Plates - Mechanics of Structures: Variational and Computational Methods - page 771

FIGURE 13.8
Stress distributions.

Other useful expressions are obtained from the second and third relations of the equilib-
rium conditions of Eq. (13.31a). Substitute the expressions for m x , m y , and m xy of Eq. (13.37)
into these equilibrium relations to find q x and q y in terms of the displacement

w ,

∂

y 2

2 2

K ∂
∂ x 2 + ∂

w w

q x =− (13.40a)

x ∂ ∂

∂

y 2

2 2

K ∂
∂ x 2 + ∂

w w

q y =− (13.40b)

y ∂ ∂

Local Form of the Governing Equations
The governing fourth order differential equation, expressed in terms of the displacement
w , is obtained by substituting Eqs. (13.37) into Eq. (13.39)

4 4 4

∂ w ∂ w

y 2 + ∂

w p z

K

x 4 +

2

y 4 =

(13.41)

∂ ∂ x 2 ∂ ∂

or

p z
K

4

∇ w =

with

= ∂

x 2 + ∂

2 2

4 2 2 2 2 2 2

x

2
y

∇ =∇ ∇ = ( ∇ ) ∇

y 2 = ∂

+ ∂

∂ ∂

An alternative second order form of the governing equations is sometimes useful. To
obtain this form define M , the moment function or moment sum ,as

K ∂ w =−

m x + m y

x y 2

M =

+ ν) =−

w + ∂ K ∇ w (13.42)

( 1