# Finite Element Anslysis (Solved Problems)

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Finite Element Anslysis

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• 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