Simplified Method to Include Cumulative Damage in the Seismic Response of Single-Degree-of-Freedom...

Simplified Method to Include Cumulative Damage in theSeismic Response of Single-Degree-of-Freedom Systems

Edoardo Cosenza1; Gaetano Manfredi2; and Maria Polese3

Abstract: In this paper a consistent method including damage criteria in the seismic response of single-degree-of-freedom systems isproposed. The method allows the determination of suitably modified strength or displacement inelastic spectra through the introduction ofan equivalent damage factor pdam that accounts for earthquake damage potential; analogously capacity spectra could be obtained. Threetypes of damage indices are considered �Park and Ang index DP&A, energy index DE, and low-cycle fatigue index DF� and derivation ofpdam is pursued for all these cases. Moreover approximate simplified expressions in the function of Cosenza and Manfredi seismologicalID index, which accounts for cyclic damage potential of an earthquake, are also proposed. In this way damage capacity spectra areobtained to improve the seismic assessment of existing structures including damage effect.

DOI: 10.1061/�ASCE�0733-9399�2009�135:10�1081�

CE Database subject headings: Ductility; Fatigue; Damage; Seismic effects.

Introduction

Performance based engineering has become a well establishedconcept in all recent design and assessment provisions in seismicregions �Structural Engineers Association of California �SEAOC�1995; Comitée Européen de Normalisation �CEN� 2003; Fédéra-tion international du béton �fib� 2003a,b; ASCE 2000�; it is ac-cepted that structures may experience significant plastic behaviorunder severe ground motions, while for more frequent moderateintensity earthquakes the systems should perform at a serviceabil-ity limit state. In this framework, design/assessment methodsshould account for the different performance objectives; the mostwidespread approach adopts a multistep methodology. The firstphase allows determining the global seismic demand �base shear,target displacement� for the single-degree-of-freedom �SDOF�system equivalent to the real structure; to this aim, suitably modi-fied strength or displacement or capacity spectra are adopted.Next, analyzing the performance of complete multidegree of free-dom structural model, it is possible to determine the presumedactions on the structural members and to design/check the struc-ture for a chosen performance objective.

This paper deals with the problem of spectral reduction factorsand the proposed method is applicable to determine the seismicactions on SDOF systems, i.e., for global predesign/assessmentof the structures. In a force based approach, the smoothed inelas-tic strength response spectra are obtained through the use of a

1Professor, Dept. of Structural Engineering, via Claudio 21, 80125Naples, Italy. E-mail: [email protected]

2Professor, Dept. of Structural Engineering, via Claudio 21, 80125Naples, Italy. E-mail: [email protected]

3Assistant Professor, Dept. of Structural Engineering, via Claudio 21,80125 Naples, Italy �corresponding author�. E-mail: [email protected]

Note. This manuscript was submitted on October 17, 2006; approvedon June 4, 2009; published online on September 15, 2009. Discussionperiod open until March 1, 2010; separate discussions must be submittedfor individual papers. This paper is part of the Journal of EngineeringMechanics, Vol. 135, No. 10, October 1, 2009. ©ASCE, ISSN 0733-
9399/2009/10-1081–1088/$25.00.
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reduction factor R� that depends on the available displacementductility �.

In the displacement-based approach �Calvi and Kingsley1995�, the evaluation is based on the use of elastic deformationspectra derived for suitable values of equivalent damping; how-ever, as shown in �Chopra and Goel 2001�, the use of highlydamped spectra can be nonconservative and more reliable resultsare obtained if inelastic displacement spectra, obtained with theaid of displacement modification factor RD of the elastic spectraaccounting for the nonlinear response, are used.

These methods are introduced in current seismic provisionsand are deemed to ensure the prescribed performance objectives.However, they suffer a conceptual limitation; indeed, the inelasticstrength spectra and the inelastic displacement spectra are signifi-cant only on the maximum demand of ductility and/or displace-ment and they do not provide any information about the effect ofthe cumulative damage in the long duration earthquakes. In thecapacity spectrum method �Applied Technology Council �ATC�1996� it is possible considering the effects of system dissipativecharacteristics through a kappa coefficient that influences equiva-lent damping. However, the number of plastic cycles is a morerobust indicator of the destructive capacity of the shaking withrespect to the earthquake duration; Hancock and Bommer �2005�recently made an extensive review of the existing definitions ofthe number of plastic cycles.

Cyclic damage plays a significant role in varying the seismicresistance of the systems and it should be accounted for if seismicbehavior has to be evaluated �Iervolino et al. 2006�. The idea ofintroducing cyclic damage potential in the usual spectral evalua-tion procedure is not new �Krawinkler and Nassar 1992; Cosenzaet al. 1993; Fajfar 1992; Cosenza and Manfredi 1996�, and thereare also very recent proposals �Malhotra 2002; Kunnath and Chai2004; Chai 2005� for the development of cyclic demand spectrathat can integrate classic amplitude spectrum for a more completedefinition of the seismic action. Teran-Gilmore and Jirsa �2005�showed that available ductility decreases with the increase ofplastic energy demand, and underline the importance of the trans-parent use of damage functional in an evaluation procedure that
accounts for structural damage; while Lu and Wei �2008� incor-
L OF ENGINEERING MECHANICS © ASCE / OCTOBER 2009 / 1081

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porated performance considerations in the definition of a damagebased strength reduction factor. Bozorgnia and Bertero �2003�proposed the adoption of damage spectra based on a combinationof normalized hysteretic energy and deformation ductility and,through the development of suitably derived attenuation laws,prefigure the use of near real-time contour maps for damage spec-tra that can provide useful information on the spatial distributionof the damage potential of ground motions.

In this paper a consistent method to account for damage in theseismic design or assessment is proposed. Aiming to introducecyclic damage criteria in a simplified procedure, damage con-trolled strength response spectra are defined; this approach gen-eralizes the inelastic spectra methods, introducing the definitionof “equivalent ductility” and allowing the use of the well-knownconcept of structural ductility, modified on the basis of some fun-damental data regarding the earthquake and the structure�Cosenza and Manfredi 1997�. In the same way, the displacement-based method can be modified to take into account the damagephenomena; damage controlled inelastic displacement spectra areintroduced, that represents the modified demand of displacementdue to the cyclic damage.

Modified strengths �ordinates Sa� and displacements �abscissasSd� allow also defining damage controlled inelastic capacityspectra, whose coordinates are joined at the given limit state�Sd=Sa /�2, where � is the angular frequency�. The explicit for-mulation of the method in the function of damage indexes Di

has the advantage of linking directly the performance objective tothe accepted damage level for the system response. Appropriatedamage factors pdam, representing the reduction of available duc-tility due to different sources of structural damage, are derived.Structural damage is introduced by means of the well-knowndamage indexes of Park and Ang �Park and Ang 1985�, hystereticenergy, and plastic cycle fatigue; correspondingly three differentdamage factors pdam, are defined. Finally, an approximate formu-lation for pdam, based on the assessment of the number of plasticcycles and on their average value �Manfredi 2001; Manfredi et al.2003�, is proposed for both far field and near source conditions.

Inelastic Capacity Spectra

The seismic-resistant assessment of structures that is not sensitiveto cumulative damage phenomena can be performed using thewell-known criterion of ductility

D� = �� − 1�/��1,alw − 1� = �xmax/�x1,alw � 1 �1�

where D�=ductility index; �xmax=xmax−xy =maximum plastic ex-cursion �xmax maximum displacement and xy displacement at theelastic limit�; and �x1,alw=x1,alw−xy =ultimate plastic excursionthe system can accommodate when it is subjected to a state ofincreasing lateral deformation. This approach links the structuraldamage level with the maximum demand of displacement and isappropriate for well-detailed structures with a stable hystereticbehavior. Accordingly, it is possible to evaluate the maximumallowable plastic displacement �xmax for a prefixed level of ac-cepted damage �D�=1 represents the collapse, while values ofD��1 represent lower damage levels related to required perfor-mances levels�

�xmax = �x1,alw · D� �2�

In terms of allowable ductility, if the structure shows ductility formonotonic loading at the collapse �1,alw, it is possible to define a
reduced ductility for lower level of accepted damage
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�dam,alw = �xy + �xmax�/xy

= 1 + ��x1,alw/xy� · D� = 1 + ��1,alw − 1� · D� �3�

At the collapse the ductility damage index D� assumes a valueequal to 1 and the reduced allowable ductility is equal to themaximum allowable ductility for monotonic loading

�dam,alw = �1,alw �4�

obtaining the standard design method based on the ductility col-lapse criteria.

The introduction of the reduction factor R�, representing theratio between the elastic and the inelastic spectrum for fixed val-ues of elastic period and inelastic ductility, allows the definitionof the ductility inelastic spectra

R� = �msSa,e/Fy� �5�

where Fy =inelastic strength; Sa,e=elastic acceleration spectra;and ms=structural mass.

The introduction of a function that allows determining R�,given allowable ductility �, can avoid performing nonlinearanalysis. The definition of reliable expressions of R� is still anopen question �Miranda and Bertero 1994; Krawinkler and Nassar1992� and different proposals exist.

In this paper a simple relation for R�, determined through astatistical analysis �Cosenza and Manfredi 1997� and suitable inthe case of rock and low depth alluvium grounds, is used

R� = 1 + 1.5 · �3/4 · �� − 1�4/5 �6�

with �=T /T1 for T�T1, and �=1 for T�T1 �T=elastic period, T1

initial of medium periods range in Newmark-Hall spectral repre-sentation �Newmark and Hall 1982��.

This formulation provides a useful tool because of its indepen-dence from the period in the field of the medium and long periodrange, and provides results similar in average with the results ofmore complex formulations �Miranda and Bertero 1994�. Themean and standard errors of the estimate obtained with Eqs. �6�are 0.085 and 0.4, respectively.

In the displacement-based approach it is required the definitionof inelastic displacement spectra Sd,i. The latter can be relatedwith the elastic spectra Sd,e by Eq. �7� that gives a first orderapproximation of Sd,i �Miranda 2001�:

Sd,i = RD · Sd,e � ��/R�� · Sd,e �7�

where RD=modifying factor taking into account the nonlinear be-havior �RD�� /R��. Accounting for different performance levels

Sd,dam = RD,dam · Sd,e � ��dam,alw/R�,dam� · Sd,e �8�

where R�,dam is evaluated with expression �6� considering allow-able ductility �dam,alw.

The assessment of seismic performance of a SDOF sys-tem may be alternatively pursued evaluating reduced allowabledisplacement

xmax,dam = xy · �dam,alw

and comparing this modified displacement capacity with inelasticdemand Sd,i.

By combining the “reduced” inelastic strength and displace-ment spectra the capacity spectra are obtained. In Fig. 1�a�the reduced inelastic capacity spectra at the collapse �D�=1� ob-tained with the proposed expression of R� �6� are represented forEurocode 8 �Comitée Européen de Normalisation �CEN� 2003�—
Class B subsoil �ag=0.35 g� and for �1,alw=2 to �1,alw=6. Elastic
9


spectrum has Sa,e as abscissa and the corresponding displacementSd,e=Sa,e /�2 as ordinate, while the inelastic spectra representyield strengths and the corresponding limit displacements giventhe allowable ductilities. Analogously, in Fig. 1�b� inelastic re-duced capacity spectra are represented for different performanceobjectives �i.e., calculating �dam,alw in Eq. �3� for D�=0.2 and 0.5and considering this reduced value for the evaluation of spectralreduction factor R�� and for �1,alw=4.

Allowable Ductility in Presence of Cyclic Damage

The use of ductility as collapse criterion is very common in theseismic design, although it is acceptable only when the crisis isdue to a limit of displacement. For this reason it seems useful toextend this concept to the cases in which the cyclic damage hasa significant influence on the structural collapse, introducing thedefinition of equivalent ductility already proposed by Fajfar�1992�. Hence, it is fundamental to define effective criteria ofcollapse for cyclic actions in the development of a consistentseismic evaluation approach. In particular, it is necessary to intro-duce appropriate damage indices that, in their normalized form,assume the value 0 in the absence of plastic damage and the value1 at the collapse �Cosenza et al. 1993�.

Apart from the already mentioned index of �cinematic or cy-clic� ductility, the most used damage indices are the Park and Angdamage index �Park and Ang 1985; Park et al. 1987�, the index ofhysteretic energy, and low-cycle fatigue index. In the following,those damage indexes will be introduced and relative damagefactor pdam for reduction of available ductility will be found. Elas-toplastic behavior is assumed; therefore, the formulation proposedis applicable only to systems that show negligible or low strain

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30Sd (cm)

S a(g)

Elastic spectrum

μ1,alw=6

a)

μ1,alw=2

μ1,alw=4

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30Sd (cm)

S a(g)

Elastic spectrum b)

D4=0.2

D4=0.5

D4=1.0

Fig. 1. Capacity spectra for Eurocode 8—Class B subsoil �ag

=0.35 g�: �a� at the collapse �D�=1�; �b� for different performancelevels ��1,alw=4�

hardening in the nonlinear field. The influence of system’s strain

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hardening and/or degrading patches can be included �FEMA 306regulations �ATC 1998� already proposed some practical expres-sions to take into consideration the effect of degrading behavior�and have to be specifically addressed in further studies.

Park and Ang Damage Approach

The analytical expression of the Park and Ang index DP&A is

DP&A =xmax

x1,alw+ �

EH

Fyx1,alw�9�

where x1,alw=ultimate displacement; xmax=maximum displace-ment; EH=hysteretic energy; and Fy =elastic strength. � is an em-pirical constant that depends on structural characteristics. Analternative expression of Eq. �9� can be considered

DP&A =1

�1,alw�1 + ��1,alw − 1�� xmax

�x1,alw+ �

EH

Fy�x1,alw��

�10�

with symbols already defined.In the case of elastoplastic behavior, a different form of the

index could be considered

DP&A =1

�1,alw�1 + ��1,alw − 1�

�xmax

�x1,alw�1 + � + �

i=1

n−1

�i��

�11�

where the dimensionless ratio �i�=�xi /�xmax is introduced.

Hence, if the effect of the cumulative damage �expressed bythe second term in Eq. �9�� is considered, only a fraction of theultimate plastic excursion �x1,alw may be used

�xmax =�DP&A�1,alw−1��x1,alw

��1,alw−1��1 + � + � i=1

n−1

�i��

=�DP&A�1,alw−1�

��1,alw−1�· �x1,alw · pdam,P&A �12�

with

pdam,P&A = �1 + � + ��i=1

n−1

�i��−1

� 1

The introduction of the damage factor pdam,P&A allows toconsider the damage concept in the seismic performance assess-ment in an easy manner. In fact, in terms of allowable elementcharacteristics, if the element shows an allowable ductility �1,alw

under monotonic action, it is possible to define, by means ofpdam,P&A, a reduced equivalent ductility �dam,P&A that the structurecan supply under a cyclic displacement history

�dam,P&A = 1 + �DP&A · �1,alw − 1� · pdam,P&A �13�

Therefore, for example, the common strength based seismicevaluation can be performed considering a reduced allowableductility capacity.

Introducing the number of plastic cycles n and their averagevalue m �defined as the mean value of the amplitudes of then−1 cycles smaller than the plastic cycle of maximum amplitude,
scaled to �xmax�


m = �n − 1�−1 · i=1

n−1

�i� �14�

it can be shown that apart from the Park and Ang parameter �, theequivalence factor pdam,P&A depends on the cyclic characteristicsof the ground motion and a simplified expression for its formula-tion can be found. In fact, from Eqs. �12�–�14�, it follows:

pdam,P&A = �1 + ��1 + �n − 1�m��−1 �15�

and therefore the assessment of pdam,P&A is related to the evalua-tion of n and m.

To obtain a reliable assessment of n and m, an extensive sta-tistical study has been conducted both in the far field �f f� and inthe near fault �nf� �Manfredi 2001; Manfredi et al. 2003�. Resultsallowed numerically determining n and m; for equivalent damp-ing =5% the expressions are

nnf = 1 + 1.25 · �−2/3 · �R� − 1�1/2 · Id4/5 �16�

mnf = 0.19 · �1/6 · �R� − 1�−1/10 · Id1/5 �17�

nff = 1 + 1.05 · �−2/3 · �R� − 1�4/5 · Id4/5 �18�

mff = 0.17 · �1/6 · �R� − 1�−1/5 · Id1/5 �19�

where Id=dimensionless index defined as

Id = 0

tE

a2�t�dt · �PGA · PGV�−1 �20�

�PGA=peak ground acceleration; PGV=peak ground velocity;a�t�=ground acceleration time history�.

The mean and standard errors of the estimate �versus themean� obtained with Eqs. �16� and �17� are 0.46 and 0.53 forn, and 0.36 and 0.50 for m, respectively. The same values forEqs. �18� and �19� are 0.11 and 0.46 for n, and 0.15 and 0.47for m, respectively. The errors are slightly greater in the nearfault, probably due to the major complexity of the phenomenonnear the source and to the difficulty describing it.

By introducing Eqs. �16�–�19� in Eq. �15� it is possible toobtain the following relation for pdam,P&A, for far field and nearfault regions, respectively ��−0.05�1�:

pdam,P&A,ff = �1 + � · �1 + 0.23 · Id�� − 1�0.5��−1 �21�

pdam,P&A,nf = �1 + � · �1 + 0.28�−0.2Id�� − 1�0.32��−1 �22�

Energy Approach

In general terms the energy criterion for damage evaluationrequires

EH � EH,1,alw

where EH=hysteretic energy, and EH,1,alw=maximum hystereticenergy that can be dissipated in a single cycle test.

Introducing the hysteretic ductility �E

�E = 1 + EH/�Fyxy� �23�

energy damage index may be expressed as

DE = ��E − 1�/��E,1alw − 1� �24�

In the case of elastoplastic behavior and recalling the definition of
average amplitude of plastic cycles �Eq. �14��


�E,1alw − 1 = �1alw − 1 = �x1alw/xy

�E − 1 = �Fyi=1

n

�xi · �Fyxy�−1 = ��xmax�1 + i=1

n−1

�i�� · xy

−1

= ��xmax�1 + �n − 1� · m�� · xy−1 �25�

It follows:

DE = ��E − 1�/��E,1alw − 1� = ��xmax�1 + �n − 1� · m�� · �x1alw−1

�26�

�xmax = �x1alw · �1 + �n − 1� · m�−1 · DE = �x1alw · pdam,E · DE

�27�

As Eq. �27� shows, the introduction of the energy damage indexleads to an expression of the damage equivalence factor equal to

pdam,E = �1 + �n − 1�m�−1 �28�

By substituting Eqs. �18� and �19� or Eqs. �16� and �17� inEq. �28� ��−0.05�1� an expression of pdam,E in the function of �and ID, both for the far field and near fault conditions, is derived

pdam,E,ff = �1 + 0.23 · ID�� − 1�0.5�−1 �29�

pdam,E,nf = �1 + 0.28�−0.2ID�� − 1�0.32�−1 �30�

By equating Eqs. �15� and �28�, it is possible to obtain the valueof coefficient � of Park and Ang damage index that provides thesame result of energy method

� =�n − 1�m

1 + �n − 1�m=

0.23ID�� − 1�0.5 − 1

0.23ID�� − 1�0.5 �31�

Varying ID between 7 and 20, � varies �for �=4� between 0.73and 0.88, confirming the values suggested in Cosenza et al.�1993�. In terms of allowable structural characteristics, the re-duced equivalent ductility �dam,E for an elastic perfectly plasticsystem is

�dam,E = 1 + ��1,alw − 1� · pdam,E · DE �32�

Low-Cycle Fatigue Approach

The analytical expression of the low-cycle fatigue index DF is

DF = i=1

n � �i − 1

�1,alw − 1b

= i=1

n � �xi

�x1,alwb

�33�

where b=damage parameter. The case DF=0 represents the ab-sence of damage and the case DF=1 the collapse under cyclicactions.

The value assumed by this damage index is defined throughthe only constant b, depending on the structural material and ty-pology, and on the amount of the different plastic displacements,independently from the exact order. Typical values of b, obtainedby experimental data for steel structures �Krawinkler and Zohrei1983� and RC structures �Stephens and Yao 1987�, are 1.5 to 2.0and 1.8, respectively; in the damage analysis, sometimes the con-servative value 1.5 is assumed. For b=1 the index gives the sameweight to each plastic displacement, independently from itsamount; hence, DF coincides with the energy based index DE forthe EPP model. For high values of b �greater than 5� the low-
cycle fatigue index provides results similar to the ductility index.
9


In this case too, considering the effect of the low-cycle fatigue,it is possible to use just a fraction pdam,F�1 of the ultimate plasticexcursion �x1,alw

�xmax = pdam,F · �x1,alw · DF1/b with pdam,F = �1 +

i=1

n−1

�i�b−1/b

�34�

where, again, �i�=�xi /�xmax.

In terms of allowable structural characteristics, the reducedequivalent ductility �dam,F in presence of low-cycle fatigue is

�dam,F = 1 + ��1,alw − 1� · pdam,F · DF1/b �35�

As previously shown for pdam,P&A, it is possible to express pdam,F

in a simpler manner if the statistical characterization of plasticdisplacements is known. More in detail, the optimal distributionof the random variable ��i

��b= ��xi /�xmax�b, with �i�� 0,1�, has

to be defined.In Cosenza and Manfredi �1997� it is shown that the following

simplified expression may be used:

pdam,F = �1 + �n − 1�mb�−1/b �36�

where mb, that represents the mean of the function of randomvariable y=xi

b truncated to 1, may be evaluated with approximateformulation

mb = e−0.45�b−1� · m �37�

In cases characterized by n→1, m→0, or still b→, a value ofpdam,F=1 is obtained by Eq. �35�, therefore the criteria of theplastic fatigue and of the ductility are coincident.

By introducing Eqs. �37�, �18�, and �19� or Eqs. �37�, �16�, and�17� in Eq. �36� the following expression for pdam,F is obtainedboth for the far field and the near fault conditions:

pdam,F,ff = �1 + 0.23 · ID�� − 1�0.5e−0.45�b−1��−1/b �38�

pdam,F,nf = �1 + 0.28�−0.2ID�� − 1�0.32e−0.45�b−1��−1/b �39�

Capacity Spectra Including Damage Criteria

Considering one of the damage criteria introduced it is possible tosuitably modify elastic spectral ordinates to account for inelasticstructural behavior and to obtain a reliable evaluation of the in-elastic force or displacement in relation with the expected groundmotions. Using the definition of equivalent allowable ductility�dam and the related reduction factor Rdam for the different dam-age criteria �Eqs. �13�, �32�, and �35��, the effect of the damagecan be included into the inelastic strength or displacement spec-tra, or equivalently, in the corresponding capacity spectrum.

In the following, the construction of damage spectra based onthe Park and Ang index in the far field condition is described asexample, other methods being analogous.

The damage factor pdam,P&A can be evaluated by Eq. �21� forthe far field case, introducing the structural damageability bymeans of � and the earthquake cyclic effect by means of theindex ID.

By introducing into Eq. �6� the values of the allowable equiva-lent ductility �dam,P&A reduced by the effect of the cyclic damage�Eq. �13��, the expression of the damage reduction factor Rdam,P&A

becomes

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Rdam,P&A = 1 + 1.5 · �3/4 · ��dam,P&A − 1�4/5 �40�

In this way the strength spectra can be obtained at the collapseintroducing in Eq. �13� a value of the damage index equal to 1and at different performance levels assuming the value of thedamage index is lower than 1, that assure limitations of the struc-tural and nonstructural damage

Sa,P&A = Sa,e/Rdam,P&A �41�

where Sa,P&A and Sa,e=reduced inelastic and the elastic strengthspectra, respectively.

Similarly, inelastic displacement spectrum Sd,P&A is evaluatedas

Sd,P&A = ��dam,P&A/Rdam,P&A� · Sd,e = RD,P&A · Sd,e

where the elastic displacement spectrum Sd,e is modified by thefactor RD,P&A accounting for cyclic damage.

Capacity spectra �Sd,P&A,Sa,P&A� are easily obtained withthe simultaneous reduction of elastic displacement and strengthspectra.

As an example, the capacity spectra for Eurocode 8—Class Bsubsoil �ag=0.35 g� at the collapse limit state �DP&A=1� andfor allowable ductilities �1,alw ranging from 2 to 6 are shown inFig. 2�a�; Fig. 2�b� shows capacity spectra for �1,alw=4 and fordifferent performance objectives �DP&A=0.2 and 0.5�. ParameterId, that represents the cyclic content of an earthquake, is assumedto be Id=35 �high cyclic content; Id for 1985 Chile earthquake,Llolleo, was 35.8�. By comparing reduced capacity spectra atthe collapse and for intermediate limit states for ductility criterion�Figs. 1�a and b�� with those obtained with P&A criterion�Figs. 2�a and b�� it can be easily noticed that the latter allows a

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30Sd (cm)

S a(g)

Elastic spectrum

μ1,alw=2

μ1,alw=4

μ1,alw=6

DP&A=1

a)

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30Sd (cm)

S a(g)

Elastic spectrumb)

μ1,alw=4DP&A=0.5 DP&A=0.2

DP&A=1.0


=0.35 g� including Park and Ang damage effect ��=0.15� and as-suming high cyclic content �ID=35� �a� at the collapse �DP&A=1�; �b�for different performance levels ��1,alw=4�

lower spectral reduction because it accounts for cumulative dam-



age that is neglected by the ductility criterion; the P&A criterionis more reliable for structures that may be affected by low-cyclefatigue phenomenon.

However, when the cyclic content of an earthquake is negli-gible, it becomes less important to consider explicitly its influenceon damage spectra; as it can be seen in Figs. 3�a and b�, where thesame representation of Figs. 2�a and b� is shown, but consideringthe low cyclic content �Id=5� that may be representative of animpulsive type earthquake.

Figs. 4�a and b� show the capacity spectra obtained with thelow-cycle fatigue index �Id=35 and b=1.8� for the same elasticinput spectrum as for Figs. 1�a and b�, 2�a and b�, and 3�a and b�.

Figs. 5�a and b� compare the two criteria at the collapse �dam-age indices equal to 1� and for intermediate and service limitstates �damage indices equal to 0.5 and 0.2� with �1,alw=4; highcyclic content �Id=35� is considered and for the case of DP&A

=0.2 a lower bound of �dam,P&A=1 is assumed. It can be notedthat generally low-cycle fatigue criterion is more conservative atcollapse; however, for noncollapse limit states �DF�1� the low-cycle fatigue criterion is less conservative with respect to Parkand Ang because it gives lower weight to the smaller cycles withrespect to the larger ones.

In Fig. 5�a� the comparison of the proposed approach withother existing proposals is also shown. In particular the equivalentductility for the Park and Ang approach was determined as sug-gested by Fajfar �1992�; the medium to high periods range, whereequal displacement hypothesis apply, is considered. In Fajfar�1992� the reduction of ductility due to low-cycle fatigue in thePark and Ang model is explicitly controlled by the parameter � ofthe P&A damage index and by a factor � that mainly depends onhysteretic energy demand. In particular, high cyclic content earth-

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30Sd (cm)

S a(g)

Elastic spectrum

μ1,alw=4

μ1,alw=2

μ1,alw=6

DP&A=1

a)

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30Sd (cm)

S a(g)

Elastic spectrumb)

μ1,alw=4

DP&A=0.5

DP&A=0.2

DP&A=1.0


=0.35 g� including Park and Ang damage effect ��=0.15� and as-suming medium cyclic content �ID=5� �a� at the collapse �DP&A

=1�; �b� for different performance levels ��1,alw=4�

quakes are characterized by higher � values and vice versa �� was



found to be approximately 1.3 in medium and high period rangesfor the Chile 1985 earthquake, and this value was assumed as therepresentative of high cyclic content�. As it can be seen, the pro-posed method gives results that are comparable with other exist-ing approaches.

To test the effectiveness of the method, a number of SDOFsystems were designed for collapse and intermediate performancelevels �DP&A=1 and 0.5� according to the proposed formula-tion for Park and Ang index and with reference to selected earth-quake spectra. In particular, two far field earthquakes with highand medium-high cyclic content were chosen ��a� Chile 1985,Llolleo—LLN record with Id=35.8 and �b� Kobe 1985,Okayamon—OKA record with Id=19.9� and a near fault one��c� Kobe 1985, University record—KBU with Id=4.85�, seeFig. 6�d�. The Park and Ang damage effectively occurring in the

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30Sd (cm)

S a(g)

Elastic spectrum

μ1,alw=4

μ1,alw=2

μ1,alw=6

DF=1

a)

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30Sd (cm)

S a(g)

Elastic spectrumb)

μ1,alw=4

DF=0.5

DF=0.2

DF=1.0


=0.35 g� including low-cycle fatigue damage effect �b=1.8� and as-suming high cyclic content �ID=35� �a� at the collapse �DF=1�; �b�for different performance levels ��1,alw=4�

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30Sd (cm)

S a(g)

DF=1DP&A=1

μ=2μ=4μ=6

a)

DP&A,Faj=1

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30Sd (cm)

S a(g)

μ1,alw=4

b)

DFDP&A

Di=0.2Di=0.5Di=1.0

Fig. 5. Comparison of capacity spectra for low-cycle fatigue �b=1.8� and Park and Ang ��=0.15� criteria for Eurocode 8—Class Bsubsoil �ag=0.35 g� and assuming high cyclic content �ID=35� �a� atthe collapse �DF=1, DP&A=1, and DP&A,Faj=1�; �b� for different per-formance levels ��1,alw=4�

9


systems was then determined with nonlinear time-history integra-tion of the elastoplastic oscillators. Figs. 6�a–c� show the DP&A

damage spectra for the three considered earthquakes and the twolevels of seismic design. The resulting damage indices have meanvalues of 0.93, 1.2, and 1.02 and coefficient of variation �CoV�0.40, 0.43, and 0.33 for �a�, �b�, and �c� cases, respectively, in thecase of the design based on DP&A=1. For intermediate perfor-mance level �design for DP&A=0.5� the damage indices havemean values of 0.45, 0.61, and 0.55 and CoV 0.19, 0.45, and 0.33for �a�, �b�, and �c� cases, respectively. It is possible to observethat the proposed simplified method provides average resultscomparable to an explicit nonlinear analysis without the compu-tational effort.

In relation to the scattering of the proposed formulation ofdamage indices, note that, analogous to the methods for the de-termination of constant ductility spectra implemented in recentinternational codes, e.g., Eurocode 8 �Comitée Européen de Nor-malisation �CEN� 2003�, the SDOF systems are designed consid-ering the mean value of the damage based reduction factor.Nevertheless, in the framework of classical approaches of thestructural safety, other representative fractiles of this variablecould be introduced in design to increase the accepted safety ofstructures.

Conclusions

In this paper, a unified method able to introduce the damage effect

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.T (s)

DP&A

μ1,alw=4(a)

CoV=0.40

CoV=0.19

design for DP&A=1design for DP&A=0.5

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.T (s)

DP&A

μ1,alw=4(b)

CoV=0.45

CoV=0.43


Fig. 6. Damage spectra for SDOF systems designed according to tmonotonic ductility �1,alw=4. �a� Damage spectra for Chile 1985 earrecord; �c� damage spectra for Kobe 1995 earthquake, KBU record;quakes. Dashed and dashed-dotted heavy lines in �a�–�c� represent deof damage index evaluated with nonlinear time-history analysis for t

in the seismic response of SDOF systems is presented. The effect

JOURNA


of the cumulative damage on the structural response is consideredby explicitly including the Park and Ang, the energy, and theplastic fatigue damage indexes into the formulation of corre-sponding damage factors pdam,P&A, pdam,E, and pdam,F; these factorsare used to reduce elastic spectra to damage controlled inelasticdesign spectra. The expression of the proposed damage factors aresupported by a consistent theoretical background; moreover, sim-plified “design” expression of pdam factors are proposed both forfar field and near fault conditions �Eqs. �21�, �22�, �29�, �30�, �38�,and �39�� that are derived by extensive statistical analysis and, asit is shown, are correctly dependent on the influence of the dam-age on the structure. In fact, the three pdam factors depend on thecyclic work required by the ground motion by means of the ID

factor; the availability of attenuation laws of the duration factor ID

�Iervolino et al. 2008� combined to the PGA allows an easy ap-plication of the proposed methods to the structural assessment anddesign.

The common seismic design methods, then, can be easilymodified through the damage factor pdam, as described in thefollowing:• By using a strength based method, damage inelastic strength

spectra can be obtained considering a reduced allowable duc-tility accounting for the damage.

• By using a displacement-based method, the displacement ca-pacity of a system may be reduced to account for reducedavailable ductility due to the low-cycle fatigue.

0

5

1

5

2

5

3

0 0.5 1 1.5 2 2.5 3 3.5T (s)

DP&A

μ1,alw=4(c)

CoV=0.33

CoV=0.33


0

.5

1

.5

2

.5

3

0 0.5 1 1.5 2 2.5 3 3.5T (s)

S a(g)

(c) Kobe-KBU(b) Kobe-OKA(a) Chile-LLN

μ1,alw=4(d)

posed method with design DP&A=1 and design DP&A=0.5 and fore, LLN record; �b� damage spectra for Kobe 1995 earthquake, OKAd� pseudoacceleration elastic spectra for the three considered earth-mage index �0.5 and 1�, whether thin lines represent the mean value

OF systems designed; relative CoV is also shown.

5

5

0.

1.

2.

0

1

2

he prothquakand �

sign dahe SD

• By using capacity spectrum method the capacity of a SDOF



system may be directly compared to the demand correspond-ing to different performance levels.The proposed unified approach allows an easy extension of the

equivalent elastic seismic design to the large class of the struc-tures influenced by cumulative damage. Further studies areneeded to evaluate the effectiveness of the proposed method forhardening and/or degrading SDOF systems.

Acknowledgments

The writers thank very much the reviewers who contributedgreatly, with their comments and recommendations, to improvethe overall quality of the work.

References

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