Plates - Mechanics of Structures: Variational and Computational Methods - page 765

FIGURE 13.4
Displacements and stresses.

From Chapter 1, Eq. (1.20) the strains for an elastic solid are given by

∂ 00

x x

0 ∂ 0

y y

u x
u y
u z

00 ∂

z z

= (13.21)

γ ∂ ∂ 0

xy y x

γ ∂ 0 ∂

xz z x

γ 0 ∂ ∂

yz z y

D

Note that u z = w is the deflection of the middle surface and does not vary with z . Hence,

z = ∂ z u z = 0 . Also, it is assumed that the middle surface transverse displacement, the

deflection , is small compared to the thickness of the plate. Hence, the rotation or slope

of the deformed surface is small. The square of the slope is then negligible in comparison
to unity, so that, as in Chapter 1, Eq. (1.97), the curvature is equal to the rate of change of
the rotation. Substitute the displacements of Eq. (13.20) into Eq. (13.21) to find the strains

w

z θ x,x z κ x

x

z θ y, y

0

κ y
0
z 2

z

y

z

= = (13.22)

γ

z (θ x, y + θ y, x )

(θ x + w ,x )
(θ y + w ,y )

κ xy
γ xz
γ yz

xy

γ

xz

γ

yz

From Eq. (13.22), the kinematical relations for the transverse deformation of a plate are

κ x
κ y

∂ x 00

0 ∂ y 0 θ x

θ 2 κ = ∂ ∂ 0

xy y x y

w

(13.23)

γ 10 ∂

xz x

γ 01 ∂

yz y

= D u u