EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 19,971-976 (1990)

A NOTE ON THE FAST COMPUTATION OF RESPONSE SPECTRA ESTIMATES

EDUARDO REINOSO Centro de Investigacibn Sismica, Fundacibn Javier Barros Sierra. Carretera a1 Ajusco 203, TIakan 14200. Mixico, D.F.. Mexico

AND

MARIO ORDAZ AND FRANCISCO J. SANCHEZ-SESMA Centro de Investigacibn Sismica and Instituto de Ingenieria, Ciudad Universitaria, Apdo. 70-472, Coyoacan 04510, Mexico, D.F., Mexico

SUMMARY An algorithm is presented for fast computation of estimates of response spectra. It is based on the formalism of random vibration theory. Numerical results show that estimates are reliable and useful for practical applications.

INTRODUCTION

Computation of response spectra by exact methods requires considerable computer resources. This is particularly true when massive computations of response spectra are needed. In many cases, the increase of strong ground motion digital recordings is not necessarily followed by an increase in the computer capabilities to reduce and analyse the data. Exact computation of response spectra is well established but if the analysis must be done in small computers, calculations become lengthy and faster algorithms could be welcomed. Random vibration theory (see e.g. Cartwright and Longuett-Higgins,’ Boore,2 Vanmarcke3) provides an elegant, reliable and economic alternative for the approximate computation of response spectra.

In this note we present an algorithm based on the formalism of random vibration theory (RVT) and closely follow the format proposed by Boore.’ We use as equivalent duration of strong ground motion the empirical definition by Trifunac and brad^.^ This, together with the correction for the equivalent duration of the oscillator response proposed by Boore and Joyner,’ allows a direct application of the main RVT results.

In what follows, we present the algorithm and show that RVT estimates for response spectra are reliable and, therefore, useful in earthquake engineering applications.

PROPOSED ALGORITHM

The necessary steps are as follows. 1. Compute the equivalent duration of the strong ground motion, T,. This duration does not have an exact

definition but it has been found that the results are reasonably good when T, is the duration of Trifunac and Brady4 which is defined in terms of the Arias intensity of the signal,6 given by

where T, is the total duration of the accelerogram and g the acceleration of gravity. Trifunac and Brady’s duration is the time interval which comprises between 5 and 95 per cent of the Arias intensity.

2. Obtain the Fourier amplitude spectrum I A ( w ) I of the accelerogram.

0098-8847/90/07097 146$05.OO 0 1990 by John Wiley & Sons, Ltd.

Received 5 December 1989 Revised 19 April 1990

972 E. REINOSO, M. ORDAZ AND F. J. SANCHEZ-SESMA

3. Calculate the response for each desired frequency R of the oscillator:

where

is the modulus of the transfer function of the elementary oscillator (acceleration to pseudoacceleration), and 5 is a fraction of the critical damping.

4. Calculate the moment integrals Mo and M, given by 1 r m

M k = - J ok I Y(R, w ) 1’ dw, k = 0, 2 n -02

5. Compute the root mean square pseudoacceleration y,,, ,

Yrms = [z] (4)

(5)

where T,,, is the response duration. To obtain T,,,, Boore and Joyner’ proposed the next empirical result, calibrated for California earthquakes:

(6)

in which f = R f 2 ~

approximation’ 6. Obtain the peak factor F , which, according to random vibration theory,’ is given by the asymptotic

Y (2 1n ~ y . 5

F , = (21n N)”’ + (7)

where y ( = 0.577 . . . ) is Euler’s constant and N = number of extrema in the time interval, T,,,, estimated by

Table I. Earthquakes whose strong ground motion records in the Valley of Mexico are analysed in this study (based on Singh et d9)

Depth Distance* Azimuth* Date Lat (“N) Long (“W) (km) Magnitude (km) (“)

19 09 1985’ 21 09 1985’ 30 04 1986’ 12 03 1987 07 06 1987’ 15 07 1987’ 08 02 1988’ 10 03 1989’

18.14 102.71 17.62 101.82 18.40 102.97

LOCAL 16.83 98.69 17.56 97.185 1750 101.14 17.51 100.75

16 22 21

40 68 20 ?

-

8.1 7.6 7.0 < 3 4.8 5.9 5.8 5.0

394.2 3665 411.8 f 10.0 281.3 288.2 289.3 270.0

69.9 55.4 74.9

? 349.4 313.1 45.3 40.0

*With respect to University City (CU) in the south of Mexico City. Thrust faults.

*Normal fault.

FAST COMPUTATION O F RESPONSE SPECTRA ESTIMATES 973

7. Finally, the expected value of the pseudoacceleration A,,, is given by

(Amax) = Y,,, F , (9) Summarizing, with expressions (1) to (9) it is possible to obtain, in a fast way, reliable estimates of

pseudoacceleration response spectra of recorded signals.

SCl 19.1X.85 NS EW CDAF 19.1X.85 NS EW

c 800

200

-Z 600

s 400

0 1 2 3 4 5

' \ 400

300

200

'. .._..

0 1 2 3 4 5 "

0 1 2 3 4 5 0 1 2 3 4 5

VIV 19.1X.85 NS EW

g 120 h

Y

..*' .. '

40

0 1.0 2.0 3.0 0 1.0 2.0 3.0

CU 19.1X.85 NS EW

140 -

20 n

- " " '

0 1.0 2.0 3.0 0 1.0 2.0 3.0

CDAF 21.1X.85 NS EW

0 1 2 ? , 4 5 0 1 2 3 4 5

D74 12.111.87 NS EW

00 .10 .20 .?Po Period (sec) Period (sec)

TAC 19.1X.85 NS EW

0- - 0 1.0 2.0 3.0 4.0 0 1.0 2.0 3.0 4.0

CU 21.1X.85 NS 60

50 - 40 - 30 -

10 - 0 I , ' , ' -

0 1.0 2.0 3.0 0 1.0 2.0 3.0

CDAO 30.1V.86 NS EW

50 ' O 0 M

0 1 2 3 4 5 0 1 2 3 4

TO2 7.V1.87 NS EW

.. ........_

0 1.0 2.0 3.0 Period (sec)

Figure 1. Comparison between response spectra for 5 per cent damping of 20 accelerograms. The dotted line shows the spectrum calculated with random vibration theory while the continuous one shows that computed by the fast Fourier transform method. All

charts show pseudoacceleration in gals and period in seconds

974 E. REINOSO, M. ORDAZ AND F. J. SANCHEZ-SESMA

RESULTS AND CONCLUSIONS

In order to calibrate this method, 40 accelerograms recorded at various sites in Mexico City were processed.' The considered earthquakes cover a large range of magnitudes and focal distances, and various fault mechanisms as well (Table I). In all cases the exact response spectra were compared with RVT results. Some typical results are shown in Figures 1 and 2. In general, the comparison is satisfactory. In the computations we observed that the proposed algorithm is about 150 times faster than the exact one."

TO5 7.V1.87 NS EW

[l

0 1.0 2.0 3.0

T10 15.V11.87 NS EW

12 l ' 1

v) 4 !l!u 2

TO3 15.V11.87 NS EW

T 1 1 15.V11.87 NS EW

0 1.0 2.0 3.0 0 1.0 2.0 3.0

046 8.11.88 NS EW

0 1.0 2.0 3.0 0 1.0 2.0 3.0

074 8.11.87 NS EW

I

0 1.0 2.0 3.0 0 1.0 2.0 3.0

074 10.111.89 NS EW

0 1 .o 2.0

050 8.11.88 NS

0 1 .o 2.0

EW

0 1.0 2.0 3.0

084 8.11.88 NS

0 1.0 2.0 3.0

EW

.. 0 1.0 2.0 3.0

084 10.111.89 NS

35 - 30 - 25 - 20 -

5 -

-0 .6 1.2 1.8 0 .6 1.2 1 8 Period (tec) Period (scc)

0 1.0 20 3 0 Period (sec)

0 1.0 2.0 3.0

EW

0 1 0 2 0 3 0 Pertod (sec)

Figure 2. Same as Figure 1 but for 20 other records

FAST COMPUTATION OF RESPONSE SPECTRA ESTIMATES 975

To measure the discrepancies between proposed and exact results, the mean relative error and the standard deviation of this error were computed. We obtained 6 and 18 per cent, respectively. In the context of building code drafting, response spectra are used basically to propose design spectra. Hence, we considered the errors in the prediction of both spectral peaks and the period of these peaks (Table 11). We found that, on average, the peaks are overestimated by a factor of 1.06, and the period of their occurrence is underestimated in 0-013 sec.

Table 11. Comparison of the position and amplitude of the peaks between exact and random vibration theory spectra of the strong ground motion records analysed in this study

Natural Exact RVT Error period T* Sat T* Sa' Tt Say

Station T,(sec) Date (gal) (set) (gal) (set) W)

SCT

CDAF

SXVI

TAC

cu cu CDAF

CDAO

D74

TO2

TO5

TO3

T10

T11

D46

D50

D74

D84

D74

D84

2-00

3-10

054

-8

-4

4

3.10

3.53

9

1.42

1.47

-8

?

1.96

0.9 5

0.42

- I

1.43

9

1.43

850919 NS EW NS EW NS EW NS EW NS EW NS EW NS EW

860430 NS EW

870312 NS EW

870607 NS EW NS EW

870715 NS EW NS EW NS EW

880208 NS EW NS EW NS EW NS EW

890310 NS EW NS EW

2.10 2'00 2.90 1.90 078 0.58 0.83 087 092 1.70 1.50 0.80 1.95 1.85 3.40 3.25 0.07 0.10 1.30 1.33 0.55 1.34 0.30 0,38 0.47 057 1.02 0-62 0.90 090 055 0.50 0.35 0.50 1-40 1-40 0-40 0-30 1 -40 1-35

601.2 972.3 432.0 323.1 158.0 164.0 113.3 85.9

137.6 106.1 47.3 45.5

177.6 96.5 75.0

239.7 13.1 12.3 12.6 15.4 6 1

12.6 2.0 5.5

12.2 10.6 101 7.4

26.9 20.8 13.8 12.5 8.7

11.9 48.6 32.6 6 1 6.6

13.4 25.2

2.03 2.03 3.00 2.10 053 0.58 083 087 0.92 1.70 1.50 080 1.90 1.85 3.25 3.15 006 0.1 1 1.40 1.35 0.55 1-35 0-30 040 0.47 057 1.02 0.55 090 090 055 0.50 0.40 045 1.35 1.35 0.30 0.30 1.35 1.30

71 1.9 L 209.6 465.1 3603 175.3 163.2 102.8 90.9

11 1-8 98.4 490 47.9 163-8 9 0 1 79.5

3 15.2 9-8

11.2 14-7 16.1 7- 1

157 1-7 5-5

14.7 10-6 11.7 6.9

25.8 21.0 13-3 13-4 8-4

11.4 49.5 41.8 7.8 6- 1

17-2 32.8

- 0.07 0-03 0.10 0.20

- 0.25 0 0 0 0 0 0 0

- 0.05 0

- 0.15 - 0.10 - 001

0.0 1 0.1 0.02 0 0.01 0 0.02 0 0 0

- 0.07 0 0 0 0 0.05

- 0.05 - 0.05 - 0.05 - 0.10

0 - 0.05 - 0.05

18-4 24.4 7.6

11.5 10.8 - 0.6 - 9.3

5.9 - 18.9 - 7.6

3.6 5.4

- 7.8 - 6.6

6.0 31.5

- 25.1 - 9.0

17.0 5.2

165 24-6

- 13.4 1.2

20.5 0

14.9 - 6.5 - 3.9

1 .o - 3.3

7.0 - 3.0 - 4.1

2.0 28.1 20 1 - 7.6 28.6 30.2

~ ~

*Period where the peak of the spectrum occurs. 'Peak pseudoacceleration. *Difference on the period prediction, in seconds.

'Hill zone site. Difference on the amplitude prediction, in per cent.

976 E. REINOSO, M. ORDAZ AND F. J. SANCHEZ-SESMA

It should be noted that these errors only measure the uncertainties associated with the method itself. This kind of uncertainty has been called professional uncertainty.’ ’ When using the proposed algorithm for decision making purposes, other sources of dispersion should be taken into account.

As we can see, this method works well, but is not an exact computation of response spectra. If a serious analysis of the data is required, it should be done with the exact methods.

Some other applications8* 1 2 * l 3 of this method have been the prediction and postdiction of seismic response spectra.

ACKNOWLEDGEMENTS

We had stimulating discussions along the way with D. M. Boore, M. A. Bravo, F. J. Chavez-Garcia, I,. E. Perez-Rocha, E. Rosenblueth and S. K. Singh. Their comments and suggestions, as well as those of two anonymous reviewers, are greatly appreciated. Most of the data used in this study were recorded by the network operated by Centro de Instrumentacibn y Registro Sismico A.C. This network is funded by Departamento del Distrito Federal through Consejo Nacional de Ciencia y Tecnologia, Mexico. The remaining accelerograms were recorded by the network operated by Instituto de Ingenieria, UNAM. This research was partially supported by Consejo Nacional de Ciencia y Tecnologia and Departamento del Distrito Federal.

REFERENCES

1. D. E. Cartwright and M. S. Longuett-Higgins, ‘The statistical distribution of the maxima of a random function’, Proc. roy. soc. London A237, 212-232 (1956).

2. D. M. Boore, ‘Stochastic simulation of high-frequency ground motions based on seismological models of the radiated spectra’. Bull. seism. SOC. Am. 73, 1865-1894 (1983).

3. E. H. Vanmarcke, ‘Structural response to earthquakes’, in Seismic Risk and Engineering Decisions (Eds. C. Lomnitz and E. Rosenblueth), Elsevier, Amsterdam, 1986, pp. 287-337.

4. M. D. Trifunac and A. G. Brady, ‘A study of the duration of strong earthquake ground motion’, Bull. seism. soc. Am. 65, 581426 (1975).

5. D. M. Boore and W. B. Joyner, ‘A note on the use of random vibration theory to predict peak amplitudes of transient signals’, Bull. seism. soc. Am. 74, 2035-2039 (1984).

6. A. Arias, ‘A measure ofearthquake intensity’, in Seismic Designfor Nuclear Power Plants (Ed. R. Hansen), Massachusetts Institute of Technology Press, Cambridge, Massachusetts, 1969.

7. A. G. Davenport, ‘Note on the distribution of the largest value of a random function with application to gust-loading’, Proc. inst. civil eng. 28, 187-196 (1964).

8. M. Ordaz and E. Reinoso, ‘Us0 de la teoria de vibraciones aleatorias para la determinacion de espectros de disefio del Reglaniento para las Construcciones del D.F.’, Proc. VII natl. con$ earthquake eng. Queretaro, Qro, Mexico A155-Al67 (1987).

9. S. K. Singh, J. Lermo, T. Dominguez, M. Ordaz, J. M. Espinosa, E. Mena and R. Quass, ‘The Mexico earthquake of September 19, 1985. A study of amplification of seismic waves in the Valley of Mexico with respect to a hill zone site’, Earthquake spectra 4,

10. E. Reinoso, ‘Un algoritmo para el calculo approximado de espectros de respuesta basado en la teoria de vibraciones aleatorias’, Civil

1 1 . E. Rosenblueth, ‘Optimum design for infrequent disturbances’, J . struct. diu. ASCE 102, 1807-1825 (1976). 12. M. Ordaz, S. K. Singh, E. Reinoso, J. Lermo, J. M. Espinosa and T. Dominguez, ‘Estimation of response spectra in the lake bed zone

13. M. Ordaz, E. Reinoso, S. K. Singh, E. Vera and J. M. Jara, ‘Espectros de respuesta en sitios del valle de Mexico ante temblores

653-674 (1988).

Engineer Thesis, School of Engneering, Universidad Nacional Aut6noma de Mexico, 1988.

of the Valley of Mexico’, Earthquake spectra 4, 815-834 (1988).

postulados en la brecha de Guerrero’, Proc. VIII narl. con$ earthquake eng. Acapulco, Gro, Mexico, A187-Al98 (1989).