Plates - Mechanics of Structures: Variational and Computational Methods - page 762

TABLE 13.1
Transfer and Stiffness Matrices of an Annular Plate with In-Plane Loading

a —Radial coordinate of the inner surface of the disc element. That is, the element
begins at a and continues to r for the transfer matrix and to b for the stiffness
matrix.

α —Coefficient of thermal expansion.

ρ —Mass density (mass/area).

—Angular velocity of rotation of disk that leads to centrifugal loading force
(radians/time).

p r ( r ) —Arbitrary loading intensity in r direction (force/length 2 ).

T ( r ) —Arbitrary temperature change.

T 1 —Constant temperature change.
Set 0 if only a constant temperature change is present.

u r and n r —Displacement and radial in-plane force per unit length.
E —Modulus of elasticity.
ν

T (ξ) =

—Poisson's ratio.
t —Thickness of disk.

r
a ξ

r 2 a 2

= (

− )

T 1 + (

1 + ν)

r

u 0 ( 1 + ν)α α T (ξ) d ξ

2 r

r η d

− (

r 2 − a 2 )

2

( 1 − ν

2

) ρ

2

− (

1 − ν

2

)

η p r (ξ) d ξ η

r 8 E Et

a a

r 2 r

r 2 a 2

t ( − ) tE α

n r =− E α T 1 −

a ξ

T (ξ) d ξ

2 r 2

) ρ

2 t

+ ν) + (

1 − ν)

2

( r 2 + a 2 )

r 2 a 2

− ( − ( 1

4 r 2

r
a η

η d r

+ (

1 − ν)

r 2

p r (ξ) d ξ η − p r (ξ) d ξ

a a

Transfer Matrix (Sign Convention 1)

a 1

r 2 Et

r 2 a 2

2

r 2 a 2

+ 2 (

− ) ( − )

r

−

1 1 − ν

u 0

u r
n r
1

2 r

a

U i z a =

r 2 a 2

− 2 (

− )

Et
2 ar 2

(

r 2

−

a 2

) −

1

n r

1

r 2

0 0 1

Stiffness Matrix (Sign Convention 2)

P a
P b

k aa u a

u b

P a
P b

k ab

= +

k ba k bb

p i k i v i p i = −

where

Et β + ν) HH ) β 1 k aa = 2 π

2
0

( 1 − ν) + ( 1 = ( 1 − ν

2 2

0

−

k ab = k ba =− 4 π Eh β 0 / H β 0 = b / a

π β − ν) H

0

P a =− k ab u 0 P b = bn r − k bb u 0 k bb = 2 ( 1 + ν) + ( 1 2 π