Structural Acoustics

58 Structural Acoustics: Deterministic and Random Phenomena

© 2011 by Taylor and Francis Group, LLC

The energy of bending of a uniform plate is derived as follows:

Using the expression for the bending strain that was derived for the beam,

it can be seen that

ε

x

z w/ x= − ∂ ∂

2 2

(4.19)

where z is the distance from the middle surface of the plate to any point in

the thickness of the plate and w is the lateral deection. Also,

ε

y

xy

z w/ y

Y z w/ x y

= − ∂ ∂

= − ∂ ∂ ∂

2 2

2

2 2( )Reference

(4.20)

The stresses are obtained from Hooke’s law:

σ ν ε νε

σ ν ε νε

x x y

y y x

E/

E/

= − +

= − +

[ ( )]( )

[ ( )]( )

1

1

2

2

(4.21)

τ ν

xy Y

xy

G E/ )= = +G 2 1(

The bending energy is

U / / Y / dx dy dz

B x x y y xy xy

= ∫ ∫ ∫ + +( )σ ε σ ε τ2 2 2

(4.22)

Thus,

U / Eh / w/ x w/ y

B