These include peak ground acceleration (PGA), peak

ground velocity (PGV), peak ground displacement (PGD),

(a) Response to arbitrary ground motion input

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

expressed as a fraction of the earth' gravity (note that one

s

]

∫o

′

(*τ*

gravity unit, or 1g, is equal to 980.7 cm/sec2 or 32.2

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

1

sin[ (t - *τ *) dτ

]

∫o x′ )

given level (0.05 g and 0.10 g as used by Bolt (1973)).

′

(*τ*

(D-8)

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.

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) *= *sin*Ω *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

(D-9)

at distances close to the earthquake fault, the onset of the

(

)

1/ 2

1 - (Ω / *ω *) 2

Ω / *ω *)2

(2*β*

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-

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=2π*ω*. 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

Integrated Publishing, Inc. |