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

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