FEM- Farsi

7/22/2019 FEM- Farsi

1/34

1-1 .

n. 1-1 n n n.n .

1-1


2/34

n. .""

. :.

1-2 . . .

. . . .


3/34

2-1 . 1-3. F2F1 ". .))1()2."

1-3 .X :)1-1(F2=K(u1-u2) :)1-2(F1+F2=0 :)1-3 (F1= -K(u1-u2)

))1-1()3-1 :)1-4(

=

2

1

2

1

u

u

kk

kk

F

F

F1F2 F1eF2e. :

)1-5(

=

2

1

2

1

u

u

kk

kk

F

Fe

:)1-6] (u[

e]K[=

e]F[


4/34

]]u1-6 ]e]F ]e]K ( .) .

.1-4. . :

)1-7(

EA

PL=

:)1-8(

K

Fx=F=kx

1-4

)1-9(KL

AE

:)1-10([ ]

L

AE

L

AEL

AE

L

AE

K

=

:

1-5


5/34


6/34

1-6 1-6: . :

K13=13:-k4K12=12:-k1-k2:

+

++

++

4334

323121

421421

KKKK

KKKKKK

KKKKK

3 .33 . .

: 1000A.1-7 .((E=200e7 Pa


7/34

1-7

: . . . . . :A

1-8

:i

ii

L

EAk =

Ki ..]66]k :

]] [K]=[F[ 2345 ))16F:.F

1-9


8/34

.


9/34

.-A2-i .3-1 " --." . .

.

- - .. . . . .1-11

1-11Courant1943 (

...) . ""Finite ElementClough1960 . "" " ."


10/34


11/34

: .:

0=== xzyzzz .

1-12

: :.zx=zy= zz=0 .

.)1-12()(5-1 ))Xi,Yii.1-13 ))ui,vi)(Fxi,Fyii. ..


12/34

1-13 . . ]]k :

)1-13([ ]

=

3

3

2

2

1

1

66

3

3

2

2

1

1

vu

v

u

v

u

K

F

F

F

F

F

F

xij

y

x

y

x

y

x

:

]][[][ kF =

:.)1-14(

++=

++=

yxyxv

yxyxu

654

321

),(

),(

12,.... ())xi,yiiui,vi.i) :


13/34

)1-15(

=

6

5

4

3

2

1

33

33

22

22

11

11

3

3

2

2

1

1

1000

0001

1000

00011000

0001

yx

yx

yx

yxyx

yx

v

u

v

uv

u

]: ] [ ] [ ] 166616 = A

.]]A)1-16([ ] [ ] [ ] 16

1

66 = A

(14-1u(x,y)v(x,y]] . :)( 2321 =++

=

= yx

xx

ux

6354 )( =++=

=

= yx

yy

vy

)()( 654321 yx

xyx

yx

v

y

uy

+++++

=

+

=

53 +=xy

xy

y

x

:


14/34

)1-17(

=

6

5

4

3

2

1

010100

100000

000010

xy

y

x

:)1-18([ ] [ ] [ ] 166313 = C

:1-16]])1-19([ ] [ ] [ ] [ ] 16

1

666313 = AC :]]

)1-20([ ] [ ] [ ] 1666363 = ACB

:

EE

yxx

=

E

v

E

xy

y

=

xy

xy

xyEG

)1(2 +

==

:

)1-21(

+

=

xy

y

x

xy

y

x

E

vEE

vE

v

E

)1(200

01

01

:

)1-22(

=

xy

y

x

xy

y

x

E

2

100

01

01

1

:


15/34

]][[][ D= D .))Stiffness . :

)1-23([ ] [ ] [ ] [ ] 166333 = CD ))0=z :EEE

zyxx

=

EEE

zyxy

+=

0=z

xy

xy

xyEG

)1(2 +

==

:

)1-24(

+

=

xy

y

x

xy

y

xE

)1(2

2100

011

01

1

)21)(1(

)1(

:133313 ][][][ = D . . "". . Fi

W :.

)1-25 (i

vol

n

i

iextvol FWdW ===

=1int


16/34

)1-26([ ] [ ] [ ]{ [ ]}[ ][ ][ ] == BDB TT :

=

)1-27( dvol .dvol:

)28-1(

)1-28(t

yx

yx

yx

Vol

33

22

11

1

1

1

2

1=

t . :)1-29 ()](][[][][ VolBDBK

T= . - :)1-30(]][][[][ BD= ][]][[ BDH :=

][][][ = H

6-1 . :0=

+

+

ZYX

xzxyxx

0=

+

+

ZYX

yzyyyx

vol

eT dvolBDB }]{]][[][[

y

x

y

x

y

x

F

F

F

F

FF

3

3

2

2

1

1


17/34

0

2 =

+

+

ZYX

zzyxx

-- . . . . . -...- . .

7-10

2

2

=++ cdx

dub

dx

udR2

. . (FiniteDifference) . .

~)0(u.~)x(u c

dx

udb

dx

uda ++

~~

2

2

:.)(

~~

2

2

xRcdx

udb

dx

uda =++

"(R(x )")Residual term )(~ xuu(x). :)(~ xu .(R(x .. ~)()1(1 += xxxu . ~)()1(1 += xxxu

.1)12(1221

~~

2

2

)( +=+=+= xxdx

ud

dx

udx


18/34

.(R(x n])])1,0. R(x):

R(xi)=0 i=1,2,,n

1:2

2

=+udx

ud)0()1(1 uu == . ~)(.: xu

1~~

2

2

)( += udx

udR x

. . n .Collection Method.~u

~)( xu dxxR .)( :

=+1

0

0]1)12([ dxx

][ 0)( 102 =+ xxx

022 =

2

1=

:)1(

2

11~ )( += xu x

= 0)( dxxR :....)1()1(1)(

~ 22

2

1 +++= xxxxxu ,..., 21 .


19/34

. ~)( xu 12 ,,..., n. 12 ,,..., wwwn i:

0)(~

= dxuW xb

a i

win n 12 ,,..., n . iw . ))Galerkin . i iw . x )1(1)(~)1( +== xxuxxw ~)()1()1(1 221 xxxxxu ++= :

)1(1

xxw =)1(22

xxw = :)1(1)(~ += xxxu

)1(1 = xxw

=1

00)( dxxWR

01)12()1(

1

0=+ dxxx :

8

5=

:)1(

8

51)(~ += xxxu

:101)( = xxU .%6/15%5/12

1-7-1 . :1-31.

)1-31(0)( )()()( =+

xxx qcdx

dua

dx

du

:


20/34

)(@,)0( 0)(0 Lxqdx

duauu x ===

1-14

(Q.)(q)a(u)

EA K

0((

128

4

D


21/34

V 2):)1-1

1-31 )( . ))1-1 .1-14 .

1-7-2 i~1-31iu . -n][0,L .- ejueju . ~eu :)1-32()(

1x

m

j

eju

e

j

e u =

=

.ej xBxA. :)1-33( =+

0)](~([

~

)( dxxguCdx

uda

dx

dw

xx

x

x

B

A

~u u .)(w u~uu . :

~u . .


22/34

)1-34(udx

dvv

dx

du

dx

uvd+=

)(

:)1-35(dxdudx

dwa

dx

dua

dx

dw

dx

duwa

dx

d/)()( =

uwu :.1

)1-36(= dxdxdu

adx

dW

x

x

A

B)]([

+=+ dxdxdu

dx

dwa

dx

duwadx

dx

du

dx

dwadx

dx

duwa B

A

B

A

B

A

B

A

x

x

x

x

x

x

x

x ][)(

:))1-37dx

dx

duaw

dx

duawdxwqcwu

dx

du

dx

dwa

AABB

B

Axxxx

x

x

+= )()( )()(

dx

dua

. w

dx.du

a Q.Q:)(

)(2dx

duaQ

e = dx

duaQ

e =1

((Free Body Diagram Qi ( Qi .) :.(-

.)uX.- ( 2 0axa122xa .ai


23/34

ui- ( .)ui :

)1-38(bxaue +=

e

x

ee

x

e uuuuBA

21 , == :Abxau e +=1

)1-39(B

ebxau +=2

=

b

a

x

x

u

u

B

A

e

e

1

1

2

1

:ue)1-40(

e

AB

Ae

AB

Beeeew

e uxx

xxu

xx

xxuuxu 212211 )(

+=+=

AB

B

xx

xx

AB

A

xx

xx

))shape Function. .)(xeu ~u .

n )2)n )(w :)1-41()()( xx

e

ii =

.in :n +

==

ej

ej

n

j

eeej

n

j

ej

eeie

jdx

da QxdxqxuC

dx

du

n

e

x

xB

A )(]))(()1

111

1([ 1

...

)42-1(ej

e

j

n

j

e

n

e

n

e

j

e

n

e

jn

e

j

e

nx

x Qdxqxej

Cdx

du

dx

da xuBA )(])(()([0

1

=

+=

:1-42 ==

ne

i

e

i

e

j

e

ij niQFuk ),...,2,1(0

+= dxCdxddxdaK ejeie

i

e

ix

x

e

ijB

A)(


24/34

)1-43(= dxqF eixxei BA e

ijK

e

iF :

)1-44(

+

=

e

n

e

e

e

n

e

e

e

n

e

e

nnn Q

Q

Q

F

F

F

u

u

u

KK

K

KK

2

1

2

1

2

1

1

21

1212

.........

...

]]F ]]QK . .KKij :

:-1

1

1

+= een uu

(e .)e)(-2

.)(:Q0:Q0.{01 0Qeen QQ =+

eeen Uuu == +11 :

=

+

+

+

E

EE

EEE

u

u

u

KK

KKK

KKK

KKK

K

.

.

.

2

1

2221

1211

1

32

3

11

2

22

2

21

2

12

2

11

1

22

2

21

1

12

1

11


25/34

)45-1(

++

+

+

++

EE

EE

Q

QQQQ

Q

f

ff

ffff

f

2

3

1

2

2

2

1

1

2

1

1

2

1

1

2

3

1

2

2

2

1

1

2

1

1

...

...

))Assembled Model Equations . ij K .1-7-3

0)( =+ kCudx

dua

dx

d

:q

dr

dTkr

dr

d

r 2)2(

1=

)46-1(

00 )(@ TRT = , 0)0(@2 =

dr

dtkr

1-2 .350.00

346.09

334.38

314.84

287.50252.34

209.38

158.59

100.00

352.63

347.42

335.27

315.48

287.95252.65

209.56

158.68

100.00

358.73

348.31

337.90

313.59

289.29249.70

210.12

155.06

100.00

377.78

356.24

335.11

315.28

249.44254.83

197.22

148.61

100.00

433.33

391.67

350.00

308.33

266.67225.00

183.33

141.67

100.00

R

S

0

0,1250.250

0.3750.500

6.625

0.750

0.875

0.000 .))1-2


26/34

))Shape Function8-1 . ))Local Coordinate Variables . . ANSYS.

.1-8-1 .)15-1(

1-15 .71[].1-15 . .)(xu (u(x . )(. . (u(x (u(x . )(. . n )(xu

(u(x n-1 . .n-1 . .


27/34

1-8-2 . . 715-1 .71 . - . . .

1-8-3 ~u 1~u1 2~u2 :1x2x

)1-47(2

12

2

1

12

1 ~~)( uxx

xxu

xx

xxxu

+

=

1-16:L2L1

L

xx

xx

xxxL

=

= 2

12

21 )(


28/34

)1-48(

L

xx

xx

xxxL 1

12

12 )(

=

=

:2211 ~)(~)(~ uxLuxLu += 1-17 L2L1. .

1-17

1-8-4:(L1(x)L2(x

1)( 11 = xL 0)( 11 =xL0)( 11 = xL 1)( 22 =xL L1 L21 .2

: ) 21 xxx


29/34

dxdxxdL

dx

xdL.

)(.

)( 21 dxxLxL )()( 21

. . Local Coordinate System x1. :x2-x1

1-18

)1-49(

,

12

12 0)(,1)(xx

xxx

===

.1-49 .

1-19:

)1-50()1(2

1)(1 =L , )1(

2

1)(2 +=L :

)1-51()()( 2211 LxLxx += :. == 2

1)()(

2

1

2

1

dd

dxLdxxL

3

1])1([

24

1)1(

4

1 11

32 == + d :)1-52( = 3

1)(

2

2 dxxL :])-]1,1(


30/34

)1-53( == 8

1)()(

2

1)()(

2121 dLLdxxLxL

61)1( 2 = d

.

: dxu x)(2.:)()()( 2211 xLuxLuxu +=

uu2,u1 : +=++=+ dxLudxLLuuLuLudxLuLu

2

1

2

12121

2

2

2

2

2

1

2

1

2

2211 =][)2( ++=++=+ ][3

1

6

12

3

12 21

2

2

2

121

2

2312

12121

2

2

2

2 uuuuuuuudxLLuuLu

.21-8-6 2 u :2

)1-54(2210)(

~ xaxaaxu ++=21 xxx .u ~)( xu . . x :

)1-55(2321

~ xxu ++=


31/34

)1-56(

==

==

==

3

~

2

~

1

~

1

2

1

0

uu

uu

uu

x

x

x

:)1-57(

++=

++=

=

3213

2

2212

11

)2

1(

2

1

u

u

u

:)1-58(

332211 )()()(~ uxQuxQuxQu ++= :

)1-59()

11)(

1

21()(1

xxxQ

=

)1

1(1

4)(2

xxxQ

=

)11

2(

1)(3

=

xxxQ

:)1-60(

=

===

1,0'11

1',2

1'3,20

)(1xNodeat

xxNodeatxQ

)1-61(

=

xNodeat

NodeatxQ

1

3,10)(2

)1-62(

= 31

2,10)(3

Nodeat

NodeatxQ

)1-63(10 x1)()()( 321 =++ xQxQxQ )1-16(. :

)1-64()1)(1()(2 +=Q

)1(2

)(1 =

Q


32/34

)1(

2)(3

+=Q

:1-20Q2Q3Q1332211 )()()(

~ uQuQuQu ++=

1-20

7-8-1 .Qi

2. :.)1(

2)(1 =

Q

)(1.1iQi Q )23(0=1=))1= Q1.)1-66(= )()()(1 GFQ

)(1 Q1=.:)1-67(1)( = F )(1 Q0=.:)1-68( =)(G


33/34

n n .2Q1. )(1 Q . )(F)(G.2)1-69()1()(1 = KxQ 11==)(1K Q:

2

1=K

))1-70)1(

2

1)(1 = Q .

.4"":

1-21)(2: Q

)()()()()(2 IHGKFQ =1)1()(01 +=== F )(F === )(0)( 0 GG

2

1)(0)(

2

1 ===

HH

1)(0)( 1 === II

12

12 )(=

=Q:

1)1)(1)(2

1

)(( 21=+

=K

3

81)

2

1)(

2

3)(1)(

2

1(

==

KK

:


34/34

)1)(1)(

2

1(

3

8)(2 +=Q

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