Displacement Ductility Demand Spectra

Engineering Structures 23 (2001) 1319–1330www.elsevier.com/locate/engstruct

Displacement ductility demand spectra for the seismic evaluationof structures

A. Tena-Colunga*

Departamento de Materiales, Universidad Autonoma Metropolitana, Av. San Pablo No. 180, Dol. Reynosa Tamaulipas, DelegacionAzcapotzaico, Mexico, DF, 02200 Mexico

Received 26 April 2000; received in revised form 14 February 2001; accepted 16 February 2001

Abstract

One of the major concerns in the structural engineering practice is to assess the nonlinear response of new or existing structuresfor moderate or strong earthquake motions in a simple way. This can be done using what the author defines as displacement ductilitydemand spectrum (DDDS). This DDDS is a variation of the well-known nonlinear response spectrum for single degree of freedom(SDOF) systems with fixed displacement ductility demands. The proposed spectra and their applications for the seismic evaluationof structures are illustrated for different structural systems that the author has previously evaluated using other methods. 2001Elsevier Science Ltd. All rights reserved.

Keywords: Seismic evaluation; Nonlinear response spectra; Displacement ductility demands; SDOF systems; Hysteretic models

1. Introduction

Damaging earthquakes occurred during the last 16years in Chile, Mexico, Armenia, The United States,Japan, Peru, Bolivia, Egypt, Turkey, Iran, Philippines,Colombia, Greece, Taiwan and India, among other affec-ted nations, have warned the engineering communityworldwide about the vulnerability of existing structures.

Therefore, the seismic evaluation of existing struc-tures is an issue of paramount importance in earthquakeengineering practice. The evaluation of the existingstructures is not only important to assess the vulner-ability of specific structures, but also to complement stra-tegic plans directed to mitigate the seismic hazard in thebuilt environment of a given region. However, availablemethods for the seismic evaluation of existing structureshave not evolved significantly during the past decade,particularly when the expected nonlinear dynamicresponse of structures for moderate or strong earth-quakes and the uncertainties associated to it have to beassessed in a simple way. This paper presents an integralmethod for the seismic evaluation of the existing struc-

* Tel.: +52-5-318-9460; fax:+52-5-318-9085.E-mail address: [email protected] (A. Tena-Colunga).

0141-0296/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.PII: S0141-0296 (01)00025-6

tures, using what the author defines as displacement duc-tility demand spectrum (DDDS). This DDDS is equival-ent to the constant strength response spectrum (CSRS)formerly studied by others with other purposes [1,2] anddiscussed in greater detailed in following sections.

2. Inelastic design spectra (IDS)

The concept of inelastic design spectra (IDS) can betraced back to the 1960s and it has been used for manyyears for the design of special structures such as nuclearpower plants [3]. In fact, these spectra and their vari-ations (i.e. strength spectra) have also been used todefine the design spectra for building structures of manyseismic codes worldwide, where a basic elastic pseudo-acceleration spectrum can be reduced for inelasticbehavior to primarily account for tolerated ductilitydemands and overstrength, based upon studies conductedfor nonlinear SDOF systems, in addition to the experi-ence and judgment of building code developers.

Inelastic spectra can be understood as a family orcurves, and depending on the parameters that are fixed,these spectra have been named in different ways bymany authors. When peak nonlinear response quantitiesare primarily assessed for a target displacement ductility

1320 A. Tena-Colunga / Engineering Structures 23 (2001) 1319–1330

demand, the resulting spectra have been called constantductility response spectra (CDRS). On the other hand,CSRS are obtained when maximum displacement duc-tility demands and displacements are primarily assessedfor a constant strength or strength ratio.

Few researchers have studied the use of CSRS [1,2].The works of reference have primarily used CSRS toevaluate IDS proposed in the literature at the time, suchas the well-known Newmark–Hall and ATC methods[1], and/or to study the variation of the ductility demandusing a set of ground motions [1,2].

On the other hand, the concept of CDRS has beenused widely by most researchers interested in inelasticspectra. Based upon the concept of CDRS, pseudo-accel-eration design spectra have been defined for differentseismic building codes worldwide and strength reductionfactors (R, Rm, etc) have been computed and proposed bydifferent researchers using SDOF systems consideringdifferent hysteretic models, primarily bilinear hystereticmodels [4–9]. In fact, is the relationship between m andR the one that has caught the interest of most researchers.

There is a direct relation between CDRS and CSRS;thus, one can convert one type of spectrum to another byusing suitable m–R relationships. Therefore, it is alwayspossible to adapt procedures developed for CDRS toCSRS or vice versa, extending the applicability of eithertype of spectrum for the seismic design or evaluation ofstructural systems.

3. Discussion on IDS and the seismic evaluation ofexisting structures

The concept of CDRS is widely accepted and has beenuseful in design practice, however, it is quite debatablethat the structural community (primarily outside aca-demic practice) would prefer to assess the global duc-tility capacity of a structural system rather than othersignificant parameters, i.e. strength capacity. In addition,the use of a constant ductility value is not as practicalas one may think from the computational viewpoint.

For example, it is well-known that there are somecomputation deficiencies with this approach, amongthem, that it is possible to have multiple yield strengthsthat produce the same target ductility, as illustrated, forexample, in Miranda [10]. Besides, it has also beenshown that there could be important variations in thestrength demands required for structural systems for aconstant ductility in the period range where most struc-tural systems are designed in practice, independently ofthe soil conditions [6,10]. These variations could beparticularly important for the design of buildings locatedin soft soil sites, such those found in Mexico City. Thesevariations in the strength demands for constant ductilityvalues are not necessarily well represented with the Rmcurves presented by others [11].

Despite the shortcomings mentioned above associatedwith the use of CDRS, and the fact that the nonlinearresponse of structural systems is not always well rep-resented by equivalent SDOF systems (particularly forirregular or special structures), the concept of IDS basedupon CDRS and the study of more rational strengthreduction factors for the design of structures are veryvaluable, because it is easier and faster to study generaltrends with this approach than using more complex mod-els. This is particularly true for the design of new struc-tures. However, the use of CDRS is not practical for theseismic evaluation of existing structures and/or thedesign of suitable retrofit schemes for such structures,as the dynamic characteristics of the site (site effect) andthe nature of earthquakes that affect the region where thestructure is located are seldom included in the smooth m–R curves presented in the literature, except the onerecently proposed by Ordaz and Perez-Rocha [9].

For the evaluation of existing structures it would beconvenient to provide a version of inelastic spectra thatpractising engineers can use with confidence and wherethey can get a good feeling of the parameters that areinvolved. For this purpose, constant strength responsespectra (CSRS, called here displacement ductilitydemand spectra, DDDS) are closer to the needs of engin-eering practice to evaluate existing structures thanCDRS, as structural engineers are used to, can computeand feel the required parameters to build a DDDS foreach specific structure. Whereas the DDDS is a variationof the CSRS studied by other with other purposes, to theauthor’s knowledge, no one has used a DDDS (CSRS)for the seismic evaluation of existing structures beforein the literature.

4. Displacement ductility demand spectra (DDDS)

4.1. Concept

A DDDS relates peak displacement ductility demands(and other important response quantities, i.e.displacements) with structural periods of nonlinearSDOF systems with given yield strengths, as shown inFig. 1 for structural systems with an elastic–perfectlyplastic (ELP) hysteretic behavior for a base shear ratioV/W=0.10 for the well-known SCT-EW accelerogramrecorded during the 1985 Michoacan earthquake (Fig.2). Thus, the DDDS are CSRS.

The main difference between a DDDS and a CDRSis that the strength is fixed rather than the displacementductility. This variation offers some advantages from thecomputational viewpoint. The computation of a DDDSis simpler and faster as no iterations are needed to targetthe fixed strength value, as is needed, for example, inthe computation of CDRS to achieve the target ductilitydemand. In addition, there are no uniqueness problems

1321A. Tena-Colunga / Engineering Structures 23 (2001) 1319–1330

Fig. 1. Displacement ductility demand spectra (DDDS) for an elastic–perfectly plastic system with a base shear ratio V/W=0.10, subjected to the1985 SCT-EW acceleration record.

Fig. 2. Recorded 1985 SCT-EW accelerogram for the 1985 Michoacan Earthquake.

in the definition of DDDS, as there are for CDRS,because the yield strength is defined and fixed a priori.

4.2. Application for the seismic evaluation of existingstructures

The concept of a DDDS offers some advantages forthe seismic evaluation of existing structures with respectto CDRS. It is “easier” for most structural engineers toestimate the lateral load capacity of an existing structurerather than defining its ductility demand capacity,although none of these parameters can be assessed withany precision. Nevertheless, the lateral load capacity ofa structure could be estimated using conventionalmethods of analysis together with the information fromblue prints and/or experimental data, or by assuming thatthe structure was designed to comply with the require-ments of a ruling building code. In addition, the uncer-tainties associated with the assessment of the lateralstrength capacity can be evaluated by computingadditional curves for other strength values consideringoverstrength sources and/or the possibility that the com-puted strength was overestimated.

A suitable hysteretic model can be selected for thestructural material to define DDDS without facing theadditional numerical problems associated with theuniqueness of a solution related to CDSR. Therefore, the

DDDS can be easily defined assuming not only an ELPbehavior for all structures, but also using other suitablehysteretic models for the structural system under study.For example, for structural steel systems, the DDDS canbe computed using hysteretic models that would takeinto account representative post-yield stiffnesses, amongothers, the bilinear or the Ramberg–Osgood hysteresismodels; for reinforced concrete structures one may pickstiffness degrading hysteresis models such as Clough,Takeda, Sina [12] or Kunnath et al. [13] models; andfor masonry structures one can use stiffness and strengthdegrading models such as those proposed by Kwok andAng [14] or Tomazevic and Lutman [15], among others.

Estimates of dominant periods (frequencies) ofresponse for subject structures could be done from adhoc analytical models or experimental methods. Theimplications on the uncertainties associated with the esti-mates of dominant structural periods can be easily evalu-ated with the DDDS, as the impact of underestimatesand overestimates in structural periods on peak ductilitydemands and displacements can be directly evaluatedwith the DDDS curves (Fig. 1).

Therefore, once the lateral load capacity and the domi-nant structural periods for the structure are estimated, asuitable hysteretic model or a set of hysteretic modelshave been chosen, and a set of representative or “criti-cal” ground motion records have been selected for the


site where the structure is located, then, DDDS can bedefined for simplified seismic evaluations of the struc-ture. Peak ductility demands and displacements can beenassessed with the DDDS, as well as the uncertaintiesthat one may have on the estimates of strength, stiffness(period) and hysteretic characteristics. Then, one mayjudge if the displacement ductility demands obtainedfrom the DDDS can be developed by the structural sys-tem depending on its characteristics and its seismicdetailing, if the lateral displacements could be accommo-dated without damaging nonstructural components, fav-oring structural pounding with neighboring structures orcreating panic in the users of the building. In addition,from the peak inelastic displacement defined by theDDDS one can compute the lateral displacements of abuilding (and, by extension, story drift ratios) using pro-cedures already available in the literature [16]. With apreliminary evaluation of a structure with a DDDS, onecould decide whether further detailed analyses for a sub-ject structure are needed or not.

4.3. Some conceptual differences with other methodsfor the seismic evaluation of existing structures

The DDDS involves strength concepts in a consistentway and, in that fashion, it offers several advantagesfrom some old evaluation strategies that were done inthe past. For example, a common evaluation procedureused in the past was to verify if an existing structuresatisfied the requirements and criteria of the ruling build-ing code (intended for new construction) and, basedupon these studies, decide whether the structure needed:(a) no retrofit; (b) to be retrofit or; (c) to be demolished.In many instances, the retrofit plan must be designed tosatisfy the strength and deformation requirements of abuilding code that did not have specific provisions forexisting buildings. Many structural engineers worldwidein earthquake-prone areas consider this procedure an oddstrategy, as there is no warranty that it would lead togood retrofit plans. Fortunately, this old practice is nolonger accepted as “good practice” .

Some efforts have been directed to develop code pro-cedures to evaluate existing structures in the past twodecades, particularly after the 1989 Loma Prieta Earth-quake. It is worth noting the efforts made by the ABKgroup for the evaluation of existing masonry structuresoutlined in the ABK methodology [17], that impactedthe appendix C of the UCBC code [18] and the effortsdirected to develop the NEHRP Guidelines for the Seis-mic Rehabilitation of Buildings (FEMA-273) [19],where specific procedures are described for the seismicevaluation of structures introducing performance-basedconcepts.

The ABK methodology [17] presents a methodologyfor analysis/retrofit that mitigates seismic hazards inexisting unreinforced masonry (URM) structures. This

methodology is based on research that includes: (a) cate-gorization of URM buildings; (b) seismic input; (c)dynamic testing of full-scale URM walls, out-of-plane;(d) static and dynamic testing of URM, in-plane; (e)static and dynamic testing of full-scale diaphragms, in-plane; (f) the performance of URM buildings in pastearthquakes; and (g) analysis methods that have beencorrelated with the described tests.

FEMA-273 document provides systematic guidanceenabling design professionals to formulate effective andreliable rehabilitation approaches that will limit theexpected earthquake damage to a specific range for aspecified level of ground shaking [19]. Among othermethods of evaluation, FEMA-273 document addressesspecific linear and nonlinear procedures for seismicanalysis. The nonlinear static procedure (NSP) proposedby FEMA-273 is based in part on the assumption thatthe response of a building can be related to the responseof an equivalent SDOF system (FEMA-274) [20]. Theproposed modifications factors of the NSP outlined inFEMA-273, C1, which relates the expected maximuminelastic displacements to displacements calculated forlinear response, and C2, that represents the effect of stiff-ness degradation and strength deterioration on maximumdisplacement response, are based on studies conductedprimarily in SDOF systems and in experimental investi-gations [20]. It should be noted, as done in FEMA-274,that the proposed values and curves for C1 and C2 aremean relations, and that considerable scatter exist aboutthe mean. Therefore, FEMA-273 does not includedirectly nonlinear dynamic analyses as the proposedDDDS does. In addition, it is clear that site effects can-not be addressed directly when using the NSP of FEMA-273, as it can be done with DDDS.

5. Seismic evaluation of structures using DDDS:case studies

The concept of a DDDS and a procedure suggestedfor the seismic evaluation of existing structures is illus-trated in following sections for two structures that theauthor has evaluated using other procedures, in orderthat the interested reader could judge the advantages andthe shortcomings of the DDDS for simplified seismicevaluations of structures. More structures have beenstudied using this procedure, studies that the interestedreader may consult elsewhere [21].

AE2 building. The subject building, located near Ala-meda Park in downtown Mexico City, was a ten-storyoffice building that was built in the 1950s according tothe provisions of Mexico’s 1942 Federal District Code,the details of which are presented elsewhere [22]. Theoriginal steel structure consisted of ordinary momentresisting frames (OMRF) in both orthogonal directions.All original connections are riveted. The original foun-


Fig. 3. Plan view of the existing MBF retrofit for AE2 building (dimensions in meters).

dation system is mixed and consists of a 4.8 m deep boxfoundation over point-bearing piles.

At the time of the 1985 Michoacan earthquake, thestructure consisted of 13 stories and a total height of44 m. The structure under these conditions experiencedmoderate structural damage during the earthquake, dueto its flexibility and torsional response.

Because of the poor performance during the 1985Michoacan earthquake, the building was retrofitted in1990 by removing the three-story addition and by addingstiff, “macro” braced frames (MBF), as depicted in theplan in Fig. 3 and in the elevation in Fig. 4.

The seismic evaluation of the original structure(ORIG), the building in its condition at the time of the1985 Michoacan earthquake (APEN), the existing retro-fit with MBF (MACRO) and an alternate retrofit planwith Added Damping and Stiffness energy dissipationdevices (ADAS) have been subject of detailed studieswhich are summarized in Tena-Colunga and Vergara[22]. The following analyses were conducted: (a) three-

Fig. 4. Elevations of the existing MBF retrofit, AE2 building (dimensions in meters).

dimensional elastic analyses; (b) limit analyses; and (c)nonlinear dynamic analyses for postulated site groundmotions for a Ms=8.1 earthquake. The artificial N–Saccelerogram for the Alameda Park site (ALAM-NS)and its associated response spectrum for 2% viscousdamping used in the study of reference are depicted inFig. 5.

For this building, it will be assessed what could beconcluded from DDDS computed for the ALAM-NSrecord for a 2% viscous damping to evaluate the globalresponse of the four models (ORIG, APEN, MACROand ADAS) in the N–S direction. For this study it wasassumed for all models that: (a) their lateral loadcapacity was computed from limit analyses [22]; (b)their elastic dynamic characteristics were defined from3D frequency analyses [22]; and (c) their ductilitycapacity could be assessed from visual inspections of thebuilding and our own engineering knowledge andexperience.

The computed DDDS are depicted in Fig. 6 for the


Fig. 5. Artificial ALAM-NS acceleration record for Alameda Park site in Mexico City for the September 19, 1985 Michoacan Earthquake.

Fig. 6. Displacement ductility demand spectra (DDDS) for AE2 building (“original” models) when subjected to the ALAM-NS record.

unretrofit models (ORIG and APEN) and in Fig. 7 for theretrofit models (MACRO and ADAS). Two hystereticmodels have been considered in the study, an ELP modeland the Ramberg-Osgood model (ROS). The lateral base

Fig. 7. Displacement ductility demand spectra (DDDS) for AE2 building (retrofit models) when subjected to the ALAM-NS record.

shear capacities computed from limit analyses in the N–S direction were [22]: V/W=0.108 (ORIG model);V/W=0.074 (APEN model); V/W=0.352 (MACROmodel); and V/W=0.191 (ADAS model), where W is the


weight of each model as shown in Table 1. The dynamiccharacteristics of each model are summarized in Table1 of which are presented elsewhere [22]. It can beobserved from Table 1 that, for models ORIG andAPEN, the first mode shapes are coupled and, therefore,one cannot define a dominant N–S first mode shape, soit will be more rational to define a period range, asmarked with broken vertical lines in Fig. 6. The naturalperiods in the N–S direction for the MACRO and ADASmodels are also depicted with broken vertical lines inFig. 7.

It can be observed from Fig. 7 that the maximum dis-placement ductility demands for model ORIG with theELP model are high (m�5). In addition, the lateral dis-placements are high and, given the characteristics ofdetailing of the original connection plus the failuremechanism observed from limit analyses, then, theobservation of the DDDS could lead us to conclude thatthe original structure could have collapsed during aMs=8.1 earthquake similar to the 1985 Michoacan earth-quake, as significant deformation and ductility demandsare detected. This observation correlates well with theresults obtained from a nonlinear dynamic analysis ofrepresentative frame models [22]. If one considers thatthe structure possess a better ductility characteristics andthus, a ROS model is more representative, then, the dis-placement ductility demands and lateral displacementsobtained from the DDDS are significantly reduced andsuggest an important nonlinear response for the struc-ture, but perhaps in acceptable bounds. However, Theoriginal steel elements and the seismic detailing of theirconnections cannot warrant that a ROS model is rep-resentative of the global hysteretic behavior of this build-ing.

It can also be observed from Fig. 6 that perhaps itwas beneficial for AE2 building to have three additional

Table 1Dynamic characteristics for the 3D models of AE2 building

Model W (Ton) Mode Period (s) Modal mass (%)

N–S E–W Rotation

Original 8784 1. Coupled 1.96 26.19 37.82 13.732. Coupled 1.83 25.00 42.46 11.073. Coupled 1.17 26.95 0.03 47.89

Appendix 11,198 1. Coupled 2.64 37.97 17.09 19.552. Coupled 2.43 11.26 62.36 4.783. Coupled 1.61 25.67 0.02 47.07

MBF 9015 1. N–S 0.90 78.12 0.44 0.652. E–W 0.81 0.42 89.21 0.023. Torsion 0.54 0.92 0.05 72.00

ADAS 8907 1. N–S 1.19 55.16 13.49 7.192. E–W 1.24 10.85 64.44 2.033. Torsion 0.86 10.11 0.05 63.01

stories (APEN model) in order to survive a Ms=8.1 earth-quake, because the dominant periods of the structurewere augmented, leading the structure to a region werethe displacement ductility demands are reduced despitethe fact that the relative lateral strength diminished, evenwhen an ELP hysteresis model is considered (3�m�4). However, the displacement ductility demands are stillimportant for a structure with the described seismicdetailing and a failure mechanism that is not completelyductile [22], in addition to the fact that lateral displace-ments are still important. Therefore, structural damagecould be expected for this building. In fact, structuraldamage was observed for the AE2 building during the1985 Michoacan earthquake. APEN model correspondsto AE2 building at the time of this earthquake. Theresults computed for APEN model with the DDDS corre-late well with the results obtained from a nonlineardynamic analysis of a representative frame also [22]. Ifthe structure may have possessed a better hystereticbehavior (ROS model), then, the displacement ductilitydemands and the lateral displacement demands wouldhave considerably been reduced and the performance ofAE2 building could have been more satisfactory.

If one studies the expected nonlinear dynamicbehavior with the DDDS for the existing retrofit usingmacro braced frames (MACRO model, Fig. 7), then, itcan be concluded that the substantial increment in thelateral stiffness and strength is beneficial to improve theoverall seismic behavior of AE2 building, as the buildingis led to reduced ductility and deformation demands,with a small nonlinear action. The predicted globalresponse with the DDDS correlates well with the resultsobtained from a nonlinear dynamic analysis of represen-tative frames. However, the DDDS does not show apotential problem that can occur with the MBF retrofit,as the nonlinear action is due primarily by column yield-


ing and dynamic brace buckling [22], effects that canonly be observed using more detailed nonlineardynamic analyses.

If the alternate retrofit plan is evaluated using theDDDS (ADAS model, Fig. 7), it can be concluded thatthe ADAS retrofit is also a suitable solution, but that thissolution leads to higher displacement and displacementductility demands with respect to the MACRO retrofit.

If the results of the DDDS presented in Fig. 7 arecompared with those computed from nonlinear dynamicanalyses of representative frames presented in Tena-Col-unga and Vergara [22], the following observations canbe made :

1. Certainly, the story displacement ductility demandsfor the ADAS retrofit are higher than for the MACROretrofit (Fig. 8); however, the ductility demands forthe ADAS model are associated almost exclusively tothe yielding of the ADAS energy dissipation devices,whereas in the MACRO retrofit the ductility demandsare associated to column yielding and brace buckling(Fig. 9). These yielding mapping cannot be deductedwith the DDDS alone.

2. Although the story drift angles are slightly higher forthe ADAS retrofit than for the MACRO retrofit formost stories, the peak story drift angle occurs for theMACRO retrofit (Fig. 8).

3. The peak story displacement ductility demand for theADAS retrofit (m=3.8) is underestimated using aDDDS with an ELP model (m=3.19, Fig. 7) and sub-stantially underestimated using a ROS model(m=2.44, Fig. 7). In addition, the DDDS cannot evalu-ate that the ADAS devices work in the desired rangesof deformation.

Fig. 8. Maximum response envelopes for the retrofit models of AE2 building.

4. The shape of the global hysteresis curves associatedto the DDDS differ from those obtained using rep-resentative frames, where singularities associated tothe distribution of structural members affect the shapeof the hysteresis curves, particularly for the MACROretrofit model, where brace buckling occurs.

5. The DDDS do not show that the ADAS retrofit leadsto lower shear (Fig. 8) and axial forces at the basethan the MACRO retrofit.

Therefore, it can be concluded that the DDDS are usefulto estimate, with an acceptable level of approximation,peak displacement ductility demands and lateral dis-placements of complex structural models such as theADAS and MACRO retrofits, however, one cannot inferthe details of how the nonlinear action takes place and,sometimes, these details can be very important. Thus, ifone takes a decision for a given structural system basedexclusively on the results of a DDDS, then, there is arisk that this decision might not be the best from thestructural viewpoint.

5.1. Firehouse in Gilroy

The seismic response of this two-story, former fire-house (Fig. 10), located in Gilroy, California, during the1989 Loma Prieta Earthquake, has been the subject ofextensive studies that can be consulted elsewhere [23–26]. The firehouse is a box-type structure, where the lat-eral force resisting elements are URM walls tied togetherwith flexible timber floor and roof diaphragms. Detailson the structural components of the firehouse can be con-sulted in the references cited above.

The building was instrumented by the California


Fig. 9. Yielding mapping of a representative frame for the retrofit models of AE2 building.

Fig. 10. Southeast view of the firehouse in Gilroy.

Strong Motion Instrumentation Program (CSMIP) withsix accelerometers and, during the Loma Prieta Earth-quake, recorded accelerations in the east–west directionwere as high as 0.29 g at the ground level (Fig. 11),0.41 g at the top of the central wall and 0.79 g at themidspan of the roof diaphragm. The structure withstoodthe Loma Prieta Earthquake with little damage.

The detailed studies reported in Tena-Colunga andAbrams [24] showed that the dynamic response of the

firehouse was essentially elastic so it could be closelyrepresented using different elastic models that take intoaccount the diaphragm flexibility. Estimates of themaximum shear stresses using suitable 2D and 3D finiteelement models [24,25] showed that peak shear stresseswere considerably larger than the allowable shear stressvalues of the UCBC code [18] and, except a reducedarea where mild cracking was observed, these peakstresses were lower than the ultimate shear strength com-


Fig. 11. Recorded E–W ground accelerogram at the firehouse in Gilroy during the October 17, 1989 Loma Prieta Earthquake.

puted with the ABK methodology [17] using shove testdata [24]. Peak shear stresses surpassed larger wall areasconsidering the allowable shear stress values of the ACI530-92 code [27]. Thus, one may conclude that, accord-ing to the ABK methodology, the elastic responseobserved during the Loma Prieta Earthquake wasexpected, whereas the UCBC code (conservative innature) would suggest to some the possibility of animportant nonlinear response. The ACI 530-92 codecould suggest a moderate nonlinear response.

Therefore, it is interesting to evaluate the seismicresponse of the firehouse of Gilroy using a DDDS anda suitable hysteresis model for URM. The Kwok-Ang(KANG) model [14] was used for this purpose. The E–W direction of the firehouse was the most severelyshaken during the earthquake; thus, DDDS (Fig. 12)were computed using the E–W ground motions recordedat the site (Fig. 11). Base shear capacities for the E–Wdirection was computed from the shear stresses pre-scribed for the ABK methodology, the UCBC and theACI codes and summarized in Table 2. The computednormalized strengths were V/W=0.972 for the ABKmethodology, V/W=0.724 for the ACI code andV/W=0.180 for the UCBC code, W=1060 kips(481.2 Ton). The natural period of the firehouse in theE–W direction is T=0.453 s, defined from Fourier ampli-tude spectra computed from measured acceleration rec-ords and depicted in Fig. 12 with a broken vertical line.The assumed viscous damping for the structure was

Fig. 12. Displacement ductility demand spectra (DDDS) for the firehouse in Gilroy when subjected to the recorded E–W ground motions duringthe Loma Prieta Earthquake.

Table 2Shear stresses and strengths for the resisting walls in the E–W directionfor the firehouse of Gilroy (from Tena-Colunga and Abrams [24])

Wall Wall area (in.2) Shear stress/strength : psi (MPa)

ABK ACI 530-92UCBC code

methodology code

South 3264 81 (0.559) 55 (0.380) 15 (0.104)Central 4620 78 (0.539) 55 (0.380) 15 (0.104)North 6060 67 (0.463) 55 (0.380) 12 (0.083)

taken as x=0.10, in agreement with previous studies[23,24].

The results plotted in the DDDS for the firehouse inGilroy (Fig. 12) can lead to the following observationsregarding its seismic behavior in the E–W direction:

1. If the base shear capacity is computed with the ABKmethodology (ABK, KANG), then, the globalresponse should be elastic for the natural period of thestructure and its neighborhood. These results correlatewell with the observed behavior and with the detailedfinite element analyses of the firehouse describedabove. Also, the peak structural displacement com-puted with the DDDS is 3.04 cm. The peak relativelateral displacements between the roof and the baseat the central wall and at the center of the south dia-phragm computed from the double integration of the


acceleration records were 1.42 cm (0.56 in.) and3.30 cm (1.30 in.), respectively. Thus, the DDDS pre-dicts reasonably well the peak diaphragm displace-ment, as the structural response of the firehouse iscontrolled by the diaphragm response as reported inthe studies of reference. However, the wall displace-ment is substantially smaller and a DDDS cannot giveus insight on this, as the dynamic response of struc-tures with flexible diaphragms is complex andrequires of suitable models as those used and reportedin previous studies. Nevertheless, it is remarkable thatthe predicted global responses with the DDDS corre-late well with the observed and recorded response,and with analyses that are more complex.

2. If the base shear capacity is computed with the ACI530-92 code (ACI, KANG), then, the global responseshould also be elastic for the natural period of thestructure and its neighborhood. Although the globalagreement with respect to the detailed finite elementanalyses is good, the overstressed regions definedwith finite elements suggest the possibility of somenonlinear response. The displacement curve coincideswith the one computed for the ABK methodology, asin both cases the response is elastic.

3. If the base shear capacity is computed with the UCBCcode (UCBC, KANG), then, the global responseshould be strongly nonlinear for the natural period ofthe structure and its neighborhood. For this case, itwould have been more dangerous that the structurewould have a shorter natural period. Peak ductilitydemands would suggest substantial shear cracking inthe masonry walls, as also suggested with the over-stressed regions defined with the finite element analy-ses. As the firehouse did not experience such damageduring the Loma Prieta Earthquake, one may con-clude again that the allowable shear criterion of theUCBC code is very conservative, as is intended forthe evaluation of existing URM buildings. The shearstress criterion of the UCBC is mostly based on theuse of the shear strength computed from shove testsconsidering that 80% of the data exceeds the definedshear strength, that is a conservative statisticalapproach already. In fact, the equation provided bythe UCBC code to estimate allowable shear stressesis based upon the equation proposed by the ABKmethodology to estimate the shear capacity dividedby a safety factor of �5 [24].

Therefore, the DDDS was also helpful to visualize theexpected global response of the firehouse in Gilroy,although the details of the response of structures withflexible diaphragms can only be assessed with morecomplex models. Nevertheless, the DDDS can be usedto decide when an elastic model is sufficient to studythe response of masonry structures with flexible dia-phragms and when a more complex nonlinear model is

required. Thus, DDDS has a potential application forassessing the vulnerability of the large inventory ofURM structures that exist in the eastern and centralUnited States, and other nations worldwide, such asChina, India, Mexico, Colombia, Turkey, Italy, etc.

6. Conclusions

The concept of a DDDS and some procedures pro-posed for the seismic evaluation of existing structureswere illustrated with two structures that the author haspreviously evaluated using more detailed and complexdynamic analysis procedures, in order that the interestedreader could judge the advantages and the shortcomingsof the DDDS for simplified seismic evaluations of struc-tures.

The case studies presented herein allow one to con-clude that the DDDS are very useful to obtain reasonableestimates of peak displacement ductility demands andlateral displacements of structural systems where onecan assess, with some precision, their lateral strength,natural periods (or range of periods) and hystereticcharacteristics. If these can be done, one should carefullyevaluate the implications of uncertainties in the estimateof strengths, stiffnesses and hysteretic characteristics andthose of the ground motions in the global response ofstructures (particularly peak displacements) and relatethese responses with the seismic detailing of structures.Nevertheless, the DDDS do not allow one to visualizethe details of the nonlinear response of structures andoften, these details can be very important, particularlyfor complex structures or structural systems, so oneshould not rely on the use of a DDDS alone for the seis-mic evaluation of an existing structure, in order to avoidthe possibility of arriving to wrong solutions.

In fact, the DDDS should be regarded as an additionaltool in the seismic evaluation of existing structures thatcould lead us to good solutions based upon our engineer-ing background, judgment and ethical conduct. With theDDDS, one can decide when more complex nonlinearanalyses should be done and when this is not necessary,for example, when elastic responses are detected withthe DDDS. For the latter case, one can use, dependingon the complexity of the structure, suitable elastic mod-els with confidence.

Acknowledgements

Luis Eduardo Perez-Rocha generated all artificialacceleration records presented in this study. The Califor-nia Strong Motion Instrumentation Program provided theacceleration records for the firehouse of Gilroy. Alejan-dro Vergara participated in the detailed nonlinear studiesof building AE2 as a part of his BS thesis directed by


the author. The additional comments of anonymousreviewers of the paper were very helpful to improve itand are gratefully acknowledged.

References

[1] Mahin SA, Bertero VV. An evaluation of inelastic seismic designspectra. ASCE J Struct Engng 1981;107(ST9):1777–95.

[2] Pal S, Dasaka SS, Jain AK. Inelastic response spectra. ComputStruc 1987;25(3):335–44.

[3] Newmark NM, Hall WH. Earthquake spectra and design. 1st ed.Oakland, CA: Earthquake Engineering Research Institute, 1982.

[4] Riddell R, Newmark NM. Statistical analysis of the response ofnonlinear systems subjected to earthquakes. Structural ResearchSeries No. 468, Department of Civil Engineering, University ofIllinois at Urbana-Champaign, 1979.

[5] Iwan WD. Estimating inelastic response from elastic spectra.Earthquake Engng Struct Dyn 1980;8:375–88.

[6] Krawinkler H, Nassar A, Rahnama M. Damage potential of LomaPrieta ground motions. Bull Seismol Soc Am 1991;81(5):2048–69.

[7] Miranda E. Site-dependent strength-reduction factors. ASCE JStruct Engng 1993;119(12):3503–19.

[8] Vidic T, Fajfar P, Fischinger M. Inelastic design spectra: strengthand displacement. Earthquake Engng Struct Dyn 1994;23:507–21.

[9] Ordaz M, Perez-Rocha LE. Estimation of strength-reduction fac-tors for elastoplastic systems: a new approach. Earthquake EngngStruct Dyn 1998;27:889–901.

[10] Miranda E. Evaluation of site-dependent inelastic seismic designspectra. ASCE J Struct Engng 1993;119(5):1319–38.

[11] Tena-Colunga A. Espectros inelasticos de demandas de ductilidadpara la evaluacion de estructuras existentes ante sismos. In: Pro-ceedings, XI Congreso Nacional de Ingenierıa Sısmica, Veracruz,Mexico, November, vol. I, 1997. p. 671–682 [in Spanish].

[12] Saiidi M, Sozen MA. Simple and complex models for nonlinearseismic response of reinforce concrete. Civil Engineering Studies,Structural Research Series No. 465, University of Illinois atUrbana-Champaign, August, 1979.

[13] Kunnath SK, Reinhorn AM, Park YK. Analytical modeling ofinelastic seismic response of R/C structures. ASCE J StructEngng 1990;116(4):996–1017.

[14] Kwok YH, Ang AH-S. Seismic damage analysis and design ofunreinforced masonry buildings. Civil Engineering Studies,Structural Research Series No. 536, University of Illinois atUrbana-Champaign, June, 1987.

[15] Tomazevic M, Lutman M. Seismic behavior of masonry walls:modeling of hysteretic rules. ASCE J Struct Engng1996;122(9):1048–54.

[16] Miranda E. Simplified method to estimate maximum seismicinterstory drift demands in multi-story buildings. In: Proceedingsof the Sixth US National Conference on Earthquake Engineering,Seattle, June, CDROM media, 1998.

[17] Methodology for mitigation of seismic hazards in existing unre-inforced masonry buildings: the methodology. ABK TechnicalReport 08, El Segundo, California, December, 1984.

[18] UCBC. Seismic strengthening provisions for unreinforcedmasonry bearing wall buildings. Structural Engineers Associationof California (SEAOC), June, Appendix chap. 1, 1990.

[19] FEMA-273. NEHRP guidelines for the seismic rehabilitation ofbuildings. FEMA Publication 273. Washington, DC: FederalEmergency Management Agency, October, 1997.

[20] FEMA-274. NEHRP commentary on the guidelines for the seis-mic rehabilitation of buildings. FEMA Publication 274. Wash-ington, DC: Federal Emergency Management Agency, October,1997.

[21] Tena-Colunga A. Evaluacion sısmica simplificada de estructurasexistentes. Rev Ingenierıa Sısmica SMIS 1998;59:1–36 [in Span-ish].

[22] Tena-Colunga A, Vergara A. Comparative study on the seismicretrofit of a mid-rise steel building: steel bracing vs energy dissi-pation. Earthquake Engng Struct Dyn 1997;26(6):637–45.

[23] Tena-Colunga A. Seismic evaluation of unreinforced masonrystructures with flexible diaphragms. Earthquake Spectra1992;8(2):305–18.

[24] Tena-Colunga A, Abrams, DP. Response of an unreinforcedmasonry building during the Loma Prieta earthquake. StructuralResearch Series No. 576, Department of Civil Engineering, Uni-versity of Illinois at Urbana-Champaign, December, 1992.

[25] Tena-Colunga A, Abrams DP. Simplified 3D dynamic analysisof structures with flexible diaphragms. Earthquake Engng StructDyn 1995;24(2):221–32.

[26] Tena-Colunga A, Abrams DP. Seismic behavior of structureswith flexible diaphragms. ASCE J Struct Engng1996;122(4):439–45.

[27] Specifications for masonry structures (ACI 530-92, TMS 602,ASCE 6-92). The Masonry Society, 1992.

Displacement Ductility Demand Spectra

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