A Methodology for the Probabilistic Assessment of Behaviour Factors

Bull Earthquake Eng (2010) 8:4764DOI 10.1007/s10518-009-9126-5

ORIGINAL RESEARCH PAPER

A methodology for the probabilistic assessmentof behaviour factors

Anbal Costa Xavier Romo Carlos Sousa Oliveira

Received: 8 August 2008 / Accepted: 10 May 2009 / Published online: 28 May 2009 Springer Science+Business Media B.V. 2009

Abstract Given the importance that traditional force-based seismic design still currentlyexhibits, studies addressing issues related to the definition of the behaviour factor values areconsidered to be of most interest. A probabilistic methodology is proposed for the calibrationof the q-factor relating its value with two fundamental parameters, the displacement ductilitycapacity measured at a relevant location of the structure and the failure probability Pf . Thegeneral foundation of this procedure is based on the probabilistic quantification of the seis-mic action and, by applying a transformation procedure, of the structural seismic demandin terms of displacement ductility. By recalling well established structural reliability proce-dures and by making use of nonlinear analysis methods, both static and dynamic, a generalprobabilistic framework, which is able to relate the ductility capacity, the failure probabilityPf and the behaviour factor, is defined. In order to illustrate some of the potentialities of themethodology, an application example is presented, addressing the q-factor assessment for aset of regular and irregular reinforced concrete frame structures, enforcing a given Pf andtwo different ductility levels.

Keywords Behaviour factor Ductility Probabilistic analysis Seismic design

A. CostaCivil Engineering Department, Universidade de Aveiro, Campus Universitrio de Santiago,3810-193 Aveiro, Portugal

X. Romo (B)Civil Engineering Department, Faculdade de Engenharia da Universidade do Porto,Rua Dr. Roberto Frias, 4200-465 Porto, Portugale-mail: [email protected]

C. S. OliveiraCivil Engineering and Architecture Department, Instituto Superior Tcnico, Avenida Rovisco Pais,1049-001 Lisboa, Portugal

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48 Bull Earthquake Eng (2010) 8:4764

1 Introduction

Although the development of seismic design methodologies based on deformations is aconcept that has been gaining considerable interest over the past few years (Priestley et al.2007), seismic design methodologies of current codes still follow the traditional force-baseddesign approach. The essential feature of force-based seismic design is the behaviour factorq , hereon simply termed q-factor. Fundamentally, the q-factor is, according to Eurocode 8Part 1 (EC8-1) (CEN 2004), a factor used in design to reduce forces obtained from a linearanalysis in order to account for the nonlinear response of the structure associated with thematerial, the structural system and the design procedures. Hence, structures are designed forequivalent seismic lateral forces that are smaller than those of an elastic structure, assumingthat the structure possesses adequate ductility in order to dissipate energy through inelasticdeformations. In more practical terms, the force-based seismic design procedure currently setby EC8-1 defines lateral design forces based on an inelastic design spectrum that is obtainedfrom the elastic design spectrum scaled by the selected q-factor. For the purpose of the studypresented herein, it is considered that scaling elastic forces based on an elastic response spec-trum with the q-factor, and obtaining design forces based on an inelastic design spectrum asset in EC8-1, are equivalent approaches.

Since seismic design forces have a direct relation to the value adopted for this parame-ter, research issues related to the q-factor are of great interest and have been the focus ofconsiderable attention over the years. Comprehensive reviews of research on the q-factorcan be found, for example, in Miranda and Bertero (1994), CEB (1997), Kappos (1999),Mwafy and Elnashai (2002), or Maheri and Akbari (2003). Furthermore, research studiesfocusing on the influence that structural parameters, such as strain-hardening, strength orstiffness degradation, pinching effects or other hysteretic characteristics, have on the q-factorcan be found in Lee et al. (1999), Borzi and Elnashai (2000) or Miranda and Ruiz-Garcia(2002). The effects of structural regularity and of the ductility class were also addressed inZeris et al. (1992) and in Salvitti and Elnashai (1996). Moreover, it has been also seen thatestimating q-factors based on simplified empirical relationships accounting for aspects suchas structural regularity and the expected ductility demand may lead to significantly differentoutcomes depending on the design codes (Booth et al. 1998). Although the referred studiesaddress the q-factor issue deterministically, probabilistic studies have also been carried out.Examples of such q-factor calibration studies can be found in Colangelo et al. (1995), whereconcepts of stochastic linearization were applied, in Bento and Azevedo (2000), where thevulnerability function methodology (Duarte 1991) was considered, in Chryssanthopoulos etal. (2000), where a simulation study accounting for material variability, confinement uncer-tainty, and local and global failure criteria uncertainty was carried out for several limit states,and in Thomos and Trezos (2005), where a simulation study based on pushover analysis andaccounting for material variability was developed.

Following the line of the referred probabilistic studies, the present work proposes a prob-abilistic methodology to calibrate the value of the q-factor that includes two fundamentalparameters: the displacement ductility capacity measured at a relevant location (e.g. at theroof level) and the failure probability Pf .

2 Brief review of general concepts

Before presenting the proposed q-factor calibration methodology, a short review of some ofthe concepts entering the referred methodology is exposed in the following. Aspects such

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Bull Earthquake Eng (2010) 8:4764 49

as the definition of the q-factor and its relation with other relevant parameters, the relationbetween earthquake demand and structural demand in the considered method and the selectedapproach to characterize structural safety are addressed herein.

2.1 The q-factor and its relation with other important parameters

For the purpose of the proposed methodology, the q-factor is considered to be generallydefined by the following expression, similar to the EC8-1 proposal:

q = q (1)where q represents a basic value of the q-factor, dependent on the type of structural systemand on its regularity, and represents a general overstrength factor. In a design situationgoverned by seismic strength demand, the considered overstrength is equivalent to theu/1 factor defined by EC8-1, in which 1 is the multiplicative factor of the seismic designaction leading to the first yield in any structural member, Fy,1, with all other design actionsremaining constant, and u is the multiplicative factor of the seismic design action leadingto the failure mechanism of the structure, FN L , with all other design actions remaining con-stant. Hence, for this design situation, Fy,1 corresponds to the design action Fd , which isobtained by scaling the elastic base shear FL , obtained from the elastic response spectrum,by the q-factor. However, design is in many cases governed by other considerations likegravity loading scenarios or other deformation-related criteria such as 2nd order effects orserviceability drift limits. In these cases the value of Fy,1 is usually larger than the seismicdesign action Fd (Elnashai and Mwafy 2002; Elghazouli et al. 2008).

Within the proposed methodology, it is considered that Fig. 1a represents the globallateral response curves of a structure exhibiting linear elastic and nonlinear behaviour. In thisschematic representation dL represents the elastic control displacement corresponding to theelastic base shear FL , considering that the structure possesses an elastic stiffness KL . Fur-thermore, dy,1 represents the control displacement corresponding to the first yield base shearFy,1, dN L represents the expected nonlinear control displacement demand for the designseismic action, considering that the structure possesses an equivalent secant post-first-yieldstiffness KN L . The referred nonlinear curve is purely conventional for the purpose of theproposed method and must not be confused with the possible bi-linearization of the capacitycurve obtained from pushover analysis. Figure 1b presents the curves of Fig. 1a overlappedwith a capacity curve obtained from pushover analysis and the corresponding idealized bi-linearization. It must be emphasized that the referred bi-linearized capacity curve is that ofthe original structure and is not assumed to correspond to the behaviour of an equivalentsingle of degree of freedom system as in the Capacity Spectrum method (ATC 2005) or inthe N2 method (CEN 2004). Observation of Fig. 1b shows that a conventional yield displace-ment dy is now introduced as a result of the pushover curve bi-linearization. The setting ofdy is fundamental as it will be the basis for the definition of the ductility capacity of thestructure. In the example of Fig. 1b, the selected bi-linearization approach is that of EC8-1(CEN 2004), i.e. considering the criterion of equal energy at maximum displacement with azero post-yield stiffness. Nonetheless, other codes might suggest different approaches, e.g.(ATC 1996, 2005; FEMA 2000), hence leading to different definitions of dy . By observationof Fig. 1, and considering the q-factor definition set by Eq. 1, it is possible to relate the severalrelevant parameters according to the following relations

FLFd

= q (2)

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FL

KLF

dd d dy,1

FNL

Fy,1

L NL

KNL

Fd

FL

KLF

dd d dy,1

FNL

Fy,1

dy L NL

KNL

bi-linearized capacity curveoriginal capacity curve

Fd

(b)

(a)

Fig. 1 a Global lateral response curves of a structure considering linear elastic and nonlinear behaviour withb the capacity curve obtained from pushover and its bi-linearization

FN LFd

= (3)FL

FN L= q (4)

= dN Ldy

(5)

2.2 The relation between earthquake demand and structural demand

The adequate definition of the evolution of the average chosen structural demand parameter,i.e. the displacement ductility D at the level of the control displacement, for increasing val-ues of the seismic intensity measure (IM) is an important feature of the proposed method. Inthe past, this functional relation has been termed vulnerability function (Duarte 1991; Bentoand Azevedo 2000), while, in the course of more recent research carried out on seismic riskassessment, the term incremental dynamic analysis (IDA) curve (Vamvatsikos and Cornell2002) has been extensively used and is also considered hereon.

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Fig. 2 Graphical interpretationof the transformation of f I M intofD

Shaded areas are equal

IDA curve

im

f (im)IM

f () D

Assuming that the probability density distribution (PDF) of the earthquake action f I M interms of a given earthquake IM is known and considering also that the average IDA curverelating the IM and the control displacement ductility D obtained from structural analysiscan be defined, the PDF of D , fD , can then be derived by standard probability transforma-tion. Figure 2 presents a graphical interpretation of the referred probability transformationwhile details can be found e.g. in Benjamin and Cornell (1970). The reader is referred toCornell et al. (2000) and Romo et al. (2008) for further applications of this procedure for thecharacterization of the probabilistic distribution of a chosen demand parameter. Furthermore,Romo et al. (2008) present a study on the adequate mathematical forms of the IDA curvesuitable for analytical treatment of the probability transformation, according to the type ofdemand parameter.

2.3 The selected structural safety approach

As previously referred, one of the central parameters within the proposed methodology is thefailure probability Pf . Given that the probabilistic distribution of the displacement ductilityD at the level of the control displacement fD , representing the structural demand, can beobtained by the procedure previously outlined, and assuming that the probabilistic distribu-tion representing the capacity C in terms of that ductility can be estimated, the value of Pfcan then be obtained by the classical reliability formulation expression:

Pf =

0(1 FD ()) fC () d (6)

in which FD () is the cumulative distribution function (CDF) of the maxima of the ductilityD , fC () is the PDF of C , assuming that the ductility demand and C can be consideredindependent variables (a commonly considered assumption though sometimes approximate(Pinto et al. 2004)).

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3 Description of the proposed methodology for q-factor calibration

A detailed description of the proposed probabilistic methodology for calibration of theq-factor is presented in the following. As previously referred, the two fundamental parame-ters of the methodology are the displacement ductility capacity at the level of the controldisplacement and the failure probability Pf .

The proposed methodology comprises a total of six steps that are generally described inthe following. Next, a discussion is presented concerning the different analyses that can beperformed with the proposed approach, depending on the type of assessment that is chosen.

3.1 General steps of the proposed methodology

The first step corresponds to a standard force-based design of the structure, based on linearanalysis values and considering a q-factor selected on the basis of code prescriptions. Thisdesign must define all the necessary data in terms of structural member dimensions anddetailing in order to be able to develop a numerical nonlinear model of the structure.

In the second step, the pushover analysis of the structure is carried out, using the previ-ously referred nonlinear model. The fundamental objective of this analysis is to define thecapacity curve of the structure at the selected control level that will lead to an estimate of itsconventional yield displacement dy . It is emphasized that the control level is not necessarilylocated at the roof level of the structure since the role of pushover analysis within the pro-posed framework does not include the application of methods such as the Capacity Spectrummethod (ATC 2005) or the N2 method (CEN 2004) where this requirement is enforced. Inthe proposed framework, the control level is that which leads to the largest displacementductility demand. Since the location of such level may not be known beforehand, capacitycurves must, therefore, be defined for all the possible levels. Moreover, according to mostcodes that allow the use of this type of analysis, there is the need to perform it for differentlateral load patterns. Namely, EC8-1 enforces the use of the uniform and of the modal loadingpatterns, each one applied for both positive and negative loading directions, which leads tothe computation of four capacity curves. An idealized bi-linearization of each capacity curveis then computed according to a chosen approach and a value of dy is extracted from eachcurve. Since the proposed procedure requires a single value of dy for a given control level i ,dy,i , and given that large variations between the several values are not expected, the averageof the four values is therefore considered.

The third step corresponds to the computation of the IDA curve of the average structuraldemand defined in terms of the control displacement ductility. To obtain this curve, a nonlinearmodel of the structure that is able to represent with sufficient detail the dynamic characteris-tics of the structural response must be defined. Then, a collection of adequate accelerogramsmust be set, either selected from an existing ground motion database or based on an arti-ficial generation procedure. For the case where the seismic action is defined according toEC8-1, the number of accelerograms must be at least seven and these must fulfil a number ofrestrictions in terms of spectral matching (CEN 2004). Nonlinear dynamic analyses are thencarried out to determine the structures peak response in terms of the control displacement forincreasing scaled intensities of those accelerograms. Again, since the location of the relevantcontrol displacement level may not be known beforehand, displacement demand must beconsidered at all the possible levels. For each IM level, the average of the absolute maximaof the displacements at the control levels resulting from each accelerogram must be recorded.To obtain the ductility demand at each level i , the referred average displacements must thenbe divided by the corresponding value of dy,i . At this point, the relevant control level, i.e.

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that which exhibits the highest average peak displacement ductility demand D , is able tobe identified. This step ends with the fitting of a mathematical expression to the evolution ofD for increasing values of the ground motion IM, which yields the required IDA curve.

The fourth step corresponds to the characterization of the probability distribution repre-senting the displacement ductility capacity. Several approaches can be selected to define suchdistribution ranging from more comprehensive and detailed methods to more simplified ones.For an example of a more comprehensive methodology that can be extended to the presentframework, the reader is referred to Thomos and Trezos (2006) where a simulation studyon the probabilistic response of reinforced concrete (RC) frames carried out using pushoveranalysis is presented. Alternatively, a more simplified approach such as the one proposed nextcan also be considered. Following the indications of Ferry Borges and Castanheta (1985), it isassumed that, from the capacity side, the main contribution to the computation of Pf comesfrom the lower tail of the probabilistic distribution of the capacity, and that such lower tail canbe well represented by the tail of a Normal distribution with a relatively small coefficient ofvariation. Therefore, for a chosen ductility level C , the corresponding Normal probabilitydistribution fC representing the capacity in terms of ductility is assumed to be defined by thelower characteristic value C,5% considered to be equal to C and by a standard deviation Cequal to 0.5 according to studies found in (Costa 1989). Such approach leads to coefficientsof variation ranging between 10 and 6% for values of C,5% between 4 and 7.

The fifth step corresponds to the computation of Pf by solving Eq. 6. To perform thiscalculation, the PDF of the earthquake action f I M must be transformed into the PDF of thecontrol ductility fD using the IDA curve, as previously referred. On the capacity side, theprobabilistic distribution fC is defined as in the previous step.

Finally, the sixth step is a general result assessment step based on which a revision of thedesign might be required or not. Figure 3 exhibits a general summary of the proposed meth-odology in which the more relevant parameters are represented, where the ductility value of 1represents the conventional yield and where SE,L stands for the elastic earthquake intensitylevel for which a certain return period is prescribed by the design code. Furthermore, anassessment of the basic q-factor value q and of the overstrength factor (see Eq. 1) mayalso be carried out in order calibrate the design code proposals. A more detailed analysis ofthis last step is presented in the following where the different analyses that can be performedwith the proposed approach are addressed.

3.2 Assessment analyses that can be performed with the proposed methodology

Three different q-factor assessment approaches that can be carried out with the proposedmethodology are addressed in the following. These approaches involve the same variablesbut differ in the parameters that are set and in those that are results of the approach.

In the first approach, the ductility level C , which is required for the definition of theprobability distribution fC representing the capacity, is not defined as a single ductility levelbut as a family of levels. Hence, in this approach, a family of probability distributions fC,irepresenting different levels i of capacity in terms of displacement ductility values C,i ischaracterized. Then, Pf is obtained for each ductility level by considering the different fC,idistributions and solving Eq. 6 for each case. Having obtained the Pf values for the wholerange of ductility values, the most adequate ductility level can be defined as that whichleads to an acceptable value of Pf . If is too high when compared to the design ductilityclass, this means that the structure is too flexible and the design should be based on a lowervalue of the q-factor. When observing Fig. 3, this redesign situation is equivalent to a shiftof fD towards the origin of the graph.

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f (im)IM

imF

f ()C

FL FNL Fy,1 Fd

1

C

q

SE,L

f ()D

Fig. 3 Graphical sketch of the proposed methodology

Instead of simulating a family of probabilistic distributions representing different levelsof ductility, a variant of this first approach can be defined by computing the ductility level directly leading to the desired Pf by solving a nonlinear equation. This equation is repre-sented by Eq. 6 where the value of Pf is set to a desired value and the terms 1 FD (.) andfC (.) must be defined analytically, the latter being a function of the objective ductility level. Eq. 6 is then solved using a standard Newton-Raphson method to obtain the ductilitylevel . For details on analytical forms of the term 1 FD (.) the reader is referred to Romoet al. (2008).

In the second approach, the ductility level C is defined according to the probability ofthe design seismic action. More precisely, the considered design code sets a return period forthe design level of the seismic action which corresponds to a certain probability of exceed-ance PC that can be associated to a life safety limit state. By integrating the PDF of theearthquake action f I M up to the value of 1 PC , one is able to define the ground motion IMvalue, I MC , corresponding to this exceedance probability. Next, by entering the IDA curvewith I MC , the resultant displacement ductility capacity C is obtained. By calculating thevalue of Pf for this ductility capacity, one obtains a failure probability associated to the lifesafety performance level which should be compared to an adequate limit value. Following asimilar line of reasoning, a further refinement of this approach can be developed, for example,to address a ductility level which corresponds to an ultimate displacement dult associatedto a near collapse limit state. By considering the proposal found in EC8-1 (CEN 2004)associated to the capacity curve obtained from pushover analysis, it is possible to see that an

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estimate of dult can be seen to correspond to what is defined as the mechanism point ofthe capacity curve, where the mechanism point is expected to represent a state of incipientcollapse of the structure. Alternatively, in a situation where 2nd order effects are importantand the capacity curve starts to exhibit a negative slope, the estimate of dult can be set tocorrespond to an admissible drop of the maximum strength, e.g. 15%. In any case, once dultis set, the corresponding value of Pf can be obtained and compared to an adequate limitvalue. In any of these two situations, if the computed Pf is too high, this means that thestructure is too flexible and the design should be based on a lower value of the q-factor.

The third approach is an optimization variant of the second one where a selected ductility, based on a design criterion other than the one related to the probability of the groundmotion is enforced and must lead to a specified Pf . Based on the initial design of the structurewith a chosen q-factor and considering the selected ductility , the value of Pf is computed.If Pf does not match the required value, the structure is redesigned for a new value of theq-factor (lower or higher than the initial one) and the process is restarted. The process stopswhen Pf matches the desired value, hence leading to the corresponding q-factor.

In any of the approaches, the end results are a ductility level that leads to an acceptableor specified value of Pf and a corresponding value of the q-factor. Based on this resultantq-factor and on the ductility level , a further assessment can also be performed. By enteringthe previously obtained capacity curves resulting from the pushover analysis with the dis-placement associated to , the nonlinear force level FN L can be obtained. Based on Eq. 3and on the design base shear Fd , the apparent overstrength can be obtained. The termapparent is considered herein assuming that the first yield base shear Fy,1 does not coincidewith Fd , hence the real overstrength r would be

r = FN LFy,1 (7)

and the term Fy,1/Fd would represent a measure quantifying the amount of design strengththat is not a direct result of seismic strength demand. Moreover, based on the value of FN Land on the value of the elastic base shear FL , an estimate of q , the basic value of the q-factor,can also be obtained (see Eq. 4). For the case of structures designed according to EC8-1, thisassessment is of considerable importance since this component is usually defined based onsimplified rules.

4 Example application of the proposed methodology

In order to illustrate some of the potentialities of the proposed methodology, an exampleapplication is presented herein for a set of regular and irregular RC frame structures. Forconciseness sake, only a sample of the results is presented herein. Hence, based on previousanalyses developed for buildings similar to those considered herein (Pinto 1994), only oneof the main directions of the buildings is analysed.

In terms of the referred q-factor assessment approaches that can be performed with theproposed methodology, the present application corresponds to the third one, where two duc-tility levels of 4 and 7 are enforced and must lead to a Pf with a value of 105. This Pfvalue is seen to be adequate for ordinary buildings, considering the structures lifetime to be50 years (Costa 1993).

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Fig. 4 Plan view of theconsidered frame buildings

5 x 4.0 = 20.0m3

x 5.

0 =

15.0

m

X

4.1 General description of the structures

The considered set of structures comprises ten different RC frame structures with four andeight storeys. Three of these structures correspond to building typologies that are regularin elevation and that served as the basis for the definition of the structures having irregularprofiles in elevation. All regular cases have the same in-plan dimensions according to theplan view showed in Fig. 4 and have a constant inter-storey height of 3.0 m. The severalbuilding types considered are identified according to the following terminology, which is setbased on their height and first mode period:

ED4F1building with four storeys and with a first mode period corresponding to theconstant acceleration branch of the response spectrum;

ED4F2building with four storeys and with a first mode period corresponding to theconstant velocity branch of the response spectrum;

ED8building with height storeys.The irregular structures were defined so as to represent commonly found irregular eleva-

tion profiles. A schematic view of their profiles is presented in Fig. 5 for the longitudinaldirection (the longer of the two in-plan dimensions identified as direction X in Fig. 4), whichis the direction of analysis considered for the present application example, along with theregular building profiles. Although some of the lateral profiles are termed irregular, it shouldbe noted that their geometry still complies with the criteria for regularity in elevation set byEC8-1 (CEN 2004). The identification of the type of lateral profile is defined according tothe following terminology:

Rbuilding with a regular profile, Fig. 5a, for the case of the four-storey structure, andFig. 5d, for the case of the eight-storey structure;

I1building with an irregular profile according to Fig. 5b, for the case of the four-storeystructure, and Fig. 5e, for the case of the eight-storey structure;

I2building with an irregular profile according to Fig. 5c, for the case of the four-storeystructure, and Fig. 5f, for the case of the eight-storey structure;

I3building with an irregular profile according to Fig. 5g for the case of the eight-storeystructure only.

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R I1 I2

R I1 I2 I3

(a) (b) (c)

(d) (e) (f) (g) Fig. 5 Side view of the regular and irregular profiles of the considered buildings

Table 1 Structural member cross sections for the considered buildings

Building Column cross section by storey h b (cm cm) Beam cross sectionsh b (cm cm)04 (storey levels) 48 (storey levels)

ED4F1 80 80 25 60ED4F2 30 30 25 60ED8 80 80 60 60 25 60

In terms of materials, a concrete of class C20/25 and a steel of class S400 were selected.For the nonlinear analysis cases, mean values of material strength are considered rather thanthe values used in the design. With respect to the loading, outside vertical loads correspond-ing to the self-weight of the structural members, vertical loads were considered to simulatethe weight of a slab with a thickness of 15 cm, the weight of the finishes and of the masonryinfills (2.5 kN/m2), and a live load of 2.0 kN/m2 considered with its quasi-permanent value.A summary of the cross section dimensions of the several structural members of the wholeset of buildings is presented in Table 1 while Table 2 presents the first mode period of theconsidered buildings for the longitudinal direction. With respect to the data presented inTable 1, it should be emphasized that the columns of structure ED4F1 were considered withlarger cross section dimensions than required for this type of structure in order to obtain afirst mode period that falls in the constant acceleration branch of the response spectrum.

4.2 Structural modelling assumptions

In terms of numerical modelling, an equivalent planar structural model for the direction ofinterest of the considered buildings was developed. The several frame components of eachbuilding were lined up in a two-dimensional plane with consecutive components linked by

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Table 2 First mode period of theconsidered buildings for thelongitudinal direction

Building Period (s)

ED4F1R 0.313ED4F1I1 0.282ED4F1I2 0.278ED4F2R 0.764ED4F2I1 0.676ED4F2I2 0.662ED8F1R 0.699ED8F1I1 0.662ED8F1I2 0.649ED8F1I3 0.641

rigid members at each storey level. The horizontal displacements of each floor are, thus,slaved so as to have only one horizontal degree of freedom per floor, which accounts for theassumed rigid slab behaviour of the floors.

Nonlinear response analysis of the numerical models under earthquake loading was car-ried out using an analysis programme developed during previous research studies, (Varum1997; Rodrigues 2005). The programme is a two-dimensional analysis platform for the studyof the nonlinear response of multi-storey buildings. Beam-column elements are modelled asmember-type nonlinear macro-models with three zones: one internal zone with linear elasticbehaviour and two plastic hinges, located at the member ends, where inelastic flexural behav-iour is considered. Nonlinear dynamic analyses are carried out using the standard Newmarkintegration method and considering an event-to-event strategy with modification of the struc-tures stiffness matrix at each event. Prior to the dynamic analysis, a static analysis is carriedout for the vertical loads corresponding to the gravity loads acting on the structure, the resultsof which become the initial conditions for the dynamic analysis. Damping was assumed tobe of Rayleigh type with parameters computed for the first and second mode periods of thestructures and a fraction of critical damping equal to 3% for both periods.

Inelastic behaviour of the beam-column elements is represented at the member level bymomentcurvature relations. Trilinear skeleton curves associated with monotonic loading,considering asymmetric bending for beams and axial load effects for columns, were obtainedfollowing the work presented in Arde and Pinto (1996). Hysteretic flexural behaviour of themembers was modelled by the piecewise linear hysteretic Costa-Costa model, (CEB 1996),which is a generalized Takeda-type model. Plastic hinge length values were considered equalto the depth of the member cross section for beams and equal to half of the depth of themember cross section for columns.

4.3 Seismic scenario

Both the current Portuguese code (RSA 1983) and the upcoming National Application Doc-ument of EC8 enforce the consideration of intraplate and interplate earthquake scenarios.The intraplate scenario is characterized by earthquakes having a smaller epicentral distance,a higher intensity and a shorter duration, while the interplate scenario is defined by earth-quakes having a larger epicentral distance, a moderate intensity and a longer duration. Forconciseness sake, the current application only refers to the intraplate scenario, considering

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a soil of type B and a seismic intensity level for the city of Lisbon which corresponds to apeak ground acceleration of 0.17g.

A set of seven artificial spectrum-compatible accelerograms with 10 s each was defined toevaluate seismic demand using nonlinear dynamic analysis and to compute the IDA curvesof the several structures. The artificial accelerograms were computed in order to meet theseveral spectral-matching requirements defined by EC8-1.

In terms of probabilistic characterization, the PDF of the earthquake action f I M can beseen to be well defined by an Extreme type probabilistic distribution (Costa 1993; Bento andAzevedo 2000; Romo et al. 2008). For the present case, f I M was considered to be wellrepresented by a Gumbel distribution with a PDF defined by

f I M (x) = e(e(xu)(xu)) (8)

where and u are the parameters of the distribution taken with values of 0.02249 and 87.36,respectively, based on hazard studies for the Lisbon area (Costa 1993).

4.4 Seismic demand results and IDA curves

For completeness, the general behaviour of the considered structures under earthquake load-ing and the resulting IDA curves are briefly addressed herein. In terms of seismic response,ductility demand was generally observed to be more concentrated at the lower storeys. Hence,for the generality of the considered cases structures, the control displacement was defined asbeing located at the first storey level of each building. An example of this ductility concen-tration at the lower storeys can be observed in Fig. 6 where the distribution of the average ofthe maximum lateral displacements, Fig. 6a, and the average of the maximum displacementductility demand, Fig. 6b, over the height of the ED8R structure are represented for severalvalues of the q-factor. Furthermore, in buildings of similar first mode periods but differentheights, the larger ductility demand is usually observed on the taller structure. With respectto the behaviour of irregular structures in comparison to that of the corresponding regularones, the effects of the vertical irregularities was found to be more relevant for the four-storeystructures. In these structures, ductility demand is larger than for the corresponding regularstructure and a concentration of ductility demand is usually present in the vicinity of theirregularity. With respect to the eight-storey structures, the considered cases were found tobe less sensitive to the effects of regularity.

With respect to the computation of the IDA curves, it is referred that the selected groundmotion IM was the PGA and that these curves were obtained for IM levels up to a value forwhich the exceedance probability is less than 0.1%, according to the considered f I M . As anexample of the type of IDA curves that were obtained, Fig. 7 presents the IDA curves for allthe considered structures, for the case where the q-factor was set to 2.5 and up to an IM levelthat is five times the design PGA, a level in agreement with the previously set exceedanceprobability limit. It can be seen from these sample curves that the behaviour of the two sets offour-storey structures is quite different. While the ED4F1 structures appear to have reacheda maximum ductility limit for the maximum represented IM level, the ductility demand ofthe ED4F2 structures still appears to be increasing linearly. For the case of the eight-storeystructures, it is also interesting to note the noticeable difference in ductility demand from themore regular structures (ED8R and ED8I3) to the more irregular ones (ED8I1 and ED8I2).For the maximum represented IM level, ductility demand appears to be still increasing lin-early for the latter, while for the former, the behaviour of the curve appears to indicate theexistence of a maximum ductility limit near that IM level.

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qqqqqqqqq

q q q q q q q q q

(a) (b)

Fig. 6 a Average of the maximum lateral displacements over the height of the ED8R structure and b averageof the maximum displacement ductility demand over the height of the ED8R structure, for several values ofthe q-factor

4.5 Behaviour factor calibration results

According to the q-factor assessment approach selected for the current application, ductilitylevels of 4 and 7, and a Pf with a value of 105 are enforced, hence leading to a certainvalue of the q-factor for each building structure under analysis. Given the optimization natureof the selected approach, a large amount of analyses and results was therefore produced. Itis also noted that, for simplicity, q-factor steps of 0.5 were considered for the optimizationprocess of its value. For conciseness sake, only final results are presented herein.

The results obtained by the proposed methodology in terms of q-factors leading to thedesired levels of and Pf value are presented in Table 3 for the considered building struc-tures. Moreover, results of the basic value of the q-factor q and of the apparent overstrength, computed according to the previously defined proposals, are also presented.

As can be seen from the results, there is a general consistent trend for the values of thecomputed parameters. Observation of these results with more detail shows that the four-sto-rey buildings with larger first mode periods require larger q-factor values to fulfil the required and Pf values. Moreover, for buildings with different heights but exhibiting similar firstmode periods, it is seen that taller structures require smaller q-factor values. With respectto the irregularity effects, it can be seen that, on average, for a given regular structure, thevalues of q , q and obtained for the corresponding irregular structures are not considerablydifferent than those obtained for the regular structure. Such result reflects the fact that bothregular and irregular structures exhibit maximum demand at the same level, which indicatesthat their global behaviour is not much different and that the EC8-1 regularity in elevation

123


Fig. 7 IDA curves for theaverage of the maximum ductilitydemand for the a ED4F1structures; b the ED4F2structures and c the ED8structures

PGA/PGA design0.0 1.0 2.0 3.0 4.0 5.0

PGA/PGA design0.0 1.0 2.0 3.0 4.0 5.0

PGA/PGA design0.0 1.0 2.0 3.0 4.0 5.0

ED4F1I2 ED4F1I1

ED4F1R 2.0

4.0

6.0

8.0

10.0

12.0

14.0

(a)

(b)

(c)

ED4F2R

ED4F2I2ED4F2I12.0

4.0

6.0

8.0

10.0

12.0

14.0

ED8R

ED8I3

ED8I2ED8I1

2.0

4.0

6.0

8.0

10.0

12.0

14.0

123


Table 3 Results obtained for q,q and for the consideredstructures and ductility levels

Building = 4 = 7q q q q

ED4F1R 3.5 2.92 1.20 5.0 4.38 1.14ED4F1I1 3.5 2.93 1.19 5.0 4.39 1.14ED4F1I2 3.5 2.94 1.19 5.0 4.40 1.14ED4F2R 4.0 3.46 1.16 5.5 4.34 1.27ED4F2I1 4.5 3.98 1.13 6.5 5.89 1.10ED4F2I2 4.5 3.98 1.13 6.5 5.89 1.10ED8R 3.5 2.93 1.19 5.5 4.82 1.14ED8I1 3.0 2.32 1.29 4.5 3.86 1.17ED8I2 3.0 2.29 1.31 5.5 4.91 1.12ED8I3 3.5 2.93 1.19 5.5 4.87 1.13

criteria appear to be adequate. Nonetheless, the importance of the irregularity appears to bemore relevant when the value of the required is larger.

5 Final remarks

Given the importance that traditional force-based seismic design still currently exhibits,studies addressing issues related to the definition of the behaviour factor (q-factor) valuesare considered to be of most interest. Towards this purpose, a probabilistic methodology forthe calibration of the q-factor value was proposed, which relates its value with the displace-ment ductility capacity measured at a relevant location of the structure and the failureprobability Pf .

The general basis of the proposed procedure is the probabilistic quantification of the seis-mic action and, by applying a transformation procedure, of the structural seismic demand interms of displacement ductility. The procedure makes use of nonlinear analysis methods, bothstatic and dynamic, relating, within a general probabilistic framework, the ductility capacity, the failure probability Pf and the q-factor. Moreover, the proposed methodology providesthree different analysis approaches, depending on the parameters that are enforced. Further-more, since the q-factor is considered to be defined by the product of two other parameters (thebasic q-factor value q and the overstrength factor ), the methodology is also able to addressthe assessment of these parameters, hence providing an additional source of calibration forthe values proposed in design codes such as the Eurocode 8.

In order to illustrate some of the potentialities of the proposed methodology, an applica-tion example was presented that addresses the q-factor assessment for a set of regular andirregular RC frame structures, enforcing a given Pf and two different ductility levels .Results of this application indicated a general consistent trend for the values of the computedparameters. Observation of the results also showed that structures of the same height withlarger first mode periods require larger q-factor values to fulfil the required and Pf val-ues. Moreover, for buildings with different heights, but exhibiting similar first mode periods,results also showed that taller structures require smaller q-factor values. With respect to theirregularity effects, it was seen that results obtained for the irregular structures were notsignificantly different than those obtained for the corresponding regular one.

123


Acknowledgments Financial support of the Portuguese Foundation for Science and Technology, throughthe PhD grant of the second author (SFRH/BD/32820/2007) is gratefully acknowledged.

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A methodology for the probabilistic assessment of behaviour factorsAbstract1 Introduction2 Brief review of general concepts2.1 The q-factor and its relation with other important parameters2.2 The relation between earthquake demand and structural demand2.3 The selected structural safety approach

3 Description of the proposed methodology for q-factor calibration3.1 General steps of the proposed methodology3.2 Assessment analyses that can be performed with the proposed methodology

4 Example application of the proposed methodology4.1 General description of the structures4.2 Structural modelling assumptions4.3 Seismic scenario4.4 Seismic demand results and IDA curves4.5 Behaviour factor calibration results

5 Final remarksAcknowledgments

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A Methodology for the Probabilistic Assessment of Behaviour Factors

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