Plates - Mechanics of Structures: Variational and Computational Methods - page 799

where the definitions of E B and E V are evident in Eq. (13.87). Also, the strain-displacement
relation of Eq. (13.23) can be divided into two expressions

θ

κ x
κ

∂ x 0

=

x

0 ∂

y y

θ

y

2 κ ∂ ∂

xy y x

κ = D κ θ

γ xz
γ yz

∂ x

w
θ x
θ y

10

=

∂ y 01

γ = D u (13.95)

γ

Finally, the internal virtual work can be expressed as

T EDu dA

− δ W i =

A δ(

Du )

T E B D κ θ dA T E V D γ

=

A δ(

D κ θ ) +

A δ(

D γ u ) u dA

A δ θ T

D T u T D T = E B D κ θ dA +

A δ

E V D γ u dA (13.96)

κ γ

which leads to the desired principle of virtual work expression

y ] x ∂∂ x + y ∂

θ dA

K

1 − ν

2

1 − ν

2

∂ y ∂ν∂ y + y ∂ ∂ x x

δ W =−

A δ

[ θ θ

x

1 − ν

2

1 − ν

2

∂ν∂ x + x ∂ ∂ y ∂∂ y + x ∂ ∂ x

θ

y y

δ θ T k B θ

∂∂ x + y ∂∂ y ∂ ∂ w

θ x
θ y

x x

5 Gt
6

dA

−

A δ

[ w x θ y ] ∂ y 10

∂ x 01

u T k V δ u

p z
0
0

p s
0
0

dA ds

+

A δ

[ wθ θ

y ]

+

S p δ

[ wθ θ

y ]

x x

u T u T δ p V δ p (13.97)

K u dA

6 k s ( 1 − ν)

A δ θ T k B θ dA

u T k V

=− +

A δ

t 2

u T u T p ds

+

A δ

p V dA +

S p δ

= 0

where k s = 5 / 6 is the shear correction factor, or

δ W =−

A δ

[ wθ θ

y ]

x

5 Gt
6

5 Gt
6

5 Gt
6

5 Gt
6

∂ ∂ + ∂ ∂ ∂ ∂

w
θ x
θ y

x x y y y x

5 Gt K ( 1 − ν)

2

5 Gt
6

K ( 1 − ν)

2

×

6 ∂ y

∂ K ∂ x + y ∂ ∂ y + ∂ K ν∂ y + y ∂ ∂ x dA

x x

K ( 1 − ν)

2

K ( 1 − ν)

2

5 Gt 5 Gt

6 6 ∂ x

∂ν K ∂ x + x ∂ ∂ y ∂ K ∂ y + x ∂ ∂ x +

y y

p z
0
0

p s
0
0

S p δ

+

A δ

[ w x θ y ] dA + [ w x θ y ] ds = 0 (13.98)