be evaluated by the procedure illustrated in figure 17-6.

relative particle amplitude as a function of inclination

from vertical.

b. Layered media.

(1) In a layered medium, the energy

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

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

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