A Damage Model for Practical Seismic Design That Accounts for Low Cycle Fatigue

A Damage Model for Practical SeismicDesign That Accounts for LowCycle Fatigue

Amador Teran-Gilmorea… and James O. Jirsa,b… M.EERI

The structural properties of a structure deteriorate when deformationsreach the range of inelastic behavior. A possible consequence of deteriorationof the hysteretic behavior of a structure is failure of critical elements atdeformation levels that are significantly smaller than its ultimate deformationcapacity. Seismic design methodologies that account for low cycle fatigue canbe formulated using the concept of target ductility. The practical use of onesuch methodology requires the consideration of simple low cycle fatiguemodels that consider the severity of repeated loading through a normalizedplastic energy parameter. The inconsistencies inherent to the use of suchindices can be corrected through simple empirical rules derived from anunderstanding of the effect of the history of energy dissipation in theassessment of the level of structural damage.�DOI: 10.1193/1.1979500�

INTRODUCTION

During the last few decades, there has been a considerable increase in understandingthe dynamic response of earthquake-resistant structures. Improvement in modeling andanalysis make it possible to consider a new approach to seismic design. In particular, theperception that the engineer has to design a structure against given and unchanging con-ditions imposed by nature has shifted towards educating the engineering profession tofocus on performance-based design.

Current philosophy for seismic design of typical residential or commercial structuresaccepts the possibility that significant inelastic behavior will occur during severe seismicexcitations. The mechanical characteristics of a structure deteriorate when deformationsreach the range of inelastic behavior. A possible consequence of deterioration of the hys-teretic behavior of a structure is failure of critical elements at deformation levels that aresignificantly smaller than its ultimate deformation capacity under unidirectional loading.In this paper, this failure mode will be termed low cycle fatigue.

This paper discusses the concept of low cycle fatigue, and some of the tools devel-oped to account for it during seismic design. Emphasis is placed on the information re-quired for the design of reinforced concrete structures. Then the convenience of usingenergy demands as representation of the severity of cumulative plastic cycling, and theissues that as a consequence arise for seismic design, are explored. Finally, a simple

a� Universidad Autonoma Metropolitana, Av. San Pablo 180, Col. Reynosa Tamaulipas, Mexico 02200, D.F.b�
University of Texas at Austin, 10100 Burnet Road, Bldg. 177, Austin, TX 78758
803Earthquake Spectra, Volume 21, No. 3, pages 803–832, August 2005; © 2005, Earthquake Engineering Research Institute

804 A.TERAN-GILMORE AND J. JIRSA

energy-based low cycle fatigue model is formulated. The applicability and reliability ofthe model is assessed through the comparison of design results obtained from the modeland other well-known damage indices.

A PARAMETRIC APPROACH TO SEISMIC DESIGN

Traditionally, earthquake-resistant design has been formulated as a demand-supplyproblem. First, all relevant seismic demands in the building have to be estimated, andthen they must be satisfied with adequate seismic supplies in terms of stiffness, strength,and maximum and cumulative deformation capacity. Although the demand-supply equa-tion should be formulated explicitly for different design objectives, this paper will focuson structures that undergo severe plastic cycling when subjected to intense ground mo-tion. Within a numerical performance-based methodology, structural properties shouldbe supplied to the structure so that the response of its structural and nonstructural mem-bers is limited within response threshold levels established as a function of the requiredseismic performance. Teran-Gilmore �1998� has observed that recently proposed designmethodologies contemplate this check at three different steps:

1. Global Predesign. Quick and reasonable estimates of global seismic demandsshould be established and checked against global threshold levels. Within thiscontext, the judicious use of response spectra provides information that allowsthe determination of a set of global structural properties �base shear, period, de-formation capacities� that can adequately control and accommodate, withintechnical and cost constraints, the global response of the structure.

2. Preliminary Local Design. Once the global structural properties have been de-termined, it is necessary to establish the member properties and detailing at thelocal level. This step contemplates the analyses of complex analytical models ofthe structure.

3. Revision of the Preliminary Design. Recommendations have been formulatedfor the revision of the preliminary design through a series of dynamic structuralanalyses that address the global and local performance of the structure.

In this paper, seismic design will be approached from the Global Predesign step. Aparametric approach will be used �Teran-Gilmore 1998�; that is, each one of the relevantstructural properties will be handled during the design process through a structural pa-rameter: base shear �Vb� and fundamental period of translation �T� to define the globallateral strength and stiffness, respectively; and the ultimate displacement ductility �µu�and a fourth structural parameter �b� to characterize the global maximum and cumula-tive deformation capacities, respectively. Although the values of these four parametersare interrelated, the relations between the parameters are complicated and hard to char-acterize for design purposes in a manner that assessment of low cycle fatigue requiresexplicit and independent consideration of each one.

LOW CYCLE FATIGUE

Experimental and field evidence indicate that the strength, stiffness, and ultimate de-formation capacity of reinforced concrete elements and structures deteriorate during ex-

A DAMAGE MODEL FOR PRACTICAL SEISMIC DESIGN THAT ACCOUNTS FOR LOW CYCLE FATIGUE 805

cursions into the plastic range of behavior. Excessive hysteretic degradation may lead toan accumulation of deformation and degradation of the ultimate deformation capacitythat may lead to failure at plastic deformation levels that are significantly smaller thanthe ultimate plastic deformation capacity of the structure under uni-directional loading.

The importance of plastic cycling on the deformation capacity of reinforced concretestructures has been known for some time. Several experimental studies have been car-ried out since the 1970s to study the cyclic response of ductile members and beam-column subassemblages, and even of models of entire buildings �Bertero and Popov1977; Gosain et al. 1977; Yamada 1992; El-Bahy et al. 1999a, 1999b�. The results ob-tained from such studies have emphasized the need to account for the effect of cyclingduring performance evaluation and seismic design of earthquake-resistant structures.

Several researchers have established analytical models to capture the degradation ofthe structural properties of elements and subassemblages, and to assess the occurrenceof low cycle fatigue. Particularly, efforts have been devoted to characterize the stabilityof the hysteretic cycle of the elements, and to study the impact that the number, ampli-tude, and, in some cases, even the sequence of plastic cycles has on the occurrence oflow cycle fatigue. Among many other researchers, relevant work in this field has beencarried out by Banon et al. �1981�, Park and Ang �1985a�, Stephens and Yao �1987�,Chung et al. �1989�, Kunnath et al. �1990�, Daali and Korol �1996�, and Krätzig andMescouris �1997�. Williams and Sexsmith �1995� and Mehanny and Deierlein �2000� of-fer excellent discussions on the use of damage models, and offer an integrated study ofa wide collection of damage indices that can be used to assess low cycle fatigue.

Since the 1980s, the engineering profession confronted the need to design structureswith predictable performance. Performance-based seismic design became a fundamentalconcept for the formulation of seismic design methodologies. As a consequence, pro-posals for design against low cycle fatigue began focusing on deformation control ratherthan relying exclusively on detailing recommendations to ensure stable hysteretic behav-ior. A key issue during the development of design methodologies to control low cyclefatigue was the recognition that the lateral strength of a structure plays an instrumentalrole in controlling the seismic demands that eventually induce this type of failure.

The end result of considering low cycle fatigue during seismic design is the need todesign structures to undergo maximum deformation demands that may be considerablysmaller than their ultimate deformation capacity. Several proposals have been made inthis direction �Fajfar 1992, Cosenza and Manfredi 1996�. To illustrate this, consider that,while Panagiotakis and Fardis �2001� observe that the deformation at failure of rein-forced concrete elements subjected to typical load histories applied in laboratory testscan be estimated as 60% of their ultimate deformation capacity, Bertero �1997� recom-mends that the maximum ductility demand a structure undergoes during ground motionshould be limited to 50% of its ultimate ductility.

Different levels of evaluation can be considered for the assessment of structural dam-age and low cycle fatigue. A local evaluation refers to the damage state of a single mem-ber or subassemblage. Global evaluation deals with the whole structure as a single entity.Attempts have been made to assess damage at the global level through the use of dam-


age indices developed at the element level. Although different weighted averages ofdamage indices in the different structural elements have been proposed to assess globaldamage �Park et al. 1985b, Powell and Allahabadi 1987, Mehanny and Deierlein 2000�,significant inconsistencies may arise �Ghobarah et al. 1999�.

Depending on one’s perspective, a damage index calculated at a story level can beconsidered as a global damage indicator when contrasted to the member level �Mehannyand Deierlein 2000�. Considering the damage indicator at the critical story as a globaldamage index, Teran-Gilmore �2004� found a high correlation between the values of thePark and Ang damage index �Park and Ang 1985a� established from equivalent single-degree-of-freedom �SDOF� systems, and the global value of that index corresponding toregular frames. For this purpose, he estimated the damage index at the critical story asthe mean value of the damage index in the beams located at that story. Although Teran-Gilmore �2004� concludes that a local damage index can be used in a global sense withthe aid of an equivalent SDOF system to adequately conceive and globally predesignregular frames, he discusses the importance of acknowledging the limitations of the useof SDOF systems during global predesign, and emphasizes the need for carrying out adetailed revision of the seismic performance of the preliminary design of the structure.

ENERGY AS DESIGN REPRESENTATION OF CUMULATIVE LOADING

For several decades, researchers have discussed the importance of considering theeffect of repeated plastic cycling during seismic design. An option that has been consid-ered attractive due to its simplicity has been the characterization of cumulative loadingthrough energy concepts. Housner �1956� offered one of the earliest discussions regard-ing the need to consider explicitly the effect of plastic cycling through energy concepts.Later, several attempts were made to estimate the energy demands in simple systems,and to offer insights on how to use these demands for design purposes �Zahrah and Hall1984, Akiyama and Takahashi 1992, Leelataviwat et al. 2002�.

Design for low cycle fatigue was advanced with the formulation and calibration ofdamage indices �Powell and Allahabadi 1987, Cosenza et al. 1993�, and the formaliza-tion of an energy balance equation �Uang and Bertero 1990�. Based on these concepts,several design methodologies that account for low cycle fatigue have been formulated�Fajfar 1992, Bertero and Bertero 1992, Krawinkler and Nassar 1992, Cosenza andManfredi 1996�. As for today, there are still significantly different approaches towardsthe formulation of a design representation for the energy demands. Some researcherssuggest that energy spectra could be formulated and used for design purposes �Akiyamaand Takahashi 1992, Chou and Uang 2000, Manfredi 2001�. Other options include ac-counting for cumulative loading through indirect measures of the plastic energy �Fajfar1992, Bertero and Bertero 1992�, and deriving the plastic energy demands from otherrelevant seismic demands �Teran-Gilmore 1996, Decanini and Mollaioli 2001�.

The plastic energy, EHµ, can be interpreted physically by considering that it is equalto the total area under all the hysteresis loops a structure undergoes during the groundmotion. Although EHµ provides a rough idea of the cumulative plastic deformations de-


mands, this response parameter by itself does not provide enough information to assessstructural performance so that it is convenient to normalize it as follows:

NEHµ =EHµ

Fy�y�1�

where NEHµ is the normalized plastic energy, and Fy and �y are the yield strength andyield displacement, respectively. For an elasto-perfectly-plastic system subjected to asingle plastic excursion �Figure 1a�:

EHµ = ��c − �y�Fy = ��c

�y− 1��yFy = �µc − 1��yFy �2�

where �c is the cyclic displacement, and µc, equal to �c /�y, is the cyclic ductility. TheNEHµ for the plastic excursion is a direct measure of the amplitude of the plastic dis-placement:

NEHµ =EHµ

�yFy= µc − 1 �3�

For an elasto-perfectly-plastic system subjected to multiple plastic excursions, NEHµ

is the sum of all plastic displacements reached in the different cycles normalized by �y,in such way that

NEHµ =

�i=1

Nexc

��ci − �y�

�y= �

i=1

Nexc

�µci − 1� �4�

where �ci and µci are the cyclic displacement and ductility, respectively, associated withthe ith excursion, and Nexc is the total number of plastic excursions during the groundmotion. Note that NEHµ is a direct measure of the cumulative plastic displacement de-mands. For a system with degrading hysteretic behavior, NEHµ could be defined to in-

Figure 1. Definitions of strength and deformation quantities.


clude all plastic excursions for which the capacity does not degrade to a value less thana specified fraction of Fy �say 0.75�. Such a definition allows for rational evaluation ofstructural damage in reinforced concrete structures.

Several researchers have used NEHµ to develop recommendations for the design anddetailing of ductile reinforced concrete elements �Gosain et al. 1977, Scribner and Wight1980, Darwin and Nmai 1985�. Krawinkler and Nassar �1992� formulated a designmethodology that explicitly considers the effect of plastic cycling through NEHµ.

IMPLICATIONS OF USE OF PLASTIC ENERGY DURING SEISMIC DESIGN

Although using energy-derived parameters as a representation of repeated cyclicloading allows the formulation of relatively simple seismic design methodologies thataccount for low cycle fatigue, this approach should be carefully assessed. Particularly,the plastic energy dissipating capacity of a structure does not depend exclusively on itsmechanical characteristics. It has been repeatedly observed that the plastic energy dis-sipated up to failure by an element or structure can change significantly as a function ofthe amplitude of the plastic cycles, in such way that the plastic energy dissipated by alarge number of small amplitude cycles can significantly exceed that dissipated up tofailure through the application of a few large amplitude cycles �Chen and Gong 1986,Chung et al. 1989, Teran-Gilmore et al. 2003�. A second issue, which will not be dis-cussed herein, is the feasibility of deriving a simple design representation for the plasticenergy demands. In the previous section, references were cited to document the workcarried out around this issue.

GROUND MOTIONS

Four sets of ground motions were established to study the impact of consideringNEHµ as the design representation of the severity of repeated cyclic loading. Three ofthese sets correspond to the Los Angeles �LA� urban area and one corresponds to thelake zone of Mexico City. The ground motions for LA, established as part of the FEMA/SAC Steel Project �Somerville et al. 1997�, were grouped in sets of 20 motions as fol-lows: design earthquake for firm soil with 10% exceedance in 50 years �LA 10in50�,design earthquake for firm soil with 50% exceedance in 50 years �LA 50in50�, and de-sign earthquake for soft soil with 10% exceedance in 50 years �LA Soft�. The set ofMexican motions �Mexico Soft� was formed of seven narrow-banded, long-durationground motions recorded in the Lake Zone of Mexico City. The Mexico Soft motionswere scaled up in such way that their peak ground velocity was equal to that correspond-ing to the EW component of the motion recorded at SCT during 1985. Figure 2 showsmean strength spectra for the four sets of motions. All spectra shown were obtained forelasto-perfectly-plastic behavior and 5% of critical damping.

In Figure 2, the circles identify the corner period, defined as the period at which thestrength spectra decreases after peaking either at a single point or at a plateau. As a ref-erence, the corner period is denoted Ts in FEMA-273 �ATC 1997�. Note that LA 10in50has a corner or dominant period around 0.3 sec, while those of LA 50in50, LA Soft, andMexico Soft are around 0.4, 1.0, and 2.0 sec, respectively. Figure 3 shows mean relativeinput energy �E � spectra. Mexico Soft has the largest input energy demands, followed, in
I


order, by LA Soft, LA10in50, and LA50in50. With the exception of Mexico Soft, the cor-ner periods defined according to Figure 2 do not correspond to the period at which theinput energy for 5% damping peaks.

Figure 4 shows mean NEHµ spectra. For constant ductility, Mexico Soft has the larg-est plastic cumulative demands, followed, in order, by LA 50in50, LA Soft, and LA10in50. Note the significant difference in the relative damage potential established forthe different sets of motions from Figures 3 and 4. If, as suggested before, NEHµ is as-sumed to be a direct measure of the plastic cumulative deformation demands in a struc-ture, it follows that input energy and even plastic energy �without normalization� spectrashould be used carefully when assessing the energy content of a ground motion. Themaximum NEHµ demands for Mexico Soft are about two to three times larger than thosecorresponding to the LA motions.

There is a distinctive feature in the NEHµ spectra corresponding to the sets of LAmotions: starting from very small T, the NEHµ demand tends to increase until T reachesthe value of the corner period, after which it remains fairly constant. For the Mexico Softset, NEHµ tends to increase until T reaches the value of the corner period. After that, ittends to decrease with a further increase in T. Note that the corner period defined ac-cording to Figure 2 delimits two distinctive zones in the NEHµ spectra.

Figure 5 shows the coefficient of variation �COV� associated with mean spectrashown in Figures 2 and 4. The COV is presented for two purposes: �1� to provide an ideaof the uncertainty and variability involved in establishing mean spectra, and �2� to pro-

Figure 2. Mean strength spectra, 5% damping.


vide reference values against which the COV associated to the use of the low cycle fa-tigue model developed here can be assessed. While the COV of the strength spectra cor-responding to the four different motion sets does not seem to follow a well-establishedpattern, the COV of the NEHµ spectra does show a surprising similitude for all sets ofmotion, and is characterized by values usually ranging from 0.3 to 0.7.

DISTRIBUTIONS OF EXCURSIONS AND DAMAGE

To clarify the possible effect of using NEHµ to characterize the severity of cumulativecycling during the assessment of low cycle fatigue, the distribution of plastic excursionsaccording to their amplitude was studied. For this purpose, the response of elasto-perfectly-plastic SDOF systems having 5% of critical damping was studied when sub-jected to the different sets of motions.

The plots included in Figures 6–8 are divided into 10 intervals along the horizontalaxis. These correspond to 10 intervals that equally divide the range of possible values ofthe cyclic ductility demand. Figure 6 shows, for LA 10in50, the mean number of plasticexcursions �n� as a function of their amplitude for different values of µmax and T. Thefollowing trends are observed:

• The number of excursions increases as their amplitude decreases.• As T increases �departs from the value of the corner period�, the number of small

excursions �intervals 1–3� decreases with respect to that of large excursions.

Figure 3. Mean relative input energy spectra, 5% damping.


• As the value of µmax increases, the number of small excursions increases withrespect to that of large excursions.

• There is a “bump” in the distributions at intervals 5 and 6, which correspond toexcursions with cyclic ductility demands close to µmax. The bump tends to de-crease and even disappear as the value of µmax increases.

Figure 6 also shows damage distributions for LA10in50. Particularly, the fraction ofoverall damage induced by excursions of different amplitude is plotted. Damage was es-timated using Equation 10 �discussed in detail later� with b equal to 1.5. The damagedistributions tend to peak at intervals 5 and 6, and the peak value tends to increase asµmax decreases and T increases.

The excursion distributions for LA 50in50, shown in Figure 7, exhibit similar ten-dencies to those of LA 10in50. Nevertheless, the former distributions exhibit a largernumber of small amplitude excursions. As the plastic energy content of the ground mo-tion increases, a larger percentage of the plastic energy tends to be dissipated in excur-sions of smaller amplitude. Although the damage distributions for LA 10in50 and LA50in50 are similar, the peaks at intervals 5 and 6 tend to be significantly smaller for thelatter set. As a consequence, the damage distributions corresponding to LA 50in50 tend,as µmax increases and T approaches the value of the corner period, to be fairly constantwith respect to the amplitude of the plastic excursions.

The excursion distributions for Mexico Soft, shown in Figure 8, are similar to thoseestablished for the LA motions. The different NEHµ content and the different dependence

Figure 4. Mean normalized plastic energy spectra, 5% damping.


Figure 5. COV of strength and normalized plastic energy spectra, 5% critical damping.

Figure 6. Distribution of plastic excursions and damage, LA 10in50.


that NEHµ exhibits with respect to T for the Mexico and LA motions, help explain thefollowing particularities for Mexico Soft:

• As T approaches the value of the corner period, the number of small excursionsincreases significantly with respect to that of large excursions.

• The “bump” located at intervals 5 and 6 tends to disappear, even for small valuesof µmax.

• As the NEHµ demands increase from LA 50in50 to Mexico Soft, a slightly largerpercentage of the plastic energy tends to be dissipated in smaller excursions.

Figure 7. Distributions of plastic excursions, LA 50in50.

Figure 8. Distributions of plastic excursions and damage, Mexico Soft.


From the results summarized in Figures 6–8, it can be concluded that as the NEHµ

demand on a structure increases �product of a larger energy content of the ground mo-tion, a larger µmax demand and/or the similarity between the value of T and the cornerperiod�, its relative number of small excursions increases. Because the plastic energydissipating capacity of a structure increases as the amplitude of the plastic cycles de-creases, it can be said in qualitative terms that using NEHµ to characterize the severity ofcumulative cycling during the assessment of low cycle fatigue implies, in general terms,an increased level of conservatism as the energy content of the motion increases.

LOW CYCLE FATIGUE MODELS

Damage indices that account for low cycle fatigue should explicitly consider the ef-fect of cumulative loading. Two significantly different approaches have been used. Onthe one hand, some damage indices consider plastic energy as a measure of the severityof plastic cycling. On the other hand, some damage indices consider the number andamplitude of the different plastic cycles, and in some cases, even their sequence.

In this section two damage indices, considered representative of each one of the twoapproaches mentioned before, will be discussed. Then an energy-based low cycle fatiguemodel that uses NEHµ to characterize the severity of cumulative cycling is derived. Thesethree damage indices will be used to define, in quantitative terms, the impact of usingNEHµ as sole representation of ground motion intensity.

PARK AND ANG DAMAGE INDEX

Park and Ang �1985a� have formulated a damage index to estimate the level of dam-age in reinforced concrete elements and structures subjected to cyclic loading:

DMIPA =�max

�u+ �

EHµ

Fy�u�5�

where �max is the maximum deformation demand during the ground motion, �u is theultimate deformation capacity, and � is a structural parameter. DMIPA less than 0.4 im-plies repairable damage; from 0.4 to 1.0, irreparable damage; and greater than 1.0, fail-ure of the element. Under the presence of repeated cyclic loading, 1.0 represents thethreshold value at which low cycle fatigue is expected to occur. While � of 0.15 corre-sponds to systems that exhibit fairly stable hysteretic behavior, values of � ranging from0.2 to 0.4 should be used for systems exhibiting substantial strength and stiffness dete-rioration �Stephens and Yao 1987, Cosenza et al. 1993, Williams and Sexsmith 1997,Silva and Lopez 2001�. DMIPA can be written as

DMIPA =µmax

µu+ �

NEHµ

µu�6�

where µmax, equal to �max /�y, is the maximum ductility demand during the ground mo-tion, and µu, equal to �u /�y, is the ultimate ductility. Within a seismic design methodol-ogy that accounts for low cycle fatigue, DMIPA can be used as follows �DMIPA=1�:


µmax = µu − �NEHµ or µu = µmax + �NEHµ �7�Within Equation 7, NEHµ becomes the ground motion parameter that quantifies the

severity of cumulative loading, and � the structural parameter that characterizes the cy-cling capacity of the structure. An increase in NEHµ implies larger plastic demands,while a decrease in � implies a more stable hysteretic behavior �better detailing�.

LINEAR CUMULATIVE DAMAGE THEORY

A damage index that takes into account the change in energy dissipating capacity ofa structure as a function of its displacement history can be formulated from Miner’s hy-pothesis, which considers that damage induced by each plastic excursion is independentof the damage produced by any other excursion. Since the deformation history is un-likely to consist of regular, complete cycles, this history is usually divided in half-cycles�termed excursions herein� rather than full cycles, using methods such as the RainflowCounting Method �Powell and Allahabadi 1987�.

Once the displacement history is separated into Nexc plastic excursions, the linearcumulative damage theory requires these excursions to be classified into intervals ac-cording to their amplitude. In this paper, Ndif denotes the number of different intervalsinto which all plastic excursions are classified, and �ci is the cyclic displacement �am-plitude� associated with the ith interval. For earthquake loading, the linear cumulativedamage theory can be formulated as

DMIMH = �i=1

Ndifni

Ni�8�

where Ni is the number of plastic excursions the structure can actually undergo beforefailure when cycled to excursions with amplitude �ci, and ni is the number of plasticexcursions of amplitude �ci resulting from the ground motion demands on the structure.DMIMH equal to one implies incipient failure.

There is a difference between the traditional definitions of deformation and the onesneeded for application of Equation 8. Figure 1b illustrates the concepts of maximumductility and maximum cyclic ductility demand. While µmax is measured with respect tothe underformed position, the maximum cyclic ductility demand �µmaxc� is measuredwith respect to the point in which the largest half-cycle initiates. Note that the largestpossible cyclic ductility demand the structure can undergo is µmaxc=2µmax−1. Similarconcepts as those discussed before should be applied to the ultimate deformation capac-ity of a structure. In this sense, �uc is defined as the ultimate cyclic displacement. Thenormalization of �uc by �y yields µuc �ultimate cyclic ductility�. Equation 8 can be re-formulated as follows �Cosenza and Manfredi 1996�:


DMIMH = �i=1

Nexc � �ci − �y

�uc − �y�b

�9�

where �ci is now the cyclic displacement of the ith excursion, and b is a structural pa-rameter. Within seismic design against low cycle fatigue �DMIMH=1�,

��uc − �y�b = �i=1

Nexc

��ci − �y�b ⇒ �µuc − 1�b = �i=1

Nexc

�µci − 1�b �10�

Within Equation 10, �i=1Nexc��ci−�y�b characterizes the severity of cyclic loading, and

b the stability of the hysteresis loops. While typical values of b range from 1.6 to 1.8, bof 1.5 is a reasonable conservative value to be used for seismic design and damageanalysis �Powell and Allahabadi 1987, Baik et al. 1988, Cosenza and Manfredi 1996�.An increase in b implies a more stable hysteretic behavior. Regarding its limitations,DMIMH does not take into account the sequence of plastic excursions, and inconsisten-cies arise when applied to systems that develop permanent plastic deformations.

AN ENERGY-BASED MODEL TO PREDICT LOW CYCLE FATIGUE

Consider the case in which ni and Ni can be related, for all i in Equation 8, throughthe same proportionality constant �.

ni = �Ni �11�If Equation 11 is substituted into Equation 8, the value of Ni cancels out for each

term in the summation. Under the assumption of proportionality, the level of damage ina structure depends exclusively on its NEHµ demand �Teran-Gilmore et al. 2003�. In thissection, a simple low cycle fatigue model is developed. Basically, this model representsa simplification of the linear cumulative damage theory through the assumption of afixed shape for the distribution of plastic excursions. If Equation 11 holds up to failurea structure can dissipate normalized plastic energy equal to

NEHµ = �i=1

Ndif

ni�µci − 1� = �i=1

Ndif

�Ni�µci − 1� = �i=1

Ndif

��µuc − 1

µci − 1�b

�µci − 1� �12�

Equation 12 can be formulated in closed form as follows:

NEHµ = �0

µuc−1

n�µc − 1�dµc = �0

µuc−1

��µuc − 1

µc − 1�b

�µc − 1�dµc = ��µuc − 1�2

2 − b�13�

where n is the number of plastic excursions of amplitude �c demanded by the groundmotion, and µc, equal to �c /�y, is the plastic cyclic ductility. The value of DMIMH cor-responding to Equation 13 can be estimated according to the closed form of Equation 8:

DMIMH = �0

µuc−1 n

Ndµc = ��

0

µuc−1

dµc = ��µuc − 1� �14�


By considering the right-hand sides of Equations 13 and 14, a simplified estimate ofDMIMH can be obtained:

DMIMHS = �2 − b�

NEHµ

�µuc − 1��15�

In Equation 15, NEHµ quantifies the severity of ground motion, and µuc and b theultimate and cumulative deformation capacities of the structure. Although the analyticalupper limit for the value of µuc is given by 2µu−1, the physical upper limit of µuc will besomewhat less than this because a plastic excursion close to µu will damage significantlythe capacity of a structure to accommodate plastic deformation in the opposite direction:

µuc − 1 = 2r�µu − 1� �16�

where r is a reduction factor �less than one�. For incipient collapse, Equation 15 can bereformulated in terms of µu as �DMIMH=1�:

µu =�2 − b�NEHµ

2r+ 1 �17�

Figure 9a compares, for LA 50in50, damage estimates derived from Equations 9 and15 �b=1.5 and µuc=8.5�. The value of µuc was established from Equation 16 by assum-ing µu of 6 and r equal to 0.75. The discontinuous lines correspond to Equation 9. Equa-

Figure 9. Estimates of damage from Equations 9 and 15, LA 50in50.


tion 15 yields higher estimates of damage for µmax of 2, slightly higher estimates for µmax

of 3, and slightly lower estimates for µmax of 4.

The energy dissipating capacity of a structure increases as the amplitude of its plas-tic excursions decreases. In the case of µmax of 2, the amplitude of the majority of theplastic excursions is small with respect to µuc. While Equation 9 accounts for an in-creased energy dissipation capacity, Equation 15 does not, so that the latter yields higherestimates of damage. As the value of µmax increases, the mean amplitude of the plasticexcursions increases with respect to the ultimate deformation capacity. Because the en-ergy dissipating capacity of a system will tend to decrease under these circumstances,Equation 15 yields similar estimates of damage than Equation 9 for µmax of 3 and 4.

As illustrated in Figure 9b, the COV of the damage estimates obtained from bothequations is practically equal. If the structural parameters involved in Equations 9 and15 are considered deterministic, the uncertainty in the estimation of the level of damageis similar to that involved in the determination of the energy demands �see Figure 5c�.

Figure 9c shows the mean ratio of the damage estimates obtained from Equations 9and 15. The ratio shows a strong dependence on µmax and a weak variation with respectto T. Figure 9d shows the COV associated with the damage ratio is very small. Similarresults as those shown in Figures 9c and 9d were obtained for the other sets of groundmotions.

As the amplitude of the plastic cycles decreases, a structure is able to accommodatelarger plastic energy demands before failure due to low cycle fatigue. Within seismicdesign, this implies that the amplitude of the plastic excursions should decrease withrespect to the ultimate deformation capacity of the structure as the energy content ofmotion increases. Under these circumstances, Equation 15 yields �with respect to Equa-tion 9� the following:

• Slightly lower estimates of damage when applied to structures subjected to mo-tions with low energy content.

• Similar or slightly higher estimates of damage when applied to motions withmoderate and large energy content.

• Higher estimates of damage when applied to motions with very large energy con-tent.

Because of the above, it was considered convenient to adjust Equation 15 by intro-ducing a parameter a that accounts for the energy content of the motion �and thus, in-directly, for the manner in which energy is expected to be dissipated�:

DMIMHS = �2 − b�

aNEHµ

µuc − 1�18�

For incipient collapse, Equation 18 can be reformulated in terms of µu as �DMIMH=1�:

µu =�2 − b�aNEHµ

2r+ 1 �19�


CONSIDERATIONS FOR THE USE OF DAMAGE INDICES IN PRACTICALSEISMIC DESIGN

Urgent issues that need to be addressed to improve the application of damage indiceswithin practical seismic design are �1� the harmonization of definitions used and resultsobtained by different researchers, and �2� a clear understanding of the implications of theapplication of a damage index to the design of a particular structure.

Regarding the first issue, the practical use of damage indices requires a precise andreliable definition of failure. After identifying important differences in the reportedDMIPA threshold values corresponding to failure, Williams and Sexsmith �1997� suggestthat a source of such differences is the different definition of failure used by differentresearchers. Iemura �1980� observes that the linear damage cumulative theory yields dif-ferent results depending on the criteria used to define failure.

The second issue emphasizes the need to have a clear understanding of the expectedbehavioral and failure modes of a given structure subjected to ground motion. Thesemodes, which can change from structure to structure, can even change in a particularstructure as a function of the characteristics of the seismic excitation �Krawinkler andZohrei 1983, Chung et al. 1989, Manfredi and Pecce 1997�. In this respect, the charac-terization of the structural parameter �b�, and the uncertainty involved in the use of adamage index does not only vary according to the structural properties of the structure,but also according to the type of loading it is subjected to.

PRACTICAL CONSIDERATION OF LOW CYCLE FATIGUE INSEISMIC DESIGN

The impact of using NEHµ to quantify the severity of ground motion will be assessedthrough the comparison of the results obtained from the three damage models consid-ered in this paper. The results obtained from Equation 10 will be considered benchmarkvalues against which the reliability of Equations 7 and 19 will be assessed.

TARGET DUCTILITY

Target ductility is defined herein as the maximum ductility �µmax� the structure canreach during the design ground motion to prevent failure due to low cycle fatigue. Eventhough the concept of target ductility has obtained acceptance from researchers andpracticing engineers, the direct use of energy demands to establish the severity of seis-mic cyclic loading is still an unresolved issue. While Fajfar �1992� suggests that the tar-get ductility can be reasonably estimated by using the plastic energy demand, Cosenzaand Manfredi �1996� discuss the importance of the manner in which this energy hasbeen dissipated. In general, it has been agreed that as the plastic energy demand in-creases, µmax should decrease with respect to µu. How much smaller µmax should be withrespect to µu �or how much bigger µu with respect to µmax� depends on three variables:the value of the known ductility �either µmax or µu�, a ground motion parameter thatquantifies the severity of the plastic demands, and a structural parameter that character-izes the cycling capacity of the structure.


Using the concept of target ductility, two approaches can be considered for the for-mulation of a performance-based design methodology:

1. Approach A requires estimating µmax given that µu is known. The second columnof Table 1 summarizes the steps involved in Approach A. First, a decision needsto be made about the type of detailing to be used for the structure �ductile vs.nonductile�, and values of µu and b should be established according to it. Next,the value of T is established. The determination of T within the context ofperformance-based design is discussed elsewhere �Bertero and Bertero 1992,Priestley 2000�. Then the value of µmax is established as a function, among otherthings, of µu and b. Once µmax is known, it is possible to establish the designbase shear that will allow the structure to control its maximum plastic demandwithin this threshold. Approach A has been used by Fajfar �1992�, Bertero andBertero �1992�, and Cosenza and Manfredi �1996� to formulate design method-ologies.

2. Approach B requires estimating µu given that µmax is known. The third columnof Table 1 summarizes the steps involved in Approach B. First, a decision needsto be made about the type of detailing to be used for the structure, and a valueof b established accordingly. Next, the value of T is established. Once a prelimi-nary value of µmax is assumed, the values of µu and Vb can be established as afunction of it. Approach B has been used by Arroyo and Teran-Gilmore �2002�and Ridell and Garcia �2002�.

Recommendations for the use of Equations 18 and 19 in practical seismic design aresummarized in Table 2. These recommendations were obtained from extensive studies ofthe seismic performance of SDOF systems designed according to Equation 19 and sub-jected to the sets of ground motions considered in this paper. Two different values of aare suggested for practical application: 0.75 for the use of Approach B with motions

Table 1. Target ductility-based design approaches

Step Approach A Approach B

1 Assume µu ,b= f�detailing� Assume µmax= f�judgment�, and b= f�detailing�2 Determine T Determine T3 Estimate µmax= f�T ,µu ,b� Estimate µu= f�T ,µmax ,b�4 Estimate Vb= f�T ,µmax� Estimate Vb= f�T ,µmax�

Table 2. Considerations for the practical use of Equations 18 and 19

Energy Content Approach A Approach B

Low a=1,µmax�µu a=1,µu�µmax

Moderate or High a=1 a=1Very High �Mexico City� a=1 a=0.75


with very large energy content �e.g., Lake Zone of Mexico City�, and 1.0 for any othercase. The unsafe nature of Equations 18 and 19 with a=1 for motions with low energycontent can be taken into consideration by establishing maximum or minimum values,respectively, for µmax and µu �such as those indicated in Table 2�.

APPROACH A

Table 3 shows implementation details for the use of Approach A with the three dam-age indices considered herein. Once the values for b , µu, and T are established, designagainst low cycle fatigue implies estimating the design base shear. The determination ofVb is carried out in two steps: �1� µmax is determined, and �2� the design Vb correspondsto the minimum strength required to control the global plastic response of the structurewithin the threshold established by µmax.

Approach A requires the target ductility to satisfy the conditions formulated in thelast column of Table 3. Due to inconsistencies in their formulation, the following con-dition was imposed on the value of µmax derived from DMIMH and DMIMH

S :µmax�µu.

Figure 10a shows, for LA 50in50 and µu of 5, a general comparison of the values ofµmax obtained with DMIMH and DMIPA. The three values of � used with DMIPA are con-sidered to characterize a wide range of structural behavior. For DMIMH ,r was set equalto 0.75 and b was set equal to 1.2 and 1.8. There is similarity between the values of µmax

obtained from DMIPA with � of 0.05 and DMIMH with b of 1.8, and those obtained from� of 0.15 and b of 1.2. Figure 10b shows values of µmax corresponding to structures withstable hysteretic behavior ��=0.15 and b=1.5�. Note the similitude of the results de-rived from the three models.

For firm soil motions, the value of µmax is not particularly sensitive to the values of� and b. The results suggest that µmax should be limited to about 0.65 µu in structuresthat exhibit significant deterioration of their hysteresis loop �b=1.2�, and to about0.75 µu for structures with stable hysteretic behavior �b=1.5�. The conservatism in-volved in the deformation thresholds suggested for displacement-control methodologies,such as FEMA-273, appears enough to protect stable structures from low cycle fatigue.

Figures 10c and 10d show the minimum strength required to control, for LA 50in50,the maximum ductility demand within the values shown in Figures 10a and 10b, respec-tively. Considering that the lateral strength is the actual structural property to be de-signed within Approach A, Figure 10 suggests that the impact of using one or anotherlow cycle fatigue model during seismic design would be minimal.

Table 3. Use of Approach A with the three different models

Model Known Unknown Target ductility should satisfy:

DMIMH b ,µuc−1=2r�µu−1� �i=1Nexc�µci�µmax�−1�b Equation 10

DMIPA � ,µu µmax , NEHµ�µmax� Equation 7DMIMH

S b ,µuc−1=2r�µu−1� NEHµ�µmax� Equation 19


Figure 11 shows that the COV of the strength demands is fairly insensitive to thevalues of b and �, and that this COV is similar to that shown in Figure 5a. If the struc-tural parameters involved in the three models are considered deterministic, the uncer-tainty involved in determining the minimum strength required to avoid failure due to lowcycle fatigue is similar to that involved in the determination of constant ductilitystrength spectra. Note also that the values of COV associated with the estimation of µmax

are very small, particularly in the case of DMIPA.

Although not shown, results obtained for other values of µu �3, 4, and 6� and for LA10in50 and LA Soft are similar to those summarized in Figures 10 and 11. Nevertheless,

Figure 10. Design values obtained from three damage models and LA 50in50, µu=5.

Figure 11. COV of Sa and µmax obtained from three damage models and LA 50in50, µu=5.


it was observed that as the energy content of the ground motions decreases, DMIPA be-comes more conservative with respect to DMIMH.

Figure 12 shows values of µmax obtained for Mexico Soft. Note that the correspon-dence between the values of � and b not only changes with respect to that observed infirm soil, but becomes sensitive to the value of T. This can be explained by the largerNEHµ demands of Mexico Soft, and by the dependence of these demands with respect toT �Figure 4d�. While Figure 12a shows that � of 0.30 yields results similar to b of 1.2 ina wide period range, Figures 12b and 12d suggest that DMIPA remains conservative withrespect to DMIMH in period intervals where the NEHµ demand is small, and becomesslightly unsafe around the corner period. In contrast, DMIMH

S yields slightly higherstrength requirements than DMIMH at the corner period and slightly lower strength re-quirements as T departs from it. As the value of µu increases, the strength requirementsderived from DMIPA around the corner period become progressively smaller than thoseobtained from DMIMH, while the opposite occurs, under the same circumstances, to thestrength requirements derived from DMIMH

S .

For structures subjected to Mexico Soft, µmax should be limited under certain circum-stances to about 0.40 µu for rapidly degrading structures and to about 0.50 µu for struc-tures with stable hysteretic behavior. Under these circumstances, the conservatism in-volved in the deformation thresholds suggested for displacement-control methodologieswould not seem enough to protect adequately structures having T close to the cornerperiod from the occurrence of low cycle fatigue.

Considering that within Approach A the base shear of the structure is the structural

Figure 12. Design values obtained from three damage models and Mexico Soft, µu=5.


property to be designed, the results shown in this section suggest that, independently ofthe energy content of the ground motion, the impact of using one or another low cyclefatigue model would be minimal.

APPROACH B

Table 4 shows the implementation details for Approach B. Once values of T ,b, andµmax are established, design against low cycle fatigue implies the determination of thedesign values for µu and Vb. This determination is carried out in two independent steps:while Vb corresponds to the minimum strength required to control the global plastic re-sponse of the structure within the threshold established by µmax ,µu should satisfy theconditions formulated in the last column of Table 4. Due to the unsafe nature of DMIMH

S

as the NEHµ content of the motion decreases, the following condition was imposed onthe value of µu derived from this index: µu�µmax.

Figure 13 shows, for LA 50in50, values of µu obtained from Approach B. Systemswith rapidly deteriorating structural properties ��=0.30,b=1.2� and with stable hyster-esis ��=0.15,b=1.5� are considered. An r of 0.75 was considered to evaluate DMIMH

and DMIMHS . Note that the estimates of µu derived from the three models are very simi-

lar.

Figure 13 shows the COV corresponding to µu. If the structural parameter used tocharacterize the stability of the hysteretic behavior is considered deterministic, the COVis comparable or smaller than that involved in estimating the NEHµ demands for constantductility �Figure 5c� The COV associated with the use of DMIPA tends to be smaller thanthat of DMIMH, which in turn is smaller than that of DMIMH

S . Results obtained for LA10in50 and LA Soft suggest that, as the NEHµ content of the motion decreases, the esti-mates obtained from DMIPA tend to become slightly higher than those derived fromDMIMH; while those obtained from DMIMH

S , tend to become slightly smaller.

Figure 14 shows values of µu for Mexico Soft. As specified in Table 2, the resultscorresponding to DMIMH

S and Mexico Soft were obtained for a=0.75. Considering thatwithin Approach B the ultimate deformation capacity of the structure is the structuralproperty to be designed, the results that have been obtained �Figures 13 and 14� suggestthat the impact of using one or another low cycle fatigue model would be minimal.

Table 4. Use of Approach B with the three different models

Model Known Unknown Ultimate ductility should satisfy:

DMIMH b ,µmax ,�i=1Nexc�µci�µmax�−1�b µu Equation 10

DMIPA � ,µmax ,NEHµ�µmax� µu Equation 7DMIMH

S b ,µmax ,NEHµ�µmax� µu Equation 19


FINAL CONSIDERATIONS

Empirical considerations have been made in this paper to establish a simple energy-based model, DMIMH

S , to assess the occurrence of low cycle fatigue. First, it has beenconsidered that the ni and Ni curves for typical earthquake-resistant structures are pro-portional. In this respect, previous work by Teran-Gilmore et al. �2003� has shown that ni

curves, such as those shown in Figures 6–8, have shapes that are similar to those of Ni

curves derived from field and experimental research. From the results obtained by Teran-Gilmore et al., it can be concluded that in general, the assumption of proportionality

Figure 13. Values of µu obtained from three damage models and LA 50in50.

Figure 14. Values of µu obtained from three damage models and Mexico Soft.


between the ni and Ni curves has a small impact in strength design relative to the un-certainty introduced to the design process from the definition of the design input and thedetermination of the deformation capacity of the structure.

In general terms, quantifying the severity of ground motion solely through the use ofNEHµ �proportionality between the ni and Ni curves� implies �1� slightly unsafe estimatesof damage for motions with low energy content, �2� reasonable estimates for motionswith moderate and large energy content, and �3� conservative estimates for motions withvery large energy content. Although this conclusion can be considered general, some ofthe details discussed herein could be related to the specifics of the sets of ground mo-tions. Of particular concern is the fact that the EW component of the motion recorded atSCT during 1985 is the only available large intensity motion recorded in the Lake Zoneof Mexico City. With the exception of SCT EW, all motions in Mexico Soft were re-corded during low-intensity events and were scaled up an order of magnitude to conformthis set. As a consequence, the Mexico Soft set shows, on average, a larger energy con-tent and larger content of small cycles than SCT EW.

The values of some parameters involved in DMIMHS , such as a and r, have been cali-

brated in such way that, for the sets of ground motions considered in this paper, DMIMHS

yields similar assessment of low cycle fatigue as other well-known damage indices. Al-though on the one hand, extensive calibration of DMIMH

S is required before it can be usedfor practical seismic design, on the other hand, the analytical results derived from it formotions with very different frequency and energy content are encouraging.

Simple seismic design methodologies, based on avoiding low cycle fatigue throughsolely controlling the energy demands in the earthquake-resistant structure, can be for-mulated with the aid of DMIMH

S . In particular, a three-step methodology is currently be-ing developed and calibrated for seismic design:

1. Determine T.2. Establish the maximum NEHµ demand the structure can accommodate before

failure due to low cycle fatigue. The threshold value of NEHµ is established fromEquation 19 once the ultimate and cumulative deformation capacities of thestructure �µu and b, respectively� are established according to the detailing to beprovided to the structural elements.

3. Establish, as a function of T, the lateral strength required to control the NEHµ

demands within the threshold established in step 2.

Preliminary results derived from the use of the three-step methodology suggest thatseismic design of structures with stable hysteretic behavior and located in firm soilshould focus on controlling their maximum ductility demand. It should be mentionedthat this methodology has been successfully applied to the seismic design of simple sys-tems located in the Lake Zone of Mexico City, and that the values of a and r recom-mended in this paper have worked fine so far for this purpose �within the limitationsdiscussed before for the conformation of sets of motions representative of large intensityseismic events�.

The value of b affects the level of conservatism involved in the use DMIMHS . Particu-


larly, as b increases, the level of conservatism of its damage estimates tends to decrease,in such way that the use of DMIMH

S with values of b larger than 1.6 seems unsafe fordesign purposes. Although DMIPA neglects the way in which the plastic energy has beendissipated, it can not be considered equivalent to DMIMH

S . In fact, the conservatism ofthe damage estimates derived from DMIPA exhibit opposite tendencies than those ob-tained from DMIMH

S .

The design level derived from the use of DMIMHS and DMIPA does not exhibit a high

sensitivity with respect to the values of b and �. As suggested before by Cosenza et al.�1993� and the results obtained herein, the uncertainty in the determination of these val-ues would appear not to impact significantly the design results obtained from Ap-proaches A and B. Within this context, b of 1.5 and � of 0.15 seem to yield reasonableresults for the seismic design of systems exhibiting fairly stable hysteretic behavior.Conservative values, such as b of 1.2 and � of 0.30, may be considered for structuresexhibiting rapid deterioration of their hysteresis loop.

Although design against low cycle fatigue has been approached in global terms inthis paper, design considerations should also be made at the local level. Detailing con-siderations of structural members against low cycle fatigue, such as those discussed byDutta and Mander �2001�, require a clear understanding of the relationship existing be-tween the plastic deformation demands at the local and global levels.

An issue that has not been considered in this paper is the effect of the hysteretic be-havior on the seismic demands and capacities of an earthquake-resistant structure. Insome cases, the response of a structure becomes sensitive to the specifics of its hystereticbehavior, particularly for systems that exhibit pinching. Another issue not considered ex-plicitly is the multi-degree-of-freedom effect. Results obtained by several researcherssuggest that the response of an SDOF system can be used to obtain reasonable estimatesof displacement, energy dissipation, and structural damage in regular structures �Qi andMoehle 1991, Tso et al. 1993, Teran-Gilmore 2004�. Nevertheless, there are still issuesto be addressed within this context, such as the effect of higher modes and of layout andstructural irregularities.

Finally, one of the challenges involved in formulating a damage control methodologythat accounts for low cycle fatigue is the estimation of the energy demands in theearthquake-resistant structure. In particular, it is of interest to formulate methodologiesthat can be easily adapted to current seismic design formats.

CONCLUSIONS

Low cycle fatigue is, in many cases of practical interest, an issue during seismic de-sign. Displacement-control seismic design methodologies, in particular, seem to provideadequate level of safety for the design of structures with stable hysteretic behavior andsubjected to “typical” firm soil motions. Nevertheless, the use of low cycle fatigue mod-els should be considered for the design of structures exhibiting rapid and excessive de-terioration of their hysteresis loop, and for any type of structure subjected to long-duration, narrow-banded ground motion.


The concept of target ductility complemented with the use of simple damage indicesseems to provide a robust set of tools for seismic design against low cycle fatigue. Thisis particularly true for well-conceived regular structures that exhibit stable hysteretic be-havior and controlled response during severe ground motion. The application of the prin-ciples of capacity design and performance-based design are instrumental to achieve thistype of behavior. As for structures that exhibit irregularities and/or exhibit rapidly dete-riorating hysteretic behavior, this set of tools becomes sensitive to the specifics of thelocal and global hysteretic behavior, and thus its application becomes less reliable. Ashas been done in other contexts, the use of the tools discussed herein can be applied todetermine the strength and ultimate deformation requirements of ductile structures withstable hysteretic behavior, while a more stringent application should be considered forstructures with erratic seismic behavior.

Urgent issues that need to be addressed to make possible the use of damage indicesin practical seismic design are the harmonization of definitions used and results obtainedby different researchers, and the development of a clear understanding of the implica-tions of the application of a damage index to the design of a particular structure.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of Universidad Autonoma Metro-politana, University of Texas, Fulbright Scholar Program and Consejo Nacional de Cien-cias y Tecnologia �CONACyT�, during Dr. Teran-Gilmore’s stay at the University ofTexas at Austin as a visiting researcher.

NOTATION

The following symbols are used in this paper:

a structural parameter that accounts for the energy content of the groundmotion

b structural parameter that characterizes the stability of the hysteretic cycle

COV coefficient of variation

DMIMH damage index based on linear cumulative damage theory

DMIMHN DMIMH

S /DMIMH

DMIMHS Teran and Jirsa damage index

DMIPA Park and Ang damage index

EHµ plastic energy demand

EI relative input energy

Fy strength at yield

n ,ni number of plastic excursions of amplitude �c ,�ci the ground motion demandsfrom a system


N ,Ni number of plastic excursions of amplitude �c ,�ci a system is able to undergobefore failure

Ndif number of intervals into which all plastic excursions are classified accordingto amplitude

Nexc total number of plastic excursions

NEHµ normalized plastic energy

r reduction factor that characterizes the cyclic deformation capacity of asystem

Sa spectral acceleration �pseudoacceleration�

T fundamental period of vibration

Ts corner period

Vb base shear

� proportionality constant between n and N

� constant in Park and Ang damage index that characterizes the stability of thehysteretic cycle

�c ,�ci cyclic displacement, subscript indicates ith excursion or ith amplitudeinterval

�max maximum displacement demand during a ground motion

�u ultimate displacement capacity

�uc ultimate cyclic displacement capacity

�y displacement at yield

µ maximum ductility demand associated to a spectral value

µc ,µci cyclic ductility, �c /�y, subscript indicates ith excursion or ith amplitudeinterval

µmax maximum ductility demand during a ground motion, �max /�y

µmaxc maximum cyclic ductility demand

µu ultimate ductility capacity, �u /�y

µuc ultimate cyclic ductility capacity, �uc /�y

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�Received 5 May 2003; accepted 10 April 2004�

A Damage Model for Practical Seismic Design That Accounts for Low Cycle Fatigue

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