)
6561
8,12 = 1
6561. (
18 )( )(9 + )(81
2)
9 =134679
1(9)3(9)4(9)6(9)7(9)9(9)
(9) {
= 0
= 9
= 0
9 =
(18 + )(1
18 )( )(9 )(9 )(9 + )
(18)(118 )(9)(9 + 9)(9)(9)
9 =
(18 + )(18 )
( )(9 )(81 2)
13122
9,13 =1
13122. (18 + )
(18 )( )(9 + )(81
2)
-
ASSIGNMENT 6-8
Find Jacobi matrix for the following matrix.
Shape functions of the above element are as follow:
1 =1
4(1 )(1 )
2 =1
4(1 + )(1 )
3 =1
4(1 + )(1 + )
4 =1
4(1 )(1 + )
[
.
] = [
.
] [
.
] [] = [
11
12
.
21
22
]
X (27, -1)
2 1
3 4
(7,-1)
(7, 18) (27, 18)
Y
-Y
-
1
= 1
4(1 )
1
= 1
4(1 )
2
= 1
4(1 )
2
= 1
4(1 + )
3
= 1
4(1 + )
3
= 1
4(1 + )
4
= 1
4(1 + )
4
= 1
4(1 )
11
= 1
4(1 )(7) +
1
4(1 )(27) +
1
4(1 + )(27)
1
4(1 + )(7)
= 7
4(1 ) +
27
4(1 ) +
27
4(1 + )
7
4(1 + )
=20
4(1 ) + +
20
4(1 + )
= 5(1 ) + + 5(1 + )
= 5 5 + 5 + 5
= 10
12
= 1
4(1 )(1) +
1
4(1 )(1) +
1
4(1 + )(18)
1
4(1 + )(18)
=1
4(1 )
1
4(1 ) +
9
2(1 + )
9
2(1 + )
= 0
-
21
= 1
4(1 )(7)
1
4(1 + )(27) +
1
4(1 + )(27) +
1
4(1 )(7)
= 7
4(1 )
27
4(1 + ) +
27
4(1 + ) +
7
4(1 )
= 0
22
= 1
4(1 )(1)
1
4(1 + )(1) +
1
4(1 + )(18) +
1
4(1 )(18)
=1
4(1 ) +
1
4(1 + ) +
18
4(1 + ) +
18
4(1 )
=19
4(1 ) +
19
4(1 + )
=19
419
4 +
19
4+19
4
=19
4+19
4
=19
2
[] = [
10 0 .
0
] Therefore, det [J] = 95
19
20
Dimension
X= 27-7 = 20
Y= 18-(-1) = 19
A = 20 x 19 = 380 Therefore, J =
4=
380
4= 95
-
ASSIGNMENT 6.9:
Determine the Jacobi matrix for the following element?
Shape functions for the above element in each node are as
follow:
1 =1
8(1 )(1 )(1 + ) 2 = 18 (1 + )(1 )(1 +)
3 =1
8(1 )(1 + )(1 +) 4 =
1
8(1 + )(1 + )(1 +)
=1
8(1 )(1 )(1 ) 6 =
1
8(1 + )(1 )(1 )
7 =1
8(1 )(1 + )(1 ) 8 =
1
8(1 + )(1 + )(1 )
[ ]
=
[
]
[ ]
X Z
Y
8
7
6
5
4
1
3
2
1
Point (X, Y, Z)
Point 1 (-9,-18, 27)
Point 2 (18,-18, 27)
Point 3 (-9, 27, 27)
Point 4 (18, 27, 27)
Point 5 (-9,-18, 0)
Point 6 (18,-18, 0)
Point 7 (-9, 27, 0)
Point 8 (18, 27, 0)
-
[] =
[ 11
12
13
21
22
23
31
32
33
]
1
=
1
8(1 )(1 + )
2
=
1
8(1 )(1 +)
3
=
1
8(1 + )(1 +)
4
=
1
8(1 + )(1 +)
5
=
1
8(1 )(1 )
6
=
1
8(1 )(1 )
7
=
1
8(1 + )(1 )
8
=
1
8(1 + )(1 )
1
=
1
8(1 )(1 + ) 2
=
1
8(1 + )(1 +)
3
=
1
8(1 )(1 +)
4
=
1
8(1 + )(1 +)
5
=
1
8(1 )(1 )
6
=
1
8(1 + )(1 )
7
=
1
8(1 )(1 )
8
=
1
8(1 + )(1 )
1
=1
8(1 )(1 )
2
= 1
8(1 + )(1 )
3
=1
8(1 )(1 + )
4
= 1
8(1 + )(1 + )
= 1
8(1 )(1 )
6
= 1
8(1 + )(1 )
7
= 1
8(1 )(1 + )
8
= 1
8(1 + )(1 + )
-
11
=
= 1
8(1 )(1 + )(-9)+
1
8(1 )(1 +)(18)
1
8(1 + )(1 +)(-9)+
1
8(1 + )(1 +)(18)
1
8(1 )(1 )(-9)+
1
8(1 )(1 )(18)
1
8(1 + )(1 )(-9)+
1
8(1 + )(1 )(18)
= 9
8(1 )(1 + )+
9
4(1 )(1 +)+
9
8(1 + )(1 + )+
9
4(1 + )(1 +)+
9
8(1 )(1 )
+9
4(1 )(1 )+
9
8(1 + )(1 )+
9
4(1 + )(1 )
= 9
4(1 + ) [
1
2(1 ) + (1 ) +
1
2(1 + ) + (1 + )]
+9
4(1 ) [
1
2(1 ) + (1 ) +
1
2(1 + ) + (1 + )]
= 9
4(1 + ) [
3
2(1 ) +
3
2(1 + )] +
9
4(1 ) [
3
2(1 ) +
3
2(1 + )]
= 9
4(1 + ) [
3
2
3
2 +
3
2+
3
2] +
9
4(1 ) [
3
2
3
2 +
3
2
3
2]
= 9
4(1 + ) [
3
2+
3
2] +
9
4(1 ) [
3
2+
3
2]
= 9
4(1 + )[3] +
9
4(1 )[3]
= 27
4(1 + ) +
27
4(1 )
= 27
4+
27
4 +
27
4
27
4
= 27
4+
27
4
= 4
4
-
12
=
13
=
= 1
8(1 )(1 + )(27)+
1
8(1 )(1 +)(27)
1
8(1 + )(1 +)(27)+
1
8(1 + )(1 +)(27)
1
8(1 )(1 )(0)+
1
8(1 )(1 )(0)
1
8(1 + )(1 )(0)+
1
8(1 + )(1 )(0)
= 27
8(1 )(1 + ) +
27
8(1 )(1 +)
27
8(1 + )(1 +)(27)+
27
8(1 + )(1 +)
0 0 0+ 0
= 0
21
=
= 1
8(1 )(1 + )(-9)
1
8(1 + )(1 +)(18)+
1
8(1 )(1 +)(-9)+
1
8(1 + )(1 +)(18)
1
8(1 )(1 )(-9)
1
8(1 + )(1 )(18)+
1
8(1 )(1 )(-9)+
1
8(1 + )(1 )(18)
=9
8(1 )(1 + )
9
4(1 + )(1 +)
9
8(1 )(1 +) +
9
4(1 + )(1 +)
+9
8(1 )(1 )
9
4(1 + )(1 )
9
8(1 )(1 )+
9
4(1 + )(1 )
= 0
= 1
8(1 )(1 + )(-18)+
1
8(1 )(1 +)(-18)
1
8(1 + )(1 +)(27)+
1
8(1 + )(1 +)(27)
1
8(1 )(1 )(-18)+
1
8(1 )(1 )(-18)
1
8(1 + )(1 )(27)+
1
8(1 + )(1 )(27)
= 9
4(1 )(1 + )
9
4(1 )(1 +)
27
8(1 + )(1 + )+
27
8(1 + )(1 +)+
9
4(1 )(1 )
9
4(1 )(1 )
27
8(1 + )(1 ) +
27
4(1 + )(1 )
= 0
-
22
=
= 1
8(1 )(1 + )(-18)
1
8(1 + )(1 +)(-18)+
1
8(1 )(1 +)(27)+
1
8(1 + )(1 +)(27)
1
8(1 )(1 )(-18)
1
8(1 + )(1 )(-18)+
1
8(1 )(1 )(27)+
1
8(1 + )(1 )(27)
=9
4(1 )(1 + )+
9
4(1 + )(1 +)+
27
8(1 )(1 +) +
27
8(1 + )(1 +)
+9
4(1 )(1 )+
9
4(1 + )(1 )+
27
8(1 )(1 )+
27
8(1 + )(1 )
=9
4(1 + ) [(1 ) + (1 + ) +
3
2(1 ) +
3
2(1 + )]
+9
4(1 ) [(1 ) + (1 + ) +
3
2(1 ) +
3
2(1 + )]
=9
4(1 + ) [
5
2(1 ) +
5
2(1 + )] +
9
4(1 ) [
5
2(1 ) +
5
2(1 + )]
=9
4(1 + ) [
5
25
2 +
5
2+5
2] +
9
4(1 ) [
5
25
2 +
5
2+5
2]
= 9
4(1 + ) [
5
2+
5
2] +
9
4(1 ) [
5
2+
5
2]
= 9
4(1 + ) [
10
2] +
9
4(1 ) [
10
2]
= 9
4(1 + )(5) +
9
4(1 )(5)
= 45
4(1 + ) +
45
4(1 )
= 45
4+
45
4 +
45
445
4
= 45
4+45
4
= 90
4
-
23
=
= 1
8(1 )(1 +)(27)
1
8(1 + )(1 +)(27)+
1
8(1 )(1 +)(27)+
1
8(1 + )(1 +)(27)
1
8(1 )(1 )(0)
1
8(1 + )(1 )(0)+
1
8(1 )(1 )(0)+
1
8(1 + )(1 )(0)
= 27
8(1 )(1 + )
27
8(1 + )(1 +)+
27
8(1 )(1 +)+
27
8(1 + )(1 +)
= 0
31
=
=1
8(1 )(1 )(9) +
1
8(1 + )(1 )(18) +
1
8(1 )(1 + )(9) +
1
8(1 + )(1 + )(18)
1
8(1 )(1 )(9)
1
8(1 + )(1 )(18)
1
8(1 )(1 + )(9)
1
8(1 + )(1 + )(18)
=9
8(1 )(1 ) +
9
4(1 + )(1 ) +
9
8(1 )(1 + ) +
9
4(1 + )(1 + )
9
8(1 )(1 )
9
4(1 + )(1 )
9
8(1 )(1 + )
9
4(1 + )(1 + )
= 0
32
=
=1
8(1 )(1 )(18) +
1
8(1 + )(1 )(18) +
1
8(1 )(1 + )(27) +
1
8(1 + )(1 + )(27)
1
8(1 )(1 )(18)
1
8(1 + )(1 )(18)
1
8(1 )(1 + )(27)
1
8(1 + )(1 + )(27)
= 9
4(1 )(1 )
9
4(1 + )(1 ) +
27
8(1 )(1 + ) +
27
4(1 + )(1 + )
+9
4(1 )(1 ) +
9
4(1 + )(1 )
27
8(1 )(1 + )
27
4(1 + )(1 + )
= 0
-
33
=
=1
8(1 )(1 )(27) +
1
8(1 + )(1 )(27) +
1
8(1 )(1 + )(27) +
1
8(1 + )(1 + )(27)
1
8(1 )(1 )(0)
1
8(1 + )(1 )(0)
1
8(1 )(1 + )(0)
1
8(1 + )(1 + )(0)
=27
8(1 )(1 )(27) +
27
8(1 + )(1 )(27) +
27
8(1 )(1 + )(27) +
27
8(1 + )(1 + )(27) 0
=27
8(1 )[(1 ) + (1 + )] +
27
8(1 + )[(1 ) + (1 + )]
=27
8(1 )[1 + 1 + )] +
27
8(1 + )[1 + 1 + )]
=27
8(1 )[1 + 1)] +
27
8(1 + )[1 + 1]
=27
8(1 )(2) +
27
8(1 + )(2)
=54
8(1 ) +
54
8(1 + )
=54
8+54
8 +
54
854
8
=54
8+54
8
=27
2
-
[] =
[ 27
2 0 0
0 4
2 0
0 0 27
2 ]
Therefore, || = 4100.625
Volume of cube element = (XYZ) =27 4527 = 32805
=
8=32805
8= 4100.625
45
Cube element
dimensions:
X= 18-(-9) =27
Y= 27-(-18) = 45
Z= 27-0=27
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