Civil Engineering Reference
In-Depth Information
wave propagation will occur, i.e. longitudinal stresses will produce lateral strains on the
outer free surfaces. This is called the Poisson contraction phenomenon. The associated
wave type is therefore called
quasi-longitudinal
and
Figure 3.18
a) may serve as an
illustration.
a)
b)
c)
Figure 3.18
Wave types in solids. a) Quasi-longitudinal wave. b) Shear wave. c) Bending wave.
The solid lines in the figure represent elements of the structure at rest whereas the
broken lines illustrate the deformations of these elements both in the direction of wave
propagation and laterally. Using the particle velocity as the variable, we may show that
for free waves (one-dimensional case) the following differential equation applies
2
2
∂
v
∂
v
x
x
′
E
−
ρ
=
0,
(3.94)
2
2
∂
x
∂
t
where
v
x
is the velocity in the x-direction (the direction of propagation) and ρ is the
density of the medium.
E
' is a property of the material which depends on the actual
lateral displacements. Taking a plate as an example, we get lateral displacements or
contractions in one direction giving
E
E
′ =
,
(3.95)
2
1
−
υ
where
E
is the modulus of elasticity (Young's modulus) of the material and υ is the
Poisson's ratio. The latter is defined as the ratio of the magnitudes of the lateral strain to
the longitudinal strain. As seen from
Table 3.1
, this ratio varies between 0.2 and 0.35 for
common building materials.
