the effective number of cycles of earthquake ground motion

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2005; 34:637–664Published online 15 December 2004 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.437

The e�ective number of cycles of earthquake ground motion

Jonathan Hancock and Julian J. Bommer∗;†

Department of Civil and Environmental Engineering; Imperial College London; London SW7 2AZ; U.K.

SUMMARY

The seismic response of any system that accumulates damage under cyclic loading is dependent not onlyon the maximum amplitude of the motion but also its duration. This is explicitly recognized in methodsfor estimating the liquefaction potential of soil deposits. Many researchers have proposed that the e�ec-tive number of cycles of the ground motion is a more robust indicator of the destructive capacity of theshaking than the duration. However, as is the case with strong-motion duration, there is no universallyaccepted approach to determining the e�ective number of cycles of motion, and the di�erent methodsthat have been proposed can give widely varying results for a particular accelerogram. De�nitions of thee�ective number of cycles of motion are reviewed, classi�ed and compared. Measurements are found todi�er particularly for accelerograms with broad-banded frequency content, which contain a signi�cantnumber of non-zero crossing peaks. The key seismological parameters in�uencing the number of cyclesof motion and associated equations for predicting this quantity for future earthquakes are identi�ed.Correlations between cycle counts and di�erent duration measures are explored and found to be ratherpoor in the absence of additional parameters. Copyright ? 2004 John Wiley & Sons, Ltd.

KEY WORDS: earthquake ground motions; strong-motion duration; e�ective number of cycles

1. INTRODUCTION

The destructive capacity of earthquake ground motion depends on both the amplitude and thenumber of cycles of the motion. The number of cycles of motion is widely recognized as beingof fundamental importance in geotechnical earthquake engineering. Many researchers assertthat the number of cycles of motion is also an important factor in aspects of the seismic designand damage assessment of structures, although there is currently no consensus on this issue.In geotechnical engineering, the number of cycles of loading is of critical importance in

the analysis of cohesionless granular soil. During an earthquake, pore water pressure increaseswith each cycle and under su�ciently severe loading conditions the soil can reach a statewhere the e�ective stress is reduced to very low levels. This results in a dramatic loss ofstrength through liquefaction of the soil, which constitutes a serious hazard to any overlying

∗Correspondence to: Julian J. Bommer, Department of Civil and Environmental Engineering, Imperial CollegeLondon, London SW7 2AZ, U.K.

†E-mail: [email protected]

Received 26 May 2004Revised 11 August 2004

Copyright ? 2004 John Wiley & Sons, Ltd. Accepted 18 October 2004

638 J. HANCOCK AND J. J. BOMMER

or buried structures and to slopes. Current techniques used to assess liquefaction potentialare based on results from laboratory testing of soils under uniform cycles of loading and onobservations of soils in the �eld after earthquakes, which produce motion containing irregularamplitude cycles [1–3]. Several di�erent techniques have been used to convert the irregularground motions from real events into equivalent uniform amplitude cycles in order to enablethe results of laboratory tests to be used to interpret �eld observations of liquefaction [4, 5].The number of cycles of loading is important for the assessment of low-cycle fatigue dam-

age in structural earthquake engineering applications [6–10]. Fatigue-based damage measuresaccumulate damage with each cycle of structural displacement, recognizing that failure can becaused by a single large amplitude motion or several smaller amplitude motions. De�nitionsof e�ective peak ground motion have also been developed by several authors based on theamplitude of secondary cycles of ground motion [11–13]. The e�ective peak of the motionis reported as a better indicator of the damage capacity of the shaking than the maximumamplitude, because it does not give undue importance to single, isolated peaks as encoun-tered, for example, in near-source ground motions from small magnitude earthquakes. Thehigh acceleration peaks of such events are sustained for a few cycles, their energy content islow, and properly designed structures experience little or no damage under such shaking. TheHAZUS methodology [14] for assessing potential losses in building stock due to earthquakeshaking also recognizes that the number of cycles of motion is important and makes useof a magnitude-dependent duration correction to reduce the in�uence of small magnitudeevents [15].This paper complements the work of Bommer and Martinez-Pereira [16, 17] who reviewed,

classi�ed and compared de�nitions for the duration of earthquake strong-motion. Measuresof the e�ective number of cycles of motion are more likely to convey useful indicationsof the in�uence of the duration of shaking on the response of structures and soils becausethey also provide information on the amplitude of the motion in a more explicit way than isachieved with most de�nitions of duration. A review is conducted of the di�erent de�nitionsof number of cycles proposed in the literature. Methods for converting irregular amplitudecycles into a single number of equivalent constant-amplitude cycles are also discussed. Thein�uence of predictive variables, such as earthquake magnitude and site-to-source distance,on the number of cycles is investigated and predictive equations for estimating the numberof cycles of ground motion in future earthquakes are reviewed. Measures of strong-motionduration can be considered as surrogates for the number of cycles of motion and there aremany more predictive equations available for duration than for the number of cycles. Theextent to which duration can be used as an indirect measure of the number of cycles ofmotion is explored by examining the degree of correlation between the two parameters. Thepaper concludes with recommendations regarding the selection of cycle counting de�nitionsfor di�erent applications.

2. DEFINING THE EFFECTIVE NUMBER OF CYCLES

2.1. Procedures for determining e�ective number of constant amplitude cyclesAlthough measures of ground-motion duration provide an indication of the number of cycles ofmotion, cycle-counting techniques are usually used because they also convey a measure of theamplitude of the motion. Whilst this feature provides additional and important information,

Copyright ? 2004 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2005; 34:637–664

EFFECTIVE NUMBER OF CYCLES 639

it also presents some extra challenges. Most applications require the number of equivalentcycles of constant amplitude. However, this information is not obtained directly from mostcycle-counting de�nitions as they produce tallies of the cycles with di�erent amplitudes withina time history. For this tally of irregular cycle amplitudes to be converted to a single numberrepresenting the e�ective number of cycles at a constant amplitude an assumption has to bemade regarding the relative importance of di�erent cycles. The transformation to constant-amplitude cycles is usually conducted assuming linear damage accumulation, which was �rstproposed by Miner [18] for the calculation of fatigue damage, FD, in aluminium:

FD=∑ ni(Si)Ni(Si)

(1)

where ni is the number of load cycles applied at stress range Si, and Ni is the number of cyclesrequired to cause failure at stress range Si. The key to the method is that failure occurs whenFD=1. This method has been adapted to several di�erent applications simply by replacingthe stress range with other measures of damage, such as strain range for low-cycle fatigueassessment and cyclic stress ratio (which is directly proportional to ground acceleration) inliquefaction analysis. The principle can be illustrated graphically; consider a time historycontaining four cycles with a range (peak-to-trough) of 1 followed by one cycle with a rangeof 2 (Figure 1). For illustrative purposes, a failure line is prescribed which states that failurewill occur with eight uniform amplitude cycles with a range of 1 or two uniform amplitudecycles with a range of 2. Therefore the total fatigue damage for this loading regime is:

FD= 48 +

12 =

12 +

12 =1 (2)

Half of the fatigue life is used by the �rst four cycles and the remaining half used by the�nal cycle, implying that the component is expected to fail at the end of the loading.A common practice is to assume that the damage caused by each cycle is proportional to

its amplitude raised to a power, which expressed in terms of the range is:

S=(Na

)− 1k

(3)

where a and k are constants dependent on the material properties and, if used in a structuralapplication, the geometry of the component under investigation. Constants a and k representa linear scale factor and the gradient of a straight line on the S–N curve when plotted onlog–log axes (e.g. Figure 1). Although the constant a may vary considerably for di�erentapplications and is dependent on the units used, the constant k is more stable and, as shownbelow, is of greater importance.The procedure illustrated above can be generalized for any excitation. Consider a time

history with n1 cycles with a range S1 followed by groups of ni cycles each with a range Si.The damage caused by these irregular amplitude cycles is:

FD=n1(S1)aS−k1

+∑ ni(Si)aS−ki

(4)



Figure 1. Illustration of Miner’s linear damage accumulation. Applied cyclic loading(upper), and number of cycles to failure represented by a S–N curve with linear axes

(lower left) and log–log axis (lower right).

The number of uniform amplitude equivalent cycles Neq at range Sref , which cause the samedamage as the irregular amplitude cycles is:

Neq(Sref )aS−kref

=n1(S1)aS−k1

+∑ ni(Si)aS−ki

(5)

which simpli�es to:

Neq(Sref )= S−kref

(n1(S1)S−k1

+∑ ni(Si)

S−ki

)(6)

Equation (6) shows that coe�cient a cancels out, and it is coe�cient k that is of primeimportance as it determines the relative damage caused by di�erent amplitude cycles. Thereference range Sref is also di�erent depending on the cycle-counting de�nition: for relativecounting measures Sref is expressed as a fraction of the maximum cycle amplitude in the timehistory, whereas absolute counting de�nitions use a �xed reference amplitude for all timehistories.



Miner’s damage accumulation theory assumes that the sequence of the loading cycles andthe direction of the load (positive or negative) have no in�uence on the ensuing damage.Whilst this is a reasonable assumption for many applications, more complex damage measureshave been proposed which do not make these assumptions [19, 20]. However, increasing thecomplexity of the damage measure leads to a dramatic increase of the di�culty in reliablydetermining the coe�cients required for the damage measure. This has led to the relativelysimple models, such as that detailed above, being the most widely used.

2.1.1. Reference level. Di�erent cycle-counting de�nitions express the number of equivalentcycles at di�erent reference amplitudes: for example, liquefaction studies frequently use thenumber of equivalent cycles at 65% maximum amplitude [21]. This is not a signi�cant problemas conversion between di�erent reference amplitudes can be made if the relative damageassumed to be caused by each cycle amplitude is known [4]:

Nnew =Nref

(NfnewNfref

)(7)

where Nnew is the number of cycles at the new reference amplitude, Nref is the number ofcycles at the original reference amplitude, Nfnew is the number of cycles to cause failureat the new reference amplitude and Nfref is the number of cycles to cause failure at theoriginal reference amplitude. However, it is not possible to make such a conversion from acycle-counting de�nition based on relative amplitudes to another based on absolute amplitudes(see Section 2.3).

2.1.2. Non-zero crossings and response spectra. Non-zero crossings in any time history, in-cluding ground and structural motions, occur when the signal has broad-banded frequencycontent. As discussed in the following sections, the presence of non-zero crossings is par-ticularly important as it causes disagreement in the number of cycles estimated by di�erentcycle-counting de�nitions, as some count non-zero crossing peaks whilst others do not. Thetreatment of non-zero crossing peaks is particularly important when dealing with motions fromlarge magnitude earthquakes as they can create motions with broad-banded frequency content,containing many non-zero crossing peaks.Several researchers have developed cycle response spectra that show the amplitude of the

motion sustained for a given number of cycles [22–24]. As the response spectra are producedfrom the motion of elastic single-degree-of-freedom (SDOF) systems, some authors have sug-gested that the motion will be narrow banded, containing only zero crossing peaks. However,examination of the actual motion shows that only the response of SDOF systems with periodslower than the dominant period of the ground motion are narrow banded. This is illustratedin Figure 2, which shows the displacement time-histories of 2:5s and 0:5s oscillators with 5%damping under the action of ground motions recorded at Shelter Cove 2 during the 1992 CapeMendocino earthquake. The response of the 0:5 s SDOF system has few non-zero crossingpeaks, while the 2:5 s SDOF system has many. The Fourier transform of the motions showsthat the response of the 2:5 s SDOF system retains the broad-banded frequency content of theground motion whilst the 0:5 s SDOF system e�ectively �lters out the broad-banded contentof the ground motion.



Figure 2. Displacement time-histories and Fourier amplitude spectra of 2:5 s and 0:5 s SDOF systemswith 5% damping under the action of 1992 (Mw =7:1) Cape Mendocino (Petrolia) earthquake ground

motions recorded 32 km from the fault rupture at Shelter Cove 2 (sti� soil).

Cycle counting de�nitions that count the numbers of cycles in the response of SDOFsystems are classi�ed as structural de�nitions. These measures are reviewed in more detail inSection 2.2.5.

2.1.3. Combination of axes. All ground-motion parameters calculated directly from accelero-grams, including the number of e�ective cycles, need to address the problem of how to com-bine the di�erent components of motion. Most cycle-counting de�nitions ignore the verticalcomponent of the motion. This leaves the two horizontal components, which are commonlytreated as independent motions. However, some researchers use the vector sum of the com-ponents, which calculates the resultant amplitude with time Ares(t) from the amplitudes of thetwo horizontal components AH1(t) and AH2(t):

Ares(t)=√AH1(t)2 + AH2(t)2 (8)

One problem with using the vector sum is that the absolute sign of the peaks is lost preventingthe use of range-counting de�nitions. Furthermore, if the two horizontal components are outof phase, with the amplitude in one direction increasing while the other is decreasing, theamplitude of the resultant tends to remain constant. Therefore, even though there are distinctcycles of motion in both of the orthogonal horizontal components, their resultant displays



Figure 3. Orthogonal components of a single cycle of a harmonic signal (top right, bottom left),where the solid lines represent in-phase motions and the dashed line represents the case of out-of-phasetransverse component. The top left plot shows the resultant vector for the two cases and the lower right

plot their variation of amplitude with time.

an amplitude that is more or less constant and hence there appears to be no cycling of themotion.Figure 3 illustrates the di�erence, for the ideal case of harmonic signals, between the cases

of in-phase and out-of-phase motions: the resultant of the in-phase components displays cycleswith the same period as the two components, whereas the out-of-phase components (illustratedby the dashed line) result in circular motion around the origin and no variation of amplitudein the resultant. In Figure 4, which shows the same information as Figure 3 except for thetwo horizontal components of a real accelerogram, it can be seen from the top left-hand graphthat there are both in-phase and out-of-phase cycles, the latter producing the arcs observablein the plot.Another problem with combining the horizontal components of accelerograms through vec-

tor resolution is that for digitized records from analogue instruments, the precise start timesof the two components, and hence their relative phase, are uncertain [25].Another method that can be used to account for both horizontal components is to count

the equivalent cycles in each component separately, and then add the number of equivalentcycles together to �nd the total e�ective number of cycles. If relative measures are employed



Figure 4. As for Figure 3 but using the horizontal acceleration components of the 1994 Northridgeearthquake recorded 20 km from the fault rupture at Castiac Old Ridge Route.

both components should be counted relative to the largest peak to prevent undue in�uencefrom the minor axis [21].

2.1.4. Low-amplitude cut-o�. Some authors propose that a minimum cycle amplitude shouldbe used to prevent many small amplitude cycles unduly in�uencing results. For example,Bolt and Abrahamson [12] discard acceleration peaks less than 0:005g whilst Koliopoulosand Margaris [26] only use the 20 largest cycles. However, the low-amplitude cut-o� isonly required by studies, such as these, which use a low-amplitude–importance relationshipso that, for example, one cycle with amplitude 5 may be considered equal to �ve cycles ofamplitude 1. A low-amplitude cut-o� is not required for studies using cycle counting forassessing liquefaction potential or fatigue damage as they use an amplitude–importance re-lationship that explicitly gives greater weight to large amplitude cycles. For example, theimportance of each cycle may be assumed to be proportional to the square of its amplitude,so a single cycle of amplitude 5 is equivalent to 25 cycles of amplitude 1. Seed et al. [21]use a similar amplitude–importance relationship for assessing the potential of a time historyto trigger liquefaction and conclude that cycle amplitudes less than about 30% of the peakamplitude make no signi�cant contribution to the number of e�ective cycles or the potentialto trigger liquefaction.



Figure 5. Accelerogram recorded for the Mw =3:9 Coalinga earthquake recorded at Sulphur Baths,5:7 km from the epicentre. Box shows section of motion used for cycle counting examples.

2.2. Generic de�nitions of number of cycles

From an extensive review of the literature, it is found that there are many di�erent cycle-counting de�nitions in use. However, these can be divided into �ve generic groups, looselybased on categories used for classifying cycle-counting de�nitions developed for fatigue testing[27, 28]:

• Peak counting• Level crossing counting• Range counting• Indirect estimation• De�nitions based on structural response

Some de�nitions count cycle ranges (from peak to trough), whilst others count cycle ampli-tudes (from zero baseline to peak). An approximate conversion between ranges and amplitudescan be made assuming cycle peaks are equally distributed either side of the baseline, althoughmany accelerograms do not display such symmetry. This assumption is employed herein whereconversion is required, so that two half-cycle ranges are taken to be equal to one peak, andone full cycle is equal to two peaks, each with amplitude equal to half the range of the fullcycle. All of the de�nitions may, or may not, count motions that do not cross the base linebetween cycles.The following �ve sub-sections describe the main generic techniques used for direct cycle

counting. Cycle-counting de�nitions are illustrated using a 1 second section of an accelero-gram recorded during an aftershock of the 1985 Coalinga (California) earthquake (Figure 5).To allow ease of comparison the number of e�ective cycles (Ncy) and cyclic damage



parameter (D), as used by Malhotra [9], are calculated for each de�nition:

D=C2tn∑i=1uci (9)

Ncy =12

2tn∑i=1

(uiumax

)c(10)

where ui is the amplitude of the i-th half cycle, umax is the amplitude of the largest half cycle,and tn is the total number of cycles. C and c are application-dependent damage coe�cients:C is a linear scale factor and c determines the relative importance of di�erent amplitudecycles. Jeong and Iwan [29] used c=2 to represent the fatigue damage of wide �ange steelcolumns and c=6 for rectangular reinforced concrete columns. Malhotra [9] proposed c=2as a reasonable value. Values of c=2 and C=1 are adopted for this study.

2.2.1. Peak counting de�nitions. Peak counting de�nitions are the most commonly used inearthquake engineering. They are employed in a wide range of applications including structuraldamage estimation and assessment of liquefaction potential. A cycle-counting de�nition isclassi�ed as peak counting if it explicitly counts the number of peaks in the recorded motion.There are several di�erent variations of peak counting, the two main di�erences betweende�nitions being:

• Cut-o� level: to prevent many small peaks dominating the results some de�nitions donot count peaks below a low-amplitude cut o� (Section 2.1.4).

• Zero-crossing counting: some de�nitions only count the largest peaks between zero cross-ings, whereas others count all peaks.

Figure 6 shows an example of a peak-counting de�nition with no low-amplitude cut-o� andwhere only zero crossing peaks are counted. The damage parameter, D, given by Equation (9),evaluated using this de�nition is 0.127; if non-zero crossing peaks are included D increasesby about 25% to 0.160.

2.2.2. Level crossing de�nitions. Level crossing de�nitions are not normally used to measureamplitudes of motion but are commonly used to measure the period or frequency content ofthe motion. The most common measure is the zero mean crossing period [30], but Sarmaand Srbulov [31] propose that other levels can be used. They de�ne the frequency as halfthe number of excursions above a set level, divided by the duration spent above the levelusing the uniform duration de�nition as proposed by Sarma and Casey [32]. The conventionin cycle counting [27] is to count only level crossings with positive gradient on the positivey-axis and crossings with a negative gradient on the negative y-axis (Figure 7).

2.2.3. Range-counting de�nitions. Range-counting de�nitions are commonly used for assess-ment of fatigue damage. In earthquake engineering this is related to problems such as theassessment of failure probability of welds or bolted connections in the beam–column jointsof steel framed buildings under earthquake loading [7, 33–35].



Figure 6. Peak counting example, applied to segment of accelerogram in Figure 5, with non-zerocrossings (shown by circles) excluded.

Figure 7. Level crossing example, applied to segment of accelerogram in Figure 5.

The most popular range-counting method is the rain�ow-counting method as it counts bothhigh- and low-frequency cycles in broad-banded signals. The rain�ow technique, as describedby ASTM [27], is illustrated in Figure 8 and its application to an accelerogram shown inFigure 9. This yields a value of the damage parameter, D, of 0.199, a factor of 1.57 greaterthan that predicted by zero-crossing peak counting de�nitions.



Figure 8. Rain�ow-counting procedure.

2.2.4. Indirect counting methods. Indirect counting methods are those that do not count thenumber of cycles directly from a time series, but use statistical techniques to approximatethe number of cycles. These are usually based on random vibration theory and use spectralmoments to determine the predominant frequency and hence number of extreme peaks perunit time [36]. Most of these de�nitions assume that the peaks are not strongly correlated andare part of a stationary process.The method presented by Boore [36] to determine the number of extreme responses, Ne,

in time interval T is:

Ne = 2f̃T (11)

where f̃ is the predominant frequency, given by:

f̃=12�

√m4m2

(12)



Figure 9. Range counting example, using the rain�ow-counting technique, applied to segment ofaccelerogram in Figure 5. Solid lines indicate full cycles, dashed lines indicate half cycles.

m2 and m4 are the second and fourth spectral moments, respectively, given by:

m2 =1�

∫ ∞

0!2|A(!)|2 d! (13)

m4 =1�

∫ ∞

0!4|A(!)|2 d! (14)

where ! is the circular frequency and A(!) is the Fourier amplitude spectrum of the accel-eration.Amini and Trifunac [22], Gupta [37, 38] and Gupta and Trifunac [39, 40] point out that

real accelerograms are non-stationary and use order statistics to derive the e�ective peakacceleration of non-stationary accelerograms. Elghadamsi et al. [41] overcome issues of non-stationarity by modelling strong ground-motion with a stationary component plus a non-stationary envelope function.

2.2.5. De�nitions based on structural response. Structurally-based cycle-counting de�nitionsare similar to those detailed above for counting the number of cycles of ground motion, withthe di�erence that they count the cycles of motion from the response of the structure. Thesede�nitions can usually be subdivided into peak, range, level crossing or indirect countingde�nitions applied to the structural response. These methods are particularly important forstructural analysis, as it is the number of structural displacements that ultimately governsresponse. For structures with complex geometry, the number of displacement cycles of indi-vidual elements can be counted. However, it is also possible to count the number of cycles ofequivalent SDOF systems, such as the displacement of the elastic SDOF shown in Figure 2.



Figure 10. Illustration of parameters used to determine the number of equivalentcycles for an elasto-plastic system.

For a speci�c structure, cycle counts determined from the structural response are clearly asuperior indicator of the destructive potential of the motion than the number of cycles in theground acceleration. The advantage in the latter, however, is that it conveys the destructivecapacity of the motion in a general sense, not tied to a speci�c structure, and this lends itselfto the development of general predictions. Whilst for liquefaction assessment it is simply thenumber of cycles that is of primary concern, it could be argued that for structural applicationsa measure of the number of cycles without an indication of the frequency content, however,is of limited usefulness.Other authors use the hysteretic energy of yielding structures to determine the number of

e�ective cycles of motion. A typical example is that suggested by Zahrah and Hall [42]who determine the equivalent number of yield cycles, neq, by dividing the total hystereticenergy, EH, by the equivalent energy that would be absorbed had the structure been loadedmonotonically to the same maximum displacement. This de�nition of the number of equivalentcycles can be de�ned using an elasto-plastic model and used as a comparative index to assessthe severity of ground motion:

neq =EH

Fy · (xmax − xy) (15)

where Fy is the yielding force, xmax is the maximum displacement and xy is the yield dis-placement (Figure 10).

2.3. Relative vs absolute measures

One of the most important distinctions amongst di�erent de�nitions is if they are calculatedusing the absolute amplitude of the cycles or using levels relative to the cycle with the largestamplitude. The damage parameter given in Equation (9) is typical of an absolute measure andthe de�nition of the number of e�ective cycles given in Equation (10) is typical of a relativemeasure. Most de�nitions of the e�ective number of cycles used in earthquake engineeringemploy relative amplitude measures.



To illustrate the range of de�nitions employed in the technical literature a summary of dif-ferent cycle-counting de�nitions proposed by di�erent authors is presented in Table I. Beyondthe broad generic categories, the main di�erences amongst the de�nitions are whether theyuse relative or absolute amplitudes and whether or not they count non-zero crossing peaks.

3. PREDICTING THE NUMBER OF CYCLES OF MOTION

Whilst many authors have attempted to estimate the duration of the ground motion fromfuture earthquakes [16], far fewer studies provide equations for predicting the number ofcycles of motion. Predictive relationships for the number of cycles predominantly originatefrom researchers interested in the prediction of liquefaction potential [5, 21], whereas structuralapplications have historically relied on the ground-motion duration for providing an indicationof the number of cycles of motion [43]. This section reviews the few published equationsfor predicting the number of cycles of motion in order to identify the implied dependency ondi�erent predictor variables.

3.1. Earthquake magnitude

Earthquake magnitude is universally acknowledged as the parameter that exerts the greatestin�uence on the number of cycles of motion. The greater number of ground motion cyclesfrom larger magnitude earthquakes has led to the magnitude scaling factor used in liquefactionassessment [3, 44]. However, Figure 11 shows that there is no conclusive agreement on theexpected number of cycles for any given earthquake magnitude. All of these studies wereconducted for the assessment of liquefaction potential and use relative cycle-counting de�ni-tions that count the number of e�ective acceleration cycles at a reference amplitude equal to65% of the maximum peak amplitude. The more recent studies employ moment magnitude,whilst the magnitude scale used for earlier studies is not known and may be one source forthe di�erences amongst the predictions. Other predictive parameters such as the source-to-sitedistance and site classi�cation are only used in some of the studies, which is another potentialsource of di�erence. These issues, in addition to the use of di�erent functional forms, are acommon source of disagreement amongst predictive equations for any ground-motion param-eter. Di�erences amongst these studies could additionally be caused by the following factorsrelated to the de�nition of the number of cycles:

• Di�erent cycle-counting de�nitions used: most studies count peaks but do not de�ne hownon-zero crossing peaks are handled. The study by Liu et al. [5] is unusual as it usesthe vector sum of the horizontal components and counts both peaks and troughs of themotion.

• Di�erences in the amplitude–importance relationship used to convert the numbers ofcycles into equivalent cycles.

There appears to be considerable di�erence in the functional forms used by di�erent re-searchers. Exact di�erences are di�cult to establish as many studies do not state the func-tional form used but simply give numbers of e�ective cycles expected for di�erent magnitudes.Haldar and Tang [45] conduct extensive regression analysis for the number of e�ective cycles



Table I. Summary of cycle-counting de�nitions used by di�erent authors.

Level crossing Peak count Range countIndirect=structural

Generic category Rel. Abs. Rel. Abs. Rel. Abs. method

Ambraseys and Sarma [61] ◦Lee and Chan [4] +Seed et al. [21] +Trifunac and Westermo [62] +Annaki and Lee [2] +Joannon et al. [63] +Mortgat [11] +Vanmarcke and Lai [30] ◦Araya and Saragoni [64] ◦Amini and Trifunac [22] +�Bolt and Abrahamson [12] •Boore [36] +Zahrah and Hall [42] �Pauschke [65] ◦Saragoni [66] ◦ASTM [27] •◦ •◦ •Kawashima and Aizawa [23] + +Sarma and Yang [67] ◦Jeong and Iwan [29] ◦ ◦Joyner and Boore [68] +Siddharthan [69] •Saragoni [70] ◦Gupta [37] +Gupta [38] +Manzocchi et al. [71] ◦Mander et al. [7] +Gupta and Trifunac [72] +Safak [24] •Basu and Gupta [73] +�Calado et al. [33] •Lu et al. [49] •Sarma and Srbulov [31] +Gupta and Trifunac [39] +Gupta and Trifunac [74] •◦ •◦Schwarz [13] +Bursi and Caldara [75] •Perera et al. [76] •Taucer et al. [34] +Liu et al. [5] •Manfredi [77] �Stewart et al. [59] •Calado et al. [78] •Erberik and Sucuoglu [79] ◦ ◦Tremblay and Bouatay [35] •Malhotra [9] + + �Manfredi et al. [50, 51] �Osinov [80] +Teran-Gilmore et al. [81] •Kunnath and Chai [10] �Green et al. [82] �

•= including zero crossing peaks, ◦=excluding zero crossing peaks,+= zero crossing counting method not speci�ed, �=structural method.



Figure 11. Predicted median values of the number of e�ective cycles at 65% maximum amplitude as afunction of magnitude: (1) Haldar and Tang [45], mean weighting; (2a) Green [46] 60 km distance; (2b)Green [46] 40 km distance; (2c) Green [46] 20 km distance; (2d) Green [46], 0 km distance; (3a) Liuet al. [5] 10km distance, �eld-based weighting; (3b) Liu et al. [5] 10km distance, averaged weighting;(3c) Liu et al. [5] 10 km distance, laboratory-based weighting; (4) Arango [44]; (5) Seed and Idriss[47]; (6a) Seed et al. [21] using the stronger component of motion; (6b) Seed et al. [21] using both

components of motion, evaluated separately; (7) Valera and Donovan [48].

at di�erent reference levels using a second-order polynomial functional form:

Neqx =�+ �M + �M 2 (16)

where Neqx is the number of e�ective cycles at a particular reference level, M is the earthquakemagnitude, �, � and � are coe�cients determined by regression. The functional form used byLiu et al. [5] is the most complex and includes magnitude, source-to-site distance and siteclassi�cation.

3.2. Source-to-site distance

Only three of the studies reviewed develop predictive relations for the number of cycles ofmotion using the source-to-site distance as a predictive parameter. These are Lu et al. [49],who derive predictive relations of the total number of ground velocity cycles for use in thefatigue assessment of pipelines, and Green [46] and Liu et al. [5] who determine predictiveequations for the number of e�ective acceleration cycles for use in liquefaction analysis. All ofthese studies use relative cycle-counting de�nitions that normalize the cycles to the maximumamplitude, which results in the number of cycles increasing with distance due to scattering,



Figure 12. Predicted median values of the number of e�ective cycles at 65% maximum amplitude as afunction of distance, according to the equations of Liu et al. [5].

Figure 13. Predicted total number of velocity cycles as a function of source-to-site distance, accordingto the equations of Lu et al. [49].

di�erent travel paths and wave velocities (Figures 12 and 13). Manfredi et al. [50, 51] also�nd that the number of equivalent cycles, de�ned using a structural measure based on theaccumulated hysteretic energy normalized to the maximum response displacement, increases



with distance by a factor of about 1.3 up to a distance comparable with the length of therupture and a further factor of about 1.1 over greater distances.All of the aforementioned studies employ relative de�nitions of the number of cycles, and

�nd that the number of e�ective cycles increases with distance. However, de�nitions basedon absolute measures, where the same absolute reference level is employed for all groundmotions, would be expected to result in a decrease in the number of cycles with increasingdistance.

3.3. Rupture directivity

The duration of strong-ground motions is in�uenced by the rupture duration, which is shorterfor earthquakes with bi-lateral rupture than for those with uni-lateral rupture [16]. However, itis the direction in which the rupture propagates that is of importance to the number of cyclesof ground motion. If the rupture propagates towards the site, waves generated on di�erentparts of the fault can arrive together leading to constructive interference resulting in a fewlarge-amplitude cycles of ground motion. If the rupture propagates away from the site thenthe ground motions would be expected to contain many cycles of lower amplitude as theseismic waves arrive out of phase [5, 52].

3.4. Site conditions

The regression analysis conducted by Liu et al. [5] shows that motions at sites classi�ed assoil have up to about 10% more cycles of motion than at those classi�ed as rock, a smalldi�erence also suggested by a number of studies reporting recordings from downhole arraysand site response analyses [4, 53, 54]. These show that the soil can change the amplitude ofthe motions, but the number of cycles remains approximately the same with only the high-frequency cycles �ltered out by increased soil damping. An exception to this is sites locatedin sedimentary basins: body waves are converted to surface waves at basin edges, which thenpropagate across the basin [55]. Soil damping reduces the number of high-frequency cyclesin the motion leaving the large amplitude, longer period basin waves to dominate the motion.The �ltering of high-frequency waves at such sites would be expected to reduce the number ofe�ective cycles measured by a relative counting de�nition, whilst the larger amplitude surfacewaves increase absolute measures of e�ective cycles.

4. NUMBER OF CYCLES AND DURATION

For most engineering applications, it is likely that the number of e�ective cycles of motionis a physically more meaningful parameter than the duration of the strong shaking. However,duration is more widely used and many more predictive equations have been derived forduration than for the number of cycles. The use of duration as an indirect estimate of thenumber of cycles of motion assumes that the two parameters are strongly correlated, andhere some simple analyses are performed to explore the validity of such an assumption. Forthis exploratory analysis, the databank of almost 500 strong-motion accelerograms employedby Bommer et al. [56] for investigating the in�uence of duration on damage in masonrystructures is used.



Figure 14. Correlations between di�erent cycle-counting de�nitions.

The �rst step, in order to reduce the number of analyses of correlation between cycle countsand duration measures, was to examine the degree of correlation between di�erent measuresof cycle counts. For this, three measures were used:

• Peak counting, including non-zero crossing peaks• Peak counting, excluding non-zero crossing peaks• Rain�ow cycle counting, with ranges converted to amplitudes

All of the measures were converted to equivalent numbers of cycles relative to the maximumamplitude of the motion using an exponent c of 2.0 in Equation (10). The correlations betweenthe number of cycles obtained by rain�ow counting with each of the two peak countingmeasures are shown in Figure 14.The three measures are seen to be well correlated and in particular the number of cycles

from rain�ow counting and from peak counting including non-zero crossing peaks are, onaverage, almost equivalent. The �gure also serves to highlight the di�erences that result fromincluding or excluding non-zero crossing peaks. Since there is a reasonably good correlation inboth cases, for this exploratory analysis it is assumed that any one of the three measures canbe adopted to represent the e�ective number of cycles. Since advantages have been identi�edin including non-zero crossings in peak counting, the number of cycles from rain�ow counting,which is the method the authors of this paper recommend for general applications, is adoptedherein. Before analysing the correlation of cycle counts with duration, it is worth noting that



the correlations shown in Figure 14 should not be taken to imply that the use of di�erentcycle-counting measures yields consistent results: were the numbers of cycles to be determinedusing de�nitions based on di�erent values of absolute thresholds, very di�erent results wouldbe obtained.For the same suite of accelerograms, the durations are calculated for six di�erent de�nitions;

for the exact de�nitions of these duration measures the reader is referred to Bommer andMart��nez-Pereira [16]. Figure 15 shows the number of cycles calculated using the rain�ow-counting method against the values of duration obtained using each of the de�nitions. Lineartrend lines, constrained to pass through the origin, are �tted to each duration and cycle countpair. The coe�cients of correlation are consistently low, as indicated on the plots. The poorestcorrelations are obtained using the two duration de�nitions based on absolute thresholds ofmotion: the bracketed duration for a threshold of 0:1g and the e�ective duration proposedby Bommer and Mart��nez-Pereira [16]. For both of these durations, it can be seen that alarge number of records yield very small values, including zeroes, for motions that do notsigni�cantly exceed the thresholds used to measure the duration. These two cases illustratethat the cycle-counting technique employed here does not take account of absolute amplitudeof the motion, simply determining the equivalent number of constant-amplitude cycles of therecorded motion. Using a relative threshold of 10% of the peak ground acceleration (PGA),in place of an absolute level of 0:1g, results in a much-improved correlation, although thereare some data points that are very far from the best �t line.The results obtained using signi�cant duration, which is the interval between the times at

which certain proportions of the total Arias intensity are reached, show correlations compa-rable with those obtained using the bracketed duration at 10% PGA. However, it is notablehow using the 5–75% limits, employed by Somerville et al. [52] in place of the limits of5–95% originally proposed by Trifunac and Brady [57], improves the correlation. The uni-form duration, measured as the total time during which the motion is above the threshold of10% PGA, is the measure providing the best correlation with the cycle count.The better correlation with uniform duration is not a surprising result since this duration

measure e�ectively identi�es the strong cycles of motion and neglects the rest of the record.However, even for this duration measure the correlation can be considered rather poor, sug-gesting that the assumption of duration being an indirect measure of the e�ective number ofcycles may warrant re-evaluation.That there is large scatter in the plots shown in Figure 15 is perhaps not surprising. For

harmonic motions, regardless of the de�nitions used, the number of cycles would be directlyproportional to the duration and frequency: short-period motions would result in steep trendlines in these plots, long-period motions in trend lines with low gradients. An insight intopossible in�uences on the ratio of number of cycles to duration can be obtained using medianvalues obtained from the equation of Liu et al. [5] for number of cycles and the equationof Abrahamson and Silva [58] for duration (Figure 16); the coe�cients of the latter equationare reported by Stewart et al. [59]. Some of the features of the curves in Figure 16 maybe artefacts of the functional forms or the data sets employed in the derivation of theseequations, but general trends can also be noted for which there are clear physical explanations.Firstly, the ratios decrease with distance, a result of the faster attenuation of high-frequencymotion (reducing the number of cycles) and the separation of the cycles (increasing theduration) due to di�erent wave propagation velocities. Secondly, rock site motions producehigher ratios than soil sites: the durations on the latter are marginally longer but due to the



Figure 15. Correlations between number of e�ective cycles and di�erent duration de�nitions, lineartrend lines constrained to pass through the origin shown by dashed lines.



Figure 16. Ratios of e�ective number of cycles to signi�cant duration (5–75% of Arias intensity)calculated using median values from the equations of Liu et al. [5] and Abrahamson and Silva [58].

higher frequency of motions at rock sites, the number of cycles may be signi�cantly higherthan at soil sites. Finally, the ratio of cycles-to-duration decreases with magnitude, presumablythe result of the greater proportion of long-period waves in the ground motions produced bylarger earthquakes: long-period cycles, by de�nition, contribute more to the duration of themotion. This is consistent with the work of Bondi et al. [60] who also �nd that the number ofe�ective cycles is dependent on the frequency content of the motion. The curves in Figure 16also indicate, however, that the in�uences of magnitude, distance and site classi�cation areinter-dependent.

5. DISCUSSION

The e�ective number of cycles of ground motion is widely used as an indicator of the potentialof earthquake shaking to cause damage, particularly in systems with degrading strength andsti�ness characteristics. The most widespread application of the number of cycles is in theassessment of liquefaction potential. As with strong-motion duration, discussions regarding theimportance and in�uence of the number of cycles are hampered by the variety of approachesthat have been proposed for its determination from accelerograms.The �rst stage in calculating the e�ective number of cycles of motion is the selection

of the process used to convert many irregular amplitude cycles to a number of equivalentcycles of constant amplitude, and this is the �rst point of divergence amongst published ap-proaches. The di�erences arise from the weighting assigned to the lower-amplitude cyclesrelative to the largest cycles in the ground motion. The conversion process can be relative(where the cycles are normalized to the amplitude of the maximum cycle) or absolute (where



they are assessed relative to an absolute amplitude measure). Many de�nitions of cycle count-ing used in engineering seismology are based on relative amplitudes, so that they can be usedin conjunction with other measures of amplitude. However, absolute measures of cyclicdamage are also employed as these can be directly related to the cyclic demand of the groundmotion [10].Some techniques do not count the number of cycles directly but estimate its value using ran-

dom vibration theory. Amongst those de�nitions that determine the number of cycles directlyfrom the accelerogram it is possible to de�ne three generic categories depending on whetherthe quantity measured is related to the number of peaks, the number of times a speci�edlevel is crossed, or the number of ranges between peaks and troughs. Within the category ofpeak-counting de�nitions, a critical di�erence is whether or not non-zero crossing peaks arecounted. This is particularly important for motions from large-magnitude earthquakes whichhave broad-banded frequency content and contain a signi�cant number of non-zero crossingpeaks.Some de�nitions of the numbers of cycles are based on structural response, essentially

applying one of the three generic categories described above to the response of an elasticor inelastic SDOF system. Clearly for a speci�c structural application, these de�nitions arelikely to produce the most meaningful estimates of the number of e�ective cycles, but itis cumbersome to develop predictive equations for a wide range of structural models (withdi�erent natural period and, for inelastic response, hysteretic models). The choice of methodfor counting the cycles in the structural response is important: it has been shown that if theperiod of the SDOF system is lower than the dominant period of the ground motion, theresponse �lters out broad-banded response. If, however, the structure has a period longer thanthe dominant period of the ground motion, as would be the case for tall buildings and forbridges, the structural response is also broad-banded. For these situations, it is particularlyimportant to use a cycle-counting de�nition, such as the rain�ow-counting de�nition, whichis designed to count the number of cycles in a broad-banded signal.There are relatively few published equations for predicting the number of cycles of earth-

quake motion, and amongst these there is appreciable disagreement in the results obtained, inpart due to di�erent approaches to counting the number of cycles from each accelerogram.Nonetheless, the existing equations serve to demonstrate the strong in�uence of magnitude onthe number of cycles, and the lesser dependence on distance and site classi�cations. Althoughthe empirical adjustments for near-source forward directivity e�ects proposed by Somervilleet al. [52] are speci�cally for the resulting reduction of duration, forward directivity can alsoclearly reduce the number of cycles of motion.

6. CONCLUSIONS

De�nitions of strong-motion duration, which e�ectively serve as a surrogate for the number ofcycles of motion, are more widely used and a larger number of predictive equations have beendeveloped for this parameter than for the number of cycles, although the number is very smallwhen compared to the number of equations available for the prediction of peak ground-motionparameters and response spectral ordinates. The validity of the assumption that duration maybe interpreted as an indirect estimate of the number of cycles has been explored throughcorrelations between cycle counts and di�erent de�nitions of duration. The results show that



the correlations are generally weak, the best results being obtained with the uniform duration,for which no predictive equations have been developed. Theory suggests that the correlationbetween duration and cycle counts should depend on the dominant frequency of the motionand preliminary results indicate that the ratio of cycles to duration varies with magnitude,distance and site classi�cation.The universal adoption of a single de�nition for the e�ective number of cycles of motion is

unlikely and possibly undesirable, since di�erent de�nitions will be better suited to di�erentapplications. Amongst the generic de�nitions of the number of cycles of motion, the rain�ow-counting de�nition has the merit of including all peaks in the time series without puttingundue emphasis on non-zero crossing peaks, and in the absence of criteria for selecting adi�erent de�nition, the authors of this paper recommend that this technique be employedto determine the number of cycles from accelerograms. In view of the large number ofde�nitions encountered in the literature, it is important that all references to the number ofcycles should—as for strong-motion durations—explicitly state the de�nition employed toavoid ambiguity and to allow meaningful comparisons and interpretations.

ACKNOWLEDGEMENTS

We are deeply indebted to two anonymous reviewers for their very thorough and insightful commentson the �rst version of this manuscript, which were instrumental in removing a number of errors andambiguities as well as contributing very directly to making major improvements to the paper. We wouldlike to thank Dr Russell Green for interesting discussion on the subject and for bringing to our attentionseveral publications referenced in this paper. The authors would also like to thank Dr John Douglasfor reviewing an early version of the paper. The work of the �rst author is supported by a doctoraltraining grant from the EPSRC.

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the effective number of cycles of earthquake ground motion

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