Plates - Mechanics of Structures: Variational and Computational Methods - page 757

x

n y

n yx

a

n x

n xy

a

t

p t

p a

FIGURE 13.2
Surface forces.

y

or in matrix form,

∂ p Vx

p Vy

n x
n y
n xy

0 ∂

x y

+

= 0

(13.4b)

0 ∂ ∂

y x

D T

s + p V = 0

Surface Forces and Boundary Conditions
The displacement boundary conditions on S u are

u x = u x

(13.5a)

u y = u y

where letters with bars over them represent prescribed or applied quantities.
The force boundary conditions occur on S p = For a surface with normal a and

tangent t (Fig. 13.2), with the surface forces p a (normal) and p t (tangent), the force boundary
conditions are

S − S u .

p a = p a

(13.5b)

p t = p t

The surface forces and stress resultants are related by
p a
p t

n x
n y
n xy

sin 2

α cos 2 α 2 sin α cos α

=

(13.6)

cos 2 sin 2

− sin α cos α sin α cos α − α + α

13.1.2 Circular Plates
A circular plate with in-plane loading is traditionally assumed to be in a state of plane stress.
The displacements and strains, as well as stresses (stress resultants), should be expressed
with polar coordinates. The relationships between the Cartesian and polar coordinate sys-
tems are (Fig. 13.3a)

x = r cos φ

(13.7a)

y = r sin φ

It follows that

r 2 x 2 y 2 = +

(13.7b)

y
x

tan − 1 φ =