Plates - Mechanics of Structures: Variational and Computational Methods - page 782

with

= w θ V m m m ] T

z [

m m

U ww U wθ U w V U w m

U
θw

U U U

θ θθ θ V m

U i

=

U V w U V θ U VV U Vm

U m w U m θ U mV U mm

and

= w m 0 m T z i

0
m

0
m

V m

θ

or

0
m

w ( x )

U U U U

w ( ) w ( )

x 0

ww wθ w V w m

m m

= +

0
m

U U U U θ ( x )

θ ( x ) θ ( 0 )

θw θθ θ V θ m

m m

(3)

U V w U V θ U VV U Vm

V m ( x ) V m ( 0 )

V m (

x )

U m w U m θ U mV U mm

m m ( x ) m m ( 0 )

m 0 m (

x )

m L y ,

in which, with β m =

U ww =− β

( − ν)

2

1

m

β + β

x sinh

m x

cosh

m x

β m x

1
2

sinh β m x

β m

U

wθ =−

( 1 + ν) + ( 1 − ν) x cosh

sinh

m x

1
2 K

β

m x

U w V = − x cosh β

β

m

β

m

x sinh β

m x

U =−

w m

2 K β m

β m x

θw =− β

m
2

sinh β m x

U ( 1 + ν) − ( 1 − ν) x cosh

β

m

U θθ = β

( 1 − ν)

m

x sinh β

m x

+ cosh β

m x

2

x sinh

β m x

U =

θ V

2 K β m

sinh

m x

1
2 K

β

m x

U = + x cosh β

θ m

β

m

β m x

m

K β sinh β m x

β m

2 2 x cosh

U V w = ( 3 − 2 ν − ν ) − ( 1 − ν)

2

m 2

K β ( 1 − ν)

U V θ =− [ x sinh β m x ]

2

U VV =− β m (

1 − ν)

2

x sinh β m x + cosh β m x

m x

m
2

= β

sinh β

m x

U Vm ( 1 + ν) − ( 1 − ν) x cosh β

β

m