Plates - Mechanics of Structures: Variational and Computational Methods - page 802

[ w w w w θ θ θ θ θ θ θ θ ]

1 2 3 4 x 1 x 2 x 3 x 4 y 1 y 2 y 3 y 4

0

0

0 0

0

A
αβ

C

αβ −

A

αβ /

2 B ν

2

− ν − ν ν

αβ 3 2 3

t 2
12

Et A

αβ

B − A

αβ /

2 ν

3

− ν − ν ν

αβ 2 3 2

=

k B

A
αβ

C − ν 2 ν 3 ν 2 − ν 3

( − ν

2

)

24 1 αβ

A

αβ − ν 3

ν 2 ν 3 − ν 2

Symmetric A B

βα −

A

βα /

2 C

βα βα

A C

βα −

A

βα /

2

βα

A B

βα βα

A

βα

(13.103)

α = b / a, β = a / b, ν = 1 − ν , ν = 3 ( 1 + ν) , ν = 3 ( 1 − 3 ν)

1 2 3

A αβ = 8 α + 4 βν

1 , αβ =−

4 α − 2 βν

1 , αβ =−

8 α + 2 βν

1

A ,B , and C are obtained by interchanging α and β in the expressions for A ,B

βα βα βα αβ αβ

and C αβ , respectively.
Turn to integral II of Eq. (13.101), which accounts for shear deformation effects. From
Eqs. (13.96) and (13.101), with t 2

ζ = 5 Gt /( 6 K ) = 6 k s ( 1 − ν)/

u T k V

T E V D

K ζ

A δ

u dA = K ζ

A δ(

D u ) u dA

γ γ

v iT

T E V D γ Nv dA

B T
γ

B γ dA v i

= K ζ

A δ(

D γ Nv ) = K ζδ (13.104)

A

or

B T
γ

k V = B dA

γ

A

5 Gt
6

with B

γ =

D N . Note that with factored out, E V of Eq. (13.87) becomes an identity

γ

matrix. We find

B 11

B 12 0

B γ = (13.105)

B 21 0 B 23

where

= − ( − η)/ ( − η)/ η/ − η/

B 11 [ 1 a 1 a a a ]

= ( − η)( − ξ) ξ( − η) ξη − η

B 12 [ 1 1 1 1 ]

B 21 = [ − ( 1 − ξ)/ b − ξ/ b ξ/ b ( 1 − ξ)/ b ]

B 23 = [ ( 1 − ξ)( 1 − η) ξ( 1 − η) ξη 1 ( 1 − η)η ]

Note that the first row ( B 11 and B 12 ) of B γ corresponds to γ xz and the second row ( B 21 and B 23 )

corresponds to The scheme of numerical integration of Fig. 13.17 will be employed to

perform the required integration. The Gaussian integration point is chosen at the center of
the element, which corresponds to n

γ yz .

= 1 of Table 6.7, i.e., at ξ = η = 0 . 5 . At this point, B

γ

becomes

1
2 a

1
2 a

1 1

2 a

1
4

1
4

1
4

1
4

−

2 a −

0

B

γ =

(13.106)

1 1

2 b

1
2 b

1
2 b

1
4

1
4

1
4

1
4

−

2 b −

0