Parametric modeling of earthquake response spectra

Parametric modeling of earthquake response spectra H. E. Febres Cedillo

Facultad de Ingenieri, Universidad de Los Andes, Mdrida, Venezuela M. Asghar Bhatti

Department of Civil and Environmental Engineering, University of lowa, Iowa City, 1,4 52242, USA A mathematical model for the response spectra is determined using statistical analysis. The form of the model is first established using fifty computer simulated accelerograms. The final form is then used on twenty-five accelerograms from fifteen past United States earthquakes. This model smooths out peaks and valleys which are characteristic of the response spectrum of any single earthquake. Thus it serves as a 'smooth design spectrum' and can be used to approximate structural response to a future seismic event. 1. I N T R O D U C T I O N The response spectrum is a diagram which gives the maximum response of any single degree of freedom system subjected to a given ground motion history. It is usually, plotted as a function of period of vibration of the structure (Fig. 1). Ground motions caused by earthquakes are random in nature as evidenced by the accelerogram records of such earthquakes. A response spectrum is therefore, unique and distinctive of each particular accelerogram record. When using these response spectra for design purposes, it is necessary to observe the following: 1) The response spectrum provides an indication of the maximum response of a structure to a single earthquake. Moreover, it is the response of the structure to the action of only one component of such earthquake motion. For most earthquakes, two horizontal and one vertical component are recorded. 2) Since the accelerogram obtained for a particular earthquake is influenced by the nature of the soil conditions at the recording site, as well as the distance to the epicenter, it is reasonable to expect different response from the same structure placed in a different location. 3) Typically, the response spectra exhibit sharp peaks and valleys (Fig. 1). This shows a large variation of the response for a very small variation of T. In fact, the precision in determining the period of vibration of a structure is not sufficient as to know with certainty whether its response corresponds to a valley or a peak of the spectrum. Therefore it is logical to use either 'averaged' response spectra or envelope spectra obtained from several real ground motion acceleration records. Idealized response spectra of this type have been proposed to be used as 'design spectra' curves t.2.3. 4) The average design spectra available in literature are based primarily on the earthquake records from the

pacific west coast of the United States. Thus they may not be suitable for other parts of the world. Furthermore, since there is no systematic procedure available for averaging the spectra, it is difficult to incorporate recent earthquake records into the design spectra.

W

T3

w

TI ~

W and ~, constants

T2

///////////////////////////////////////////; a) Simple oscillators with different natural periods of vibration Ground Acceleration

(g)

~

(secs)

b) Ground motion acceleration history Sa

(g)

!

I

I

Zl

T2

1"3

r (secs)

c) Diagram of maximum acceleration response Paper accepted April 1990. Discussion closes January 1992.

Fig. 1. A typical acceleration response spectra

© 1991 Elsevier Science Publishers Ltd.

Soil Dynamics and Earthquake Engineering, 1991, Volume 10, Number 6, August 291

Parametric modeling o f earthquake response spectra: H. E. F. Cedillo and M. A. Bhatti

Therefore, there is a need to systematically characterize the response spectra computed from a given accelerogram record. This paper develops a mathematical model of the response spectra by means of statistical analysis. New values of the coefficients in the model can readily be computed for any set of earthquake records. Thus obtaining 'customized design spectra' for any site.

2. THE RESPONSE SPECTRUM The response spectrum is a plot of the maximum response to a given ground motion history for all possible single degree of freedom systems. The equation of motion of such system (see Fig. 2) is: W

in which, o~ = x / ~ w is the undamped natural frequency, = c/c, = c/(wo~w/g) is called damping ratio, and w d = a r v c - ( - - ~ 2 is the damped natural circular frequency. The maximum value of displacement occurs at t = t m a x. Its amplitude is [y(tmax)[, and is called the Displacement Response Spectrum (Sd). The maximum force acting on the mass m is ky(tmaO, therefore, the maximum acceleration is (k/m) y(tm~x). The Acceleration Response Spectrum (So) is then: So = k__ Sa = 092 Sa = m

(1)

-- fiw + cy + k y = O

g

(4)

The Velocity Spectrum (S,) can obviously be obtained as

in which, w= g= c= k=

Sa

S v = .~max

weight of the body, acceleration due to gravity, damping constant, and spring constant.

Noting that uw = ub + y, equation (1) can be written in terms of relative displacement as: (2)

my~ +