4 MS lect5 mo id 37217 Nieznany (2)

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Computational Methods

Bar Elements & Trusses

Małgorzata Stojek

Cracow University of Technology

March 2012

MS

(L-53 CUT)

FEM

03/2012

1 / 26

Bar Member - Nomenclature & Notation

axial rigidity EA

u(x)

q(x)

x

L

cross
section

P

Quantity

Meaning

x

Longitudinal bar axis

u

(

x

)

Axial displacement

q

(

x

)

Distributed axial force, given per unit of bar length

L

Total length of bar member

E

Elastic modulus

A

Cross section area, may vary with x

P

Prescribed end load

MS

(L-53 CUT)

FEM

03/2012

2 / 26

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Bar Member - Strong Formulation

Quantity

Meaning

e

=

du/dx

=

u

Infinitesimal axial strain

σ

=

Ee

=

Eu

Axial stress

F

=

A

σ

=

EAe

=

EAu

Internal axial force

Definition

Governing Equations:

Kinematic

e

=

du/dx

=

u

Constitutive

σ

=

Ee

Equilibrium

F

+

q

=

0

Definition

(S):

− (

EAu

)

=

q

on

(

0, L

)

u

(

0

) =

0

at x

=

0

EA u

(

L

) =

P

at x

=

L

MS

(L-53 CUT)

FEM

03/2012

3 / 26

Bar Member - Weak Formulation

RECALL:

Z

L

0

EAu,

x

w ,

x

dx

=

Z

L

0

qw dx

+

EAu,

x

(

L

)

w

(

L

)

EAu,

x

(

0

)

w

(

0

)

P

0

←−

[

bar member

]

P

L

−→

global

coordinate system

x

−→

P

0

−→

[

bar member

]

P

L

−→

Z

L

0

EAu,

x

w ,

x

dx

=

Z

L

0

qw dx

+

P

L

w

(

L

) +

P

0

w

(

0

)

MS

(L-53 CUT)

FEM

03/2012

4 / 26

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c

Felippa

Introduction to FEM

FEM Displacement Trial Function

End node 1 assumed fixed

Axial displacement plotted normal to

x

for visualization convenience

(1)

(2)

(3)

(4)

u

= 0

1

u

2

u

3

u

4

u

5

u

1

f

1

u

3

f

3

u

4

f

4

u

5

f

5

u

2

f

2

x

u(x)

u

IFEM Ch 11 Slide 10

2

3

4

1

5

MS

(L-53 CUT)

FEM

03/2012

5 / 26

c

Felippa

Introduction to FEM

Element Shape Functions

1

2

(e)

1

0

0

1

N

e

N

e

L

e

IFEM Ch 11 Slide 12

1

2

=

ξ

1−ξ

ξ

MS

(L-53 CUT)

FEM

03/2012

6 / 26

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Element Stiffness Matrix

Localization & Strain-Displacement Matrix

element DOFs and shape functions,

N

e

1

=

x

x

2

x

1

x

2

=

1

ξ,

N

e

2

=

x

x

1

x

2

x

1

=

ξ

local interpolant

u

e

(

x

) =

N

e

1

N

e

2





u

1

u

2



=

N

e

u

e

strain-displacement

matrix, B

e

=

d N

e

dx

du

e

(

x

)

dx

=



dN

e

1

dx

dN

e

2

dx

  u

1

u

2



=

B

e

u

e

MS

(L-53 CUT)

FEM

03/2012

7 / 26

Element Stiffness Matrix

Natural Coordinates & Change of Variables

natural coordinates,

x

∈ (

0, L

)

,

ξ

∈ (

0, 1

)

,

h

=

l

ξ

=

x

x

1

h

,

d

ξ

=

1
h

dx

RECALL: change of variables - rules of calculus for any f

(

x

)

Z

h

0

f

(

x

)

dx

=

h

Z

1

0

f

(

ξ

)

d

ξ

;

df

(

x

)

dx

=

df

(

ξ

)

d

ξ

d

ξ

dx

=



1
h



df

(

ξ

)

d

ξ

MS

(L-53 CUT)

FEM

03/2012

8 / 26

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Element Stiffness Matrix

Strain-Displacement Matrix

chain rule

B

e

=

d N

e

dx

=



1
h



d

d

ξ

N

e

1

N

e

2



symbolic derivation

B

e

=



1
h



d

d

ξ

1

ξ

ξ



results in

B

e

=

1
h

1 1



MS

(L-53 CUT)

FEM

03/2012

9 / 26

Bar Element Stiffness Matrix I

Fact

a

(

w

e

,

u

e

) =

Z

h

0

dw

e

dx

EA

du

e

dx

dx ,

w

e

∈ {

N

e

1

,

N

e

2

}

Fact

For B

=

B

1

B

2





B

1

B

2



B

1

B

2



=



B

1

B

1

B

1

B

2

B

2

B

1

B

2

B

2



Definition

Bar Element Stiffness Matrix

K

e

2

×

2

=

Z

h

0

(

EA

)

B

T

B dx

MS

(L-53 CUT)

FEM

03/2012

10 / 26

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Bar Element Stiffness Matrix II

EA=const

RECALL:

B

=

1
h

1 1



change of variables

K

e

=

Z

h

0

(

EA

)

B

T

B dx

= (

h

)

Z

1

0

(

EA

)

B

T

B d

ξ

prismatic bar, i.e. EA

=

constant

K

e

=

EA

h



1
h



2

Z

1

0



1

1



1 1



d

ξ

K

e

=

EA

h



1

1

1

1



MS

(L-53 CUT)

FEM

03/2012

11 / 26

Element Load Vector

RECALL:

(

q, w

e

) =

Z

h

0

qw

e

dx ,

w

e

∈ {

N

e

1

,

N

e

2

}

element load vector due to internal load, q

(

x

)

:

F

e

q

=

Z

h

0

qN

T

dx

change of variables

F

e

q

=

h

Z

1

0

q



1

ξ

ξ



d

ξ

for q

(

x

) =

const

F

e

q

=

qh

Z

1

0

1

ξ

ξ

!

d

ξ

=

qh

1
2
1
2

!

MS

(L-53 CUT)

FEM

03/2012

12 / 26

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Interpretation

c

Felippa

EA

L



1

1

1

1

 

u

i

u

j



=



F
F



d

=

u

j

u

i

F

=

k

s

d

MS

(L-53 CUT)

FEM

03/2012

13 / 26

2D Bar Element

Local 2D Generalization

Fact

1D local interpolant

u

e

(

x

) =

N

e

1

N

e

2





u

1

u

2



=

N

e

u

e

Definition

2D local interpolant

u

e

(

x

) =

N

e

1

0 N

e

2

0



u

1

v

1

u

2

v

2

=

N

e

d

e

MS

(L-53 CUT)

FEM

03/2012

14 / 26

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2D Bar Element

Local 2D Generalization

Definition

2D strain-displacement matrix, B

e

=

d N

e

d x

d u

e

(

x

)

d x

=



dN

e

1

d x

0

dN

e

2

d x

0



u

1

v

1

u

2

v

2

=

B

e

d

e

Definition

2D Bar Element Stiffness Matrix

K

e
4

×

4

=

Z

h

0

(

EA

)

B

T

B d x

MS

(L-53 CUT)

FEM

03/2012

15 / 26

2D Bar Element

Local 2D Stiffness Matrix

1D

B

=

1
h

1 1



K

e

=

EA

h

Z

1

0



1

1



1 1



d

ξ

=

EA

h



1

1

1

1



2D

B

=

1
h

1 0 1 0



K

e

=

EA

h

Z

1

0

1

0
1
0

1 0 1 0



d

ξ

=

EA

h

1

0

1

0

0

0

0

0

1

0

1

0

0

0

0

0

MS

(L-53 CUT)

FEM

03/2012

16 / 26

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2D Bar Element

Local 2D Load Vector

RECALL:

F

e

q

=

Z

h

0

qN

T

dx

1D

N

=

N

e

1

N

e

2



=

1

ξ

ξ



F

e

q

=

h

Z

1

0

q



1

ξ

ξ



d

ξ

q

=

const

=

qh

Z

1

0

1

ξ

ξ

!

d

ξ

=

qh

1
2
1
2

!

2D

N

=

N

e

1

0 N

e

2

0



=

1

ξ

0 ξ 0



F

e

q

=

qh

Z

1

0

1

ξ

0
ξ
0

d

ξ

=

qh

1
2

0

1
2

0

=

F

e
q

1

0

F

e
q

2

0

MS

(L-53 CUT)

FEM

03/2012

17 / 26

Rotation of Coordinate System

α

— the angle of rotation

(

x , y

)

global

coord.

α

rotated

−→

(

x , y

)

local

coord.



x
y



=



cos α

sin α

sin α cos α





x
y



For c

=

cos α, s

=

sin α

A

=



cos α

sin α

sin α cos α



=



c

s

s

c



RECALL: rotation is an

orthonormal

transformation

A

1

=

A

T

MS

(L-53 CUT)

FEM

03/2012

18 / 26

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2-D Bar Element

Globalization: Displacement Transformation

u

xi

u

xi

c

u

yi

s

u

yi

u

xi

s

u

yi

c

u

x j

u

x j

c

u

yj

s

u

yj

u

x j

s

u

yj

c

Node displacements transform as

i

x

y

c

cos

s

sin

in which

Globalization: Displacement Transformation

Introduction to FEM

x

y

j

u

xi

u

yi

u

xj

u

yj

u

yi

u

xi

u

xj

u

yj

IFEM Ch 2 Slide 18

α

_

_

_

_

_

_

T

=

c

s

0

0

s

c

0

0

0

0

c

s

0

0

s

c

u

x

i

u

y

i

u

x

j

u

y

j

=

c

s

0

0

s

c

0

0

0

0

c

s

0

0

s

c

u

x

i

u

y

i

u

x

j

u

y

j

d

e

=

T

d

e

MS

(L-53 CUT)

FEM

03/2012

19 / 26

2-D Bar Element

Globalization: Load Vector Transformation

Globalization: Force Transformation

Node forces transform as

or

x

y

i

j

f

xi

f

yi

f

x j

f

yj

f

xi

f

yi

f

xj

f

yj

c

s 0

0

s

c

0

0

0

0

c

s

0

0

s

c

Note:

global on LHS,

local on RHS

Introduction to FEM

f

xi

f

yi

f

x j

f

yj

f

yi

f

xi

f

xj

f

yj

f

(

T

)

f

e

e T

e

_

IFEM Ch 2 Slide 20

α

_

_

_

_

T

=

c

s

0

0

s

c

0

0

0

0

c

s

0

0

s

c

f

x

i

f

y

i

f

x

j

f

y

j

=

c

s

0

0

s

c

0

0

0

0

c

s

0

0

s

c

f

x

i

f

y

i

f

x

j

f

y

j

f

e

=

T

T

f

e

MS

(L-53 CUT)

FEM

03/2012

20 / 26

background image

2-D Bar Element I

Globalization: Stiffness Matrix Transformation

RECALL:

global

&

local

coordinate systems

K

e

·

d

e

=

f

e

.

formal replacement

d

e

=

T

d

e

K

e

T

d

e

=

f

e

;

multiplication by T

T

T

T

K

e

T

d

e

=

T

T

f

e

;

replacement

f

e

=

T

T

f

e

T

T

K

e

T

d

e

=

f

e

;

leads to

K

e

·

d

e

=

f

e

;

where

K

e

=

T

T

K

e

T

MS

(L-53 CUT)

FEM

03/2012

21 / 26

2-D Bar Element II

Globalization: Stiffness Matrix Transformation

RECALL:

K

e

=

T

T

K

e

T

K

e

=

EA

h

×

c

s

0

0

s

c

0

0

0

0

c

s

0

0

s

c

T

1

0

1 0

0

0

0

0

1 0

1

0

0

0

0

0

c

s

0

0

s

c

0

0

0

0

c

s

0

0

s

c

K

e

=

EA

h

c

2

cs

c

2

cs

cs

s

2

cs

s

2

c

2

cs

c

2

cs

cs

s

2

cs

s

2

MS

(L-53 CUT)

FEM

03/2012

22 / 26

background image

2-D Bar Element III

Globalization: Load Vector Transformation

RECALL:

F

e

q

=

F

e
q

1

0

F

e
q

2

0

,

f

e

=

T

T

f

e

F

e

q

=

c

s

0

0

s

c

0

0

0

0

c

s

0

0

s

c

F

e
q

1

0

F

e
q

2

0

=

F

e
q

1

c

F

e
q

1

s

F

e
q

2

c

F

e
q

2

s

MS

(L-53 CUT)

FEM

03/2012

23 / 26

Nonhomogeneous Natural BCs & Concentrated Forces

Global Coordinate System

1D

global

coordinate system

x

−→

P

0

−→

[

bar member

]

P

L

−→

R

L

0

EAu,

x

w ,

x

dx

=

R

L

0

qw dx

+

P

L

w

(

L

) +

P

0

w

(

0

)

Kd

=

F

q

added after assembly

+

W

2D

global

coordinate system

Kd

=

F

q

added after assembly

+

W

MS

(L-53 CUT)

FEM

03/2012

24 / 26

background image

Postprocessing in Global Coordinate System

Siły przyw ˛ezłowe W

−→

W

x

1

,

W

y

1

α

−→

W

x

2

,

W

y

2

K

e

d

e

=

F

e

=

F

e

q

+

W

e

W

e

=

W

e

x

1

W

e

y

1

W

e

x

2

W

e

y

2

=

K

e

d

e

F

e

q

MS

(L-53 CUT)

FEM

03/2012

25 / 26

Finite Element Program

c

Felippa

The Direct Stiffness Method (DSM) Steps

(repeated here for convenience)

Disconnection

Localization

Member (Element) Formation

Globalization

Merge

Application of BCs

Solution

Recovery of Derived Quantities

Breakdown

Assembly &

Solution

Introduction to FE

post-processing

steps

processing

steps

conceptual

steps

IFEM Ch 3 Slide 2

MS

(L-53 CUT)

FEM

03/2012

26 / 26


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