ch14c-finite eelement modeling of structure-beam

8/13/2019 Ch14C-Finite Eelement Modeling of Structure-Beam

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Element Matrices in Global Coordinates:

Coordinate Transformation

Transformation Matrix of Axial Deformation and Bending of

Beam on Plane

Nodal Displacement in Global Coordinate System

Nodal Displacement in Local Coordinate System

3U

2U

i

jY

X

4U

x5

U 6

U

1U

2u

5u

i

j

y x 3u

6u

1u

4u


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3/19

3

}{}{

}{}{

654321

654321

UUUUUUU

uuuuuuu

T

T

66

654321

1.....

....

....

...1..

....

....

][

-

-=

CS

SC

CS

SC

R

UUUUUU

(14.28)

)()(222

ijijYYXXL -

L

YY

L

XXijij

-== sincos

Note: If ][][,0 IR .Energy and Virtual Work in Global Coordinates

]][[][][ RkRK T= ]][[][][ RmRM T=

}{][}{ pRP T

For each element

L

YYS

L

XXC

YYXXL

ij

ij

ijije

-=

-=-

sin

cos

)()(222

]][[][][ 66 eeT

ee TkTK


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]][[][][66 ee

T

ee RmRM

}{][}{ 16 eT

e pRP

}}{{

}]{[}{2

1

}]{[}{2

1

, eeenc

ee

T

ee

ee

T

ee

PUW

UMUT

UKUV

&&

Note: If 0 , then ][][ IR ][][ kK = ][][ mM = }{}{ pP = Energy and virtual wok of element in system displacements:

Axial Deformation and Bending of Beam on Plane

}]{[}{2

1 )(UKUV

eT

e neeTenne LKLK 666)( ][][][][

}]{[}{2

1 )(

UMUT

eT

e ]][[][][)(

ee

T

enn

e

LMLM enc

W,

}{}{)(eT

PU }{][}{ 1)(

eT

ene

PLP :n system DOF


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Total energy and virtual wok of system (structure)

=

=

==

NE

e

eT

NE

e

e

UMU

TT

1

)(

1

}]{[}{2

1 &&

}]{[}{2

1 UMU

T && =

= NEe

eMM1

)( ][][

=

=

==

NE

e

eT

NE

ee

UKU

VV

1

)(

1

}]{[}{

2

1

}]{[}{2

1UKU

T =

= NEe

eKK

1

)(][][

}{}{1

, ext

TNE

eencnc

PUWW =

=NE

eext

eT PPU1

)( }{}{}{

}{}{ PU T }{}{}{1

)( ext

NE

e

e PPP = }{

}{}{}{P

U

V

U

T

U

T

dt

d

-

& Lagrange Equations

}{}]{[}]{[ PUKUM && System (Structural) Equations of Motion


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Summary

Potential Energy of Element (Beam Bending)

}]{[)()()()(),(

4

3

2

1

4321v

v

v

v

v

xxxxtxv Y

= y

1441412

2

41

4321

}{)],([)]([),(

}{)]([}{),(

=-

=-

-

-

vyxBxx

yx

utx

vx

x

yv

xxxx

y

x

vytxu

x y

yy

BeamForEDvBD

txEtxxx

a][]{][][

),(),(

144111

e

dVBDBk

vkv

vdVBDBv

dVV

V

T

T

V

TT

xV

T

x

]][[][][

}]{[}{2

1

}){]][[][(}{2

1

2

1

==== s

}{}{}{)]()[,(),(),(

}]{[}{2

1}{][][}{

2

1),(

2

1

00

0

2

0

pvdxvxtxpdxtxvtxpW

vmvvdxAvdxtxvAT

TL TTL

nc

TL

TL

r

==

&&&&&


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Lagrange Equations

(1) }]{[}{

2

1ee

T

ee ukuV

]][[][][}]{[}{2

1}]{[}{)2(

e

T

eeee

T

eeeee RkRKUKUVURu

]][[][][}]{][{2

1}]{[}{)3( )()(

ee

T

e

eeT

eee LKLKUKUVULU

][][}]{[}{2

14(

1

)(

1

NEe eTNEe e KKUKUVV) V }{}{}{]{}{}{}{

1

)(

ext1

, ext

NE

e

eT

nc

TNE

eencnc PPPPUWPUWW =

(5) }{}{}{}{

PU

V

U

T

U

T

dt

d

-

&

}{]]{[}]{[ PUKUM && Principle of Virtual Work

(1) }]{[}{ee

T

ee ukuV

]][[][][}]{[}{}]{[}{)2(e

T

eeee

T

eeeee TkTKUKUVURu

]][[][][}]{][{}]{[}{)3( )()(ee

T

e

eeT

eee LKLKUKUVULU

][][}]{[}{)4(1

)(=

= NEe

eT KKUKUV (5) 0

inertianc WVW


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14.8 Finite Element Solutions for Natural Frequencies and Modes

a. A hand-crank 2-DOF solution based on the finite elementmodel

b. The effect of increasing the number of degrees-of-freedom in a

finite element model based on a consistent mass matrix.

c. The effect of reducing out freedoms by using the Guyan

reduction method.

d. The comparison of lumped mass models with consistent mass

models

Example14.7 Accuracy of 2-DOF Model

a. One Element

Element

DOF SystemDOF1 Constrained

2 Constrained

3 1

4 2

=

-

-

-

-0

0

46

612

422

22156

4202

1

23

2

1

2 U

U

LL

L

L

EI

U

U

LL

LAL (1)

),v( tx

1U

2U

L

3 4 1 2


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b. Natural Frequencies

0}]{[}]{[ uKuM && (2)ttU wcos)( = (3)

}0{}]{[ fMK EI

AL

240

42rm = (4)0]det[ MK m

- --- - 00422 2215646 612det 21

22 ffm LL LLL L (5)

0)226()44)(15612( 222 mLLLL (6)

88457.2

1097147.2

2

2

1 -

mm

=

=4

2

2

4

2

1

52.1211

4802.12

AL

EI

AL

EI

rwrw

2/1

42

2/1

41

81.34

533.3

=

=

AL

EI

AL

EI

rw

rw

2/1

42

2/1

41

03.22

516.3

=

=

AL

EI

AL

EI

exact

exact

rw

rw (7)

FE Model Continuum Model

Comment:

exactFEM)()(

11 w>

exactFEM)()(

22 w>

Why? The FEM model is a constrained model. Therefore it is stiffer

than a real structure.


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Example 14.10 Comparison of Consistent Mass Finite Element Models

of a Uniform Cantilever Beam with Continuum Model

1ILAE r

Figure 14.1. Finite element model of uniform cantilever beam.

Table 14.1 Comparison of Consistent Mass Finite Element Models of a

Uniform Cantilever Beam with Continuum Model

Ne

Mode

No. 1 2 3 4 5

Exact

(Reference 14.1)

1 3.53273 3.51772 3.51637 3.51613 3.51606 3.51602

2 34.8069 22.2215 22.1069 22.0602 22.0455 22.0345

3 75.1571 62.4659 62.1749 61.9188 61.6972

4 218.138 140.671 122.657 122.320 120.903

5 264.743 228.137 203.020 199.860

6 527.796 366.390 337.273 298.566

7 580.849 493.264 416.991

8 953.051 715.341 555.165

9 1016.20 713.079

10 1494.88 890.732

12

3

4 1

2 5 6

Element coordinates

System coordinates

)1( )1( -e

N )( eN

1L


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Example 14.11 Cantilever Beam

DOF condensed out: 3, 4

Table 14.2 Frequencies Obtained by Using Guyan Reduction to Reduce

out Rotations of Uniform Cantilever Beams

Ne

Mode

No. 1 2 3 4 5

Exact

(Reference 14.1)

1 3.56753 3.52198 3.51699 3.51628 3.51611 3.51602

2 22.2790 22.2362 22.0946 22.0573 22.0345

3 62.6685 62.9703 62.2180 61.6972

4 123.545 124.725 120.903

5 205.277 199.860

Comment

-exactFEM

l> - as the number of elements increases, the error becomes smaller- the higher modes have larger errors

4 12 3 5

5.01

=L 5.02

=L

5

5

5 5


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Lumped Mass Models

Table 14.3 Frequencies of a Uniform Cantilever Beam Based onLumped Mass Models\

Ne

Mode

No. 1 2 3 4 5

Exact

(Reference 14.1)

1 2.44949 3.15623 3.34568 3.41804 3.45266 3.51602

2 16.2580 18.8859 20.0904 20.7335 22.0345

3 47.0284 53.2017 55.9529 61.6972

4 92.7302 104.436 120.9035 153.017 199.860

Comments

-exactlumped

l orexactlumped

l- as the number of elements increases, the error becomes smaller- the higher modes have larger errors


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Finite Element ProgramsGeneral Purpose Programs

ADINA: Prof. K-J Bathe, MIT

SAP- 4, 6, 8, 2000: Prof. E. L. Wilson, UC. Berkeley

NASTRAN: NASA

ANSYS, ABAQUS, COSMOS

Special Purpose Programs

STRUDL: Structural AnalysisNEABS: Nonlnear Earthquake Analysis Bridge System

SHAKE: Prof. Schnable, EERC, U.C., Berkeley

TAB/SAP86, ALGOR, GIFTS, IMAGES-3D, MSC/PAL, RM

*MIDAS: Korea


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Comments

(1) The displacement functions for truss and beam elements are

derived from the equation of motion under free load, but those for

other elements are assumed.

((11))The static analysis of a structure consisted of truss or beamelements by the FEM gives the exact solution.

The dynamic analysis does not.

(2)Numbering nodes(3)Storage methods

Controls all sub-programs- half-bandwidth method: Building- skyline method: Bridge(2-Bay)

(4)Structural analysis programs compute ]det[aa

K the stiffness

matrix after the boundary conditions are imposed, to check the

input data.

Numerical (Approximate) Solution of Boundary Value Problem

BCce)stress(forSon0

BCntdisplacemeSon0

EquationalDifferentidomainin][

2

u1

s

==

B

B

fuL

:L Linear Differential Operator

Methods

Finite Element Method (FEM):Physical approximation in domain

Boundary Element Method (BEM):

Physical approximation on boundary

Efficient for problems in infinite domain

Finite Difference Method (FDM):

Mathematical approximation in domain


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ch14c-finite eelement modeling of structure-beam

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