Element Assembly11

8/10/2019 Element Assembly11

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The Finite Element Method

Element assembly

Element assemblyComputational Mechanics, AAU, EsbjergThe Finite Element Method


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Structure Stiffness matrix

Number of elements NELE=3

Number of FE nodes NNODE=4

Total Number of Degrees of Freedom NDOF=4x2=8

Number of Degrees of Freedom per node NDOF=2

K1 L1 K2

N4


N1 N2N3 E2E1 E3

i j

i j

i j


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1

3

4

1

2

2

x

y

5000 N

5000 N

Two Element Model with Equivalent Nodal Loads


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i = 1

1

j = 3m = 2

Element 1


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=

435130356040070

1302407010060140

3570350070601000100600

400600604000

701407000140

91.075000k

)1(



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i = 1

2

j = 4

m = 3Element 2


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=

400040060060

014070140700

4007043513035606014013024070100

0703570350

600601000100

91.075000k

)2(



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Assembly

[ ]

++

++

=

)2(

22

)2(

23

)2(

21

)2(

32

)2(

33

)1(

22

)1(

23

)2(

31

)1(

21

)1(

32

)1(

33

)1(

31

)2(12

)2(13

)1(12

)1(13

)2(11

)1(11

0

0

kkk

kkkkkk

kkk

kkkkkk

K



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Systematic Assembly of the System

Stiffness Matrix Assembly of the stiffness matrix, [K] follows a pattern

based on element-node number connectivity shown

in the table.

1 2 3

1 k1 2 k2 3 k3 4P

Element (e) Nodes 1e, 2e

1 1, 2

2 2, 3

3 3, 4

k1

-k1

-k1

k1+ k2

-k2

-k2

-k3

k3-k3

k2+ k3

[ ]=K

1 2 3 4

2

1

3

4



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Systematic Assembly of the

System Stiffness MatrixRepeat previous problem with a different

node numbering system:

1 2 3

3 k1

1 k2

4 k3

2

P



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Systematic Assembly of the

System Stiffness MatrixRepeat previous problem with different node

numbering system:

Element(e) Nodes 1e, 2e

1 3, 1

2 1, 4

3 4, 2

1 2 3

3 k1 1 k2 4 k3 2P


k1-k1

-k1k1+ k2 -k2

-k2

-k3k3

-k3 k2+ k3

1 2 3 4

2

1

3

4

[ ]=K


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Example

1 2 31 2

11 u,X 33 u,X

1k 2k22 u,X

Force equilibrium w/ load-displacement relationship:

=

3

2

1

22

2211

11

3

2

1

3222112

3223

2111

u

u

u

kk0

kkkk

0kk

X

X

X

or

uukuukX

uukX

uukX

The stiffness matrix of a system without constraints is singular.

Symmetric

+ diagonal elements

Simple way to assemble

stiffness matrix



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Example: Assembly of the Stiffness Matrix

1 2 31 2

11 u,X 33 u,X

1k 2k22 u,X

Step 1: Define the element stiffness matrices.

=

=

11

11kK,

11

11kK 2

21

1

[ ]

=

=

22

2211

11

21

kk0

kkkk0kk

110

110000

k

000

011011

kK

(Step 2: Align the element coordinates with the global coord.)

Step 3: Expand into the global DOF.

Banded:

Bandwidth

of 3

Force equilibrium and displacement compatibility is maintained.Element assembly

Computational Mechanics, AAU, EsbjergThe Finite Element Method

E l A bl f h S iff M i


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Example: Assembly of the Stiffness MatrixOrdering of the nodes and its impact

1 311 u,X 22 u,X

2

33 u,X

1 2

1k 2k

Step 1: Define the element stiffness matrices.

=

=

11

11kK,

11

11kK 2

21

1

[ ]

=

=

2121

22

11

21

kkkk

kk0

k0k

110

110

000

k

101

000

101

kK

(Step 2: Align the element coordinates with the global coord.)

Step 3: Expand into the global DOF.

Full:Bandwidth

of 5

The node ordering influences the stiffness matrix characteristics.


E l S i f l l ti


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Example: Spring force calculation

1 2 31 2

1k 2kR P

The stiffness equation:

=

=

P

0

R

X

X

X

u

u

u

kk0

kkkk

0kk

3

21

3

21

22

221111

0u1=

=

P

0

u

u

kk

kkk

3

2

22

221

21

3

1

2k

P

k

Pu,

k

Pu

Apply the prescribed displacement or the boundary condition

The stiffness equation becomes

Solve the system of equations to get (the fundamental solution)

Either displacement or load

(not both) is known at each DOF


E l S i f l l ti ( t )


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Example: Spring force calculation (cont.)

1 2 31 2

1k 2k

R P

The stiffness equation:

=

=

P

0

R

X

X

X

u

u

u

kk0

kkkk

0kk

3

2

1

3

2

1

22

2211

11

P

k

PkukukR

1

12111

=

=

=

=

P

P

kP

0

kk

kk

u

u

kk

kk

u

u

kk

kk

X

X

111

11

2

1

11

111

2

111

2

1

Spring 1:

The reaction:

Internal spring forces:

=

=

=

=

P

P

k

P

k

P

kP

kk

kk

u

u

kk

kk

u

u

kk

kk

X

X

21

1

22

22

3

2

22

222

2

122

2

1

Spring 2:

X1 X2= P

P


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Element Assembly11