Plates - Mechanics of Structures: Variational and Computational Methods - page 760

or

r
φ
γ r φ

1 − ν 0 n r

n

1
Et

=

− ν 1 0

002

(13.10b)

φ
n r φ

( 1 + ν)

E − 1 = s

If thermal loading effects are to be included, Eqs. (13.10a) and (13.10b), with 0 =

T 0] T , can be written as [Chapter 1, Eqs. (1.43) and (1.44)] [ α T α

1 ν 0 α E Tt /( 1 − ν)

n r
n φ
n r φ

r
φ
γ

= −

ν 10

00 1 − ν
2

D α E Tt /( 1 − ν)

(13.10c)

0 r φ

E 0 s = E −

and

1 − ν 0 n r

n φ
n r φ

α T

α

r
φ
γ

1
Et

= +

− ν 1 0

002

T

(13.10d)

( 1 + ν) 0

r φ

E − 1 0 = s +

Conditions of Equilibrium
The equilibrium equations, w hich p rov id e relationships between the stress resultants and
the body forces (force/area p V =

] T , are (Fig. 13.3b)

[ p Vr p V φ

∂ n r

∂

1

r ∂ ∂φ + (

n r φ n r − n φ )

r +

+ p Vr = 0

r

n r φ = n (13.11a)

φ r

1

r ∂ ∂φ + ∂

n φ n r φ

∂

2 n r φ
r

+ + p V φ = 0

r

or in matrix form,

∂ p Vr

p V φ

n r
n φ
n r φ

+ 1 / r − 1 / r ( 1 / r )∂ φ

r

+

= 0

0 ( 1 / r )∂ φ ∂ + 2 / r (13.11b)

r

D s

+ =

s p V 0

Boundary Conditions
The displacement boundary conditions are

u r = u r

on S u (13.12)

u

φ =

u

φ

and the force boundary conditions are

p r = p r

on S p (13.13)

p φ = p

φ

where p r and p are the surface forces (force/length) in the r and φ directions around the

φ

circumference of the disk.