These include peak ground acceleration (PGA), peak
ground velocity (PGV), peak ground displacement (PGD),
(a) Response to arbitrary ground motion input
x(t). For any given ground acceleration x″(t), the relative
and strong motion duration. It should be noted that these
ground motion parameters provide only gross descriptions
displacement response u(t) is
of the recorded ground motions. The PGA value, normally
t
expressed as a fraction of the earth' gravity (note that one
s
x′ ) e - ωβ (t- τ ) sin[ D (t - τ ) dτ (D-7)
]
∫o
u(t) = - 1
′
(τ
ω
gravity unit, or 1g, is equal to 980.7 cm/sec2 or 32.2
ωD
ft/sec2), has been the key parameter in the past
characterizing the level of ground shaking for engineering
purposes; while duration has been used to characterize the
the single-degree-of-freedom system and β is the damping
time duration of significant shaking during earthquakes.
ratio. For the case of zero damping, this equation
Different definitions of strong motion duration have been
simplifies to
used. Bolt (1973) defined a bracketed duration as the lapsed
time between the first and last acceleration greater than a
t
1
sin[ (t - τ ) dτ
]
∫o x′ )
given level (0.05 g and 0.10 g as used by Bolt (1973)).
′
u(t) = -
(τ
ω
(D-8)
ω
Trifunac and Brady (1975) and Dobry et al. (1978) defined
significant duration as the time needed for the integral of
where ω is the undamped natural frequency of the system.
build up between 5 and 95 percent of its total value for the
Relative velocity and acceleration responses are given by
accelerogram. The integral of (x″(t))2 is a measure of the
the time derivatives u′) and u″(t), respectively.
(t
energy of an accelerogram (Arias, 1969). There are
empirical relationships between duration and earthquake
(b) Response to sinusoidal input. If the
magnitude (e.g., Bolt, 1973; Dobry et al., 1978).
ground acceleration x″(t) were to be a single unit amplitude
sinusoid at frequency Ω , x″(t) = sinΩ t, then the
(2) In general, the recorded ground motion
corresponding response is given by u(t) = H(T) sin[Ω t + N],
consists of the three main types of seismic waves described
where N is a phase angle and
in paragraph D-1d.. Experience indicates that each
accelerogram has a variable degree of detail. For example,
1
H (ω ) =
(D-9)
at distances close to the earthquake fault, the onset of the
(
)
1/ 2
1 - (Ω / ω ) 2
Ω / ω )2
(2β
2
main S waves is often associated with a longer-period pulse
+
related to the fault slip (see Figure D-6). It is important to
take this into consideration when designing structures near
is the system' frequency-response function which either
s
an active fault.
amplifies or attenuates the response according to the
frequency ratio Ω/, and the damping ratio β, see Figure D-
ω
b. Response Spectrum. Seismic ground motion may
8. This function is most useful in the explanation of how
be characterized as the superposition of a set of harmonic
predominant harmonics in ground motion can amplify the
motions having a fairly broad range of frequencies. This
ordinates of the response spectrum.
characterization of the ground motion (called the Fourier
spectrum) is often used by seismologists and is different
from the response spectrum discussed here. Structures
(2) Response spectra. For a given ground
subjected to the input ground motion tend to amplify the
acceleration x″(t) such as shown in Figure D-5(a), and
harmonics near their own natural frequencies and filter or
given damping ratio, the absolute maximum values found
attenuate the others. The resulting structural response
from the complete time history solution of equation D-7
therefore depends upon the frequency content of the
provide the response spectrum values at the system
harmonics in the ground motion and their relation to the
frequency ω, or period, T=2πω. A response spectrum is
/
dynamic frequency characteristics of the structure. This
traditionally presented as a curve connecting the maximum
paragraph provides the definitions and discussions of the
response values for a set of prescribed frequency or period
response spectrum representation of this inter-relationship
values, such as shown in Figure D-5(b). The different
between ground motion input and structural response.
response spectra quantities are defined as:
(1) Single degree-of-freedom system response.
SD = [ u(t) ]max = Relative Displacement Response
Figure D-7 shows the system and the definition for seismic
input and response.
D-11
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