Plates - Mechanics of Structures: Variational and Computational Methods - page 769

z

x

dy

dx

w

x

y

w w

) dy

(

FIGURE 13.7
Twisting.

+

y x

x

13.3.1 Rectangular Plates

Kinematical Relationships
Since shear deformation is not taken into account,

γ xz = γ yz = 0, and the final two relations

of Eq. (13.22) lead to

θ =− w

x ,x

θ =− w (13.32)

y ,y

Then the remaining strains of Eq. (13.22) become

z ∂

2

w

z ∂

2

w

2 z ∂

2

w

=− =− γ =−

x 2 , y 2 ,

(13.33)

x y xy

∂ ∂ ∂ x ∂ y

or in terms of curvatures,

x

− ∂

κ x
κ y

=

y

− ∂ [ w ]

(13.34)

2 κ xy

− 2 ∂ x ∂ y

= D u u

The final expression of Eq. (13.34) is called the twist with respect to the x and y axes of the
middle surface (Fig. 13.7).

Material Law
The material relationship of Eq. (13.29) applies, i.e.,

κ x
κ y
2

m x
m y
m xy

1 ν 0

=

K ν 10

00 1 − ν
2

(13.35)

κ

xy

s = E

where

Et 3

K =

12 ( 1 − ν

2

)