Plates - Mechanics of Structures: Variational and Computational Methods - page 783

m 2

K β ( 1 − ν)

U m w = x sinh β m x

2

m x

m

K β sinh β

m x

2 2 x cosh

U m θ = ( 3 − 2 ν − ν ) + ( 1 − ν) β

2 β

m

m x

1
2

sinh β

m x

U mV = ( 1 + ν) + ( 1 − ν) x cosh β

β

m

= β

( 1 − ν)

m

U mm x sinh β

m x

+ cosh β

m x

2

and

1
2 K

0
m

w = p m (β m x sinh β m x − 2 cosh β m x + 2 )

β

m

sinh

β m x

1
2 K

β m x
β m

0
m

2
m

θ = p m β − x cosh

β

m

p m 1

m x

− ν
2

3 − ν

2

V m =

β − β

x cosh

m x

sinh

β

m

p m 1

1
β

− ν
2

m 0 m =− β

m x sinh

β

m x

+ ν cosh β

m x

− ν

m

The boundary conditions for the state variables are

w m ( L ) = m m ( L ) = w m ( 0 ) = m m ( 0 ) = 0 . (4)

Substitution of these conditions into (3) leads to

] x = L =− w

m x = L

0

[ U

wθ θ m (

0 ) + U

V V m (

0 )

w

(5)

] x = L =− m 0 m x = L [ U m θ θ m ( 0 ) + U mV V m ( 0 )

θ ( ) ( )

Thus, the initial parameters 0 and V m 0 can be determined as

m

− w

m 0 m U w V x = L

1

m U mV

θ ( 0 ) = +

m

(6)

−

m U m θ x = L

1

m 0 m U

0

V m ( 0 ) =

wθ + w

where

= [ U wθ U mV − U m θ U w V ] x = L (7)

Once the state variable parameters at x 0 are known, the values of the state variables,

as well as those of the stresses, at any location of the plate can be determined. These
manipulations are rather tedious for hand calculations, but very convenient for computers.
This solution leads to the deflection

=

1 β m x

2

∞

5 K
m

4 p 0 L y
π

sin m

y
L y =

π 1

m 5

tanh β m L
2

w =

1 w m (

x ) + +

m =

sinh 1 cosh

m x sin

β

m L

+ β

m x
2

tanh β

m L
2

− β

m x

− β β

m y

4 cosh 2 (β m L / 2 )

m = 1 , 3 , 5 , ... (8)

The series converges so rapidly that often the first term provides sufficient accuracy.