Ride Llon Ground Motion i i

On Ground Motion Intensity Indices

Rafael Riddell,a… M.EERI

The characterization of strength of earthquake demands for seismicanalysis or design requires the specification of a level of intensity. Numerousground motion intensity indices that have been proposed over the years arebeing used for normalizing or scaling earthquake records regardless of theirefficiency. An essential point of this study is that a ground motion index isappropriate, or efficient, as long as it can predict the level of structuralresponse. This study presents correlations between 23 ground motion intensityindices and four response variables: elastic and inelastic deformationdemands, and input energy and hysteretic energy; nonlinear responses arecomputed using elastoplastic, bilinear, and bilinear with stiffness degradationmodels. As expected, no index is found to be satisfactory over the entirefrequency range. Indeed, indices related to ground acceleration rank better inthe acceleration-sensitive region of the spectrum; indices based on groundvelocity are better in the velocity-sensitive region and, correspondingly,generally occur in the displacement-controlled region. Despite frequentcriticism, the peak ground motion parameters passed the test successfully. Aranking of indices is presented, thus providing a choice of the most appropriateone for a particular application in the frequency range of interest.�DOI: 10.1193/1.2424748�

INTRODUCTION

The question of how to characterize the strength of ground motions by using asimple index has been discussed for a long time. Housner and Jennings �1982� clearlystated that the difficulties are fundamental: “It is inherently impossible to describe acomplex phenomenon by a single number, and a great deal of information is inevitablylost when this is attempted.” While one may agree with such a premise, one is faced withthe inexorable demand for a simple specification of the level of intensity of ground mo-tions for seismic analysis and design. Probably the best specification of ground motionsone could use in engineering applications is an ensemble of real or synthetic accelero-grams with appropriate intensity, duration, frequency characteristics, and consistencywith previously recorded motions for a specific site. But to set the intensity level of suchrecords, an index is necessary, for example, as is customary for the expected peak ac-celeration at the site. While using this index is suitable to deal with stiff structures, it canbe very inconvenient for the analysis of flexible systems. The same problem arises whenthe design ground motion is specified by means of a design spectrum; while in somecircumstances it may be appropriate to anchor the short-period end of the spectrum to anestimate of the expected peak ground acceleration, in other circumstances this oversim-

a�
Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago, Chile; E-mail: [email protected]
147Earthquake Spectra, Volume 23, No. 1, pages 147–173, February 2007; © 2007, Earthquake Engineering Research Institute

148 R. RIDDELL

plification may be unacceptable, and at least two or three parameters may be required.Notwithstanding that the latter was proposed about four decades ago by Newmark-Veletsos-Hall �design spectra constructed from peak ground displacement, velocity, andacceleration�, it is surprising that some standards are still based in only one parameter.Although incomplete, the use of two parameters may be acceptable since normally thespectral velocity region is conservatively extended to the low-frequency range.

Besides the specification of intensities of ground motions or spectra for design, nor-malizing or scaling procedures are often used by researchers for a variety of studies onearthquake response and structural behavior; in these cases, normalization is done tohave a common input intensity as a basis for discussion of the level of the resulting re-sponses. Many references on scaling of ground motion records and on intensity param-eters can be found in recent papers by Kurama and Farrow �2003� and Akkar and Ozen�2005�.

It is apparent from the previous discussion that an index is efficient as long as it canpredict or compute a structural response quantity reliably, i.e., with the least dispersion.In other words, one expects an index to be correlated to a response quantity, in such away that if the index of strength of ground motion increases or decreases, the responsequantity changes accordingly. This is the reason why peak ground acceleration �PGA� isnot a satisfactory sign of the response of very flexible systems, since these are basicallyinsensitive to acceleration spikes. This can be observed in a simple experiment: cuttinga portion of the peak ground acceleration of a record will in general have very little ef-fect on the response of long-period systems. In turn, this explains why many examplescan be found in the literature that show the lack of correlation of PGA with observedstructural performance during past earthquakes. Indeed, the use of PGA has come undermuch criticism on the general premise that “it is not adequate to characterize seismicperformance of structures.” Such a statement is part right and part wrong, as inferredfrom the above discussion and as it will be shown later in the paper. In fact, the criticismbearing on PGA is extensible to all intensity indices known, for none of them is ad-equate over the entire frequency range.

The purpose of this study is to evaluate the correlation between various intensity in-dices with response quantities, in order to rank them according to quality. For this pur-pose, 90 earthquake records �Table 1� were used as input ground motions, and responseswere calculated for elastic and inelastic systems representative of the three characteristicregions of the spectrum. A detailed definition of available intensity indices is given first,then their correlation with response quantities is presented. Finally, the indices areranked, thus permitting selection of the most appropriate for specific applications.

GENERAL MEASURES OF INTENSITY OF A FUNCTION

The simplest index of intensity of a function of time f�t� is its maximum value, orpeak, often regardless of its sign:

fmax = max�f�t�� 1�

ON GROUND MOTION INTENSITY INDICES 149

Table 1. Earthquakes records used in this study

Station, Component, Date amax �g� vmax �cm/sec� dmax �cm�

CMD Vernon, USA, S08W �10/3/1933� −0.133 −29.03 −19.50

El Centro, USA, S00E �18/5/1940� −0.348 −33.45 −12.36

Olympia, USA, N86E �13/4/1949� 0.280 17.09 −9.38

Eureka, USA, N79E �21/12/1954� 0.258 −29.38 −12.55

Ferndale, USA, N44E �21/12/1954� −0.159 −35.65 14.72

Kushiro Kisyo-Dai, Japan, N90E �23/4/1962� 0.478 −20.01 5.22

Ochiai Bridge, Japan, N00E �5/4/1966� −0.276 23.66 8.36

Temblor, USA, S25W �27/6/1966� 0.348 −22.52 −5.55

Cholame 2, USA, N65E �27/6/1966� 0.489 78.08 −26.27

Cholame 5, USA, N85E �27/6/1966� 0.434 25.44 −6.89

El Centro, USA, S00W �8/4/1968� 0.130 −25.81 12.96

Hachinohe, Japan, N00E �16/5/1968� 0.269 −35.43 −9.68

Aomori, Japan, N00E �16/5/1968� −0.257 −39.12 −19.97

Muroran, Japan, N00E �16/5/1968� −0.220 30.28 7.90

Itajima Bridge, Japan, L �6/8/1968� 0.612 −22.56 −4.59

Itajima Bridge, Japan, L �21/9/1968� −0.261 −12.93 −2.80

Lima, Peru, N08E �31/5/1970� 0.409 −15.20 −11.67

Toyohama Bridge, Japan, L �5/1/1971� 0.450 15.90 3.38

Pacoima, USA, S16E �9/2/1971� 1.171 113.23 −41.92

Orion LA, USA, N00W �9/2/1971� 0.255 30.00 16.53

Castaic, USA, N21E �9/2/1971� 0.316 17.16 −5.05

Karakir, USSR, N00E �17/5/1976� −0.608 65.36 25.28

Bucarest, Romania, S00E �4/3/1977� 0.206 75.12 −19.93

San Juan, Argentina, S90E �23/11/1977� 0.193 −20.60 6.33

9101 Tabas, Iran, T �16/9/1978� 0.852 −121.19 94.53

El Centro Array # 5, USA, S50W �15/10/1979� 0.379 −90.50 63.01

6604 Cerro Prieto, México, N45E �9/6/1980� −0.621 31.57 13.24

Ventanas, Chile, T �7/11/1981� 0.268 −17.87 −8.04

Papudo, Chile, L �7/11/1981� −0.603 −18.93 −7.43

La Ligua, Chile, L �7/11/1981� −0.469 −18.83 4.49

Rapel, Chile, N00E �3/3/1985� 0.467 −21.64 −6.54

Zapallar, Chile, N90E �3/3/1985� 0.304 13.46 −1.69

Llo-Lleo, Chile, N10E �3/3/1985� −0.712 −40.29 −10.49

Viña del Mar, Chile, S20W �3/3/1985� 0.363 30.74 −5.42

UTFSM, Chile, N70E �3/3/1985� 0.176 14.60 3.11

Papudo, Chile, S40E �3/3/1985� 0.231 12.41 1.60

Llay Llay, Chile, S10W �3/3/1985� −0.352 −41.79 8.43

San Felipe, Chile, N80E �3/3/1985� 0.434 −17.77 −3.50

El Almendral, Chile, N50E �3/3/1985� 0.297 −28.58 −5.78

Melipilla, Chile, N00E �3/3/1985� −0.686 34.25 12.02

150 R. RIDDELL

Table 1. �cont.�


Pichilemu, Chile, N00E �3/3/1985� 0.259 −11.68 3.73

Iloca, Chile, N90E �3/3/1985� 0.278 15.09 1.39

SCT, Mexico, N90E �19/9/1985� −0.171 −60.61 21.16

TLHB, Mexico, N00E �19/9/1985� −0.139 64.08 −36.26

6097 Site 1, Canada, N80W �23/12/1985� −1.096 −46.05 −14.58

Corralitos, USA, N00E �18/10/1989� 0.630 −55.20 12.03

47524 Hollister-South & Pine, USA �18/10/1989� 0.371 62.33 30.27

16LCPC, USA, N00E �18/10/1989� 0.563 94.69 41.16

Saratoga-Aloha Ave., USA, N00E �18/10/1989� −0.512 −41.14 16.21

Saratoga-West Valley C., USA, N90W �18/10/1989� −0.332 61.52 36.38

14 WAHO, USA, N90E �18/10/1989� 0.638 37.93 5.85

Erzincan, Turkey, NS �13/3/1992� 0.515 84.24 26.72

89156 Petrolia, USA, N90E �25/4/1992� 0.662 −89.66 29.57

24Lucerne, USA, N00E �28/6/1992� −0.814 32.06 89.91

Coolwater, USA, T �28/6/1992� −0.417 42.33 −13.76

Yermo Fire Station, USA, N90W �28/6/1992� −0.245 −51.39 43.82

KSR Kushiro, Japan, N63E �15/1/1993� 0.725 33.59 4.73

Pacoima Dam, USA, S05E �17/1/1994� −0.415 44.68 4.65

Newhall, USA, N00E �17/1/1994� 0.591 −94.73 28.81

Pacoima-Kagel, USA, N00E �17/1/1994� 0.433 −50.88 −6.64

Sylmar, USA, N00E �17/1/1994� 0.843 −128.88 −30.67

Santa Monica, USA, N90E �17/1/1994� −0.883 41.75 −15.09

Moorpark, USA, S00E �17/1/1994� 0.292 20.28 4.67

Castaic, USA, N90E �17/1/1994� 0.568 −51.51 −9.19

Arleta, USA, N90E �17/1/1994� 0.344 −40.37 8.36

Century City-LA, USA, N90E �17/1/1994� 0.256 21.36 −6.51

Obregon Park-LA, USA, N00E �17/1/1994� −0.408 −30.86 −2.65

Hollywood-LA, USA, N00E �17/1/1994� −0.389 22.26 4.27

KJMA, Japan, N00E �16/1/1995� −0.821 81.27 −17.69

Nishi-Akashi, Japan, N90E �16/1/1995� −0.503 −36.61 11.26

Takatori, Japan, N00E �16/1/1995� 0.611 127.15 35.78

Takarazuka, Japan, N90E �16/1/1995� −0.694 85.27 16.76

Duzce, Turkey, S �17/8/1999� −0.312 58.83 44.12

Gebze, Turkey, N �17/8/1999� 0.244 50.29 42.74

Yarimca, Turkey, N60E �17/8/1999� 0.268 −65.72 −57.02

CHY024, Taiwan, EW �21/9/1999� −0.282 51.17 95.68

CHY025, Taiwan, EW �21/9/1999� −0.162 51.52 63.62

CHY028, Taiwan, NS �21/9/1999� −0.765 −84.10 28.39

CHY101, Taiwan, NS �21/9/1999� 0.398 108.60 −80.96

TCU049, Taiwan, NS �21/9/1999� −0.241 59.36 −117.74


More information about the strength of a function of time is provided by its averagevalue over a specified interval between t1 and t2:

fave =1

t2 − t1�

t1

t2f�t�dt �2�

For an alternating function, i.e., one that has positive and negative values, the averagevalue may be meaningless, since it may be close or equal to zero. The latter is the casewith ground acceleration histories; since the integral of the ground acceleration is theground velocity, it becomes zero if integration is carried out from the beginning to theend of the motion when the ground comes to rest. Similarly, the average value of theground velocity may also be near zero, this is not always the case, however, since per-manent offset of the ground �sometimes quite large� often results from earthquake mo-tions. To eliminate the possibility of zero averages, the integral of the squared functionmay be considered �as has been customary in the study of response of electric circuits�:

fave2 =

1

t2 − t1�

t1

t2f2�t�dt �3�

also called mean-square value, from which the effective value of the function, or root-mean-square value is given by

feff = frms = �fave2 �4�

which is often encountered in stochastic analysis. While the use of the mean-squarevalue �Equation 3� over the interval t2− t1 may be suitable to describe the effectiveness ofa periodic function �for which t2− t1 may simply be taken as the period of the function�,its use as a measure of total strength of earthquake motions needs caution. Indeed, an

Table 1. �cont.�


TCU051, Taiwan, EW �21/9/1999� 0.160 −51.67 123.97

TCU063, Taiwan, NS �21/9/1999� 0.133 82.24 −68.80

TCU067, Taiwan, EW �21/9/1999� −0.498 97.58 186.91

TCU071, Taiwan, NS �21/9/1999� −0.651 83.95 289.47

TCU074, Taiwan, EW �21/9/1999� 0.597 −70.20 −196.16

TCU076, Taiwan, NS �21/9/1999� 0.428 −62.93 −72.08

TCU102, Taiwan, NS �21/9/1999� 0.171 −71.50 −102.24

Lamont 375, Turkey, N �12/11/1999� −0.970 36.49 −5.84

Bolu, Turkey, N �12/11/1999� 0.728 −56.42 23.07

Duzce, Turkey, W �12/11/1999� 0.535 −83.48 51.60

152 R. RIDDELL

important attribute of ground motions is the duration of strong ground shaking. So, twoground acceleration records with the same mean-square value �Equation 3� may causedifferent amounts of structural damage if one is longer than the other. Thus the plainintegral of the squared function is often used when dealing with duration-dependent re-sponse quantities like energy terms:

fsq = �t1

t2f2�t�dt �5�

or its root-square value

frs = �fsq �6�

GROUND MOTION INTENSITY INDICES

As mentioned above, the intensity of motion cannot be satisfactorily characterized bya single parameter. Furthermore, as it will be shown herein, different intensity measuresare suitable in the three characteristic spectral regions: short period �acceleration-sensitive systems�, intermediate period �velocity controlled responses�, and long period�displacement-sensitive systems�. These three spectral regions were first identified byNewmark-Veletsos-Hall in their pioneer work on earthquake response amplification forthe derivation of design spectra. The reference to the sensitivity of the systems, coinedby Chopra �1995�, is especially lucid. In fact, if one is interested in the response of in-termediate period systems, velocity-related indices will be more significant, since suchsystems are less sensitive to ground acceleration or ground displacement histories, i.e.,their responses do not present good correlation with acceleration- or displacement-related indices. In this section, indices relevant in each of the three mentioned periodranges will be presented. It must be noted that the indices in each group are correlatedamong them, as discussed below. The basic indices defined above and others frequentlyused in the literature will be considered.

ACCELERATION-RELATED INDICES

Basic indices as defined by Equations 1 and 3–6 have been used with f�t� equal toa�t�, the ground acceleration history. The simplest acceleration-related index is the peakvalue of the ground acceleration history �PGA�, designated as amax according to Equa-tion 1. The indices introduced by Housner and Arias are quite popular. Housner �1975�argued that a measure of seismic destructiveness could be given by the mean-squarevalue of the acceleration history �Equation 3�, which he termed “earthquake power in-dex:”

Pa =1

t2 − t1�

t1

t2a2�t�dt �7�

where t1 and t2 are the limits of the strong portion of motion. To avoid arbitrariness inthe selection of t1 and t2, the definition of significant duration of motion after Trifunacand Brady �1975a� was adopted in this study, i.e., the interval between instants t5 and t95


at which 5% and 95% of the total integral in Equation 7 are attained, respectively. Arias�1970� proposed as a measure of earthquake intensity the integral of the squared accel-eration �Equation 5, but affected by a coefficient�, which he interpreted as the sum of theenergies dissipated, per unit of mass, by a population of damped oscillators of all naturalfrequencies �o��:

IA�� =cos−1 �

g�1 − �2�o

tfa2�t�dt �8�

where � is the damping of the oscillator �as a fraction of critical damping�, tf is the totalduration of the ground motion, and g is the acceleration of gravity. The plain integral ofthe squared acceleration �Equation 5�, its root-mean-square �rms� value �Equation 4�,and its root-square �rs� value �Equation 6� have been also considered as potential mea-sures of ground motion intensity. The specialized formulae in this case become, respec-tively

asq = �o

tfa2�t�dt �9�

arms = �Pa �10�

ars = �asq �11�

where arms was introduced in earthquake engineering by Housner and Jennings �1964�,and ars was also proposed by Housner �1970�.

A brief discussion about the root-mean-square or effective acceleration value is nec-essary. Such value has no relation with the effective peak acceleration �EPA� concept.The concept of EPA was introduced in codes �ATC 1978, BSSC 1984� to characterizethe intensity of design ground motions. In a certain way, it is plausible that EPA wasintroduced to justify the use of PGA, which had already come under much criticism.EPA was justified as consistent with an amplification factor of 2.5 in the accelerationplateau of the linear-elastic 5% damped pseudo-acceleration average spectrum; in otherwords, the argument was that EPA was not PGA, but a value obtained backwards �withthe 2.5 deamplification factor� from the smoothed average �or desired exceedance prob-ability� spectrum in the acceleration region. Such justification seems to be based on fal-lacy. Basically, the 2.5 factor comes from Newmark and associates’ studies on responseamplification �although the 2.5 value is somewhat larger than the 5% damping 50-percentile amplification factor�. In all those studies PGA was actually used to normalizethe ground motion records to compute the amplification factor in the acceleration region.That is indeed legitimate as long as a complete family of strong ground motions is used�covering the desired conditions regarding earthquake parameters and site location char-acteristics�. Naturally, anemic, short-duration, and unrepresentative records must be dis-regarded. In turn, normalization to PGA is legitimate as long as the response ofacceleration-sensitive systems is concerned, as has been discussed so far and as will beinsisted later on.

154 R. RIDDELL

Park et al. �1985� found that the “characteristic intensity” given by

IC = arms1.5 td

0.5 �12�

that combines the root-mean-square value with the significant duration of the record td

= t95− t5, was a reasonable representation of destructiveness of ground motions, for itcorrelated well with structural damage expressed in terms of the damage index afterPark and Ang �1985�. Riddell and Garcia �2001� found that the following compound in-dex could minimize the dispersion of hysteretic energy-dissipation spectra of inelasticsingle-degree-of-freedom �SDOF� systems with a stiffness-degrading force-displacement relationship:

Ia = amaxtd1/3 �13�

VELOCITY-RELATED INDICES

The simplest velocity-related index is the peak value of the ground velocity history,vmax according to Equation 1. Basic indices given by Equations 7 and 9–11 above can bespecialized for f�t� equal to v�t� the ground velocity history as follows:

Pv =1

t95 − t5�

t5

t95

v2�t�dt �14�

vsq = �o

tfv2�t�dt �15�

vrms = �Pv �16�

vrs = �vsq �17�

Araya and Saragoni �1980� defined the “potential destructiveness” of an earthquake as

PD =IA

� o2 �18�

where IA is Arias intensity �Equation 8� and �o is the number of zero-crossings per unitof time of the accelerogram. The significance of this index is the incorporation of thefrequency content of the ground motion through the parameter �o, which modifies its“acceleration-related” character. As �o decreases, the index accounts for shifting of therecord power to the intermediate-frequency range. Fajfar et al. �1990� proposed the com-pound index

IF = vmaxtd0.25 �19�

as a measure of the ground motion capacity to damage structures with fundamental pe-riods in the intermediate period range. Riddell and Garcia �2001� found that a similarindex


Iv = vmax2/3 td

1/3 �20�

could minimize the dispersion of hysteretic energy dissipation spectra for intermediate-frequency inelastic systems. The exponents of vmax and td varied depending on both thetype of force-deformation relationship and the level of ductility; the exponents used inEquation 20 are approximately applicable for any resistance function and for moderateductility level. Housner’s spectral intensity �1952�

SI�� = �0.1

2.5

Sv��,T�dT �21�

may also be regarded as a velocity-related index inasmuch as it weighs preferentially theintermediate-period range �approximately 0.5�T�2� against the short-period range�T�0.5� and the long-period range �T�2�. Sv in Equation 21 is the pseudo-velocityresponse spectrum. Therefore, Housner’s intensity is a response quantity since it involvesstructural properties ��, T�, i.e., it is an a posteriori index, as opposed to the other indi-ces presented herein, which are of an a priori nature, i.e., they are intended to be a mea-sure of the cause �ground motion strength� indicative of the effect �structural response�.

DISPLACEMENT-RELATED INDICES

The simplest index in this case is the peak value of the ground displacement history,dmax. Basic indices given by Equations 7 and 9–11 above can be specialized for f�t�equal to d�t� the ground displacement history as follows:

Pd =1

t95 − t5�

t5

t95

d2�t�dt �22�

dsq = �o

tfd2�t�dt �23�

drms = �Pd �24�

drs = �dsq �25�

Riddell and Garcia �2001� found that the combined index

Id = dmaxtd1/3 �26�

could minimize the dispersion of hysteretic energy dissipation spectra for low-frequencyinelastic systems.

CORRELATION AMONG INDICES

The indices in each of the groups presented above are correlated to each other. Cor-relations computed on the basis of the relationship explained below �Equations 29 and30� are presented in Table 2. The observation can be made from this table that“acceleration-related indices” in general present high correlation among them, while

156 R. RIDDELL

their correlation with “velocity-related indices” is poor, and they do not correlate at allwith “displacement-related indices.” In turn, “velocity-related indices” in generalpresent high correlation among them, while some of them also present good correlationwith “displacement-related indices.” Finally, “displacement-related indices” are stronglycorrelated among them.

CORRELATION BETWEEN INTENSITY INDICES AND RESPONSEQUANTITIES

The response of simple SDOF systems was considered for this purpose. Elastic sys-tems and inelastic systems with force-displacement relationship given by three nonlinearmodels were considered: elastoplastic, bilinear, and stiffness-degrading bilinear. Thepost-yield stiffness of the two latter was 3% of the elastic stiffness. These models covera broad range of structural behavior. They are intended to represent overall generic be-havior, rather than specific characteristics of individual systems �Riddell and Newmark1979�. A damping factor �=0.05 was used �5% of critical�. SDOF systems associatedwith three control frequencies �0.2, 1, and 5 cps� were chosen as representative of thethree characteristic spectral regions. These frequencies are roughly in the middle of thethree spectral regions of response amplification.

Ninety earthquake records were used as input ground motions �Table 1�. Theserecords represent moderate to large intensities of motion. Almost all the records satisfythe following condition: peak ground acceleration larger than 0.25 g and/or peak groundvelocity larger than 25 cm/sec. An effort was made to achieve, as much as possible,uniform distributions of the intensity indices for the data set, to make sure that the dataare not biased for some indices. Cumulative frequency distributions for six indices areshown in Figure 1; although uniformity is not perfect, it seems reasonable given the ob-jectives of the study. Uniformity is difficult to attain in the range of large values of eachindex, since a large value of one index for one record does not necessarily imply largevalues for other indices uncorrelated with the former, e.g., a large amax for one recorddoes not necessarily go in company with large vmax and dmax.

The group of records is intended to be a heterogeneous sample of records represent-ing a variety of conditions regarding tectonic environment, site geology �most of therecords are in Site Classes A, B, C, and D, and fewer in Class E�, modified Mercalliintensity �MMI�, and distance to source, or others, thus offering an ample spectrum ofintensities and frequency content. It is therefore expected that the findings of this studyenjoy some generality, i.e., they do not refer to records belonging to a family with spe-cial attributes. It would certainly be of interest to group records featuring common char-acteristics, for example, according to soil class, or to have records corresponding tonear-field pulse-like motions that have received a great deal of attention in the literature.However, forming special groups was beyond the scope of this study, and its conclusionsmay not be directly extensible to such special conditions without further examination.

Four response quantities were selected. Two response variables were considered forelastic systems: spectral ordinates �being immaterial which one in particular, since


pseudo-acceleration, pseudo-velocity, and maximum displacement are directly relatedthrough the frequency factor �=2�f� and input energy per unit of mass EI supplied tothe system by its moving base. The input energy is defined as

EI = − �o

u

a�t�du = − �o

t

a�t�u̇�t�dt �27�

where u�t� is the relative displacement of the system with respect to the ground. Moreabout this expression and that of EH defined further on, as well as more about the energybalance equation derived from the equation of motion of the system, are available else-where �Kato and Akiyama 1975, Zahrah and Hall 1982, Riddell and Garcia 2001�; in

Figure 1. Cumulative frequency distributions for intensity indices amax, vmax, dmax, Ia, IF, and SI

for the ground motion data set.

158 R. RIDDELL

turn, Akiyama �1985� has developed procedures to apply energy concepts in practicaldesign. On the other hand, it has been found that there is an exact functional relationshipbetween the input energy spectrum and the Fourier amplitude spectrum �Ordaz et al.2003�; for this reason, the Fourier spectrum, which is a significant response variable, wasnot considered in this study.

When Equation 27 is integrated until the ground motion stops �tf�, EI is equivalent tothe total energy dissipated by damping in the elastic system. It is worth noting that, ac-cording to the previous definition, EI is a response quantity, i.e., it is not a ground mo-tion record attribute. Indeed, as shown in Figure 2, the energy input across the spectrumvaries, i.e., different amounts of energy are transmitted from the ground to the varioussystems depending on their natural frequencies. As can be seen in the figure,intermediate-frequency systems take more energy, while less energy is input to low-frequency systems, and even less to high-frequency systems. In turn, similar amounts ofenergy, regardless of inelastic response level, are received by low-frequency systems andsome intermediate-frequency systems, while energy input increases as ductility in-creases in the high-frequency range. Therefore, one should be cautious about the expres-sion “earthquake energy input,” since the actual input depends on the system itself. Thusthe quoted term may be misleading, for it may give the wrong impression that there is afixed amount of energy that the ground forces into every system. In the case of inelasticsystems, two response variables were considered: the maximum deformation umax andthe hysteretic energy EH dissipated by the oscillator, both of them for a response asso-ciated with a displacement response ductility, µ=3; however, four correlation studieswere carried out since different load-deformation models were used. The specific value

Figure 2. Energy input spectra for the Sylmar N00E record, Northridge earthquake �17 January1994�, for elastoplastic systems with 5% damping and response ductilities of 1, 3, and 10.


of µ chosen is not a limitation since similar conclusions are reached if other values areused. Note, however, that a reasonably moderate value of ductility must be used, since itis meant to represent the “global response ductility” of a structural system rather thansubstantially higher “local ductilities” that may be developed by individual members orjoints. The total hysteretic energy dissipated per unit of mass is defined as

EH =1

m�

o

tfF�u�u̇�t�dt �28�

where F�u� is the hysteretic restoring-force history. Energy dissipation is a componentof structural damage �Park and Ang 1985� and thereby plays an important role as anearthquake demand parameter for the evaluation of seismic performance. It is worthpointing out here that the intensity indices are not being assessed in this study accordingto damage potential; any such correlation would have required explicit consideration ofMMI. Relationships between MMI and amax have been proposed by several researchers;among them, the studies by Gutenberg and Richter �1942� and Trifunac and Brady�1975� are classic. In a more recent study, Wald et al. �1999� presented relationships be-tween MMI and amax and vmax. In turn, some of the indices included here have beenconsidered by their authors as related to “destructiveness” �Equations 7, 12, 18, and 19�;such quality is not being evaluated in this study.

To have an objective measure of correlation, a curve of the form

R = �I �29�

was fitted to the data for all possible pairs between the intensity indices �I in Equation29� and the response quantities considered �R in Equation 29� at the three control fre-quencies �f=0.2, 1, and 5 cps�, where � and are the nonlinear regression parameters�or linear regression between the logarithms of the variables�. The goodness of fit isquantified by the correlation coefficient given by

=n � ��nI�nR� − � �nI � �nR

��n � ��nI�2 − �� nI�2��n � ��nR�2 − �� nR�2��30�

To visualize the correlation among response quantities and intensity indices, scatterplots like Figures 3–6 were made for all indices. In particular, Figure 3 illustrates therelation between umax �the maximum displacement of inelastic systems with bilinearconstitutive rule and response ductility µ=3� with indices amax �left side of the figure�and asq �right side�. Each dot corresponds to the response to each of the 90 earthquakerecords used. It can be seen that amax and umax present excellent correlation �=0.902� atthe high control frequency �f=5 cps�, but they show very poor correlation at the inter-mediate control frequency �f=1 cps with =0.296�, and no correlation at all at 0.2 cps�=0.044�. Similarly, asq, another acceleration-related index, presents better correlationwith umax for f=5 cps �=0.719� than for f=1 or 0.2 cps, where drops to 0.453 and0.127, respectively.

160 R. RIDDELL

Figure 3. Correlation between maximum inelastic displacement umax for bilinear systems withresponse ductility µ=3 and peak ground acceleration amax �left� and asq �right�. Scatter plots forasq are also valid for Arias intensity IA since these indices only differ by a constant �i.e., a shift-
ing in the log plot�.


Figure 4. Correlation between maximum inelastic displacement umax for bilinear stiffness-degrading systems with response ductility µ=3 and Riddell-Garcia’s index Ia �left� and peakground velocity v �right�.
max

162 R. RIDDELL

Figure 5. Correlation between dissipated energy �EH� for elastoplastic systems with responseductility µ=3 and Housner’s intensity S �left�, and Fajfar’s index I �right�.
I F


Figure 6. Correlation between Riddell-Garcia’s index Id and dissipated energy EH for stiffness-degrading systems with response ductility µ=3 �left�. Correlation between maximum grounddisplacement d and elastic response �right�.
max

164 R. RIDDELL

Figure 4 illustrates the relation between umax �the maximum displacement of inelas-tic systems with bilinear stiffness-degrading force-deformation relationship and re-sponse ductility µ=3� and indices Ia �acceleration-related� and vmax �velocity-related�; Ia

clearly presents very good correlation with umax in the acceleration region �f=5 cps, =0.821� and very poor in the other regions, while the vmax presents excellent correlationfor intermediate frequencies �f=1 cps, =0.910�, it is still efficient in the low-frequencyregion �f=0.2 cps, =0.827�, but it is uncorrelated with and umax at f=5 cps.

Figure 5 shows the relation between Housner’s intensity SI �Equation 21� and Fajfar’sindex IF with hysteretic energy EH �for elastoplastic systems with response ductility µ=3�; it can be seen that SI and EH present excellent correlation for intermediate-frequency systems �f=1 cps, =0.940� and good correlation for low frequencies �f=0.2 cps, =0.747�, but they are uncorrelated at high frequencies �f=5 cps, =0.154�;Fajfar’s index presents very good correlation with EH at both intermediate and low fre-quencies �=0.858 and 0.874, respectively� and no correlation at all at f=5 cps. Figure6 shows two displacement-related indices: Riddell-Garcia’s Id �Equation 26� versus hys-teretic energy EH for stiffness-degrading systems with response ductility associated to aresponse ductility factor µ=3, and dmax against and umax �elastic response deformation�;both indices show no correlation at all with responses at f=5 cps �negative �, fair cor-relation at f=1 cps, and excellent correlation at f=0.2 cps �=0.914 and 0.900, respec-tively�.

It is worth making two observations regarding Figure 6 �right� for f=0.2 cps. First,there are nine points that clearly deviate from the linear trend of the data. These pointscorrespond to seven records from Taiwan, CHY024, and Lucerne �USA�, which exhibitlarge permanent �final� displacement of the ground. It is apparent that responses to theserecords do not present the typical response amplification in the displacement region�umax�dmax� or they do so only slightly. For most of these records their elastic spectrumapproaches dmax from below �asymptotic spectral property for long period�, i.e., barelyor never exceeding dmax. The obvious explanation is that large permanent displacementsof the ground result from very long period waves that weakly excite the 0.2 cps system.The second observation is that excluding these special records, the data in the figure fol-low a linear trend associated with an amplification factor �umax/dmax� of the order of1.5-2, which is the expected value according to Newmark and associates’ studies.

The correlation coefficients for all response vs. intensity combinations, for the threecontrol frequencies, are summarized in Tables 3–8. The indices ranked in the top five foreach frequency are noted. The correlation coefficient is the same for indices that differonly by a constant or by the exponent �this is the case for example of IA, asq, and ars�.The main conclusion drawn from the results presented in these tables is that no indexshows satisfactory correlation with response in the three spectral regions simulta-neously; indeed, acceleration-related indices are the best for rigid systems �5 cps�,velocity-related indices are the best for intermediate-frequency systems �1 cps�, anddisplacement-related indices are generally better for flexible systems �0.2 cps�, although

Table 2. Correlatio

amax D IF SI Iv dmax dsq, drs Pd, drms Id

amax 1.000 22 0.176 0.351 0.040 0.051 −0.061 0.071 −0.042IA, asq, ars 04 0.433 0.514 0.422 0.197 0.143 0.158 0.207Pa, arms 66 0.263 0.461 0.083 0.140 0.003 0.174 0.017IC 73 0.391 0.524 0.310 0.186 0.095 0.174 0.142Ia 83 0.264 0.357 0.245 0.070 0.018 0.041 0.068vmax 74 0.958 0.913 0.846 0.787 0.663 0.743 0.734vsq, vrs 33 0.952 0.862 0.946 0.836 0.776 0.763 0.861Pv, vrms 66 0.875 0.882 0.739 0.812 0.680 0.791 0.737PD 00 0.703 0.767 0.677 0.448 0.374 0.385 0.455IF 1.000 0.896 0.964 0.798 0.716 0.725 0.804SI 1.000 0.812 0.666 0.541 0.603 0.634Iv 1.000 0.749 0.711 0.654 0.808dmax 1.000 0.966 0.983 0.980dsq, drs 1.000 0.962 0.975Pd, drms 1.000 0.943Id 1.000

ON

GR

OU

ND

MO

TIO

NIN

TE

NS

ITY

IND

ICE

S165

n coefficients among intensity indices

IA, asq,ars Pa, arms IC Ia vmax vsq, vrs Pv, vrms P

0.733 0.891 0.843 0.890 0.308 0.080 0.319 0.11.000 0.758 0.964 0.862 0.410 0.424 0.400 0.4

1.000 0.904 0.696 0.434 0.189 0.499 0.21.000 0.849 0.446 0.355 0.465 0.3

1.000 0.264 0.208 0.219 0.11.000 0.880 0.950 0.6

1.000 0.871 0.71.000 0.6

1.0

166 R. RIDDELL

some velocity-related indices also do well in the displacement region, particularly vsq-vrs, which rank first for energy terms, as in Tables 6–8. Several specific observations canbe made from the mentioned tables as detailed next.

CORRELATION BETWEEN INTENSITY INDICES AND SPECTRAL ORDINATES„TABLES 3–5…

• The peak ground motion parameters �amax, vmax, dmax� show very good correla-tion with elastic and inelastic spectral ordinates. In the acceleration and displace-ment regions, amax and dmax are, respectively, the best indices. In the velocity re-gion, vmax ranks second, after Housner’s intensity �SI�. Noting that SI is aresponse variable itself, and hence less appealing as a predictor variable, onecomes to realize that vmax is an excellent intensity index in the velocity region.

• In the velocity region, averaging the values in Tables 3–5, it is found that theaverage for vmax is only 8.5% smaller than that for Housner’s intensity �SI�. Inturn, Fajfar’s index �IF� presents an average only 1.5% smaller than that forvmax; from this it is concluded that IF is also an excellent index in the velocityregion.

Table 3. Correlation coefficient between spectral ordinates for elastic systems and variousground motion intensity indices

f=0.2 cps f=1 cps f=5 cps

rank rank rank

Acceleration-relatedamax 0.031 0.405 0.859 1IA and asq and ars 0.178 0.507 0.642 5Pa and arms 0.129 0.468 0.760 3IC 0.170 0.523 0.730 4Ia 0.046 0.408 0.775 2

Velocity-relatedvmax 0.814 0.833 2 0.133vsq and vrs 0.869 3 0.759 5 −0.058Pv and vrms 0.847 5 0.789 4 0.139PD 0.511 0.586 −0.046IF 0.824 0.810 3 0.026SI 0.712 0.910 1 0.158Iv 0.770 0.727 −0.077

Displacement-relateddmax 0.900 1 0.632 −0.079dsq and drs 0.840 0.508 −0.174Pd and drms 0.864 4 0.577 −0.067Id 0.881 2 0.597 −0.151


• In the displacement region, Id emerges as an excellent index, ranking second af-ter dmax.

• Considering the previous observations, and recalling that Nau and Hall �1982�tested several of the indices used herein �except PD, IC, IF, Ia, Iv, and Id� to findthat none of them provided noteworthy advantage over the peak ground motionsto reduce the dispersion of elastic and inelastic spectral ordinates, amax, vmax, anddmax must be regarded as significant intensity parameters to characterize theearthquake demand, especially because these indices can be probabilistically pre-dicted for future earthquakes as a function of magnitude, source mechanism, dis-tance, and site conditions. Nau and Hall also found that SI presented advantageover vmax in the velocity region, but it was so for low response ductilities only.

• Arias intensity, which has been widely used as a measure of ground motion se-verity and as scaling parameter, is essentially an acceleration-related index, andtherefore appropriate only when the response of acceleration-sensitive systems isconcerned. In turn, it is not as satisfactorily correlated with elastic and inelasticspectral ordinates as amax. Average for IA is 22.4% smaller than that of amax.

Table 4. Correlation coefficient between maximum displacement of bilinear systems with re-sponse ductility µ=3 and various ground motion intensity indices


rank rank rank


Velocity-relatedvmax 0.830 0.878 2 0.379vsq and vrs 0.866 4 0.832 4 0.152Pv and vrms 0.845 0.830 5 0.366PD 0.492 0.669 0.180IF 0.841 0.873 3 0.261SI 0.704 0.944 1 0.405Iv 0.787 0.802 0.131

Displacement-relateddmax 0.908 1 0.651 0.065dsq and drs 0.850 5 0.529 −0.049Pd and drms 0.870 3 0.582 0.074Id 0.890 2 0.628 −0.019

168 R. RIDDELL

• Similarly, Housner’s intensity is essentially a velocity-related or intermediate-frequency index. It is defeated by most indices in the displacement region and itdoes poorly with regard to the response of rigid systems.

CORRELATION BETWEEN INTENSITY INDICES AND ENERGY RESPONSES„TABLES 6–8…

• Housner’s intensity ranks first for intermediate frequencies. It does poorly forrigid systems. In the velocity region, SI is followed by Fajfar’s index �IF� andvsq-vrs. IF, vsq, and vrs present an average �for Tables 6–8� 9.8% smaller thanthat for SI. Iv and peak ground velocity �vmax� come closely behind IF �1.9% and3%, respectively�. Again, IF shows up as an attractive index, closely followed byvsq-vrs, Iv, and vmax.

• In the acceleration region, Riddell-Garcia’s index �Ia� is the best, followedclosely by Park et al.’s index �IC�, Arias intensity �IA�, and PGA �amax�. The dif-ferences of the latter three on average �for Tables 6–8� with respect to Ia are1%, 4.1%, and 6.8%, respectively.

• In the displacement region, the velocity-related indices vsq-vrs present the bestcorrelation with energy responses, followed by Riddell-Garcia’s index �Id� and

Table 5. Correlation coefficient between maximum displacement of bilinear stiffness degrad-ing systems with response ductility µ=3 and various ground motion intensity indices


rank rank rank


Velocity-relatedvmax 0.827 0.910 2 0.376vsq and vrs 0.848 0.850 5 0.150Pv and vrms 0.851 5 0.859 4 0.365PD 0.495 0.678 0.219IF 0.823 0.899 3 0.256SI 0.694 0.967 1 0.406Iv 0.757 0.820 0.125

Displacement-relateddmax 0.920 1 0.681 0.064dsq and drs 0.858 4 0.558 −0.049Pd and drms 0.889 3 0.615 0.070Id 0.892 2 0.653 −0.021


dmax, with average differences of 3.3% and 4.3%, respectively, with respect tovsq-vrs.

• Including td to form compound indices �IC, Ia, IF, Iv, Id� generally results in im-proved correlation coefficients. For example, considering the average for Tables6–8, in the acceleration region, IC is better than arms, and Ia is better than amax. Inthe velocity region IF and Iv are better than vmax, although not significantly, andin the displacement region Id is better than dmax, although not significantly either.Similarly, the effect of ground motion duration in energy responses is manifestedin the velocity and displacement region by the fact that root-square values arealways better correlated with energy responses than root-mean-square values do.

• Although other indices outperform the peak ground motion parameters, dmax andamax come reasonably close �respectively, only 4% and 7.7% difference in aver-age with the index ranking top in the corresponding spectral region�. Less suc-cessful is vmax in the velocity region, with an average 15.5% smaller than thatof the top index.

Table 6. Correlation coefficient between input energy EI for damped elastic systems and vari-ous ground motion intensity indices


rank rank rank

Acceleration-relatedamax −0.018 0.389 0.629 2IA and asq and ars 0.253 0.619 0.675 4Pa and arms 0.078 0.414 0.578 5IC 0.198 0.574 0.678 3Ia 0.079 0.504 0.752 1

Velocity-relatedvmax 0.811 0.748 5 −0.052vsq and vrs 0.932 1 0.766 3 −0.103Pv and vrms 0.824 0.689 −0.060PD 0.601 0.550 −0.204IF 0.871 3 0.788 2 −0.068SI 0.737 0.847 1 0.004Iv 0.860 5 0.766 3 −0.077

Displacement-relateddmax 0.863 4 0.586 −0.137dsq and drs 0.812 0.502 −0.161Pd and drms 0.800 0.518 −0.137Id 0.879 2 0.596 −0.144

170 R. RIDDELL

GLOBAL RANK CONSIDERING THE SIX TESTS „TABLES 3–8…

If the correlation coefficients in the six tables are now averaged, the best index in theacceleration region is amax, followed by Ia and IC; the latter two present global average 4.2% and 5.4% smaller than amax. In the velocity region, the best index is Housner’sintensity �SI�; however, if only a priori indices are considered, the best indices are IF andvmax, with a difference of only 1.5% in average . In the displacement region the bestindex is dmax followed by vsq-vrs and Id, with negligible difference in average amongthe three of them.

SUMMARY AND CONCLUSIONS

This study has attempted to contribute to a better understanding of ground motionintensity indices used for specification of design ground motions or for normalizing orscaling ground motions for earthquake response studies. The findings were as follows:

• No index shows satisfactory correlation with response in the three spectral re-gions simultaneously. Indeed, acceleration-related indices are the best for rigidsystems, velocity-related indices are better for intermediate-frequency systems,and displacement-related indices are better for flexible systems; some velocity-related indices do also well in the low-frequency region.

Table 7. Correlation coefficient between hysteretic energy EH for elastoplastic systems withresponse ductility µ=3 and various ground motion intensity indices


rank rank rank


Velocity-relatedvmax 0.827 0.829 4 0.090vsq and vrs 0.928 1 0.845 3 −0.044Pv and vrms 0.838 0.748 0.097PD 0.593 0.682 0.003IF 0.874 4 0.858 2 0.017SI 0.747 0.940 1 0.154Iv 0.851 5 0.820 5 −0.052

Displacement-relateddmax 0.891 3 0.635 −0.126dsq and drs 0.843 0.535 −0.197Pd and drms 0.838 0.564 −0.121Id 0.898 2 0.637 −0.173


• The peak ground motions parameters �amax, vmax, dmax� present excellent corre-lation with elastic and inelastic spectral ordinates in their corresponding fre-quency ranges. Peak ground acceleration and displacement also present goodcorrelation with input and hysteretic energies in their respective spectral regions;vmax, however, does only moderately well. Since peak ground motion parameterscan be established for future earthquakes with relative ease, on the basis of gen-erally accepted available methodologies for earthquake hazard assessment, theyare strongly recommended as intensity indices.

• Housner’s intensity is the best index in the velocity region regarding correlationwith both spectral ordinates and energy responses. It does moderately well in thedisplacement region and very poorly in the acceleration region. It should benoted as well that Housner’s intensity is a spectral response quantity itself, henceit is less appealing as a predictor variable.

• Arias intensity, a widely used measure of ground motion severity, is essentiallyan acceleration-related index and is therefore appropriate only when the responseof rigid systems is concerned. In the high-frequency range it presents good cor-relation with energy responses, but it is amply outperformed by the peak groundacceleration and other indices with regard to spectral ordinates.

Table 8. Correlation coefficient between hysteretic energy EH for bilinear stiffness degradingsystems with response ductility µ=3 and various ground motion intensity indices


rank rank rank


Velocity-relatedvmax 0.824 0.847 5 0.124vsq and vrs 0.922 1 0.888 3 0.051Pv and vrms 0.833 0.797 0.122PD 0.569 0.737 0.091IF 0.871 4 0.893 2 0.092SI 0.726 0.957 1 0.207Iv 0.848 0.868 4 0.055

Displacement-relateddmax 0.908 3 0.658 −0.082dsq and drs 0.864 5 0.570 −0.135Pd and drms 0.858 0.580 −0.090Id 0.914 2 0.670 −0.102

172 R. RIDDELL

• Fajfar’s index in the velocity region is slightly inferior to vmax with regard tospectral ordinates, but the contrary occurs with regard to energy terms.

• Riddell-Garcia’s indices Ia and Id emerge as good indices in their respective re-gions; Ia is the best in the acceleration region with regard to energy terms, whileId is the second best in the displacement region with regard to both energy termsand spectral ordinates.

• Although not very significantly, the duration of motion increases the correlationwith energy responses when combined with other simple indices in productform; compound indices proposed by Park et al., Fajfar, and Riddell-Garciapresent better correlations than the corresponding single indices in their fre-quency ranges.

ACKNOWLEDGMENTS

This study was carried out in the Department of Structural and Geotechnical Engi-neering at the Universidad Católica de Chile with financial assistance from the NationalScience and Technology Foundation of Chile �FONDECYT� under Grant Nos. 1990112and 1030598.

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�Received 16 April 2004; accepted 6 June 2006�

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