Environmental Engineering Reference
In-Depth Information
Note
: The elastic moduli are defined in
Table 3.26
and
Figure 3.69
and expressions for
the dynamic elastic parameters
E
d
,
G
d
, and
K
and Poisson's ratio given in terms of
V
p
and
V
s
are found in
Table 3.27.
When these velocities are propagated synthetically in the field
(see
Section 2.3.2)
or in the laboratory (see
Section 3.5.5),
values for
E
d
,
G
d
, and
K
can be
computed.
Characteristics
Seismic waves
(Figure 11.4)
may be described by the quantities of vibratory motion, i.e.,
amplitude, wavelength, period, and frequency. Amplitude and frequency are the two
parameters commonly used to define vibratory motion in Earthquake studies.
Amplitude
A
is the displacement from the mean position or onehalf the maximum displacement.
Wavelength
is the distance between crests.
Period T
is the time of a complete vibration, or
the time a wave travels distance (
λ
λ
), expressed as
T
1/
f
2
π
/
ω
(11.1)
where
f
is the frequency and
the circular frequency.
Frequency f
is the number of vibrations per second (or oscillation in terms of cycles per
unit of time), given normally in hertz (Hz) with units of cycles per second, expressed as
ω
f
ω
/2
π
(Hz)
(11.2)
Ground shaking is felt generally in the ranges from 20 Hz (high frequency) to less than 1
Hz (low frequency of long waves).
Circular frequency
ω
defines the rate of oscillation in terms of radians per unit of time;
2
π
rad is equal to one complete cycle of oscillations.
ω
2
π
f
2
π
/
T
(11.3)
11.2.3
Ground Motion
Elements
Ground motion occurs as the seismic waves reach the surface, and is described in terms of
several elements derived from the characteristics of seismic waves, including displace-
ment, velocity, and acceleration.
Displacement
y
at a given time
t
is a function of position
x
and time
t
in
Figure 11.4,
expressed as
y
A
sin 2
π
/
λ
(
x
vt
)
cm
(11.4)
Velocity
v
, termed the particle or vibrational velocity, is expressed as
y
.
v
λ
/
T
f
λ
, or
v
d
y
/d
t
cm/sec
(11.5)
Acceleration
a
is
α
d
2
y
/d
t
2
ÿ
cm/sý
(11.6)
If Equation 11.5 is substituted for
v
and
f
is expressed in terms of the circular frequency
ω
from Equation 11.3,
y, v,
and
a
can be expressed as
ground displacement:
y
A
sin
ω
t
(11.7)
