Elementos finitos con computadoras.pdf

8/15/2019 Elementos finitos con computadoras.pdf

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C 0

C 0


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C 0 C 0

N 1 N 1 N 2 N 2

φ

x y

N 1 N 2 N 3 N 4


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π i

u i θi M i V i


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π

π 14245

π

π = 3 ,1416


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u = N a −→i

N i xa i = [N 1 , N 2 . . . N n ]

a 1a 2

a n

i, j, m, ...etc

u u

u = u(x, y )v(x, y )

a i = uivi


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i N i , N j , N m

= Su = Ba

B = SN

S S

=x

y

z

=∂ ∂x 00 ∂ ∂y∂

∂y∂

∂x

uv

B


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σ0

σ i = D ( − 0) + σ0

σ i =σxσyτ xy

x

y

τ xy=

1E σx − vE σy1E σy −

vE σx

1G γ xy

=1E − vE 0

− vE 1E 0

0 0 1G

σxσy

σxy

a i

b u

q i =

q e1

q e2

q en

bi =bex

bey

V (δi σ)dV + l (δub)dV = i (δ (u )X

i )

V ([a e ]T B T DBa )dV ) + V ([a e ]T (N T i b)dV )) = [ a e ]T q i K i a i + f i = q i

K i = V (B T DB )dV f i = V (N T b)dV


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σ i = E = E dudx

E

V (δ σ)dV = l0 (δub)dx + pi =1 (δu i X i )

l

0 δ EAdudx =

l

0 δubdx +

p

i =1δu i X i

u i ui =u0 + a 1x

u i = N 1a 1 + N 2a 2

u1 = a0 + a1x1 u2 = a0 + a1x2

a 0 a1

a0 = u1 − u 2x1 − x2

a1 = x2u1 − x1u 2

x2x1


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C 0

N i ...N m x2 − x1 = le

x 2

x1

(dN 1dx

δu1)AE (dN 2dx

δu2) −

x 2

x1

[N 1δu 1 + N 2δu 2] = δu1X 1 + δu 2X 2

δu 1 δ 2u2

x 2x 1 ( dN 1dx (EA ) dN 1dx u1 + dN 1dx (EA ) dN 2dx u2)dx − x 2x 1 N 1bdx − X 1 = 0 x 2x 1 ( dN 2dx (EA ) dN 2dx u1 + dN 2dx (EA ) dN 1dx u2)dx − x 2x 1 N 1bdx − X 2 = 0

x 2x 1 (dN 1dx (EA )

dN 1dx ) (

dN 1dx (EA )

dN 2dx )

( dN 2dx (EA )dN 2dx ) (

dN 2dx (EA )

dN 1dx )

dxu1

u2− x 2x 1 N 1N 2 bdx =

X 1

X 2

ki a i + f i = q i

K i = x 2x 1 (dN 1dx (EA )

dN 1dx ) (

dN 1dx (EA )

dN 2dx )

( dN 2dx (EA )dN 2dx ) (

dN 2dx (EA )

dN 1dx )

dx = x 2x 1 (B T i DBi )dx f i = x 2x 1 N 1N 2 bdx

N i = (ξ − ξ 1)(ξ − ξ 2) . . . (ξ − ξ n )(ξ i − ξ 1)(ξ i − ξ 2) . . . (ξ i − ξ n )

N i =n

i =1

( ξ − ξ jξ i − ξ j

)


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C 0

N 1 = (1 − ξ

2 )

N 2 = (1 + ξ

2 )

K i f i

x2 − x1 = le

ξ s = 2(x − x1)(x2 − x1)

− 1

dξ dx

= 2le

B i D i

B i = [∂N 1∂ξ

∗ 2le

, ∂N 2

∂ξ ∗

2le

]

K I = 1− 1(∂N 1∂ξ ∗

2l e

∂N 2∂ξ ∗

2l e

(AE )[∂N 1∂ξ

∗ 2le

, ∂N 2

∂ξ ∗

2le

])(le2

)dξ

K I = AE

L 1 − 1− 1 1

f e = ble

2 11


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L (2 ,1) = (ξ − ξ 2)(ξ − ξ 3)(ξ 1 − ξ 2)(ξ 1 − ξ 3) =

ξ (ξ − 1)2

L (2 ,2) = (ξ − ξ 1)(ξ − ξ 3)(ξ 2 − ξ 1)(ξ 2 − ξ 3)

= ξ (ξ + 1)

2

L (2 ,3) = (ξ − ξ 1)(ξ − ξ 2)(ξ 3 − ξ 1)(ξ 3 − ξ 2)

= (1 − ξ 2)

ξ i

x = N 1x1 + N 2x2 + N 3x3 dxdξ

= dN 1

dξ x1 +

dN 2dξ

x2 + dN 3

dξ x3

dxdξ =

2ξ − 12 x1 +

2ξ + 12 x2 − 2ξx 3

dξ dx

= 2

2ξ (x1 + x2 − 2x3) + le

dξ dx

= 2le

1− 1( (ξ − 12 )

2l e

(− 2ξ ) 2l e(ξ + 12 )(

2l e )

AE {(ξ − 12

) 2le

, (− 2ξ ) 2le

, (ξ + 12

)( 2le

)}(le2

))dξ

A E


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K e = (EA

6 )

14 − 16 2− 16 32 − 16

2 − 16 14

f i = 1− 1ξ (ξ − 1)

21 − ξ 2ξ (ξ +1)

2

ble2

dξ

f i =141

ble6

L A I XZ

θ = dwdx

= θy = yd2wdx 2

σ = Eyd2wdx 2

dF i = ( Ey d2w

dx 2 )dA

dM i = ( E d2wdx 2

)y2dA

M i = EI d2wdx 2


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w w C 1

dwdx

C 1

w = α 0 + α 1ξ + α 2ξ 2 + α 3ξ 3

w = α 1 + 2 α 2ξ + 3 α 3ξ 2

w = N 1w1 + Ṅ 1(dwdx

)1 + N 2w2 + Ṅ 2(dwdx

)2

H 1 Ḣ 1 H 2 Ḣ 2 H 3 Ḣ 3 H 4 Ḣ 4


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N 1 N 1

1 − 1 1 − 1

0 1 − 2 3

1 1 1 1

0 1 2 3

α 0

α 1

α 2

α 3

=

1

0

0

0

1 − 1 1 − 1

0 1 − 2 3

1 1 1 1

0 1 2 3

α 0

α 1

α 2

α 3

=

0

1

0

0

1 − 1 1 − 1

0 1 − 2 3

1 1 1 1

0 1 2 3

α 0

α 1

α 2

α 3

=

0

0

1

0

1 − 1 1 − 1

0 1 − 2 3

1 1 1 1

0 1 2 3

α 0

α 1

α 2

α 3

=

0

0

0

1

H i = α0 + α 1ξ + α 2ξ 2 + α 3ξ 3


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N 1 = 14

(2 − 3ξ + ξ 3)

N 1 = 14

(1 − ξ − ξ 2 + ξ 3)

N 2 = 14

(2 + 3 ξ − ξ 3)

N 2 = 14

(− 1 − ξ + ξ 2 + ξ 3)

V B T i DB i dv x

x = 1 − ξ

2 x1 +

1 + ξ 2

x2

dxdξ

= le2

K i = 1− 1 4l2edN 1dξ

dN 1dξ

dN 2dξ

dN 2dξ

EI {dN 1dξ

, dN 1

dξ ,

dN 2dξ

, dN 2

dξ }

le2

dξ

K i =

EI

l2e

12 6le − 12 6le6le 4l2e − 6le 2l2e

− 12 − 6le 12 − 6le6le 2l2e − 6le 4l2e

K (6 x 6)

K T =

AE l e 0 0 −

AE l e 0 0

0 12EI l3e 6EI l2e

0 − 12 EI l 3e 6EI l2e

0 6EI l2e 4EI l e 0 − 6

EI l 2e

2 EI l e

−AE l e 0 0

AE l e 0 0

0 − 12 EI l 3e − 6EI l 2e

0 12EI l3e − 6EI l 2e

0 6EI l2e 2EI l e 0 − 6

EI l 2e

4 EI l e


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K i a i + f e = q f e v N T i qdv

1− 1N 1

N 1

N 2

N 2

qdx =

− ql2

− ql2

12

− ql2

ql 2

12

=

V 1

M 1

V 2

M 2

φ

θ = φ + dwdx

x = dudx

= − zdθdx

γ xz = dwdx

+ dudz

= dwdx

− θ = − φ

φ σx


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σxz

σx = E x = − zE dθdx

σxz = G(dwdx

− θ)M i = EI dθdx

Q i = GA (dwdx

− θ) = GAγ xz

τ xz = αGγ xz α A∗ = αA

V (δ x + δγ xz τ xz )dV = − l δqdx + ni =1 δ ( dwdx )i M i +n

i =1

δwi Z i

l [δ ( dθdx )EI ( dθdx ) + δ ( dwdx − θ)GA ∗ ( dwdx )]dx

θ w

C 0

w(ξ ) = N 1(ξ )w1 + N 2(ξ )w2 θ(ξ ) = N 1(ξ )θ1 + N 2(ξ )θ2


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w1 θ1 w2 θ2

χ = dθdx

= (δN 1(ξ )

δξ θ1 +

δN 2(ξ )δξ

θ2)dξ dx

γ xy = dwdx

− θ = dθdx

= (δN 1(ξ )

δξ w1 +

δN 2(ξ )δξ

w2)dξ dx

− N 1θ1 − N 2θ2

dξdx =

2l e

χ = B f a γ xz = B c a

B f = [0, 2le

δN 1δξ

, 0, 2le

δN 2δξ

]

B c = [ 2le

δN 1δξ

, − N 1 , 2le

δN 2δξ

, − N 2]

a i = [w1 , θ1 , w2 , θ2]

B f = [0, −1le

, 0, 1le

]

B c = [−1le

, ξ − 1

2 ,

1le

, − ξ − 1

2 ]

[δa e ]T l e [B T f (EI )B f + B T c (GA )B c ]dx (a e ) = − [δa e ]T l e N T (q )dx + [ δa e ]T q e [K ef + K

ec ]a

e − f e = q e

K ef = l e B T f (EI )B f dx K

e

c = l e BT

c (GA∗

)B c dx

f e = − l e N T qdx N = [N 1 , 0, N 2 , 0]

dx = le2 dξ

K f =

1

− 1B T f (EI )B f

le

2 dξ

K g = 1− 1 B T g (EI )B g le2 dξ f e = − 1− 1 N T le2 dξ


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K f

K f = (EI le

)

0 0 0 0

0 1 0 − 1

0 0 0 0

0 − 1 0 1

K c ξ

K c = (GA ∗

l )

1 le2 − 1 le

2

l e2

l2e3 −

l e2

l e2

− 1 − l e2 1 −l e2

l e2

l e2 −

l e2

l2e3

GA ∗

lGA ∗

2 −GA ∗

lGA ∗

2

GA ∗

2 (GA ∗

3 l + EI

l ) −GA ∗

2 (GA ∗

3 l − EI

l )

− GA∗

l −GA ∗

2GA ∗

l −GA ∗

2

GA ∗

2 (GA ∗

3 l − EI

l ) −GA ∗

2 (GA ∗

3 l + EI

l )

w1

θ1

w2

θ2

=

V 1

M 1

P

0

w1 = 0 θ1 = 0 w2

GA ∗

l −GA ∗

2

− GA∗

2 (GA ∗

3 l + EI

l )

w2

θ2=

P

0

w2

θ2=

γ γ + 1

( GA∗

3 l + EI

l ) l2

EI

l 2EI

lEI

P

0

γ = 12EI GA ∗ l 2

w2 = γ γ + 1 ( l

GA ∗ + l3

3EI )P w 2 = l

3

3EI P

ϕ = w2w2

= 3(4λ 2 + 3)4λ 2(λ 2 + 3)


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ϕ λ

ξ = 0

B c = [− 1le

, 12

, 1le

, −12

]

1− 1(− 1l e

12

1l e

12

GA ∗ [− 1le

, 12

, 1le

, −12

])le2

dξ


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K c

K c = (GA ∗

l )

1 le2 − 1 le

2

l e2

l 2e4 −

l e2

l 2e4

− 1 − l e2 1 − l e

2

l e2

l 2e4 −

l e2

l 2e4

K =

GA ∗

l −GA ∗

l

− GA∗

l (GA ∗

4 l + EI

l )

F =( 1GA +

l34EI ) −

l22EI

− l2

2EI ( lEI

ϕ = w2

(w2)eexacta=

3λ 2 + 34λ 2

ϕ λ λ −→ 0 ∞

λ −→ ∞


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z

x − y

x y

u(x, y ) =u(x, y )

v(x, y )


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u(x, y ) v(x, y ) x y

ξ x = ∂u∂x

ξ y = ∂v∂y

γ xy = ∂v∂x

+ ∂u∂y

γ xz = γ yz = 0

ξ z σz

ξ z σz ξ z

ξ = [ξ x , ξ y , γ xy ]T

τ xz

τ yz

ξ z σz

σ = [σx , σy , σ z ]T

σz = 0 ξ z = 0 γ xz = γ yz = 0

σ = Dξ

D

D =

d11 d12 0

d21 d22 0

0 0 d33

d12 = d21

d11 = d22 = E 1− v 2 d11 = d22 = E (1 − v )

(1+ v )(1 − 2v )

d12 = d21 = vE 1− v 2 d12 = d21 = ( E 1− v 2 )(

v1− v )

d33 = E 2(1+ v ) d33 = E

2(1+ v ) = G


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E v

D = 11− vxy vyx

vx vxy E x 0

vyx E y E y 0

0 0 (1 − vxy vyx )Gxy

D = 1ad − bc

aE x bE x 0

cE y dE y 0

0 0 (ad − bc)Gxy

1Gxy

= 1 + vyx

E x+

1 + vxyE y

a = 1 − vyz vzy b = vxy + vxz vzy

c = vyx + vyz vzx d = 1 − vxz vzx

E yE x

= vxyvyx

E yE x

= bc

x y θ x y

(P − U )u = x

(P − U )uort = y

xcos (θ) + ysen (θ) = x

− xsen (θ) + ycos (θ) = y

u v

u = ucos (θ) + vsen (θ)

v = − ucos (θ) + vcos (θ)

ε = dudx

εx = ∂u∂x

= ∂u∂x

∂x∂x

+ ∂u∂y

∂y∂x

εy = ∂v∂y =

∂v∂x

∂x∂y +

∂v∂x

∂x∂y

γ xy = ∂u∂y

+ ∂v∂x

= ∂v∂x

∂x∂x

+ ∂v∂y

∂y∂x

+ ∂u∂x

∂x∂y

+ ∂u∂x

∂x∂y


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εx = εx cos2(θ) + γ xy sen (θ)cos (θ) + εy sen 2(θ)

εy = εx sen 2(θ) − γ xy sen (θ)cos (θ) + εy cos 2(θ)

γ xy = εx − 2cos (θ)sen (θ) + εy 2cos (θ)sen (θ) + γ xy (cos2(θ) − sen 2(θ))

εxε

yεxy=

cos 2(θ) sen 2(θ) sen (θ)cos (θ)

sen 2(θ) cos2(θ) − sen (θ)cos (θ)

− 2sen (θ)cos (θ) 2sen (θ)cos (θ) cos2(θ) − sen 2(θ)

εxεy

γ xy

ε = T ε

x y

x y

σ = T T σ

σ = D ε

D

σ = T T D T ε = Dε

D = T T D T


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dij

ε εe

σ = Dε e = D (ε − ε0)

N : 1, 2, 3 (i , j ,k )

x1 , y1 x 2 , y2 x3 , y3 (i , j ,k )


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u = N 1u1 + N 2u2 + N 3u3 v = N 1v1 + N 2v2 + N 3v3

(u i , vi ) N i i

u v

u = uv = N 1 0 N 2 0 N 3 0

0 N 1 0 N 2 0 N 3

u1v1u2v2u3v3

u = N i a e

u = uv

N = [N 1 , N 2 , N 3] N i = N i 0

0 N i


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i

a e =a e1a e2a e3

a ei = uivi

i N ae N i aei

u = α 1 + α 2x + α 3y v = α 4 + α 5x + α 6y

u1 = α 1 + α 2x 1 + α 3y1 u2 = α 1 + α 2x 2 + α 3y2 u3 = α 1 + α 2x 3 + α 3y3

u = 12A

[(a 1 + b1x + c1y)u 1 + ( a 2 + b2x + c2y)u 2 + ( a 3 + b3x + c3y)u3]

N i = 12A

(a i + bi x + ci y)


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εx = ∂N 1

∂x u1 +

∂N 2∂x

u2 + ∂N 3

∂x u3

εy = ∂N 1

∂y v1 +

∂N 2∂y

v2 + ∂N 3

∂y v3

γ xy = ∂u∂y

+ ∂v∂x

= ∂N 1

∂y u1 +

∂N 1∂x

v1 + ∂N 2

∂y u2 +

∂N 2∂x

v2 + ∂N 3

∂y u3 +

∂N 3∂x

v3

ε =

∂u∂x∂v∂y

∂v∂x +

∂u∂y

=

∂N 1∂x 0

∂N 2∂x 0

∂N 3∂x 0

0 ∂N 1∂y 0 ∂N 2

∂y 0 ∂N 3

∂y∂N 1∂y

∂N 1∂x

∂N 2∂y

∂N 2∂x

∂N 3∂y

∂N 3∂x

u1v1u2v2u3v3

ε = Ba e B = [B 1 , B 2 , B 3]

B i =

∂N i∂x 0

0 ∂N i∂y

∂N i∂y

∂N i∂x


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B B i

B = 12A

b1 0 b2 0 b3 0

0 c1 0 c2 0 c3

c1 b1 c2 b2 c3 b3

B i = 12A

bi 0

0 ci

ci bi

σ = Dε = DBae

σ = D (ε − ε0) + σ0 = DBa e − Dε 0 + σ0


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A δεT σtdA = A e δu T btdA + l e δu T tds + 3i =1 δu i U i +3

i =1

δvi V i

δu i δvi U i V i

A e

δε T σtdA −

A e

δu T btdA −

δuT ttds = [δa e ]T q e

δa e = [δu 1 , δv1 , δu 2 , δv2 , δu 3 , δv3]T q e = [U 1 , V 1 , U 2 , V 2 , U 3 , V 3]

δu T = [δa T ]N T δε T = [δa T ]B T

δa T

[δa e ]T [ A e B T σtdA − A e N T btdA − l e N T tdS ] = [δa e ]T q e A e B T σtdA − A e N T btdA − l e N T tds = q e


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σ

[ A e B T DBtdA ]a e − A e B T Dε 0 tdA + A e B T σ0 tdA − A e N T btdA − l e N T tds = q e K e a e − f e = q e

K e =

A

eB T DBtdA

f e = f eε + f eσ + f

eb + f

et

f (e )ε = A e B T Dε 0 tdA f (e )σ = − A ( e ) B T σ0 tdA

f (e )b =

A e

N T btdA

f (e )t = l e N T tds

e

q (e )i = P j


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J =

∂x∂ξ

∂y∂ξ

∂z∂ξ

∂x∂η

∂y∂η

∂z∂η

∂x∂ζ

∂y∂ζ

∂z∂ζ

x = a 0 + a 1ξ = [1, ξ ]

a 0

a 1

ξ

ξ a

P a = u

a = A− 1

ux e1

x e2=

1 − 1

1 1

a 0

a 1

[1, ξ ]1 − 1

1 1

− 1 x e1

x e2

[N 1 , N 2]x e1

x e2

N 1 = 1 − ξ

2

N 2 = 1 + ξ

2


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u =2

a =1

N a (ξ )u a e

C 0


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(m = 1)

L e

∂x∂ξ

= dxdξ

=dN 1dξ

, dN 2

dξ xe1x e2

=− 12

, 12

= xe2 − x e1

2 =

Le2

C 0

u e0 = c0 + c1xea

u =2a =1 N a (ξ )u

ea

u =2

a =1N a (ξ )u ea

u =2

a =1

N a (ξ )(c0 + c1x ea )

u = 2

a =1

N a (ξ ) c0 + 2

a =1

N a (ξ )x ea c1

2

a =1

N a (ξ ) = N 1(ξ ) + N 2(ξ ) = 12

(1 − ξ ) + 12

(1 + ξ ) = 1

2

a =1(ξ ) = 1


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x y

x(ξ, η) = a0 + a1ξ + a 2η + a 3ξη = 1 ξ η ξη

a 0

a 1

a 2

a 3

y(ξ, η) = β 0 + β 1ξ + β 2η + β 3ξη = 1 ξ η ξη

β 0

β 1

β 2

β 3

x(ξ a , ηa , a 0 , · · · )


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a ξ a ηa

{x ea } =

x e1

x e2

x e3

x e4

=

1 − 1 − 1 +1

1 +1 − 1 − 1

1 +1 +1 +1

1 − 1 +1 − 1

a0

a1

a2

a3

= Aa i

{x ea } =

ye1

ye2

ye3

ye4

=

1 − 1 − 1 +1

1 +1 − 1 − 1

1 +1 +1 +1

1 − 1 +1 − 1

β 0

β 1

β 2

β 3

= Aβ i

a i = A− 1{x ea }

β i = A− 1{yea }

x = 1 ξ η ηξ a i

x = 1 ξ η ηξ A − 1{x i }

x = N 1 N 2 N 3 N 4 {x i }

N 1 = 14

(1 − ξ )(1 − η)

N 2 = 14

(1 + ξ )(1 − η)

N 3 = 14

(1 + ξ )(1 + η)

N 4 = 14

(1 − ξ )(1 + η)

C 0

C 0

(n ) ξ 0 · · · ξ n f


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(ξ ) (n − 1)

P (ξ ) = f (ξ 1)L n − 1,1(ξ ) + · · · + f (ξ n )Ln − 1,n (ξ )

Ln − 1,k (ξ ) =n

i =1

(ξ − ξ i )(ξ k − ξ i )

u =n

a =1 N a (ξ )u ae

N a (ξ ) = Ln − 1,a (ξ )

•

•

u =n

a =1N a (ξ )u a

e

N 1 = L1,1 = (ξ − ξ 2)(ξ 1 − ξ 2)

= (1 − ξ )

2

N 2 = L1,2 = (ξ − ξ 1)(ξ 2 − ξ 1)

= (1 + ξ )

2

•

•

u =n

a =1

N a (ξ )u ea


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N 1 = L2,1 = (ξ − ξ 2)(ξ − ξ 3)(ξ 1 − ξ 2)(ξ 1 − ξ 3)

= (ξ − 1)(ξ − 0)(− 1 − 1)(− 1 − 0)

= 12

ξ (ξ − 1)

N 2 = L2,2 = (ξ − ξ 1)(ξ − ξ 3)(ξ 2 − ξ 1)(ξ 2 − ξ 3)

= (ξ + 1)( ξ − 0)(+1 + 1)(+1 − 0)

= 12

ξ (ξ + 1)

N 3 = L2,3 = (ξ − ξ 1)(ξ − ξ 2)(ξ 3 − ξ 1)(ξ 3 − ξ 2)

= (ξ + 1)( ξ − 1)(+0 + 1)(+0 − 1)

= 1(1 − ξ 2)


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(n +1) ξ (m + 1) η

N a (ξ, η) = Ln,b (ξ )xL m,c (η)

•

•

u =n

a =1

N a (ξ, τ )u ea

ξ η


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L(ξ ) L(η)

N 1 = L1,1(ξ )xL 1,1(η) = 12(1 − ξ ) 1

2(η − 1) = 1

4(1 − ξ )(1 − η)

N 2 = L1,2(ξ )xL 1,1(η) = 12

(ξ + 1)12

(η − 1) = 14

(1 + ξ )(1 − η)

N 3 = L1,2(ξ )xL 1,2(η) = 12

(ξ + 1)12

(η + 1) = 14

(1 + ξ )(1 + η)

N 4 = L1,1(ξ )xL 1,2(η) = 12

(1 − ξ )12

(η + 1) = 14

(1 − ξ )(1 + η)

•

N 1


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N 2

•

u =n

a =1

N a (ξ, η)u ea

ξ η


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N 3

N 4


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L(ξ ) L(η)

N 1(ξ, η) = L2,1(ξ )xL 2,1(η) = 12

(ξ )(ξ − 1)12

(η)(η − 1) = 14

(ξη)(ξ − 1)(η − 1)

N 2(ξ, η) = L2,2(ξ )xL 2,1(η) = 12

(ξ )(ξ + 1)12

(η)(η − 1) = 14

(ξη)(ξ + 1)( η − 1)

N 3(ξ, η) = L2,2(ξ )xL 2,2(η) = 12

(ξ )(ξ + 1)12

(η)(η + 1) = 14

(ξη)(ξ + 1)( η + 1)

N 4(ξ, η) = L2,1(ξ )xL 2,2(η) = 12(ξ )(ξ − 1) 12(η)(η + 1) = 14(ξη)(ξ − 1)(η + 1)

N 5(ξ, η) = L2,3(ξ )xL 2,1(η) = (1 − ξ 2)12

(η)(1 − η) = 12

(1 − ξ 2)(η)(1 − η)

N 6(ξ, η) = L2,2(ξ )xL 2 ,3(η) = (1 − η2)12

(ξ )(1 + ξ ) = 12

(1 − η2)(ξ )(1 + ξ )

N 7(ξ, η) = L2,3(ξ )xL 2,2(η) = (1 − ξ 2)12

(η)(1 + η) = 12

(1 − ξ 2)(η)(1 + η)

N 8(ξ, η) = L2,1(ξ )xL 2 ,3(η) = (1 − η2)12

(ξ )(1 − ξ ) = 12

(1 − η2)(ξ )(1 − ξ )

N 9(ξ, η) = L2,3(ξ )xL 2,3(η) = (1 − ξ 2)(1 − η2) = (1 − ξ 2)(1 − η2)


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ξ η

N 5(ξ, η) = L1,2(ξ )xL 2,3(η)

N 5(ξ, η) = 12

(1 + ξ )(1 − η2)

N 5(ξ, η) = 0 j = 1 , 2, 3, 41 j = 6

N 1 = N 2 = 0

N 2 = N 3 = 12


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N 5

N 1 = 14

(1 − ξ )(1 − η)

N 2 = N 2 − 12

N 5

N 2 = 14

(1 + ξ )(1 − η) − 14

(1 + ξ )(1 − η2)N 2 = 14

(1 + ξ )η(η − 1)

N 3 = N 3 + 1

2N 5

N 3 = 14

(1 + ξ )(1 + η) − 14

(1 + ξ )(1 − η2)

N 3 = 14

(1 + ξ )η(η + 1)

N 4 = 14

(1 + η)(1 − ξ )

C 0

r = A 1A s = A 2

A t = A 3

A 0 ≤ r, s, t ≤ 1


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r + s + t = 1

x = N 1x e1 + N 2x e2 + N 3x e3 y = N 1ye1 + N 2ye2 + N 3ye3

N 1 = r N 2 = s N 3 = t

u = N 1u1e + N 2u2 e + N 3u3 e

t

t = 1 − r − s

J =

∂x∂r

∂y∂r

∂x∂s

∂y∂s

L I − 1,I (r ) =I

i =1( r − r ir I − r i

) I = 1

1 I = 1

a

N a (r,s,t ) = T i (r )T j (s )T k (t)


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N 1 = T 2(r ) × T 1(s) × T 1(t) = r − r ir 2 − r 1 = r

N 2 = T 1(r ) × T 2(s) × T 1(t) = s − s1s2 − s1

= s

N 3 = T 1(r ) × T 1(s) × T 2(t ) = s − s 1s2 − s 1

= t


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N 1(r,s,t ) = T 3(r ) × T 1(s) × T 1(t) = (r − r 1)( r − r 2)(r 3 − r 1)( r 3 − r 2)

= r (2r − 1)

N 2(r,s,t ) = T 3(s) × T 1(r ) × T 1(t ) = (s − s1)(s − s 2)(s3 − s1)(s3 − s2)

= s(2s − 1)

N 3(r,s,t ) = T 3(t) × T 1(r ) × T 1(s) = (t − t1)( t − t2)(t3 − t1)( t3 − t 2)

= t(2t − 1)

N 4(r,s,t ) = T 2(s) × T 2(r ) × T 1(t) = s − 00,5 − 0

× r − 00,5 − 0

× 1 = 4 rs

N 5(r,s,t ) = T 2(s ) × T 2(s) × T 1(r ) = s − 0

0,5 − 0

× t − 0

0,5 − 0

× 1 = 4 st

N 6(r,s,t ) = T 2(r ) × T 2(t) × T 1(r ) = r − 00,5 − 0

× t − 00,5 − 0

× 1 = 4 rt

x y

K ij = D A e xm yn dA K ij = Dc n +1 am +1 − (− b)m +1

m!n !(m + n + 2)!

−→

K ij = D(2a )m +1 (2b)n +1

(m + 1)( n + 1) −→

m n

K f x y

K ij = T T K ij T f i = T T f i

T =cos (xx ) cos(xy )

cos (yx cos (yy )

xx

x


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(r,s,t )

∂ (r,s,t )∂x

= ∂N 1

∂r∂r∂x

+ ∂N 1

∂s∂s∂x

+ ∂N 1

∂t∂t∂x

∂ (r,s,t )∂x =

bi

2Ae

∂ (r,s,t )∂y =

ci

2Ae

A e r k s l tm dA = 2Ae k! × l! × m !(2 + k + l + m )! l ( e ) r k s l ds = le k! × l!(1 + k + l)!


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θi

(u) (w)

u w


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ν

u = u(r, s )w(r, s )

u w θ

γ rθ γ zθ

εr = ∂u∂r εz =

∂w∂z γ rz =

∂u∂z +

∂w∂r

εr εz σrz

r + u εθ

εθ = 2π (r + u)

2πr =

ur

ε =

εr

εz

εθ

σrz

=

∂u∂r

∂w

∂z

ur

∂u∂z +

∂w∂r

σ = {σr , σ z , σ θ , τ rz }

σr, σ

z, σ

θ γ

rz

σ = D (ε − ε0) + σ0

D = E

(1 + ν )(1 − 2ν )

1 − ν ν ν 0

ν 1 − ν ν 0

ν ν 1 − ν 0

0 0 0 1− 2ν 2


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dV = ( rdθ )dr.dz = rdθdA

A 2π0 δε T σrdθdA = A 2π0 δu T brdθdA + 2π0 δu T brdθds + i 2π0 δu T i q i r i dθ

b =br

bz t =

t r

t z q i =

q ri

q zi

θ

2π A δε T σdA = 2π A δu T brdA + 2 π l δu T rds + 2 π i δa T i q i r i 2π

q i


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u =u

w=

N 1 × u1 N 2 × u2 N 3 × u3

N 1 × w1 N 2 × w2 N 3 × w3

N = [N 1 , N 2 , N 3] =N 1 0 N 2 0 N 3 0

0 N 1 0 N 2 0 N 3

a e =

a1

a2

a3

= [u1, w1, u 2, w2, u 3, w3]T

N i


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ε =3

i =1

∂N i∂r 0

0 ∂N i∂z

N ir 0

∂N i∂z

∂N i∂r

u i

wi= [B 1 , B 2 , B 3]

a 1

a 2

a 3

= Ba e

B i

N ir

B i

B i = 12A

bi 0

0 ci

(a i + bi r + c i z )r 0

ci bi

σ =3

i =1

B i Da (e ) − Dε 0 + σ0

εθ = ur

00 εθ

εθ = εr εθ εr

K e a e − f e = q e

K (e )ij = 2 π

A ( e )

B T i DB j rdrdz

K eij


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f e = 2 π A N T brdA + 2 π l e N T trds + 2 π A B T Dε 0rdA − 2π A e B T σ0rdA f eb

f et f eε f eσ


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•


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•

•

•


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•


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•u i = K

− 1D F i

•

•


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•

F y 1 Mz 1 F y 2


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••

• m4

•


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•


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•

θ2

θ3

•

•


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F x 1 F y 1 Mz 1 F x 4 F y 4

Mz 4


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(x, y )


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[ , ]


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θi

M zz


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θi


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π i


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K 11 , K 12 , K 21 , K 22

π =

1 2 3

a 1 d

b 2 e

c 3 f

1 4 4

2 5 5

3 6 6

→

K T {π i [i, n ], π i [ j, n ]} ← K ni [i, j ]

←− K nij π(i, n ) π( j, n )

[K ]

π i

π i


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π i


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K i =

AE L 0 0 −

AE L 0 0

0 12EI L 3 6EI L 2 0 − 12

EI L 3 6

EI L 2

0 6EI L 2 4EI L 0 − 6

EI L 2 2

EI L

− AE L 0 0 AE

L 0 0

0 − 12 EI L 3 − 6EI L 2 0 12

EI L 3 − 6

EI L 2

0 6EI L 2 2EI L 0 − 6

EI L 2 4

EI L


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T =

cos (θ) sen (θ) 0 0 0 0

− sen (θ) cos(θ) 0 0 0 0

0 0 1 0 0 0

0 0 0 cos(θ) sen (θ) 0

0 0 0 − sen (θ) cos(θ) 0

0 0 0 0 0 1

K g = T T K i T


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θ1 , v1 , u 1 , θ2 , v2 , u 2

ug = N 1u 1 + N 2v1 + N 3θ1 + N 4u 2 + N 5v2 + N 6θ2

N 1(x, l ) = 1 − xl

N 2(x, l ) = 2(xl )3 − 3(

xl )2 + 1

N 3(x, l ) = x(1 − x

l )2

N 4(x, l ) = xl

N 5(x, l ) = 3(xl )2 − 2(

xl )3

N 6(x, l ) = −x2

l (1 −

xl )

K − 1D × F i = u i


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u i = T u i

K i (T u i ) = F


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u i = N 1(u 1) + N 2(v1) + N 3(θ1) + N 4(u2) + N 5(v2) + N 6(θ3) + R i (x, l )

θi = ∂N 1

∂x (u1) +

∂N 2∂x

(v1) + ∂N 3

∂x (θ1) +

∂N 4∂x

(u2) + ∂N 5

∂x (v2) +

∂N 6∂x

(θ2) + R i (x, l )

M i = EI ∂ 2u∂x 2

= ∂ 2N 1

∂x 2 (u 1) +

∂ 2N 2∂x 2

(v1) + ∂ 2N 3

∂x 2 (θ1) +

∂ 2N 4∂x 2

(u2) + ∂ 2N 5

∂x 2 (v2) +

∂ 2N 6∂x 2

(θ2) + R i (x, l )

V i = EI ∂ 3u∂x 3

= ∂ 3N 1

∂x 3 (u1) +

∂ 3N 2∂x 3

(v1) + ∂ 3N 3

∂x 3 (θ1) +

∂ 3N 4∂x 3

(u2) + ∂ 3N 5

∂x 3 (v2) +

∂ 3N 6∂x 3

(θ2) + R i (x, l )


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π i

•

•

• πi

•

u i , θ i , M i , V i•


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Elementos finitos con computadoras.pdf

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Transcript of Elementos finitos con computadoras.pdf