Plates - Mechanics of Structures: Variational and Computational Methods - page 798

The governing equations can also be arranged in first order form similar to Eqs. (13.58).
For this plate theory, which includes shear deformation effects, there are six variables having
derivatives with respect to x . The first order equations using these variables as the state
variables are

∂w
∂

6
5

q x
Gt x =− θ

+

x

m x −

∂θ x
∂

1

K ∂θ y
∂ x = ( 1 + ν − ν

2

) K y

∂θ

x =− ∂θ

2

y x

y +

m xy

∂ ∂ K ( 1 − ν)

(13.92)

∂

2

∂ q x

∂

Gt 5
6 y 2 + ∂θ

w

y

x =−

− p z

∂ ∂ y

∂ m x

∂

− ∂

m yx
∂

= q x

x y

2

∂ m xy

∂

= ∂w
∂

ν

2 ∂

m x
∂

1 − ν

2 ∂

θ y

y + θ y + y −

K ν

x 1 + ν − ν 1 + ν − ν ∂ y 2

The first three of these equations come from the kinematic relations of Eq. (13.22) and the
material law of Eq. (13.29), the final three from the equilibrium conditions of Eq. (13.31).
Note that when the shear deformation effects are included, the plane normal to the middle
surface does not remain normal after deformation. This can be seen from the first relation
of Eq. (13.92). Moreover, this relation implies that the rotations are independent variables
and hence there are six state variables, including three displacements (w , θ x , θ y ) and three

forces . There is no need to define an artificial shear force on the boundary as

we did for the Kirchhoff plate theory, where it was necessary to reduce the number of forces
on the boundary from three to two. Also, compare the first equation of Eqs. (13.92) with
Chapter 1, Eq. (1.133a) and note that the factor 6/5 is analogous to 1

( q x ,m x ,m xy )

/ k s of beam theory. If

k s = 6 , the shear correction factor for plates, the first relation of Eq. (13.92) can be written

as ∂ ∂ x =− θ x +

5 /

q x

k s Gt , which is similar to beam theory. It is shown in Pilkey (2002) that it may
be wise to choose a different shear correction factor for very thin plates.

13.4.4 Variational Relationships
It is convenient to express the internal virtual work in two terms, one for bending and one
representing shear effects

A δ T A δ κ T

T

− δ W i = s dA = m dA +

A δγ

q dA (13.93)

where the stress resultants are

q y ] T [ mq ] T s = [ m x m y m xy q x =

and the strains are given by

yz ] T

[ κγ ] T

= [ κ κ 2 κ γ γ =

x y xy xz

Using this notation, the material law of Eq. (13.87) can be expressed as
m
q

E B 0
0 V

κ
γ

(13.94)

=

=

s E