TM 5-818-1 / AFM 88-3, Chap. 7
be evaluated by the procedure illustrated in figure 17-6.
relative particle amplitude as a function of inclination
from vertical.
b. Layered media.
17-4. Wave transmission, attenuation,
and
(1) In a layered medium, the energy
isolation. Vibrations are transmitted through soils by
transmitted by a body wave splits into four waves at the
stress waves. For most engineering analyses, the soil
interface between layers. Two waves are reflected back
may be treated as an ideal homogeneous, isotropic
into the first medium, and two waves are transmitted or
elastic material to determine the characteristics of the
refracted into the second medium. The amplitudes and
stress waves.
directions of all waves can be evaluated if the properties
a. Half-space. Two types of body waves may
of both media and the incident angle are known. If a
be transmitted in an ideal half-space, compression (P-)
layer containing a lower modulus overlies a layer with a
waves and shear (S-) waves; at the surface of the
higher modulus within the half-space, another surface
halfspace, a third wave known as the Rayleigh (R-) wave
wave, known as a Love wave, will occur. This wave is a
or surface wave will be transmitted. The characteristics
horizontally oriented S-wave whose velocity is between
that distinguish these three waves are velocity, wavefront
the S-wave velocity of the layer and of the underlying
geometry, radiation damping, and particle motion.
medium.
Figure 17-7 shows the characteristics of these waves as
(2) The decay or attenuation of stress
they are generated by a circular footing undergoing
waves occurs for two reasons: geometric or radiation
vertical vibration on the surface of an ideal half-space
with is = 0.25. The distance from the footing to each
damping, and material or hysteretic damping.
An
equation including both types of damping is the following:
wave in figure 17-7 is drawn in proportion to the velocity
exp[-α (r2 - r1 )]
A2 = A1 r1 C
(17-18)
of each wave. The wave velocities can be computed
r2
from the following:
ρ
where
A2 =
desired amplitude at distance r2
P-wave velocity:
A, =
known or measured amplitude at
vc= λ+2G
(17-15)
radial distance r, from vibration
p
source
S-wave velocity:
C=
constant, whichdescribes
geometrical damping
vs =
G
(17-16)
=
1 for body (P- or S-) waves
p
=
0.5 for surface or R-waves
R-wave velocity:
α
=
coefficient of attenuation, which
vR = Kvs
(17-17)
describes material damping (values
where
in table 17-3)
λ = 2G
and G are Lame's
E
c. Isolation. The isolation of certain structures
1-2 constants;
G =2(1 + j)
or zones from the effects of vibration may sometimes be
p = y/G= mass density of soil
necessary.
In some instances, isolation can be
y = moist or saturated unit weight
accomplished by locating the site at a large distance
K = constant, depending on Poisson's ratio
from the vibration source. The required distance, r2, is
0.87 < K < 0.98 for 0 <_< 0.5
calculated.from equation (17-18). In other situations,
(1) The P- and S-waves propagate radially
isolation may be accomplished by wave barriers. The
outward from the source along hemispherical wave
most effective barriers are open or void zones like
fronts, while the R-wave propagates outward along a
trenches or rows of cylindrical holes. Somewhat less
cylindrical wave front.
All waves encounter an
effective barriers are solid or fluid-filled trenches or
increasingly larger volume of material as they travel
holes. An effective barrier must be proportioned so that
outward,
thus decreasing in energy density with
its depth is at least two-thirds the wavelength of the
distance. This decrease in energy density and its
incoming wave. The thickness of the barrier in the
accompanying decrease in displacement amplitude is
direction of wave travel can be as thin as practical for
construction considerations. The length of the barrier
(2) The particle motions are as follows: for
perpendicular to the direction of wave travel will depend
the P-wave, a push-pull motion in the radial direction; for
upon the size of the zone to be isolated but should be no
the S-wave, a transverse motion normal to the radial
shorter than two times the maximum plan dimension of
direction; and for the R-wave, a complex motion, which
the structure or one wavelength, whichever is greater.
varies with depth and which occurs in a vertical plane
17-5. Evaluation of S-wave velocity In soils. The
containing a radius. At the surface, R-wave particle
key parameter in a dynamic analysis of a
motion describes a retrograde ellipse. The shaded
zones along the wave fronts in figure 17-7 represent the
17-9
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