Plates - Mechanics of Structures: Variational and Computational Methods - page 766

13.2.2 Material Law

The material law of Chapter 1, Eq. (1.32) for a three-dimensional continuum appears as

− ν − ν .

x
y
z
......

σ x
σ y
σ z
......

1

− ν .

− ν 1 0

− ν − ν 1

1
E

... ... ... ...... ...... ......

=

(13.24)

.

2 ( 1 + ν)

γ xy
γ

τ xy
τ

.

0 2 ( 1 + ν)

xz xz

.

2 ( 1 + ν)

γ τ

yz yz

E − 1 = σ

z is smaller than other stress components
and can be neglected. This approximation may be questionable in the neighborhood of
concentrated transverse loading. With

Recall from Eq. (13.22) that = 0 . The stress σ

z

σ =

0 , inversion of Eq. (13.24) gives

z

0 .

σ

x
σ y
τ xy
···

ν

x
y
γ xy
···

1

0 .

ν 1 0

00 1 − ν 2 .
··· ···

E

=

(13.25)

1 − ν

2

··· ······ ······

.

τ xz
τ yz

1 − ν

2

γ xz
γ yz

0

.

1 − ν

2

σ =

E

Frequently, the two expressions for τ

xz and

τ

yz are written as

τ = G γ

xz ,

τ = G γ (13.26)

xz yz yz

with

E

=

G

2 ( 1 + ν)

Figure 13.4 shows the stress components defined in Eq. (13.25). The stresses σ x , σ y , τ xy , τ xz ,

and τ yz are defined similarly to those in the engineering beam theory. There, it was conve-

nient to replace cross-sectional stresses by their resultant forces. For plates, we choose to
utilize forces and moments per unit length. These are the stress resultants (Fig. 13.5).

m x
m y
m xy =

σ x
σ y
τ xy = τ yx

t
2

= zdz

Bending and Twisting Moments

t
2

m yx

−

(13.27)

q x
q y

τ dz t

2

xz

= Shear Forces

τ

t
2

yz

−

The signs (directions) of the moments and forces in Fig. 13.5 correspond to those of the
stress components in Fig. 13.4.