Plates - Mechanics of Structures: Variational and Computational Methods - page 785

The other fundamental quantities become

K

1
r ∂

1
r 2 ∂

2
r

2

m r =− ∂ w + ν w +

φ w

r

K 1

1
r 2 ∂

2 2

r

m

φ =−

r ∂ r w +

φ w + ν∂

w

(13.70)

K 1

1
r 2 ∂ φ w

m r φ =− ( 1 − ν)

r ∂

∂ φ w −

r

2

q r =− K ∂ r ∇ w

K 1

2

q

φ =−

r ∂ φ ∇

w

with the equivalent shear forces

K

1
r ∂ r ∂ φ w −

1
r ∂ φ

1 − ν

r

1
r 2 ∂ φ w

V r =

2

V = q r + m r φ =− ∂ r ∇ w + ∂ φ

(13.71)

K 1

r 1

1
r 2 ∂ φ w

V φ =

2

q φ + ∂

r m r φ =−

r ∂ φ ∇

w + ( 1 − ν)∂

r ∂

∂ φ w −

r

The stresses σ

r ,

σ φ , τ are given by

r φ

12 m r z
t 3

12 m z 12 m r φ z

φ

σ = σ φ = τ

φ =

(13.72)

r r

t 3 t 3

Note that the peak stress values occur on the surfaces at z .

The first-order governing differential equations can be obtained by rearranging Eqs.
(13.70) and (13.71) and introducing the kinematic equation

=± t / 2

Alternatively, con-
vert Eq. (13.58) to a polar coordinate system, with the r axis coinciding with the x axis.
These procedures lead to the equations

∂w/∂ r =− θ.

. . .

0 − 1 0 0

w w 0

········· ········· ······

. . .

2
∂φ

) ∂ r 2 ν( 1 / − ν/ r 0 1 / K

2

. . .

θ θ 0

········· ········· ··· ······

K 1
r 4

∂
∂

4 .

. .

=

4
∂φ

+

) ∂

2

( 1 − ν

r

2

2 ) 2

∂φ

2
∂φ

K (3 − 2 ν − ν ∂

) ∂

. . .

r 2

− − 1 / r − (ν/ 2(1 − ν )

2
∂φ

V

∂

V − p z

−

r 3

2 2

r 4

········· ········· ··· ······

.

r 2 (

. .

K

2

1 − ν )

2 )

K (3 − 2 ν − ν

2
∂φ

∂

− 1 − ( 1 − ν)/ r

m r m r 0

2 .

r 3

2

. − .

2
∂φ

− ν) ∂ (

2 1

∂ z

= A z + P

∂ r

(13.73)

Including the effects of an elastic foundation of modulus k , compressive in-plane forces
per unit length n r (radial) and n φ (circumferential), and the dynamic response, the governing