Duration of strong ground motion during Mexican earthquakes in terms of magnitude, distance to the...

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2001; 30:653–673

Duration of strong ground motion during Mexican earthquakesin terms of magnitude, distance to the rupture area and

dominant site period

Eduardo Reinoso1;2;∗;† and Mario Ordaz1;21Instituto de Ingenier��a, UNAM, Ciudad Universitaria, Coyoac�an 04510, Ciudad de M�exico, M�exico

2ERN Ingenieros Consultores S. C., Mexico

SUMMARY

A study of the duration of strong ground motion using accelerometric data of subduction and normal-faulting Mexican earthquakes is presented. Duration is obtained based on the time between 2.5 and97.5 per cent of the Arias intensity. An expression to predict this duration in terms of the magnitude,distance to the rupture area and site period is proposed and compared with predictions available inthe literature. The e�ect of large duration for very distant sites and the contribution of soft soils tothe duration of strong ground motion are widely discussed. We have found that large magnitude notonly yields long duration at the source, but also proportionally longer duration with distance and withdominant site period compared to small magnitude. The duration obtained from the regression is usedas a parameter to obtain input and hysteretic energy and on the use of damage models available in theliterature. Finally, duration is used together with the random vibration theory to predict response spectra.Copyright ? 2001 John Wiley & Sons, Ltd.

KEY WORDS: duration; strong ground motion; accelerometric data; input energy; hysteretic energy;random vibration theory

1. INTRODUCTION

In newspapers of the 19th century one �nds relations of earthquakes felt in Mexico City suchas: ‘It lasted more than one minute; however, there was someone who extended it to 22,others more moderated to 15 and others a bit less, proportionally to the terror of each one’.During the 18th century, when clocks were less common and minutes less popular: ‘It lastedmore than what it takes to pray two creeds with devotion’. Catholics know that to pray twocreeds with devotion could take 2–3 min. Considering that Mexico City inhabitants are usedto earthquake shakings, these 2–3 min should be related to very intense ground motion. Atthat time, the fear of the earthquake motion forced the believer to pray saying clearly all of

∗ Correspondence to: Eduardo Reinoso, Instituto de Ingenier��a, UNAM, Ciudad Universitaria, Apartado Postal 70-472,Coyoac�an 04510, Ciudad de M�exico.

† E-mail: [email protected]

Received 12 December 1999Revised 16 August 2000

Copyright ? 2001 John Wiley & Sons, Ltd. Accepted 18 August 2000

654 E. REINOSO AND M. ORDAZ

what he was supposed to believe (the creed) in an attempt to avoid punishment from God dueto excess of sins and lack of faith. Unfortunately, there is not enough data to build maps ofequal creed duration, and the possibility to correlate it to modern parameters is a mere act ofhistoric curiosity. More recently, with reliable and useful instrumental data, reports of dramaticescapes of survivors that were inside collapsed buildings during the 1985 earthquake suggestthat modern structures take a long time to accumulate damage and eventually collapse.Nowadays, most seismic design codes recognize that displacement, and more precisely in-

terstorey drift, are essential values to design seismic-resistant structures. But forces to obtaindisplacements are generally based on maximum ampli�cation parameters such as peak andspectral acceleration. These parameters not only do not exhibit a straight correlation with lossand damage but also do not consider cumulative damage or degradation due to hysteretic be-haviour or several earthquake excitations during the lifetime of the structure. Structural damagedepends not only on the earthquake maximum intensity but also on the whole history of de-mands, before and after this maximum intensity. A lot of research is now in progress in orderto study and include such concepts. This paper deals with the study of the duration as a basisfor research relating energy and damage. Duration of strong ground motion is proposed to beused to obtain input [1; 2], dissipated [3] and hysteretic [4] energy, on the use of damagemodels [5; 6] and to obtain response spectra via random vibration theory [7; 8].The amount of digital accelerometric data recorded in Mexico during the last 15 years

allow us to formulate a regression that predicts duration in terms of the earthquake magnitude,distance to the rupture area and dominant period of the site. It is widely accepted that durationof strong ground motion at the source is longer for large magnitude. Although this durationgrows with distance, its e�ects are not of engineering importance unless site ampli�cation suchas the one observed in Mexico City is present. Peak acceleration at lakebed sites in MexicoCity is large in comparison to peak acceleration at �rm sites, but much smaller compared toother sites with shorter epicentral distances. However, motion has a very large energy contentthere, with long harmonic duration and long dominant periods.

COMPUTING STRONG MOTION DURATION FROM ACCELEROGRAMS

There are many ways to measure strong motion duration from accelerometric records. A com-prehensive recent work summarizes all ways to compute it [9]. In the present work, we use theduration between the times where the 2.5 and 97.5 per cent of the Arias intensity is developed.These limits were chosen after computing hundreds of strong motion durations where it wasclear that using 5 and 95 per cent as limits yielded results with larger scatter due to longcodas. In this way, the limits 2.5 and 97.5 pick up the part of the record that is useful forengineering purposes.In order to compare data for several stations during di�erent earthquakes, accelerograms have

been carefully selected and revised. As was done by many works, we omitted from the databasethose records with low signal-to-noise ratios, and those where length of recordings were clearlyshorter than the expected duration of strong motion. Additionally, selected accelerograms wereall set to the same threshold, �ltering them in the time domain simulating a typical digitalaccelerometer with pre- and post-event recording times. This �xed threshold was set to 20 cm=s2

at the epicentral area, 10 cm=s2 for distant stations and 4 cm=s2 for lakebed sites in Mexico City;for hill-zone sites in Mexico City no �xed threshold was used. Only horizontal components

Copyright ? 2001 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2001; 30:653–673

STRONG GROUND MOTION DURING MEXICAN EARTHQUAKES 655

Figure 1. Location of epicentres and rupture areas (for M¿ 6:9) of subduction and normal-faultingearthquakes used in this study. The plot also shows the date and magnitude of each earthquake and,

with small triangles, the accelerometric sites.

were taken into account since vertical motion is, especially for Mexico City and distant stations,less important from the engineering point of view. No di�erence was made regarding directivityof motion. This is because few earthquakes have yielded data where it is clearly seen thatsites along the waves path have large amplitude but short duration, while sites located on theother direction have considerably small amplitude and large duration [10].Data used in this study are from subduction and normal-faulting earthquakes, the former

originated at the contact of the North America and Rivera and Cocos plates, and the lattercaused by the fracture of the subducted Cocos plate. The earthquakes have magnitudes between5.2 and 8.1. Figure 1 shows, together with their date and magnitude, the location of epicentresand rupture areas of some of the earthquakes used in this study. The earthquakes have occurredat depths between 12 and 27 km, except those of 22 May, 16 June and 14 September thatoccurred at depths of 45, 47 and 60 km, respectively. With small triangles, some accelerometricsites are also shown in Figure 1. It can be seen that some of these stations are located overthe epicentral area and others as far as 520 km away from the rupture area. The stations usedin this work total about 200. They are located either over rock, �rm or lakebed zone sites.As an example of the diversity of strong ground motion in terms of amplitude, frequency

content and duration that a large subduction earthquake generates, Figure 2 shows some free-�eld accelerograms recorded during the 1995 Colima earthquake (M =8:0). This �gure exhibitsclearly the attenuation of intensity and site e�ects in Mexico City. Small circles in this �gureindicate accelerometric stations and full circles correspond to stations that triggered during theearthquake.



Figure 2. Free-�eld accelerograms recorded during the 1995 Colima earthquake (M =8:0). Notice thediversity of strong ground motion in terms of amplitude, frequency content and duration that a largesubduction earthquake generates. Small circles indicate accelerometric stations and full circles correspond

to stations that triggered during this earthquake.

PREDICTION OF THE DURATION OF STRONG GROUND MOTION

There have been some important works on the prediction of D, the strong motion duration. Afew of them are brie y reminded. Esteva and Rosenblueth [11] described D in terms of theearthquake magnitude, M , and source to station distance, R

D=0:02 e0:74M + 0:3R (1)

Using 16 measurements and several witnesses’ reports, Housner [12] proposed an upper boundfor the duration valid for M¿5

D=11:2M − 53 (2)

Bolt [13] found, for accelerations greater than 0:05g, very similar results as Housner [12].Dobry et al. [14] obtained duration based on the times between 5 and 95 per cent of Arias’intensity. They used data from rock sites located 0.1–130 km from epicentres of earthquakeswith 4:5¡M¡7:6 in the western U.S.A. and obtained the following linear regression with



respect to magnitude

D=10(0:43M−1:83) (3)

Trifunac and Brady [15] obtained the duration as a function of magnitude, distance and siteconditions for acceleration, velocity and displacement. Total duration is the sum of the earth-quake source duration, the time interval between the fastest and the slowest wave arrival at thestation, and the duration caused by repeated wave scattering from material discontinuities andsurface topography. Data employed have peak accelerations greater than 0:05g. D was obtainedas 90 per cent of the contribution to the mean-square integral of motion. They include a terms that considers site characterization (s=2; 1 and 0 for rock, �rm and soft soil, respectively).Their expression for acceleration is

D= − 4:88s+ 2:33M + 0:149R (4)

Trifunac and Westermo [16] obtained regressions of duration in terms of magnitude, epicentraldistance and site characteristics at the station for horizontal and vertical components. Thestrong motion duration was obtained with the sum of the intervals that contribute most tothe amplitude of the smoothed integrals of the history of square acceleration, velocity anddisplacement. Records were band-pass �ltered so results were presented for seven narrowfrequency bands. Their expressions for acceleration and for f=18 and 0.22Hz (Equations(5a) and (5b), respectively) are

D=1:82 + 0:317M + 0:133R (5a)

D=38:66− 3:46M + 0:087R (5b)

Novikova and Trifunac [17] obtained similar regressions as the Trifunac and Westermo [16]work for 12 frequency bands. Their expressions for acceleration and for f=21 and 0.037Hz(Equations (6a) and (6b), respectively) are

D=10:1− 4:68M + 0:62M 2 + 0:056R (6a)

D=7:8 + 0:84M + 0:191R (6b)

Trifunac and Novikova [18] studied the duration of earthquake fault motion in California.They compared their predictions with accelerometric data considering three frequencies as ahomogenous set [17] and the source duration. In their estimation (Equations (7) and (8)) thesecond term corresponds to the duration of the high-frequency radiation:

D=2:10 + 0:0031× 100:48M + 0:066R+ 0:28s (7)

D=1:28 + 0:014× 100:40M + 0:062R (8)

Frequency-dependent trends of duration [16–18] have some advantages over duration of thecomplete motion. However, since attenuation relation describes the radiation of seismic energyin earthquakes, it is more robust [9] to deal with the whole accelerogram as has been done inthis work.



Prediction of strong motion duration for Mexican earthquakes

Based on previous works [19; 20] and on the recorded accelerograms of the earthquakes shownin Figure 1, we present the following estimate of duration in terms of magnitude, distance tothe rupture area and dominant site period, Ts,

D=0:01 eM + (0:036M − 0:07)R+ (4:8M − 16)(Ts − 0:5) (9)

where the �rst, second and third term of Equation (9) represent the contribution to the durationof the source, distance and site e�ects, respectively. In the previous equation, Ts must equal0.5 s for sites with Ts ¡ 0:5 s, that is rock and �rm sites.Equation (9) was obtained computing strong motion duration as described in the previous

section of this work for more than 800 accelerometric records of 200 stations. These dataare from subduction and normal-faulting earthquakes with magnitudes between 5.2 and 8.1(Figure 1) with depths between 12 and 60 km. Distances from accelerometric stations to therupture area are between 12 and 520 km. The stations rest either over rock, �rm or lakebedzone sites. With this information, a database of all horizontal component of motion was builtin terms of computed duration, M;R and Ts.To obtain the trend of duration with respect to magnitude and distance to the rupture area

only data from rock and �rm soil sites were employed. For each earthquake, an individuallinear trend of the distance to the rupture area against duration was obtained. After comparingall trends for all earthquakes, it was observed that ordinates for R=0 (that is, the durationof strong ground motion at the source) were proportional to magnitude. The �rst term ofEquation (9) is the best �t of all these ordinates at R=0 plotted against the correspondingmagnitude.The second term of Equation (9) that relates the expected duration for a given magnitude

and distance to the rupture area was obtained with the trend of all slopes of each individualearthquake. As will be explained later, it can be observed that the expected duration growsnot only with distance but also with magnitude: large earthquakes yield proportionally longerdurations at distant sites.The third term of Equation (9) was obtained for data of the Mexico City Valley. A linear

trend of duration with respect to Ts was computed for each earthquake. We chose only thoseearthquakes with the same distance to the rupture area to Mexico City, so the di�erencesbetween trends were only due to magnitude. For these earthquakes, the ordinate at Ts = 0:5 isgiven by the contribution of the magnitude (�rst and second terms of Equation (9)) since allR were the same (roughly, R≈ 300 km). It was found that the slope of the trend computed foreach earthquake was larger for large earthquakes, which means that larger earthquakes alsoproduce longer durations for longer Ts. This is why M appears in Equation (9) to predict Din terms of Ts.Finally, the standard log (natural logarithm) error comparing measured and computed du-

ration was obtained. The average error for all records was 24 per cent. However, consideringonly the prediction for rock and �rm sites the average error was 29 per cent, while the errorfor sites over soft soils was 23 per cent. This is unusual since the trend of duration consideringsite e�ects [9] always increase considerably the error in the predictions. But for Mexico City,the duration is so well predicted that errors are smaller than those computed by the predictionat the given �rm sites M and R.



Figure 3. Accelerograms recorded over the rupture area for four of the earthquakes shown inFigure 1. Site, date, magnitude and source duration [22] are shown near each accelerogram.Notice that accelerograms of the largest event (M =8:1) exhibit low amplitude but large

duration and di�erent frequency content.

Duration of strong ground motion with respect to magnitude

Accelerograms recorded over the rupture area exhibit large acceleration and short durationcompared to sites that are more distant. It has been long recognized that neither peak acceler-ation or response spectra are strictly correlated with structural damage. Using accelerometricdata recorded over the rupture area of four recent earthquakes (Figure 1), Reinoso et al.[21] have shown that a parameter that correlates well with observed damage is the durationof strong ground motion. This can be observed in Figure 3 that shows these records andthe source duration obtained by Singh and Ordaz [22] based on Brune’s model. Notice thataccelerograms of the largest event (M =8:1) exhibit low amplitude but large duration anddi�erent frequency content. It should be said that during this event the observed damage wasrelatively large, while during small events no damage was observed.Figure 4 shows, together with results from works referenced above, the trend computed in

this work with solid line (Equation (9), with R=0 and Ts = 0:5) and the theoretical results [22]with circles. It is clear that our prediction almost matches these theoretical results. Esteva andRosenblueth11 with no strong motion data available (Equation (1)) predict very short durations,specially for large M . Because of the linear trend of duration with respect to M proposedby Trifunac and Brady [15], their prediction works �ne for M¡6 but for large magnitudesduration is underestimated by a factor larger than 3. Results by Novikova and Trifunac [17]exhibit quadratic dependence of duration with respect to M for high frequency (f=21Hz) andunderestimate the duration for large magnitude (M¿6:5). Dobry et al. [14] results (Equation(3)) overestimate duration for M¿6 up to 50 per cent. Trifunac and Novikova [18] results(Equation (8)) are very close to Equation (9) but slightly underestimate duration for verylarge M . However, the expression [18] �0 = 0:01eM , the duration of the rupture process at thesource (obtained for f→∞) is exactly the same as the �rst term of Equation (9).



Figure 4. Duration of strong ground motion with respect to magnitude. Results correspond to the pre-diction given by Equation (9) (with R=0 and Ts = 0:5) and other works available in the literature.

Theoretical results are shown with circles.

All predictions plotted in Figure 4 exhibit an increase of duration with magnitude. Anexponential variation of duration with M has been used by many authors [11; 18] and itseems to be a very e�cient way to model it. From these predictions and Equation (9), it isclear that very large rupture areas (200 × 80 km) yield large durations even at sites close tothe source.First term of Equation (9) should be considered reliable since it reproduces the observed

data and predictions by theoretical models. It also seems to work very well for the Northridgeearthquake where, in spite of very large scatter of computed duration, average duration reportedfor sites near the source [23] is very close to our prediction.

Duration of strong ground motion with respect to distance to the rupture area

Contrary to what happens to the attenuation of intensity, duration grows with distance (Figure 2).Despite this increase in duration, distant earthquakes are not of engineering importance for mosthuman settlements since the intensity is largely attenuated. Most studies [9] do not considerdistances larger than 100 km since strong motion becomes, from the engineering point of view,weak motion. Moreover, when taking into account the amplitude of motion, some works [9]predict shorter duration for larger distances. However, this phenomena becomes important forsoft soils as the seismic energy for long periods, which attenuates less than for short ones,ampli�es signi�cantly.Figure 5 shows with solid line the trend of duration with respect to distance computed for

six earthquakes. As can be observed in Figure 1, rupture areas for large earthquakes (M¿ 6:9)are also large, so for these earthquakes distances shown in Figure 4 are relative to this rupturearea. Figure 5 also shows, with dashed lines, the prediction for each earthquake of Equa-tion (9) where it can be seen that it matches reasonably well the trend obtained for eachearthquake.



Figure 5. Duration of strong ground motion with respect to distance to the rupture area.Results are shown for six earthquakes and are compared with the prediction (dashed lines)

given by Equation (9) (with Ts = 0:5).

In Equation (9), the coe�cient a�ecting R varies with M . It has a linear variation from 0.11for M =5–0.218 for M =8. As far as the authors are concern, there is no other trend relatingR also with M . Large errors could have been obtained without considering the in uence of M.Other variations have been proposed as 0.3 [11], 0.149 [15] and 0.191 [17]. All these valuesare similar if computing duration for relatively short distances (R6 30 km) which are betweenreasonable engineering limits. As has been mentioned along this work, strong ground motionat distant sites may not be important if soft soils are not taken into account. Therefore, littleattention has been paid to motion at long distances. The slope of the trend proposed in ourwork should be considered more reliable since it was computed with hundreds of records withvery large distances to the source.Data of the Manzanillo earthquake (M =8) was not employed to obtain Equation (9), so it

can be used to test it. Figure 6 shows the accelerometric record obtained over the epicentralarea. Computed duration using the 2.5–97.5 of the Arias’ intensity is for the NS and the EWcomponents 37.7 and 39.4 s, respectively. The duration obtained with Equation (9) (D=36:8 s)proves an excellent agreement between observations and prediction.The following example illustrates the accuracy of Equation (9) and some of the problems re-

lated with thresholds and types of accelerometers. Figure 7 shows two accelerograms recordedat a hill-zone site in Mexico City (CU, University City, UNAM), for the Manzanillo earth-quake (M =8). The di�erence between both CU accelerograms is that the one on the top wasrecorded by a broadband accelerometer while the one on the bottom is from a digital standard



Figure 6. Accelerometric record obtained over the epicentral area during the Manzanillo 1995 earth-quake (M =8). The �gure shows computed duration using the 2.5–97.5 of the Arias’ intensity for both

components of motion and the duration obtained with Equation (9).

Figure 7. Accelerograms recorded at a hill-zone site in Mexico City (CU, University City, UNAM), forthe Manzanillo earthquake (M =8) and at station 21. CU accelerograms were recorded by a broadband(top) and standard (bottom) accelerometers. Empirical duration is shown near the records. Duration

predicted with Equation (9) is 138 s, practically the same as station 21.

one. The record at station 21, another hill-zone site, is also shown. Observed duration is dra-matically di�erent for both CU records: 217 and 36 s. Duration predicted with Equation (9)is 138 s, four times larger than the obviously short duration of the standard accelerometer butjust 65 per cent of the broadband one. Because in this study predicted duration was obtainedwith standard accelerograms, it is expected that a broadband accelerometer would yield longer



records. On the other hand, the standard record at CU is notoriously short and, therefore, itshould not be taken as reliable to measure strong ground motion duration at CU. This is whywe also show the record from the standard accelerometer station 21, which is long enough tobe considered reliable. Measured duration for this record is 142 s, practically the same as thepredicted one.

Duration of strong ground motion with respect to dominant site period

Seismic waves from subduction earthquakes hit Mexico City after travelling more than 250 km.Because of patterns of radiation and attenuation, these waves arrive with low amplitude butlong duration, and soil ampli�cation causes very large, harmonic motion that produces dam-age and even collapse to structures. Another problem for the city is that it is a�ected byalmost all subduction earthquakes, experiencing on average an M¿7 event every two years,where cumulative damage becomes relevant and should be taken into account in design andmaintenance programs. Figure 8 shows the geotechnical zones, some reference sites and someaccelerograms recorded in the city during the 14 September 1995 earthquake (M =7:4). Inthis �gure, it is clear the dependence of duration and amplitude of motion with respect to thegeotechnical zones. Although the code classi�es the soil as �rm, transition and lakebed, inthis work the transition is also treated as lakebed zone. In other words, every site with a cleardominant period larger than 0.5 s is considered lakebed [24]. Although there is evidence thatstrong ground motion is already ampli�ed with respect to what one would expect at similarepicentral distances [25] we have no evidence that duration has also been a�ected.All attenuation relationships that include the in uence of site e�ects predict longer shaking

on soil sites than �rm or rock sites [9]. We have characterized site e�ects in terms of thelongest period of the site, so soil properties, depth of deposits and irregular valley con�gura-tions are taken into account in just one single parameter. Dominant periods at lakebed zonesin the city vary from 0.5 to 5.2 s. Figure 9 shows a map with contours of equal duration forthe Manzanillo 1995 earthquake (M =8; R=500) where strong ground motion duration longerthan 500 s was computed for sites located over the thickest soil deposits of the valley. Thecontours shown in Figure 9 are very similar to the contours of equal thickness of deposits andtherefore similar to contours of site dominant period.Figure 10 plots the computed duration for each horizontal accelerogram (symbols), the trend

computed with data of each earthquake (solid line) and the trend obtained with Equation (9)(dashed line). All nine earthquakes in Figure 10(a) have roughly the same distance to therupture area (Figure 1), so observed di�erences should be due only to magnitude. Noticethat not only the average duration is longer for large earthquakes, but also the slope ofthe regression is larger. Duration increases signi�cantly with Ts and Equation (9) matchesreasonably well the observed trends. For an earthquake with M =5 motion is 18 s longer forTs = 5 s than for Ts = 0:5 s, and for the same periods duration is 72 s longer for M =8:1 thanfor M =5. This considerable increase in duration is most probably present in the input motion[25] rather than in complicated 2D and 3D local scattering [26].Figure 10(b) shows two earthquakes with di�erent distances to the rupture area (Figure

1) than those shown in Figure 10(a). Data from these earthquakes were not used to ob-tain Equation (9). Considering that these earthquakes have such di�erences, it is remarkablethat Equation (9) predicts reasonably well the observed trend. For the 9 October 1995 earth-quake, observed duration for small periods is overestimated by Equation (9). This is because



Figure 8. Mexico City: geotechnical zones (lakebed is white, transition light gray and �rm soil isdark), reference sites and accelerograms recorded in the city during the 14 September 1995 earthquake(M =7:4; R=298 km). Seismic waves arrive to Mexico City with low amplitude but long duration,and soil ampli�cation causes very large, harmonic motion. Notice the clear dependence of duration and

amplitude of motion with respect to the geotechnical zones.

observed data were generally very short, as was shown in Figure 7 for the CU standard record,due to the small amplitude of strong ground motion at hill zone sites during this event. Aswas also mentioned in Figure 7, observed duration at site 21, one of the fewest sites thatrecorded properly this earthquake, is 138 s, practically the same duration as the predicted byEquation (9).As was mentioned above, it is remarkable that although large site e�ects that yield long

durations have been taken into account, the standard error computed is relatively small (24per cent). As far as the authors are aware, there are no other studies that predict such largechange in duration due to site e�ects and, moreover, with such small uncertainty. Even recentstudies [9] have observed no more than 20 per cent of increase in duration because of sitee�ects and, even with this small increase, the computed uncertainties are much larger.



Figure 9. Map with contours of equal duration for the Manzanillo 1995 earthquake (M =8; R=500).Notice the large variation of strong ground motion duration. Computed peak values are longer than 500s was for sites located over the thickest soil deposits of the valley. These contours are very similar tothe contours of equal thickness of deposits and, therefore, similar to contours of site-dominant period.

Variation of duration with magnitude at CD

The accelerometric station Central de Abasto (CD) has recorded almost all earthquakes felt inMexico City since 1985. This site has a dominant period of 3.3 s and has shown little non-linear behavior during very strong motions as the 1985 earthquake, where observed dominantperiod moved to 3.8 s [24]. A set of these good quality records is shown in Figure 11; thedate, magnitude, distance to the rupture area and duration from Equation (9) are written overeach accelerogram. Figure 12 is a comparison of observed and predicted duration, where astraight 45◦ line should be obtained if prediction matches observations perfectly. Di�erentsymbols correspond to NS and EW component of motion. The average error obtained forthese results, plotted with the dark dotted lines is, ±22:5 s. Although there is a pair of resultswhere di�erences are up to 50 per cent, we believe that Figure 12 re ects that Equation (9)is reliable.

DURATION OF STRONG GROUND MOTION AND ENERGY

Duration itself does not mean damage or require engineering solutions. However, if it is relatedwith some ampli�cation parameter such as maximum velocity or acceleration it is relevant forengineers.



Figure 11. Accelerometric records at station Central de Abasto. The date, magnitude, distance to therupture area and observed and predicted duration are written over each accelerogram.

The duration of strong ground motion is required by some methods [5] for the seismicdamage analysis of reinforced concrete buildings. Duration predicted with Equation (9) couldbe used as an input data for these models.Fajfar et al. [6] proposed the expression

I = vgD0:25 (10)

as a measure of earthquake ground motion capacity to damage structures with fundamentalperiods in the medium-period region; vg is the peak velocity and D is the duration of strongground motion that could be obtained with Equation (9).

←−Figure 10. Duration of strong ground motion with respect to site dominant period. The �gure showscomputed duration for each horizontal accelerogram (symbols), the trend computed with data of eachearthquake (solid line) and the trend obtained with Equation (9) (dashed line): (a) nine earthquakeswith roughly the same distance to the rupture area (Figure 1); (b) earthquakes with di�erent distances

to the rupture area (Figure 1) than those shown above.



Figure 12. Comparisons of observed and predicted duration at station Central de Abasto shown inFigure 11. A straight 45◦ line should be obtained if prediction matches perfectly the observations.

Fajfar et al. [1] and Vidic and Fajfar [2] propose Equations (11) and (12), respectively,to obtain the input energy per unit mass, that have been proved for a wide range of data,including that of Mexico City. The expressions are in terms of D; vg and peak accelera-tion, ag

EI =m=2:2v2gD0:5 (11)

EI =m=0:24a0:75g v1:25g D0:75 (12)

Nurtug and Sucuoglu [3] obtained the energy dissipated per unit mass under a groundexcitation, Ed, with respect to the pseudo-velocity, Sv, e�ective response duration, De, andmean square amplitude ratio (�2 + �2)

Ed=m= �!DeS2v (�2 + �2) (14)

where � and ! are the damping and frequency of the SDOF. To obtain De they use the sumof the duration of the strong ground motion, D, and the duration of vibration of the SDOF as

De =D+ 3:3T=(6�) (15)

where T is the period of the system.Uang and Bertero [4] propose to obtain the maximum energy input to a structure, Emazi =m,

whose fundamental period is close to the predominant excitation period of an expected earth-quake with the expected maximum ground velocity, vg(max) and the ampli�cation factor (for�=5 and �=5 per cent):

Emaxi =m= 12(vg(max))

2 (16)



Table I. Average observed parameters of records shown in Figure 3, duration given by Equation (9) andsome estimates of input energy and damage.

Earthquake M vg ag DEquation (9) EI=m(1) EI=m(2) Emaxi =m(4) I (6)

08=02=1988 5.7 6 284 6 222 660 60 925=04=1989 6.9 12 187 13 1314 1913 560 2419=09=1985 8.1 16 139 37 3513 4512 3825 39

where

=1:0 + 0:12D (17)

As was mentioned for Figure 3, neither peak acceleration or response spectra are correlatedwith structural damage. The parameter that correlates well with observed damage is the durationof strong ground motion. Using some of the above expressions, we have compared the energyand the potential to damage structures of some of the records shown in Figure 3. Table I showsthe average values computed for each earthquake. It is clear that considering these types ofparameters there is a good correlation of the size of the earthquake, intensity and damage.From these results, engineers should be careful in designing structures that will be submitted

to long duration (large energy). This will happen near active and large faults, which is obvious,but also at relatively far sites from epicentres where site a�ects amplify the input motion withlarger amplitude and longer duration. The most e�cient solution for such structures is yetto be proposed. However, there are some partial recommendations. These structures must becapable of dissipating energy without experiencing damage, so engineers are forced to chosereasonably resistant but ductile structures. Special care should be given for �nal structuraldetailing since it has been proved that this may be main reason of structural survival. Alle�ort to avoid cumulative damage should be taken into account, especially where the structurecould experience several intense earthquakes during its lifespan.

COMPUTING RESPONSE SPECTRA USING RANDOM VIBRATION THEORY ANDSTRONG MOTION DURATION

Random vibration theory (RVT) can be used to predict response spectra using only theoret-ical or empirical amplitude Fourier spectrum [7; 8; 27]. To apply RVT, an estimation of thestrong ground motion duration is needed for computing response spectra, duration that maybe predicted with Equation (9). RVT has already been tested for Mexico City [8; 19] and asummary of the main results are presented here.Figure 13 shows response spectra obtained for several accelerograms computed from the

Fourier amplitude spectra. The �gure compares exact (thick line) and RVT spectra computedin two ways: using Equation (9) (thin line) and using the duration between 2.5 and 97.5 percent of Arias’ intensity (dashed line) of each record. The idea of computing dashed spectrais to �nd out whether lack of accuracy in the prediction of RVT may be due to computingduration with Equation (9) or not. Figures 13(a)–13(d) present results for four earthquakeswith di�erent magnitude. For each earthquake, �ve stations with di�erent distances to therupture area and site-dominant period were chosen as representative of all other stations. It



can be seen that RVT and the duration predicted by Equation (9) yield response spectra verysimilar to the exact one. For the 19 September 1985 (M =8:1) earthquake, the spectra for Zi-huatanejo was underestimated by RVT by 30 per cent; however, the dashed spectra also lacksaccuracy, which means that duration is not responsible for not predicting this spectrum. Twostations in Mexico City with di�erent dominant periods prove that RVT works reasonably wellfor lakebed sites during large, distant and destructive earthquakes. For the other earthquakes,notice that stations 22, 49 and 56 located in Mexico City were chosen for three events withdi�erent magnitude and distance, and results using RVT and Equation (9) are very similar tothe exact spectra.On the other hand, Figure 13(e) plots response spectra for ten earthquakes (Figure 11)

recorded at the station CD (Ts = 3:3 s) located in Mexico City. Although these spectra camefrom a large variety of accelerograms in terms of duration, frequency content and amplitude,all spectra are very similar to the corresponding exact one.Results of Figure 13 show that predicting response spectra with RVT and the duration given

by Equation (9) yields reliable results. Errors, which in average are less than 10 per cent, arefar below the magnitude of uncertainties surrounding the prediction of strong ground motion.

CONCLUSIONS

An expression to predict the duration of strong ground motion in terms of magnitude, dis-tance to the rupture area and dominant site period has been presented. It was obtained withaccelerometric data of recent Mexican subduction and normal-faulting earthquakes with mag-nitudes between 4.9 and 8.1. Data employed were from accelerometric sites located over therupture area and as far as 520 km from it. It was found that large magnitude not only yieldlong duration at the source, but also longer duration with distance and with dominant site pe-riod compared to small magnitude. For instance, for a site with Ts = 5 s located at R=300 km,a M =5 earthquake yields a duration of 71 s, while a M =8 earthquake yields a duration of196 s, a di�erence of 125 seconds. For a �rm site located near the fault (R=15 km) di�erencesare just 30 s.Equation (9) predicts accurately strong ground motion duration for Mexican earthquakes

with average standard error of 24 per cent, and reproduces all observations concerning source,attenuation and site e�ects.

−→Figure 13. Response spectra obtained for several accelerograms. The �gure compares exact (thick line)and random vibration theory spectra computed in two ways: using Equation (9) (thin line) and usingthe duration between 2.5 and 97.5 per cent of Arias’ intensity (dashed line) of each record: (a) 19September 1985 earthquake (M =8:1). Three stations over the epicentral area (Zihuatanejo, Caleta Cand La Union) and two in Mexico City (R=305 km): Viveros (Ts = 0:5 s) and SCT (Ts = 2:0 s); (b)9 October 1995 earthquake (M =8:0). Three stations in Mexico City (22, 49 and 56; R=505 km),one station over the rupture area (Manzanillo) and one at Puerto Vallarta (R=105 km); (c) 25 April,1989 earthquake (M =6:9). Three stations in Mexico City (22, 49 and 56; R=295 km), one stationover the rupture area (Las Vigas) and one at F. Caballo (R=65 km); (d) 8 February, 1988 earth-quake (M =5:7). Three stations in Mexico City (22, 49 and 56; R=302 km) and two stations overthe rupture area (El Balcon and Papanoa); (e) response spectra computed for site Central de Abasto in

Mexico City during 10 earthquakes (Figure 11).



As was mentioned for Figure 3, neither peak acceleration or response spectra are correlatedwith structural damage. The parameter that correlates well with observed damage is the du-ration of strong ground motion. It was shown that taking into account the duration and anampli�cation parameter [1; 2; 6], a good correlation of size of the earthquake, intensity anddamage is found.Engineers should explicitly design structures that will be submitted to long duration near

active and large faults, and where site a�ects amplify the input motion. These structuresmust be reasonably resistant but mainly ductile and capable of dissipating energy withoutexperiencing damage, avoiding all types of geometrical irregularities and with special care onthe structural detailing.Finally, strong ground motion duration predicted by Equation (9) was used to obtain re-

sponse spectra via Random Vibration Theory. Exact response spectra were compared with theRVT spectra using the duration given by Equation (9). It was shown that the exact and RVTspectra are very similar for most records no matter the magnitude, distance to the rupture areaor the dominant period of the site.

ACKNOWLEDGEMENTS

Accelerometric data employed in this work were obtained by many dedicated people working at theInstituto de Ingenier��a UNAM, Centro de Instrumentaci�on y Registro S��smico A. C. and CENAPRED.Ra�ul Guerrero del �Angel collaborated in some parts of the manuscript.

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