Building Acoustics

Waves in fluid and solid media 95

EExυzzx= υ. (3.111)

The bending stiffness in the two directions will then be given by

() ()

3 3

and.

12 1 12 1

x z

xz

x zxz

Eh Eh

BB

υυ υυ

==

−−

(3.112)

Analogous to Equation (3.109), we shall give a formula for the natural frequencies of a
simply supported rectangular plate with dimensions a and b. Formulae covering cases
with other types of boundary condition may be found in the literature, e.g. Blevins
(1979). For the simply supported plate we get

1

44 22 2

, 44 22

2

,

2

in x z xz

in in

fBBB

ma b ab

π ⎡⎤

=++⎢⎥

⎣⎦

(3.113)

where Bxz is given by

3

2.

12

xz

xz x z

Gh

BB=+⋅ν (3.114)

For the isotropic case, where

EEExz== ==,,and/2(1)υxzυυ G Exz= +[ υ],

it is easy to show that Equation (3.113) simplifies to Equation (3.109). It was formerly
pointed out that the types of orthotropic plate normally found as building components,
mainly in industrial buildings, are plates with attached stiffeners or corrugated plates.
The latter may have many different shapes; from the “wavy” corrugated type to the more
sophisticated having trapezoidal corrugations denoted as cladding. When applying the
general theory of orthotropic plates on corrugated plates several assumptions must be
fulfilled. We shall not delve into these assumptions, but just point to the fact that for
many types equivalent expressions for the stiffness components Bx, Bz and Bxz exist in the
literature, expressions which one may use to calculate e.g. the natural frequencies.
One example we shall use is the “wavy” type of corrugations; a panel having
thickness h and where the “waves” have sinusoidal shape with wavelength L and
amplitude H. The total height of the panel is then 2H. Following Timoshenko and
Woinowsky-Krieger (1959)^2 we may write:

(^2) These equations are also referenced in Blevins (1979), unfortunately, with a misprint in the expression for B
z.