Civil Engineering Reference
In-Depth Information
In the literature, e.g. Blevins (1979), we find the eigenfunctions and
eigenfrequencies listed for various types of structural element having different shapes
and dimensions, and subjected to different boundary conditions. As a rule, exact data are
only found for regular-shaped elements having idealized boundary conditions. For more
complicated cases one must resort to finite element methods (FEM), but given the
versatility of modern FEM software packages this seldom gives practical problems. As
an illustration we give the results of an exact calculation on a typical isotropic element; a
thin rectangular panel
simply supported
along the edges having length
a
and
b
,
respectively. This boundary condition implies that the velocity as well as the moment is
zero along the edges. We may remark that measurement results of natural frequencies for
floors in buildings of monolithic concrete give reasonable agreement with calculations
when using this condition.
The eigenfunctions Ψ
i,n
(
x,z
) for the plate, placed in the plane x-z, must satisfy the
following wave equation (presupposing harmonic time variation e
jω
t
)
22
2
B
⋅∇ ∇ Ψ
(,)
x z
−
ω
⋅ ⋅Ψ
m
(,) 0.
x z
=
(3.107)
in
,
in
,
in
,
Imposing simply supported boundaries gives the solutions
i
n
π
π
Ψ
(,) sin
xz
=
x
⋅
sin
z
where
in
,
=
1,2,3, .
(3.108)
in
,
a
b
The associated eigenfrequencies will be given by
2
2
⎡
⎤
π
B
i
n
⎛⎞ ⎛⎞
f
=
⎢
+
⎥
.
(3.109)
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
in
,
2
ma
b
⎢
⎥
⎣
⎦
For a homogeneous plate this equation may be expressed as
2
2
⎡
⎤
π
i
n
⎛⎞ ⎛⎞
f
=
c h
⎢
+
⎥
.
(3.110)
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
in
,
L
a
b
43
⎢
⎥
⎣
⎦
Each of these eigenfunctions, or each set of indices (
i,n
), thereby defines a mode of
vibration, a
natural mode
or
eigenmode.
Any complex pattern of vibration may then be
expressed by a sum of these modes. It should be noted that none of the indices
i
and
n
may be equal to zero. The first eigenfrequency or natural frequency of a plate is therefore
f
1,1
.
Figure 3.21
gives examples of natural modes of vibration for a plate according to
Equation
(3.108)
and calculated for some of the lowest set of indices.
Figure 3.22
gives
corresponding examples on natural frequencies for a simply supported 180 mm thick
concrete slab. The edges
a
and
b
are 4.0 and 6.0 metres, respectively.
3.7.3.3
Eigenfrequencies of orthotropic plates
In contrast to the isotropic plates (or panels), the material properties for orthotropic plates
will depend on the direction. These properties are by definition, as also mentioned above,
symmetric about three mutually perpendicular axes. We will again assume that the panel
is placed in the xz-plane, furthermore that the x- and z-axis are axes of symmetry with
