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Finite Element Methods for EngineersExercise 3

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Overview

Repetition

Task 1Task 2

2 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16

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Procedure for the tasks in this exercise

1.Determine all relevant information for each element e

3 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16

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Procedure for the tasks in this exercise

1.Determine all relevant information for each element e

2.Derive/Look up the element stiffness matrices for each used element type

3 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16

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Procedure for the tasks in this exercise

1.Determine all relevant information for each element e

2.Derive/Look up the element stiffness matrices for each used element type

3.Calculate the element stiffness matrix for all elements

3 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16

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Procedure for the tasks in this exercise

1.Determine all relevant information for each element e

2.Derive/Look up the element stiffness matrices for each used element type

3.Calculate the element stiffness matrix for all elements

4.Assemble the global stiffness matrix K

3 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16

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Procedure for the tasks in this exercise

1.Determine all relevant information for each element e

2.Derive/Look up the element stiffness matrices for each used element type

3.Calculate the element stiffness matrix for all elements

4.Assemble the global stiffness matrix K

5.Prepare the whole equation system r= Ka

3 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16

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Procedure for the tasks in this exercise

1.Determine all relevant information for each element e

2.Derive/Look up the element stiffness matrices for each used element type

3.Calculate the element stiffness matrix for all elements

4.Assemble the global stiffness matrix K

5.Prepare the whole equation system r= Ka

6.Prepare the reduced equation system r= Ka (Delete rows/columns with 0-displacements)

3 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16

7/24/2019 FEM exercises

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Procedure for the tasks in this exercise

1.Determine all relevant information for each element e

2.Derive/Look up the element stiffness matrices for each used element type

3.Calculate the element stiffness matrix for all elements

4.Assemble the global stiffness matrix K

5.Prepare the whole equation system r= Ka

6.Prepare the reduced equation system r= Ka (Delete rows/columns with 0-displacements)

7.Solve the reduced equation system r= Ka

3 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16

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Element stiffness matrix of a rod

Ke = EA

L

cos()

2

cos()sin() cos()2

cos()sin()cos()sin() sin()2 cos()sin() sin()2 cos()2 cos()sin() cos()2 cos()sin()

cos()sin() sin()2 cos()sin() sin()2

4 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16

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Overview

Repetition

Task 1Task 2

5 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16

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Task 1

Calculate the displacement and force vector.

1

2

3

1 2

x

y

L

F

E =210 000 N/mm2,

A1 =25 mm2,

A2 =50 mm2,

=135,

L =400 mm,

F =250 N

6 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16

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Task 1

Solution

1.Determine all relevant information for each element eElement e e sin e cos e Le Ae Ee

1 90 1 0 L A1 E2 135

2/2

2/2

2L 2A1 E

7 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16

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Task 1

Solution

1.Determine all relevant information for each element eElement e e sin e cos e Le Ae Ee

1 90 1 0 L A1 E2 135

2/2

2/2

2L 2A1 E

5.Prepare the whole equation system r= Ka

F1,xF1,yF2,xFF3,xF3,y

=

2A1E

2L

0 0 0 0 0 0

0 2 0 2 0 00 0 1 1 1 10

2 1 1 +

2 1 1

0 0 1 1 1 10 0 1 1 1 1

000

a2,y00

7 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16

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Task 1

Solution

1.Determine all relevant information for each element eElement e e sin e cos e Le Ae Ee

1 90 1 0 L A1 E2 135

2/2

2/2

2L 2A1 E

5.Prepare the whole equation system r= Ka

F1,xF1,yF2,xFF3,xF3,y

=

2A1E

2L

0 0 0 0 0 0

0 2 0 2 0 00 0 1 1 1 10

2 1 1 +

2 1 1

0 0 1 1 1 10 0 1 1 1 1

000

a2,y00

6.Prepare the reduced equation system r= Ka

F =

2A1E

2L (1 + 2)a2,y

7 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16

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Task 1

Solution

1.Determine all relevant information for each element eElement e e sin e cos e Le Ae Ee

1 90 1 0 L A1 E2 135

2/2

2/2

2L 2A1 E

5.Prepare the whole equation system r= Ka

F1,xF1,yF2,xFF3,xF3,y

=

2A1E

2L

0 0 0 0 0 0

0 2 0 2 0 00 0 1 1 1 10

2 1 1 +

2 1 1

0 0 1 1 1 10 0 1 1 1 1

000

a2,y00

6.Prepare the reduced equation system r= Ka

F =

2A1E

2L (1 + 2)a2,y

7.Solve the reduced equation system r= Ka

a2,y = 1.1158 102 mm

7 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16

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Task 1

Solution

For the complete vectors follows:

r=

0146.45103.55

250

103.55103.55

N, a=

000

1.1158

00

102 mm

8 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16

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Overview

Repetition

Task 1Task 2

9 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16

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Task 2

Calculate the displacement and force vector.

1

2

3

4

1

23

4

5

x

yL

L

F

E=70000N/mm2,

A =50 mm2

, =30,

=60,

L =500 mm,

F =500 N

10 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16

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Task 2

Solution

1.Determine all relevant information for each element eElement e e sin e cos e Le Ae Ee

1 0 0 1

3L/2 A E2 90 1 0 L/2 A E3 30 1/2

3/2 L A E

4 90 1 0 L A E5 150 1/2

3/2 L A E

11 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16

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Task 2

Solution

6.Prepare the reduced equation system r= Ka

00F

00

=

AE

4L

5 0 0

3 10 8/

3 0 0 0

0 0 8 0 83 0 0 6 0

1 0 8 0 10

a1,ya2,xa2,ya3,xa3,y

12 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16

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Task 2

Solution

6.Prepare the reduced equation system r= Ka

00F

00

=

AE

4L

5 0 0

3 10 8/

3 0 0 0

0 0 8 0 83 0 0 6 0

1 0 8 0 10

a1,ya2,xa2,ya3,xa3,y

7.Solve the reduced equation system r= Kaand derivation of the vectors

r=

433.0100

50000

433.01500

N, a=

00.0357

00.19640.01030.1607

00

mm

12 of 12 Finite Element Methods for Engineers | Prof. Dr.-Ing. Mikhail Itskov | Department of

Continuum Mechanics | WS 2015/16