Plates - Mechanics of Structures: Variational and Computational Methods - page 776

FIGURE 13.11
Surface conditions.

while the force (static) boundary conditions on S p = S − S u are

V a

a

m a = m a , = q a + ∂ t m at = (13.56)

where (Fig. 13.11) the surface force conditions are

m a
m at

m x
m y
m xy

sin 2 cos 2 α α 2 sin α cos α

=

sin 2

cos 2

(13.57)

− sin α cos α sin α cos α − α + α

q a = q x sin α + q y cos α

First Order Form of the Local (Differential) Governing Equations
The kinematical relations, material law, and equilibrium conditions of this section, as well
as the relationships of Eqs. (13.48) and (13.37) are readily rearranged in first order form
with V x .

θ = θ

x ,V

=

∂w
∂

x =− θ
∂θ
∂

2

x = ν ∂

w m x

K y 2 + ∂

(13.58a)

4 2 m x

∂

∂ V

) ∂

y 4 − ν ∂

w

2

x =

K ( 1 − ν

y 2 −

p z ( x, y )

∂ ∂

∂ m x

∂

− ν) ∂

2

θ

x =−

2 K ( 1

y 2 +

V

∂