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Soil Dynamics and Earthquake Engineering 29 (2009) 1–16

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Free vibration of soils during large earthquakes

S. Ruiz, G.R. Saragoni�

Civil Engineering Department, University of Chile, Casilla 228/3, Santiago, Chile

Received 6 November 2006; received in revised form 15 December 2007; accepted 17 January 2008

Abstract

Free vibration of soils happens frequently during some large earthquakes, perhaps seeming like a paradox. This happens because the

energy released from seismic sources in some cases is not stationary in time, allowing relaxation intervals in between without important

seismic wave arrivals in which free soil vibration happens. Two techniques to estimate the natural period of the free vibration from

accelerograms are presented: autocorrelograms and Fourier spectra. Both techniques sometimes allow measuring higher mode

frequencies of the soil for the three first modes as well as modal damping. Free vibration modal periods satisfy the classic 1D equation S-

wave theory. The presence of free vibrations corresponds to shear wave soil energy radiation episodes rather than to energy amplification

of incoming stationary seismic shear waves suggested by the dynamic soil amplification. These results explain the discrepancies observed

between the theoretical soil dynamic amplification and the accelerographic measurement. Observation of free vibration of soils is not

always possible, it depends on the duration of the time windows without important seismic waves arrivals compared to the natural period

and damping of the soil.

r 2008 Elsevier Ltd. All rights reserved.

Keywords: Soil free vibration; Earthquake; Accelerogram; Autocorrelogram; Fourier spectra; Soil amplification

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Autocorrelograms of earthquake accelerograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3. Theoretical functions for autocorrelograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

4. Free vibration of soils in large earthquake accelerograms estimated from autocorrelograms . . . . . . . . . . . . . . . . . . . . . . . . . 5

5. Soil damping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

6. Soil fundamental period estimation using Fourier spectra and comparison with the autocorrelogram technique . . . . . . . . . . . 7

7. Soil higher modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

8. Soil degradation in free vibrations from large earthquakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

9. Influence of soil free vibrations on soil dynamic amplification during large earthquakes. . . . . . . . . . . . . . . . . . . . . . . . . . . 10

10. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

e front matter r 2008 Elsevier Ltd. All rights reserved.

ildyn.2008.01.005

ing author. Tel.: +562 978 43 72; fax: +562 689 28 33.

ess: [email protected] (G.R. Saragoni).

1. Introduction

Present seismic design practices, which incorporateinformation from strong motion accelerograms, veryseldom reconcile the differences between accelerographicmeasurements and theoretical predictions. One factor

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ARTICLE IN PRESSS. Ruiz, G.R. Saragoni / Soil Dynamics and Earthquake Engineering 29 (2009) 1–162

involved, which is recognized as being very influential, isthe effect of local conditions.

Scholars studying earthquake damage have observed themodification of earthquake motion by local soil for a longtime [1].

The earliest researchers to quantify the problem were theJapanese, the most prominent ones being Sezawa [2,3] andKanai [4,5]. These researchers obtained algebraic expres-sions in the frequency domain for the surface motion toincident wave ratio from the assumption of stationary,vertically propagating, plane SH waves. Their work islimited to one and two horizontal layers of constantvelocity for which they included the visco-elastic behaviorand predict important amplification at the natural modeperiods of the soil given by

Tn ¼4H

VSð2n� 1Þ, (1)

where n is the mode number, H the soil depth and VS is thesoil shear wave velocity.

Therefore, when Fourier spectra from earthquakeaccelerograms show important peaks at the naturalfrequencies of the soil, they are normally considered tobe a consequence of the soil amplification of stationary,incoming shear waves.

However, the unexpected collapse of structures due tosoil amplification effects, designed according to modernseismic codes during the Mexico 1985 earthquake, hastaken to review soil amplification theories by deployinghigh-density accelerograph arrays to have a better under-standing of the phenomenon.

On September 19, 1985, a large subduction MS=8.1earthquake, with epicenter in the Pacific Ocean offMichoacan state, struck Mexico. In Mexico City, 400 kmaway, large damages and collapse of modern high-risebuildings were observed despite the reduced damagesreported at the epicentral zone: Ixtapa, Zihuatanejo andLazaro Cardenas [6].

In Mexico City, for the 1985 earthquake—an ideal casewhere the most common assumptions for the soilamplification theory are fulfilled, i.e. strong contrastbetween soil impedances, almost linear soil behavior andvertical incident shear waves due to the condition of beinga far epicenter of Pacific subduction earthquakes (epicenterdistances longer than 400 km)—most accepted amplifica-tion theories are partially verified. These theories can onlyreproduce the natural soil response period but fail toestimate the observed large soil amplification ratios andlarge durations [7–10].

Bard and Tucker [11] and Geli et al. [12] have also statedthat amplifications observed in the field are systematicallylarger than the values predicted using theoretical models.

In this paper, the assumption of stationary incidentseismic waves in the soil response will be studied from theviewpoint of accelerographic measurements for largeearthquakes in the epicentral zone or for long-epicentraldistances, such as the classic Mexico City case, to try to

understand the origin of the vibration at the naturalperiods of the soil. Two techniques will be considered forthe interpretation of accelerograms: autocorrelogram andFourier spectra.

2. Autocorrelograms of earthquake accelerograms

The first researcher that considered earthquake accel-erograms as a sample of stationary random processes wasHousner [13]. He assumed that accelerograms could berepresented as a series of pulses randomly distributed intime as a white noise process. Housner’s idea does notconsider at all any filtering effect due to the soil.On the other hand, in Japan, Tajimi [14], based on a

Kanai work [5], studied the frequency content of accel-erograms. He observed that the pulse duration is similar tothe natural period of buildings, since Japanese accelero-grams have a predominant period of less than 0.8 s.Based on these observations, Tajimi proposed the

filtering of an incident random white noise through a onedegree of freedom oscillator that represents the soilresponse. This assumption leads to power spectral densityfunctions with a predominant period. Other authors havelater improved this model proposed by Tajimi, but keepingthe two major ideas, i.e. an acceleration random processand a predominant soil period.The autocorrelation function of an accelerogram is

called the autocorrelogram, which measures the expectedvalue of the correlation between two values of a time seriesseparated by a time difference t. The definition is given by

fxxðtÞ ¼ limT!1

1

T

Z T=2

�T=2xðtÞxðtþ tÞ dt, (2)

where x(t) in this case is the ground acceleration at time t

and T is the total duration of the accelerogram [15].Arias and Petit-Laurent [16] analyzed the autocorrelo-

grams of accelerograms of USA, Mexico City and Santiago,Chile considered as a random process. They found out thatthe Mexico City 1962 earthquake accelerograms haddeterministic components despite the fact that USA andChile accelerograms show high randomness in a wide bandof frequencies. For Arias and Petit-Laurent, the frequencyband was a consequence of the soil properties at the site.The autocorrelation functions analysis for different accel-

erograms shows that some of them have an importantpresence of sine wave components (deterministic), as Ariasand Petit-Laurent [16] found for the Mexico City autocorre-lograms of the 1962 Mexico earthquake. A similar situation isobserved for some autocorrelograms of the Parkfield 1966earthquake [17], Rumania 1977 and 1986 earthquakes [18]and some Chilean earthquakes [19] (see Fig. 1).

3. Theoretical functions for autocorrelograms

The shape of these autocorrelograms, i.e. the character-istic period and the attenuation constant, allows to

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Fig. 1. Normalized autocorrelograms: (a) Parque Alameda, 1962 [16], (b) Temblor S25W, USA 1966 [17], (c) Bucharest NS, Rumania 1977 [18], and

(d) San Isidro L. Chile 1985 [19].

S. Ruiz, G.R. Saragoni / Soil Dynamics and Earthquake Engineering 29 (2009) 1–16 3

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Fig. 2. Comparison between autocorrelograms and theoretical functions: (a) Autocorrelogram CDAO N00E, (b) CDAO N00E, (c) SCT N00E, (d) San

Isidro L, (e) Tarzana 90, and (f) Boeing 270.

S. Ruiz, G.R. Saragoni / Soil Dynamics and Earthquake Engineering 29 (2009) 1–164

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Table 1

Soil natural period and damping estimation using the autocorrelogram

technique

Accelerographic station Fundamental mode

Natural period TS

(s)

Soil damping

b

1. San Isidro Longitudinal, Chile

Central 1985

0.34 0.104

2. SCT N00E, Mexico 1985 2.04 0.045

3. CDAO N00E, Mexico 1985 3.57 0.050

4. Tarzana 90, Whittier Narrow 1987 0.32 0.100

5. Boeing 270, Nisqually 2001 1.22 0.070


assimilate them to the corresponding deterministic functionof the displacement of the free vibration of the one degreeof freedom oscillator with an initial displacement and zerovelocity, i.e.:

yðtÞ ¼ Ae�bð2pðt=TSÞÞ cos 2pt

TS

� �, (3)

where TS is the natural period and b is the damping ratio.Making this assumption and choosing adequate values

for the natural period and damping ratio of the oscillator,both curves coincide [19].

In Fig. 2a, the autocorrelogram of Central de Abastos(CDAO) N00E accelerogram of the Mexico 1985, MS ¼

8.1 earthquake is shown. The autocorrelogram has beennormalized by the expected quadratic value. For thisautocorrelogram, the following TS and b values wereestimated: TS ¼ 3.57 s and b ¼ 0.05. Then Eq. (3) becomes

yðtÞ ¼ e�0:05ð2pðt=3:57ÞÞ cos 2pt

3:57

� �. (4)

Fig. 2b shows an excellent match between the function ofEq. (4) and the autocorrelogram of CDAO N00E in a timerange of 20 s. This result suggests that soil at the CDAOstation mainly vibrated freely during the 1985 Mexicoearthquake as a simple damped one-degree of freedomoscillator.

Figs. 2c–f also show an excellent match between thecorresponding autocorrelograms of Ministry of Commu-nications and Transportation (SCT), N00E, Mexico 1985;San Isidro Longitudinal, Chile 1985; Tarzana 90, WhittierNarrows 1987 and Boeing 270, Nisqually 2001 and theircorresponding theoretical function of the type of Eq. (3).Autocorrelograms of Fig. 2 have been normalized to havean expected square acceleration equal to one.

The corresponding values of the soil natural period TS

and damping b of the theoretical function of the casesanalyzed in Fig. 2 are indicated in Table 1.

4. Free vibration of soils in large earthquake accelerograms

estimated from autocorrelograms

The reason why free vibration happens during somelarge earthquakes, which seems like a paradox, is due to the

fact that the energy released from the seismic source is notpermanently continuous in time; there are relaxationintervals in between without important seismic wavearrivals from the source in which free vibration of the soilhappens many times. Therefore, accelerograms can beconsidered as a random sequence of seismic episodes ofseismic wave arrivals and episodes of free vibrations of thesoil.However, the observation of soil free vibrations in

accelerograms of large earthquakes will not be alwayspossible; it depends on the duration of the time windowswithout important seismic arrivals compared to the naturalperiod and the damping of the soil and also the almostelastic soil response. Therefore, soil free vibrations aredifficult to estimate in accelerograms of short duration andin the long duration ones, but with a more permanentarrival of seismic waves.The autocorrelation function estimation is usually done

considering Eq. (2) by using only one sample accelerogrambased on the ergodic theorem; however the use of thistheorem in practice implies considering random samples.When the samples, as in the case analyzed, have strongdeterministic components, the estimator is only recognizingthe presence of many free vibrations along the record dueto random initial conditions originated by intermittentepisodes of short-duration, random seismic wave arrivals.The free vibration of a structure in a push-back test is

used to measure the natural period and damping ofstructures. Similarly, the free vibration of the soil can beused to measure the soil damping as the random decrementmethod. Other authors have also proposed the measure-ment of soil damping in a probabilistic manner [20,21].These deterministic free vibrations of the soil are

observed in different accelerograms as shown in Fig. 3:Tarzana 90, Whittier Narrow 1987; Central de AbastosOficina (CDAO) N00E, Mexico 1985; Boeing 270, Nisqu-ally 2001 and San Isidro Longitudinal, Chile 1985. Allthese accelerograms show time windows with a cleardamping of the acceleration amplitudes. This situation isreflected in their corresponding autocorrelograms. There-fore, in these cases, autocorrelograms allow estimating thenatural period and damping of the soil.In Fig. 3, in addition to the accelerograms, in the central

part there is close-up of a specific region of the strongmotion part of the accelerogram where the natural periodand the damping of the soil can be clearly observed. To theleft of the figure, the corresponding autocorrelograms areshown with similar period and damping.The most evident fact of this discovery is that they

happen during the strong motion part of accelerograms oflarge earthquakes in epicentral zones or for long distances,as the Mexico 1985 earthquake demonstrates. Further-more, for some time the peak ground acceleration of therecord corresponds to these free vibrations of the soil as thecase of Boeing 270 of the Nisqually 2001 earthquake.This result suggests that accelerograms with harmonic

autocorrelograms correspond mainly to episodes of free

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Fig. 3. (a) Free vibration in different accelerograms, (b) zoom of strong motion zones showing almost harmonic damped vibrations, and

(c) autocorrelogram with same period and damping.


vibrations of the soil and the deterministic componentsfound in autocorrelograms by other authors in the past arethese free vibrations of the soil.

In Fig. 4, the corresponding autocorrelograms for thethree components of Cauquenes station, 1985 Chile,Tarzana station, Northridge 1994 and SCT station, Mexico1985 are shown. In this figure it can be also observed thatin addition to the deterministic autocorrelograms for thehorizontal components, the vertical autocorrelograms alsoshow the same pattern.

This situation, which is not always frequent, can also beappreciated for the Tarzana vertical, Northridge 1994earthquake in the near source region of a large earthquake(PGAE1 g). Fig. 4 suggests that in some cases the verticalmode of the soil vibration can also be estimated from theautocorrelograms.

From Fig. 4 it can be appreciated that the characteristicperiod of both horizontal autocorrelograms are similar,however vertical autocorrelograms have a period that islower than the horizontal ones. In general, the verticaldamping for vertical vibrations is higher.

These episodes of vertical free vibrations of the soil thatare due to seismic waves arriving from the source are

strongly coupled in the three accelerogram components byhigh-frequency Rayleigh waves [9,22].

5. Soil damping

Soil damping b has been empirically calculated by usingthe autocorrelogram technique. The values obtained fromFig. 2 for CDAO N00E and SCT N00E, Mexico 1985, SanIsidro Longitudinal, Chile Central 1985, Tarzana 90,Whittier Narrow 1987 and Boeing 270, Nisqually 2001,are summarized in Table 1.In particular, the soil damping values estimated for

SCT and CDAO stations for the lake clay of MexicoCity are larger than the ones estimated by the resonantcolumn test of b ¼ 0.01–0.03 [23] and by the randomdecrement technique of b ¼ 0.02–0.03 [20]. The b valuesobtained from the autocorrelogram technique usuallyslightly overestimate soil damping values due to theinterference of small seismic wave arrivals during freevibrations and the interference of the other higher moderesponse.Small damping values as SCT and CDAO Mexico City

stations lead to a large number of free vibrations which

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Fig. 4. Autocorrelograms of the three accelerogram components, showing free damped vibrations in horizontal and vertical directions.

Table 2

Soil profile of Central de Abasto Oficinas (CDAO ) acelerographic

stations, Mexico City [24]

Depth (m) Soil type Shear wave velocity (m/s)

0–5 Silty sand 60

5–42 Clay 60

42–52 Sandy silt and silty clay 110

52–56 Stiff clay 110

Hard layer 900


together with the soil’s natural periods of 2.04 and 3.57 sproduce long free vibrations of 20 s or more which affectthe total duration of the accelerogram and explain theirobserved differences.

6. Soil fundamental period estimation using Fourier spectra

and comparison with the autocorrelogram technique

In this section the periods of the peaks of the Fourierspectra are compared with the periods estimated fromautocorrelation functions of the accelerograms, whichcorrespond to free vibrations of the soil.

During the 1985 Mexico earthquake, important accel-erograms were recorded at the Mexico City lake zone. In particular, the two components of the accelerogramsrecorded at CDAO accelerographic station in MexicoCity for the 1985 and other three earthquakes arestudied in this section. The stratigraphic soil of CDAOstation is presented in Table 2. In addition, in Fig. 5,the two horizontal components of CDAO stationare shown with their corresponding Fourier spectraand normalized autocorrelograms for the 1985 Mexicoearthquake.

The soil fundamental period estimated from the fourdifferent earthquakes is 3.53 s [25]. This value matches the

peaks of the Fourier spectra and also matches the period ofthe first cycle of the autocorrelograms as indicated inTable 3.From Table 3 it can be observed that the soil

fundamental period T1 obtained from the peak of Fourierspectra coincide with the one from the autocorrelogramsfor the two components of the four different earthquakesstudied. In addition, these values match the results ofEq. (1) of the elastic soil 1D model for the fundamentalperiod:

T1 ¼4H

VS. (5)

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Fig. 5. Horizontal components CDAO station, Mexico earthquake 1985 with their corresponding Fourier spectra and autocorrelograms.


It can be concluded for these cases that Fourier spectraand autocorrelograms represent mainly free vibrations ofsoils in shear mode with few episodes of forced seismicwave response. Therefore, the fundamental soil period canbe also estimated either by autocorrelogram or Fourierspectra techniques from large earthquake accelerograms.

7. Soil higher modes

The presence of different modes of soil vibration inaccelerograms can be also detected using Fourier spectra.In Fig. 6 the first, second and third horizontal modefrequencies of the soil are shown by the three Fourier

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Table 3

Comparison of soil natural period at CDAO station from different

Mexican earthquakes estimated from Fourier spectra and autocorrelo-

gram techniques

Earthquake Magnitude MS Autocorrelogram

period (s)

Fourier spectra

period (s)

Component Component

N00E N90E N00E N90E

09/19/1985 8.1 3.69 3.93 3.62 3.78

09/21/1985 7.6 3.71 3.62 3.71 3.43

04/25/1989 6.9 3.29 3.47 3.35 3.46

09/14/1995 7.4 (Mw) 3.17 2.99 3.13 2.85


spectra peaks for CDAO N00E and CDAO N90Eaccelerograms of the CDAO, Mexico City; for differentMexican earthquakes and for both components ofVentanas and Almendral, Valparaiso stations for theChile 1985 earthquake and the corresponding main after-shocks.

The frequency values of the different Fourier spectrapeaks for the CDAO, Ventanas and Almendral, Valparaisostations are presented in Table 4. These values satisfy theapproximate relationship 1:3:5 indicated in Table 5.Therefore, the frequency of the Fourier spectra peakssatisfies the relation, expressed in this case as a naturalperiod of vibration, by Eq. (1).

From the analysis of these accelerograms of largeearthquakes, it can be concluded that upper modes ofvibration of soils are also observed in accelerograms oflarge earthquake and they satisfy the shear soil vibrationcondition given by Eq. (1).

These last results can only be observed when the soil isallowed by the earthquake source to freely vibrate intime windows with enough time to show many cyclesof free vibration. These time windows are sometimepossible between intermittent wave arrivals of a largeearthquake.

Since soil vibrates in the fundamental and higher modes,the best way to estimate the modal damping is filtering theaccelerogram around the fundamental period and thenestimating the damping from the corresponding autocor-relogram.

With the assumption that Fourier spectra have a narrowband characteristic, it is possible to filter the accelerogramsavoiding the influence of the other modes and have a betterestimation of the soil modal damping.

This technique is applied to the CDAO N00E 1985Mexico, Ventanas EW and Almendral S40E, Valparaiso,Chile 1985 accelerograms. Fig. 7 shows the correspondingautocorrelograms for the filtered accelerograms around thefirst and second mode periods. The autocorrelograms havethe characteristics given by Eq. (3). Table 6 indicates thedamping obtained for the first and second soil modes.From these figures and Table 6 it can be observed that the

second mode damping is somewhat larger than the onecorresponding to the first mode.

8. Soil degradation in free vibrations from large earthquakes

The soil natural period estimated in Table 3 for CDAOsoil for Mexico September 19, 1985 earthquake comparedwith the one corresponding for the aftershock of Septem-ber 21, 1985 shows a slight degradation. The differencebetween the main event and small events such as theGuerrero 1989 and Ometepec 1995 earthquakes may be aconsequence of soil non-linearity due to high peak groundaccelerations or because small magnitude earthquakes onlyexcite the upper part of the soil stratum.On March 3, 1985 an earthquake MS ¼ 7.8 struck the

central part of Chile, and this event was recorded by manystations. In particular, accelerograms recorded at Ilocastation, at an epicentral distance of 200 km and with aPGA of 0.2 g, are studied in detail, in a similar manner asthat for the CDAO Mexican station.The period observed in the autocorrelograms and

Fourier spectra of Fig. 8 can be easily identified in theaccelerograms. Table 7 summarizes the soil natural periodsestimated for Iloca station from Fourier spectra andautocorrelograms for the main shock and four aftershocksof the 1985 Chile earthquake.The soil periods estimated for Iloca do not show

significant soil degradation from earthquakes of differentmagnitudes. Similar results were obtained for most ofthe Chilean accelerographic stations with the exception ofthe ones located on gravel and sand that show lightdegradation, despite the large magnitude of this earth-quake [26].However, the accelerographic networks that recorded

Chilean and Mexican 1985 earthquakes did not considerdown-hole arrays allowing a detailed study on the non-linear characteristics of soil response in order to confirmthat slight variation of soil natural period estimated fromfree vibrations in accelerograms is due to this effect.The observed differences between the soil natural periods

for both horizontal accelerographic components may bedue to seismic wave arrivals with a strong directivity effect,which the soil response attenuates by coupling the motionin both directions.The important Tarzana 901, Northridge 1994 accelero-

gram has the largest PGA ¼ 1.8 g of this earthquakeand no important soil degradation was apparently ob-served. In Fig. 9, the Tarzana 901 record is shown overlaidwith a sine wave of a 0.27 s period and constant amplitude.The sine wave is shown in four parts of the accelerogramillustrating that the record keeps the same characteristicperiod.However, it must keep in mind that PGA alone is not the

best indication for the degree of soil nonlinearity inducedby the quakes, and this is particularly important foraccelerograms having significant high-frequency spikes as itis the case of some of the analyzed.


9. Influence of soil free vibrations on soil dynamic

amplification during large earthquakes

The presence of free vibrations detected in accelerogramsfrom large earthquakes corresponds to episodes of soil

Fig. 6. Fourier spectra of accelerograms showing the peaks corresponding to t

station for different seisms.

energy radiation of shear waves rather than energyamplification of incoming seismic waves claimed by thedynamic S-wave soil amplification theory. However, thecoincidence of mode natural periods of free vibration andperiods of soil amplification both given by Eq. (1) leads to

he three first soil mode frequencies. Frequencies remain the same for each

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Table 4

Natural frequencies of the different modes of vibration of soil estimated from Fourier spectra peaks

Earthquake Soil mode natural frequencies (Hz)

First mode. Component Second mode. Component Third mode. Component

Mexico CDAO N00E N90E N00E N90E N00E N90E

09/19/1985 0.28 0.25 0.74 0.78 1.27 1.26

09/21/1985 0.27 0.29 0.77 0.82 1.31 1.27

04/25/1989 0.30 0.29 0.90 0.82 1.34 1.36

09/14/1995 0.32 0.35 0.88 0.84 1.42 1.41

Chile Ventanas EW NS EW NS EW NS

03/03/1985 1.00 0.99 2.72 3.02 4.38 4.53

04/09/1985 1.11 1.02 2.93 2.66 4.22 4.84

Chile Almendral S40E N50E S40E N50E S40E N50E

03/03/1985 1.21 1.07 3.59 3.44 4.74 4.26

Table 5

Ratio between the soil mode natural frequency and the first natural mode frequency estimated from Fourier spectra peak values

Earthquake Ratio between different soil natural mode frequencies

First mode/first mode. Component Second mode/first mode. Component Third mode/first mode. Component

Mexico CDAO N00E N90E N00E N90E N00E N90E

09/19/1985 1 1 2.6 3.1 4.5 5.0

09/21/1985 1 1 2.9 2.8 4.9 4.4

04/25/1989 1 1 3.0 2.8 4.5 4.7

09/14/1995 1 1 2.8 2.4 4.4 4.0

Chile Ventanas EW NS EW NS EW NS

03/03/1985 1 1 2.7 3.1 4.4 4.6

04/09/1985 1 1 2.6 2.6 3.8 4.7

Chile Almendral S40E N50E S40E N50E S40E N50E

03/03/1985 1 1 3.0 3.2 3.9 4.0

Fig. 6. (Continued)


the misinterpretation that all the energy released at thosevalues is only due to soil amplification.

These results mainly explain the discrepancy observedbetween the soil dynamic amplification theory and the ratiobetween the amplitude Fourier spectra of accelerographicrecords obtained in the soil and the bedrock.

In several works related to the amplification of thesubsoil of Mexico City, the maximum amplification factor

has been found around the observed fundamental period.Nevertheless, this predicted amplitude value is extremelysmaller than the one that was observed. Kawase and Aki[8] found great discrepancy in the spectral ratios betweenthe accelerograms of the 1985 Mexico earthquake, and thecomputed one using the 1D model.Fig. 10 shows the Fourier amplitude spectral ratios of

the accelerograms observed at SCT and CDAO with

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Fig. 7. Comparison of autocorrelograms of the two first soil modes. CDAO N00E, Mexico, Almendral S40E1, Chile and Ventanas EW, Chile.

Table 6

Autocorrelogram soil period and damping estimation for the first two

modes

CDAO N00E

Mexico 1985

Ventanas EW

Chile 1985

Almendral S40E

Chile 1985

First mode

Period (s) 3.57 1 0.83

Filtrate rank (s) 10–2 2–0.67 2–0.5

Damping 0.018 0.04 0.06

Second mode

Period (s) 1.35 0.37 0.28

Filtrate rank (s) 2–1 0.67–0.25 0.5–0.2

Damping 0.09 0.08 0.09


respect to TACY (Tacubaya), a hill zone station, for the1985 Mexico earthquake [9]. This figure also includes theamplification characteristics calculated under the 1D

S-wave excitation, evaluated with the SHAKE [27]program when the models were subjected to the Tacubayarecorded accelerogram.The models consider strain-dependent shear module

and damping ratio values for the clay according to Romoet al. [23].In theses sites, as in most of the area of the Lake Zone,

the impedance ratio is about 0.10. In both stations thecalculated amplifications factors are considerably smallerthan the observed spectral ratios: about 5 times in the SCT-EW component and about 4 times in the CDAO-NScomponent. Predominant site periods are approximatelysimilar to the ones observed.The amplification relations calculated assuming that the

soil is overlaying bedrock are also included in Fig. 10 [10].This transfer function represents the boundary limit in theS-wave amplification approach, because in this model theeffects of radiation damping are disregarded. However, this

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Fig. 8. Horizontal components of Iloca station, Chile Central 1985 earthquake, showing their corresponding Fourier spectra and autocorrelograms.


curve is also lower than the one observed, illustratingthe important effect of the free vibrations in the LakeZone. When the soil is overlaying rigid rock, the transferfunction reaches a maximum value of 20, but when themodel is assuming elastic rock, this amplification factor is

only about 8. Namely, although the base should be theconsidered elastic, a significant amount of the shear waveselastic energy (about 60%) is not effectively removed fromthe soil layer by radiation, due to the free vibration effectsfound.

ARTICLE IN PRESS

Table 7

Estimation of soil natural period at Iloca station considering the main shock and four aftershocks of the 1985 Chile earthquake using autocorrelogram and

Fourier spectra [26]

Earthquake Magnitude MS PGA (cm/s2) Autocorrelogram period (s) Fourier spectra period (s)

Component Component Component

EW NS EW NS EW NS

03/03/1985 7.8 277 209 0.31 0.32 0.29 0.30

03/03/1985R 6.4 42 44 0.29 0.30 0.30 0.31

03/25/1985R 6.3 89 101 0.29 0.30 0.29 0.31

04/03/1985R 6.0 74 52 0.30 – 0.31 0.28

04/09/1985R 7.2 110 159 0.30 0.32 0.30 0.33

R ¼ afterstock.

Fig. 9. Comparison in four parts between Tarzana 901, 1994 of PGA 1.8 g

with sine wave of the same period and constant amplitude, showing that

the soil period remains constant.


The fact that more than 60% of the amplification factorsconsidered in the analyzed cases of 1985 Mexico City aremainly due to the free vibrations found also explain theparadox why the 1D model of the soil response in the areaof the Lake Zone has rendered similar results than moresophisticated 3D models [28].

In addition, the duration of free vibrations controlled bycommon low soil damping and different soil fundamentalperiods explains the time duration differences betweendifferent accelerographic stations at the Lake Zone.Accelerogram stations with soil with longer natural periodshave longer duration, phenomena which cannot beexplained by the 1D S-wave theory [8].

10. Conclusions

1.
Free vibration in soils happens many times during somelarge earthquakes, something that seems like a paradox.The reason is because the energy released from seismicsources is not stationary in time, allowing relaxation
intervals in between without important seismic wavearrivals from the source in which free vibrations of thesoil can happen many times along the accelerograms.Therefore, accelerograms of this type for large earth-quakes can be considered as a random sequence ofseismic episodes with seismic wave arrivals and episodesof free vibrations of the soil.

Observation of free vibration of soils in accelerogramsfrom large earthquakes will not be always possible, itdepends on the duration of the time windows withoutimportant wave arrivals, compared with the naturalperiod and damping of the soil, and the non-linearresponse of the soil. Therefore, soil free vibrations aredifficult to estimate in accelerograms of short durationand in the ones with long duration, but with morecontinuous arrivals of seismic waves.

2.
Two techniques to estimate free vibrations of soils fromaccelerograms were presented: autocorrelograms andFourier spectra. Both techniques sometimes allowmeasuring up to the three first soil modes as well asthe corresponding modal damping. These techniquesalso allow measuring natural periods and the dampingof the free vertical vibrations of the soil.
These techniques allow to experimentally measure soilproperties under large deformations due to large earth-quakes and comparing them with the ones obtainedfrom laboratory measurements.

Soil damping estimates using the autocorrelogramtechnique can be overestimated due to the interferenceof small seismic wave arrivals and higher moderesponses during soil free vibrations.

3.
Free vibration modal periods satisfy the classic 1Dequation for S-waves, specially the three first modenatural frequencies satisfying the 1:3:5 ratio.
Free soil vibrations correspond to episodes of shearwave soil energy radiation rather than energy amplifica-tion of incoming seismic waves claimed by the dynamicS-wave soil amplification theory.

4.
The coincidence of mode natural periods of freevibrations and periods of soil amplification leads tothe misinterpretation that all the observed phenomenaduring large earthquakes are only due to soil amplification

ARTICLE IN PRESS

Fig. 10. Disagreement between the maximum observed and estimated values for amplitude spectral relations of SCT and CDAO accelerograms with

respect to TACY Mexico’85 [10].


of incident S-waves without including the free vibrationepisodes.

These results mainly explain the underestimation ofthe soil dynamic S-wave amplification theory andobserved ratio between Fourier spectra of accelero-graphic records obtained from the soil and bedrock.

The longer duration of accelerograms, which cannotbe explained by the S-wave amplification theory, is dueto the presence of free vibrations of the soil foundthrough the records.

Acknowledgement

Authors wish to thank Professor Jose Manuel Roesset ofTexas A&M University for the comments and reviewing ofthis paper’s draft.

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