refined force reduction factors for seismic design.pdf

Engineering Structures 22 (2000) 12441260www.elsevier.com/locate/engstruct

Refined force reduction factors for seismic designB. Borzi, A.S. Elnashai *

Imperial College of Science, Technology and Medicine, Department of Civil and Environmental Engineering, Imperial College Road,London, SW7 2BU, UK

Received 13 April 1999; received in revised form 19 July 1999; accepted 29 July 1999

Abstract

Whereas seismic design based on deformations is a concept that is gaining ground, existing codes are fundamentally force-based,with a final check on deformations. A central feature of force-based seismic design is the response modification factor (R or q).Many studies have attempted to quantify the potential of structural systems to delimit the level of force imposed by virtue of theirductility and energy absorption capacity. This paper employs a well controlled and evenly distributed earthquake data-set (in magni-tude, distance and site characterization spaces) to derive values for force reduction factors needed for the structure to reach, andnot exceed, a pre-determined level of ductility. It is observed that the force modification factors are only slightly influenced by theshape of the hysteretic model used in their derivation and even less sensitive to strong motion characteristics. A linear representationis recommended for use in a benchmark for demand considerations and given in an easy-to-use parametric form. 2000 ElsevierScience Ltd. All rights reserved.

Keywords: Force reduction factors; Seismic design; Ductility; Design spectrum

1. Introduction

Conventional seismic design, as employed in codes ofpractice, is entirely force-based, with a final check onstructural displacements. Force-based design is suited todesign for actions that are permanently (or persistently)applied. Members are designed to resist the effects ofthese actions at levels of stress constrained by their plas-tic capacity. The deformations corresponding to the plas-tic member capacity are not normally excessive, andevaluating them is not an onerous task. Since seismicdesign was developed as an extension to primary loaddesign, it followed the same procedure, noticing thoughthat inelastic deformations may be utilised to absorbquantifiable levels of energy leading to reduction in theforces for which structures are designed. This lead tothe creation of the response modification (or behaviour)factor; the all-embracing parameter that purports toaccount for over-strength, energy absorption and dissi-pation as well as the structural capacity to re-distribute

* Corresponding author. Tel.: + 44-171-594-6058; fax: + 44-171-594-6053.

E-mail address: [email protected] (A.S. Elnashai).

0141-0296/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved.PII: S0141- 02 96 (99)00 07 5- 9

actions from inelastic highly stressed regions to otherless stressed locations in the structure. Problems of eval-uating behaviour factors that are generally applicable tovarious structural systems, materials, configuration andinput motion are well documented and the inherentweakness in code-specified factors is widely accepted.However, the majority of existing studies are concernedwith the capacity of structural systems to absorb energyand hence levels of force well below the elastic valuesare recorded. In other words, studies of the supplyresponse modification factors abound. Much less so arestudies that aim at quantifying the demand imposed byearthquake motion using a verified and well distributednatural records data-set.

In this work a well controlled data-set was employedfor the definition of inelastic constant ductility acceler-ation spectra. This data-set was selected by Bommer etal. [1] and already used for the definition of attenuationrelationships for elastic displacement spectra with differ-ent damping values. Inelastic spectra were derived hereinusing two models: an elastic perfectly plastic represen-tation and another more complex system which has ayield point, a maximum force point and a post-ultimatebranch which may represent hardening as well as soften-ing. The strong motion records were further used to

1245B. Borzi, A.S. Elnashai / Engineering Structures 22 (2000) 12441260

derive values for response modification factors (q or R)for force-based seismic design. Moreover, the q-factorvalues herein obtained were compared with the formu-lation of reduction coefficient proposed in the technicalliterature. The simplified trilinear representation pro-posed hereafter is shown to provide uniform reliabilityacross the period range.

2. Input motion

The dataset employed for the definition of inelasticconstant ductility spectra was assembled by Bommer etal. [1] for the derivation of frequency-dependent attenu-ation equations for ordinates of displacement responsespectra. All records were filtered individually at ImperialCollege, London, and used to derive displacement spec-tra for different levels of damping, from 5% to 30%.The dataset has been adapted from the one employed byAmbraseys et al. [2] to derive attenuation relationshipsfor ordinates of elastic acceleration response spectra.Some weaker records were excluded whilst recordsobtained after the aforementioned work was undertakenwere added. This is a high quality dataset in terms ofboth accelerograms that have been individually correctedand information regarding the recording stations andearthquake characteristics.

The accelerograms of the dataset were recorded dur-ing 43 earthquakes of a magnitude between 5.5 and 7.9,at a distance from the nearest point on the fault of upto 260 km. While the source distance and the surface-wave magnitude are available for all the accelerograms,for three records the local site geology is unknown. Forthe remaining 180, the percentages of distribution in thethree site groupings of rock, stiff and soft soil are 25.0%,51.1% and 23.9% respectively. For two records only onecomponent of the motion is available. The total numberof used records is 364. In Fig. 1 the distribution of rec-ords comprising the data-set with regard to magnitude,distance and site classification are shown. The figuresdemonstrate that the data is well-distributed with respectto all three parameters, hence results of analysis will nothave significant bias.

The attenuation model used in this work is that ofAmbraseys et al. [2]. In this attenuation model three soiltypes are defined as a function of the shear wave velo-city. When the shear wave velocity exceeds 750 m/s thenthe soil is classified as rock. Shear wave velocity lessthan 360 m/s leads to categorising the soil as soft. Stiffsoil conditions are assumed in the intermediate range ofshear wave velocity. Further details of the dataset,attenuation relationship and the regression model aregiven in Bommer and Elnashai [3].

Fig. 1. Distribution of records for (a) rock; (b) stiff and (c) soft soil.

3. Structural models

3.1. Elastic perfectly-plastic model

In order to determine the influence of magnitude, dis-tance and soil condition on inelastic response spectra,attenuation relationships have been defined using anelastic perfectly-plastic response model (EPP). The EPPmodel was employed since it is the simplest form ofinelastic force-resistance as well as being the basis forearly relationships between seismic motion and responsemodification factors. Moreover, by virtue of its two para-meters definition: level of force-resistance and stiffness.Few structural characteristics are included, hence theinfluence of strong-motion records may be better vis-ualized. The stiffness is corresponding to the period ofvibration for which the spectral ordinate has to be calcu-lated and the resistance is derived iteratively. In thiswork inelastic constant ductility spectra were obtained.Therefore the resistance of the system corresponds to theresistance for which the system has a required ductilityequal to the target ductility. The ensuing inelastic spectrawould reflect solely the characteristics of the inputmotion.

1246 B. Borzi, A.S. Elnashai / Engineering Structures 22 (2000) 12441260

3.2. Hysteretic hardening-softening model

In order to investigate the influence of the responsecharacteristics of structures on inelastic spectra, a hyster-etic hardening-softening model (HHS) was used [4]. Thestructural model is characterised by the definition of aprimary curve, unloading and reloading rules. The pri-mary curve for a hysteretic force-displacement relation-ship is defined as the envelope curve under cyclic loadreversals. For non-degrading models the primary curveis taken as the response curve under monotonic load. Inthis model the primary curve is used to define the limitsfor member strength. Two points on the primary curvehave to be defined. It is essential to define cracking andyield loads (Vcr and Vy) and the corresponding displace-ments (Dcr and Dy), as shown in Fig. 2. If, for example,this model was used to describe the hysteretic behaviourof reinforced concrete members, the cracking load wouldcorrespond to the spreading of cracks in the concrete andthe yielding load would be related with the load at whichthe strain in bars is equal to the yield strain of steel.Unloading and reloading branches of the HHS modelhave been established through a statistical analysis ofexperimental data [5,6]. The load reversal rules arebriefly described below.

Structural members exhibit stiffness degradationunder cyclic loading. When the number of cycles or themagnitude of inelastic deformation increases, the systembecomes softer. Furthermore, the hysteretic behaviour isaffected by pinching. The axial load is an importantparameter in predicting pinching effects (due to the onsetof crack closure). The slope of reloading branchesincreases beyond the crack load.

The slopes of the lines connecting the origin to thecracking point (K1 in Fig. 2) and the yield point to thecracking point in the opposite quadrant (K2 in Fig. 2)are used to define the unloading branches under cyclicloads. The latter slope depends on deformation and force

Fig. 2. HHS model for structural members.

levels attained at the beginning of unloading. Experi-mental results indicate that if unloading starts betweenthe cracking and the yield load, and the yield load hasnot been exceeded in the relevant quadrant, thenunloading stiffness is bounded by K1 and K2. In thismodel a linear variation between these limits was pro-posed as a function of displacement ductility. If theunloading load exceeds the yield load, the unloadingcurve changes the slope to a value close to the crack-ing load.

In the current investigation, second order effects havenot been considered but the effect of axial load on pinch-ing has been accounted for. Fortunately, in most situ-ations, particularly in regions where large seismic forcesneed to be considered for design, P-D effects do notmarkedly affect the force and deformation supply of thestructures [7]. The response of well-designed structuresof intermediate periods will not be affected by P-Deffects during inelastic excursions. On the other hand,after very large displacements, due to large velocitypulse, the frame may not be restored to its original unde-formed condition. However, this is more likely to occurin structures without seismic detailing. It is thereforereasonable to ignore second order effects in the courseof deriving inelastic spectra for general use.

4. Procedural considerations

In order to define inelastic acceleration spectra andbehaviour factors, displacement ductility of 2, 3, 4 and6 are considered. Ductility levels higher than 6 are notincluded because they constitute global displacementductility and structures very rarely have a local ductilitysupply commensurate with global ductility above 6. Theinelastic spectra have been defined between 0.05 s and3 s. The following period steps have been adopted:

T = 0.05 s 0.2 s DT = 0.01 s


T = 0.2 s 0.5 s DT = 0.02 sT = 0.5 s 1 s DT = 0.05 sT = 1 s 3 s DT = 0.1 s

For the HHS model the initial elastic period can beconsidered either as the stiffness before Vcr or the secantstiffness. In the current work an initial elastic period cor-responding to the secant stiffness was considered. Thisis because the stiffness before Vcr is not representativeof the structural behaviour. A low damping value of 1%was included. This viscous damping is representative ofthe non-hysteretic dissipation, since hysteretic dampingis already included. To justify the chosen low dampingvalue, experimental results on reinforced concrete build-ings are taken into account [8]. A damping value up to2% is adequate to describe sources of damping otherthan hysteretic for reinforced concrete. Furthermore,sensitivity analyses were undertaken and have indicatedthat further refinement of this assumption is unwar-ranted.

The input parameters for the HHS model describedabove are the monotonic curve and the relationshipbetween axial compressive force and nominal concentricaxial capacity. In order to define the inelastic constantductility spectra the magnitude of the monotonic curveis not an input parameter. It is defined in an iterativeway forcing the relationship between maximum andyield displacements to satisfy the target ductility. Toobtain the inelastic spectra and response modificationfactor (R or q) an approximation of the primary curvewith three linear branches has been assumed (Fig. 3).Consequently, the input parameters defining the shapeof the primary curve are:

1. the relationship between the cracking and the yieldingload (Vcr/Vy);

2. the relationship between the stiffness before thecracking load and the secant stiffness (Kcr/Ky);

3. the slope of the post yield branch.

To select the values of parameters to be employed,

Fig. 3. Shape of primary curve used in this work.

extensive analysis of the influence of each parameter onthe inelastic spectra was undertaken. The results of para-metric investigation indicate that the parameter with thestrongest influence on inelastic spectra is the slope ofthe post yield branch. Hence fixed ratios between Vcr andVy and between Kcr and Ky were considered. From theexperimental results of Paulay and Priestley [7],Priestley et al. [9], Calvi and Pinto [10], and Pinto [11],it is reasonable to consider the secant stiffness at theyield point equal to 50% of the stiffness before Vcr; Thelatter is taken equal to 30% of Vy. The ratio betweenthe cracking and the yield load influences the pinchingbehaviour that does not occur often for structures withloads higher than approximately 30% of the yieldingload Vy. The considered representative slopes of thestructural behaviour are:

K3 = 0 (elastic perfect plastic behaviour)K3 = 10% Ky (hardening behaviour)K3 = 2 20% Ky (softening behaviour)K3 = 2 30% Ky (softening behaviour)

Only a level of axial load equal to 10% of the nominalaxial load is assumed, since the model does not accountfor second order effects and is not strongly affected bythe chosen level of axial load. The above characteristicshave been verified to cover both new structures with fullseismic detailing, and existing structures with no specialdetailing [12,13].

An iterative procedure was utilized for the definitionof spectral ordinates corresponding to a target ductility.Sometimes it has not been possible to obtain a conver-gent solution with the HHS model, due to instabilityemanating from steeply descending post-peak loadresponse. In order to define the spectral ordinates whenthe solution is not convergent, the following approachwas adopted:

O For elastic-perfectly plastic and hardening behaviourit was almost always possible to obtain ductilityvalues higher and lower than the target ductility. Incases where the difference of both values from thetarget ductility value did not exceed 2, the averagesolution was taken. Otherwise the closest solution wasassumed when it corresponded to an obtained duc-tility with a difference from the target of less thanone;

O For the softening cases, the abrupt change in theinstantaneous force capacity leads to unstable sol-utions, as depicted in Fig. 4. This is the underlyingreason why cases with descending post-ultimatebranches had a higher percentage of failed conver-gence. This is dealt with by one of the followingoptions:O If the target and obtained ductility are bounded and

differ by 1 or less, the median value is adopted;


Fig. 4. Failure of iterative procedure for softening behaviour.

O Otherwise, the closest obtained ductility is adoptedprovided that its variation from the target ductilityis 0.5 or less;

O In a case where no such solution existed, the spectralordinates were not taken into consideration when con-ducting the regression analysis.

The percentage of spectral ordinates that have notbeen considered in the regression analyses are reportedin Table 1.

It is observed that the number of spectral ordinates tobe excluded from parametric analysis for the slope ofthe third branch equal to 2 30% Ky and ductility equalto 4, is very high. The attenuation relationship for thiscombination of parameters was therefore not considered.In the softening cases it was also not possible to obtaina convergent solution for periods less than 0.1 s. Inelasticresponse spectra start therefore from 0.1 s for the soften-ing cases. The above observations (non-convergence) arefully justified by noting that highly degrading systemsare inherently of low ductility. Therefore the decisionstaken do not affect the generality of the reported spectraand response modification factors.

5. Inelastic acceleration spectra

The influence of magnitude, distance and local siteconditions have been studied for elastic response spectrain previous work. The objective of this part of the workis to investigate whether inelastic response spectra exhi-

Table 1Percentage of ordinates excluded from regression analysis

K3 m = 2 m = 3 m = 4 m = 6

0 0.16% 0.58% 1.23% 3.16%10% Ky 0.03% 0.13% 0.27% 0.80%2 20% Ky 1.67% 5.12% 12.00% 2 30% Ky 2.45% 8.16% 30.98%

bit the same dependence to the above-mentioned para-meters. In order to investigate this, the coefficients ofthe attenuation relationship used by Ambraseys et al. [2]have been calculated for inelastic constant ductility spec-tra. To isolate the influence of hysteretic behaviour andinput motion characteristics, the attenuation coefficientsof inelastic spectra defined for the EPP model were con-sidered. In order to obtain the relationship between elas-tic and inelastic acceleration spectral ordinates (responsemodification factors) a regression analysis was perfor-med to define an elastic acceleration spectrum for adamping value of 1%. Fig. 5 shows the influence of duc-tility, magnitude, distance and soil conditions on inelas-tic acceleration spectra. These figures demonstrate thestrong influence of input motion parameters on inelasticacceleration spectra. To compare the nature of influencethat the input motion parameters have on inelastic andelastic acceleration spectra, the attenuation coefficientscalculated by Ambraseys et al. [2] were invoked. Thesewere obtained using a data-set from which the data-setof Bommer et al. [1] was selected. The records have asurface wave magnitude between 4 and 7.9 and a focaldepth less than 30 km. The influence of magnitude anddistance on elastic acceleration spectra, calculated withthe attenuation law presented above, is shown in Fig. 6.Considering this figure and the one related to the inelas-tic spectra, it is possible to observe that the input para-meters have the same influence on elastic and inelasticspectra.

Fig. 7 depicts the influence of different structuralcharacteristics on inelastic acceleration spectra for all theductility levels considered. It indicates that the hystereticbehaviour does not have a strong influence on acceler-ation spectra. This confirms that in a force-based methodthe hysteretic behaviour does not significantly changethe level of force for which the structure has to bedesigned, in order to reach a fixed level of displacementductility. In this work inelastic constant ductility spectrawere derived. It was therefore assumed that the damageis only a function of the ductility requirement, neglectingthe influence of energy dissipation. If a damage indicator


Fig. 5. Influence of (a) ductility, (b) magnitude, (c) distance and (d) site condition on enelastic acceleration spectra evaluated for the EPP model.

Fig. 6. Influence of (a) magnitude and (b) distance on elastic acceleration spectra obtained by Ambraseys et al. [2].

taking into account the amount of energy that the sys-tems can dissipate is assumed, the hysteretic behaviourbecomes more important in terms of level of resistancefor which the structure has to be designed. However,the inelastic acceleration spectra can be used in terms ofequivalent or reduced ductility instead of displacementductility. By means of a reduction of the original dis-placement ductility the amount of the damage due to thedissipated energy can be taken into account [14].

6. Seismic force reduction factors (R or q)

6.1. Response modification factor supply

Knowledge of initial period and damping values areinsufficient to define the seismic force intensity for asystem exhibiting inelastic behaviour. To obtain thebase-shear force, a standard approach is in use, based onthe definition of a behaviour factor (q and R for Europe


Fig. 7. Inelastic acceleration spectra for different hysteretic behaviours (Ms = 6, d = 10 km, soft soil).

and USA, respectively). The response modification orbehaviour factor has been used to derive the designacceleration response spectrum from its linear elasticequivalent, allowing the benefits offered by the energydissipation capacity of structures to be availed of, whileensuring that the imposed ductility demand does notexceed the available supply. This factor accounts for thepresence of damping and other force reducing effects,such as period elongation (or stiffness degradation). Itis evident that there is a strict correlation between thebehaviour factor and the ductility resource of the struc-ture. In the behaviour factor both the local ductility ofplastic hinges and that of the structural system are con-sidered. This must be taken into account because in orderto guarantee a certain level of displacement global duc-tility, an adequate supply of local ductility in the plastichinge has to be provided and mechanisms of collapsecharacterised by low dissipation of energy have to beavoided. A brief review of the behaviour factor in seis-mic codes in Europe and in the United States ispresented below in order to identify areas of possibleimprovement.

The behaviour factor (q) used in Eurocode 8 (EC8)[15] is the ratio between the elastic and inelastic designspectra. In EC8 maximum allowable behaviour factorvalues are specified for different structural types,

materials and ductility classes. The behaviour factors ofEC8 have also to be interpreted as the lower bound of thereal ductility capacity of structures designed accordingto the code. The period-dependent response modificationfunction proposed by EC8 is:

q 51 +

TT1

(hb0 - 1)

1 +TT1

(hb0/q - 1)when T , T1, q 5 q (1)

when T . T1where T1 is a characteristic period of the design spectrum(lower-bound period of the constant branch of the EC8design acceleration spectrum), h relates to the equivalentviscous damping of the structure, (h is 1 when the damp-ing is 5%), b0 is the acceleration amplification factor,(set equal to 2.5), and q is the behaviour factor of thestructure. Values of behaviour factor between 1 and 5are given for different construction materials and struc-tural systems.

Until recent research work, the behaviour factor (R)given by United States codes had not been substantiallymodified since the 1950s, when it was first introduced.In 1957 a committee of Structural Engineers Associationof California (SEAOC) introduced a horizontal force


factor, the predecessor of the behaviour factor. In themid-1980s, data from experimental research to definebase shear versus roof displacement curves for steelbraced frames and for the formulation of the bahaviourfactor [16,17]. It was proposed to split the behaviourfactor into three constituents:

R 5 RSRmRx (2)where RS is the strength factor (defined as the ratiobetween the supply resistance and the design one, alsoconsidered as an over-strength factor), R

m

the ductilityfactor and R

x

the damping factor. Other research work[1821] lead to a second formulation of the behaviourfactor in which R

x

is replaced by RR. The latter coef-ficient is known as the redundancy factor, introduced toaccount for the number and distribution of active plastichinges. Due to different number of plastic hinges, struc-tures characterised by the same shear resistance have dif-ferent reliability. This is taken into account by means ofthe coefficient RR. The new relationship for the behav-iour factor is:

R 5 RSRmRR (3)

In this formulation Rx

was excluded. This was becauseof the recognition that although the damping factor canbe used to scale the displacement in inelastic structures,it cannot be used to proportionally reduce the strengthdemand.

Substantially larger behaviour factors are proposed inthe United States than in Europe. As a consequence,similar structures designed according to these codes arelikely to suffer different levels of damage during earth-quakes, provided the strength of materials side is thesame. However, it is most important to note that higherbehaviour factors do not necessarily lead to lighter struc-tures, since there are other loading scenarios that maytake precedence.

6.2. Response modification factor demand

The behaviour factor demand represents the mini-mum reduction coefficient corresponding to a specificlevel of ductility. The relationship between displacementductility and ductility-dependent behaviour factor hasbeen the subject of considerable research. A few of themost frequently used relationships reported in the techni-cal literature are discussed below.

6.2.1. Newmark and Hall [22]During the seventies the reduction factor was para-

meterised as a function of ductility [22]. It was observedthat in the long period range, elastic and ductile systemswith the same initial stiffness reached almost the samedisplacement. As a consequence, the behaviour factorcan be considered equal to the displacement ductility.

For short period structures the ductility is higher thanthe behaviour factor and the equal energy approach maybe adopted to calculate force reduction. This approachis based on the observation that energies associated withthe force corresponding to the maximum displacementreached by an elastic and inelastic systems are similar.The proposed relations for behaviour factor are:Rm 5 1 when T , 0.03 s

Rm 5 2m - 1 when 0.12 s , T , 0.5 sRm 5 m when T . 1 s (4)

6.2.2. Krawinkler and Nassar [23]A relationship was developed for the reduction factor

derived from the statistical analyses of 15 westernUnited States ground motions with magnitude between5.7 and 7.7. The records were obtained on alluvium androck site, but the influence of site condition was notexplicitly studied. The influence of behaviour para-meters, as well as yield level and hardening coefficient,was taken into account. A 5% damping value wasassumed. The equation derived is given as:Rm 5 [c(m 2 1) 1 1]1/c (5)where:

c(T,a) 5 Ta

1 + T a 1bT (6)

in which a is the hardening parameter of the employedhysteretic model and a and b are regression constants.

6.2.3. Miranda and Bertero [24]The equation for reduction factor introduced by Mir-

anda and Bertero [24] was obtained considering 124ground motions recorded on a wide range of soil con-ditions. The soil conditions were classified as rock,alluvium and very soft sites characterised by low shearwave velocity. A 5% of critical damping was assumed.It is given as

Rm 5m - 1

F1 1 (7)

where F assumes different formulations for rock, alluv-ium and soft sites as shown below:

f 5 1 11

10T - mT 21

2Texp( 2 1.5(ln(T)

2 0.6)2) for rock site

f = 1 +1

12T - mT -2

5Texp( - 2(ln(T) - 0.2)2) for alluvium site

f = 1 +T13T -

3T14T exp( - 3(ln(T/T1) - 0.25)

2) for soft site


(8)where T1 is the predominant period of the groundmotion.

6.2.4. Vidic et al. [25]The reduction coefficients introduced by Vidic et al.

[25] (Rm

) were approximated with a bilinear curve. Inthe short period range the reduction factor increases lin-early with the period, from 1 to a value that is almostequal to the ductility. In the remaining part of the periodrange the reduction factor is constant. To calculate thereduction factor, a bilinear model and a stiffness degrad-ing Q-model were investigated. A mass proportionaldamping and an instantaneous stiffness proportionaldamping were assumed as mathematical models ofdamping. In this work the standard records form Califor-nia and Montenegro 1979 were chosen as being rep-resentative for standard ground motion (i.e. severeground motion at moderate epicentral distance, with aduration between 10 and 30 seconds and predominantperiod between 0.3 and 0.8 seconds). The study ofinfluence of different structural behaviours was workedout using these records. In order to account the influenceof input motion to the groups of records from Californiaand Montenegro (20 records all together) three othergroups of records were used. These are the records fromFriuli 1976 (6 records), Banja Luka 1981 (6 records) andChile 1985 (8 records). The main characteristic of rec-ords from Friuli and Banja Luka are the short durationand the short predominant periods, whilst the durationof Chile records is long, but their predominant periodsare in the range of standard records period. It shouldbe noted that the groups are rather small. The proposedformulation of reduction factor is:

Rm 5 c1(m 2 1)cRTT0

1 1 when T , T0

Rm 5 c1(m 2 1)cR 1 1 when T . T0 (9)where T0 is the period dividing the period range into twoportions. It is related to the predominant period of theground motion T1 by means of:T0 5 c2mcTT1 (10)

The coefficients c1, c2, cR and cT depend on the hyster-etic behaviour and damping.

6.3. Constant reliability response modification factors

The above formulations were significant steps forwardat the time they were undertaken. Areas of possibleimprovement, though, are identified as:

O The data-set can be improved using a large number

of records well distributed in terms of magnitude, dis-tance and soil conditions from a wide range ofseismo-tectonic environments;

O Use of comprehensively represented hysteretic mod-els exhibiting hardening-softening behaviour;

O Using regression curves focusing on uniform distri-bution of target reliability across the period range andgiving simple code-amenable expressions.

In this work, following the definition of response modi-fication factor, regression analyses for the evaluation ofthe ratio between the elastic and inelastic accelerationspectra (q-factor) were undertaken. The influence of duc-tility and input motion parameters (magnitude, distanceand soil conditions) on the behaviour factor was studiedutilising the EPP hysteretic model. These results arepresented in Fig. 8. It was observed that the influenceof input motion parameters on elastic and inelastic accel-eration spectra is similar (and significant). However, theeffect cancels out for their ratio. Ductility is the mostsignificant parameter influencing the response modifi-cation factor. Consequently, analyses to define a period-dependent behaviour factor functions for all the ductilitylevels and all structural models were undertaken. Theaverage values and the standard deviations were calcu-lated considering various combinations of input motionparameters.

The period dependent behaviour factor functions her-ein calculated were further approximated with a trilinearspectral shape. The reduction coefficient is equal to 1 ata zero period and increases linearly up to a period T1,which is defined as the period at which the behaviourfactor reaches the value q1. A second linear branch isassumed between T1 and T2. The value of the reductioncoefficient corresponding to T2 is herein denoted q2. Forperiods longer than T2 the behaviour factor maintains aconstant value equal to q2:

q 5 (q1 2 1)TT1

1 1 when T , T1

q 5 q1 1 (q2 2 q1)T - T1T2 - T1

1 1 when T1 , T , T2

q 5 q2 when T . T2 (11)

The values q1, q2, T1 and T2 that allow the definitionof approximate spectra for all relevant ductility levelsand hysteretic behaviour, as they are obtained by a piece-wise linear regression, are reported in Table 2.

Fig. 9 shows that the simplified trilinear curves givean accurate approximation of the behaviour factor spec-tra throughout the whole period range. To demonstratethis, the standard deviation s of the ratio g between theapproximate and the original spectral values is studied.The standard deviation was calculated for all branches


Fig. 8. Influence of (a) ductility, (b) magnitude, (c) distance and (d) site condition on q-factor evaluated for the EPP model.

Table 2Constants for trilinear behaviour factors spectra

m = 2 m = 3 m = 4 m = 6

T1 T2 q1 q2 T1 T2 q1 q2 T1 T2 q1 q2 T1 T2 q1 q2

EPP 0.20 0.79 2.06 2.20 0.21 0.78 2.89 3.31 0.22 0.87 3.59 4.34 0.25 0.99 4.81 6.13K3 = 0 0.20 0.56 2.20 2.51 0.25 1.67 3.10 4.09 0.27 1.55 3.76 5.45 0.29 1.26 4.78 7.79K3 = 10% Ky 0.21 0.54 2.04 2.33 0.27 1.80 2.78 3.62 0.29 1.64 3.25 4.56 0.33 1.54 3.93 6.10K3 = 2 20% Ky 0.26 0.26 2.43 2.43 0.24 1.76 2.83 3.93 0.25 1.69 3.25 5.12 K3 = 2 30% Ky 0.26 0.26 2.42 2.42 0.24 1.85 2.76 3.81

of the approximate spectra and across the whole periodrange. These values are reported in Table 3.

It is observed that the dispersion of g is close to theglobal standard deviation. This has an important conse-quence from a practical point of view, as the behaviourfactor spectra herein proposed correspond to an almostconstant seismic design reliability over the whole periodrange, a feature not previously achieved. Finally, the co-ordinates of the points that allow the definition of theapproximate spectra were expresses as a function of duc-tility and given here as:T1 5 bT1 (12)

T2 5 aT2m 1 bT2 (13)q1 5 aq1m 1 bq1 (14)q2 5 aq2m 1 bq2 (15)where bT1, aT2, bT2, aq1, bq1, aq2 and bq2 are constantvalues. It was seen that the control periods of theapproximate spectral shape do not strongly depend onthe hysteretic behaviour. Therefore, it was possible touse common values of the control periods for all thehysteretic models in Eqs. (12) and (13). On the otherhand, it was seen that different values of aq1, bq1, aq2and bq2 correspond to the different hysteretic behaviour


Fig. 9. Simplified three-linear curve used to fit the q-factor period dependent curves.

patterns. The values of the constant in Eqs. (12)(15)are reported in Tables 4 and 5.

To demonstrate that the co-ordinates given as a func-tion of ductility in Eqs. (12)(15) still result in a goodapproximation of the original behaviour factor spectra(characterised by a constant seismic design reliability),the standard deviation s of the ratio g (approximate-to-accurate) are reported in Table 6.

By comparing the errors reported in Tables 3 and 6,it is clear that the parameterised values obtained withEqs. (12)(15) do not significantly modify the level ofapproximation of the original curves.

The results obtained in this work confirm the mainconclusions of previous studies and give comprehensiveand qualitative guidelines for response modification fac-tors (R or q) for seismic design and assessment. The


Table 3Standard deviations of g% (ratio of approximate to accurate q/R factors)

m = 2 m = 3 m = 4 m = 6

s1 s2 s3 s s1 s2 s3 s s1 s2 s3 s s1 s2 s3 s

EPP 2.1 1.9 2.0 2.0 3.4 2.1 2.8 2.8 4.7 2.5 3.5 3.6 7.0 2.8 4.2 5.0K3 = 0 3.1 1.6 2.7 2.6 4.5 3.6 3.3 3.9 6.2 3.8 3.9 4.8 8.5 3.0 6.2 6.4K3 = 10% Ky 3.8 1.6 2.8 2.9 6.4 3.4 3.5 4.7 9.0 3.4 3.4 6.1 2.8 1.7 2.5 2.6K3 = 2 20% Ky 3.6 4.7 4.4 2.6 3.2 3.7 3.2 2.9 3.1 4.3 3.4 K3 = 2 30% Ky 3.7 4.9 4.6 2.9 3.1 4.1 3.3

Table 4Values of the constants in Eqs. (12) and (13)

bT1 aT2 bT2

0.25 0.163 0.60

Table 5Values of the constants in Eqs. (14) and (15)

aq1 bq1 aq2 bq2

EPP 0.69 0.90 1.01 0.24K3 = 0 0.55 1.37 1.33 0K3 = 10% Ky 0.32 1.69 0.96 0.51K3 = 2 20% Ky 0.38 1.67 1.24 0K3 = 2 30% Ky 0.29 1.83 1.21 0

behaviour factor is only slightly dependent on the periodin the long period range and almost corresponds to theductility value An exception to this, briefly investigatedby the writers but not reported herein, is long durationor multiple earthquake events. On the other hand in theshort period range, the behaviour factor is dependent onboth ductility and period. A moderate, though not negli-gible, influence from the hysteretic behaviour isobserved throughout the period range. The behaviourfactor spectra calculated in this work were comparedwith the formulations of ductility-dependent reduction

Table 6Standard deviations of g(%) for approximate q-factor definition

m = 2 m = 3 m = 4 m = 6

e1 e2 e3 e e1 e2 e3 e e1 e2 e3 e e1 e2 e3 e

EPP 3.3 3.2 2.1 4.0 4.1 2.5 3.0 3.6 5.2 3.4 3.3 4.0 6.7 4.2 4.9 5.8K3 = 0 3.5 2.3 3.0 3.5 4.6 2.6 5.1 4.2 6.2 3.8 4.1 5.1 8.2 4.9 6.9 9.0K3 = 10% Ky 3.6 2.9 3.0 3.4 6.5 2.5 5.6 5.4 9.3 3.9 3.8 7.0 13.5 3.7 4.7 11.2K3 = 2 20% 4.4 2.1 2.8 4.1 3.5 3.3 3.3 4.0 3.4 3.4 4.2 3.7 KyK3 = 2 30% 4.5 2.2 3.1 4.5 3.8 4.2 3.7 5.1 Ky

coefficients given in previous investigations. This isshown in Fig. 10. The results obtained in the currentwork are in good agreement with the previous formu-lations of reduction coefficient. The current formulationis, however, derived using a much wider data-set withconsistent distributions in the magnitude, distance andsite condition spaces. Moreover, the idealisation pro-posed above leads to hazard-consistent or reliability-con-sistent force reduction factors. They are therefore con-sistent with the commonly used uniform hazardresponse spectra.

6.4. Comparison between supply and demand

To investigate the relationship between supply anddemand in terms of reduction coefficient of lateral seis-mic loads, two reinforced concrete regular frame struc-tures were considered. In the first case a 12-story build-ing designed according to EC8 provisions for ductilityclass H (Fardis [26]) was studied. The design acceler-ation spectrum proposed by EC8 for soil class B and apeak ground acceleration equal to 0.3 g were assumedto evaluate the seismic loads. The reinforcement (bothlongitudinal and transverse) and the concrete are classi-fied as S500 and C25/30, respectively. The 12-storybuilding was analysed by Salvitti and Elanshai [27] bymeans of modal, nonlinear static and dynamic analyses.In the dynamic analyses four acceleragrams artificiallygenerated from the EC8 design spectrum were


Fig. 10. Reduction coefficients proposed (a) in this work and (b), (c), (d) and (e) in previous works.

employed. Peaks of acceleration equal to and twice thedesign value were adopted. The aims of the dynamicanalyses were to evaluate the behaviour factor and estab-lish how the structure performed in terms of both localand global damage. The second selected case is a 7-storybuilding. The building was founded on rock. It is locatedin Los Angeles where it was affected by both the SanFernando and Northridge earthquakes of 1971 and 1994,

respectively. The building was analysed by the ATC pro-ject team (FEMA 273 and 274, 1997) [28] to demon-strate techniques for evaluation of strength and ductilityfactors by means of inelastic static collapse analysis.

The main aim of the research in the aforementionedwork [27,28] was to compare the behaviour factor pro-posed by seismic codes design with that obtained bymeans of sophisticated inelastic analyses. The aim of the


Fig. 11. Measured and idealised force-displacement curve of the structure analysed by (a) Salvitti et al., 1997 and (b) ATC project team.

current work is to investigate whether the behaviour fac-tor, as given in the codes, corresponds to an adequateductility supply. In order to define the ductilitydemand, the behaviour factors proposed by the Euro-pean and United State codes are compared with thebehaviour factor spectra corresponding to the hystereticmodel that best describes the global behaviour of thereal structure. To calculate the ductility supply of bothstructures, the force-displacement curves obtained by thenonlinear static analysis were used. The latter curveswere also employed to evaluate the hysteretic model bestdescribing the structures. In order to define the character-istics of the equivalent hysteretic model (substitutestructure), a secant stiffness approach as proposed byPaulay and Priestley [7] was adopted. Hence, the initialelastic stiffness was evaluated as the secant stiffness forthe point corresponding to a base shear value equal to75% of the maximum force (Fig. 11). The collapse cri-teria assumed by Salvitti and Elnashai [27] are based oneither a maximum interstory drift of 3% or the attain-ment of a confined concrete limiting strain. On the otherhand, the ATC research team [28] used a maximumrotation capacity in the vertical structural elements of0.005 rad to define failure. The hysteretic model associa-ted with the global structural behaviour of the structuresis an HHS model with elastic-perfectly plastic behaviour.Values for the ratio of shear force to total weight of thestructure (V/W) and for the yield and maximum top storydisplacement (Dy and Dm, respectively) are reported inTable 7. Overstrength (Rs) and ductility values were also

Table 7Results from static nonlinear analyses

Dy (mm) Dm (mm) V/W Rs m

Salvitti and 220 648 0.260 2.11 2.95Elnashai [22]ATC [27] 127 300 0.179 2.98 2.36

calculated. Herein, overstrength is defined as the ratiobetween the calculated maximum strength and thedesign strength.

Both structures have a large overstrength factor Rs.This is principally, though not exclusively, due to thefact that in the design process design material resistancevalues are assumed, while in the analyses nominal valueswere employed. For the structure analysed by Salvittiand Elnashai [27], large overstrength was also calculatedin the nonlinear dynamic analyses. Overstrength factorsof 1.93 and 2.35 were evaluated for the peak accelerationvalues equal to and twice the design peak ground accel-eration, respectively. Evidently, the ratio 1.93 at thedesign acceleration is an instantaneous overstrength,not the true value.

The definition of behaviour factors used in the workof Salvitti and Elanshai [27] are:q9 5 (SA)elc /(SA)ely (16)qD 5 (SA)elc /(SA)ind (17)

In the first expression the behaviour factor is obtainedby relating the elastic spectra at yield (subscript y) tothat at collapse (subscript c), while in the second equ-ation, the denominator changes to the inelastic designspectral value. The behaviour factor given by the codeis the ratio between the elastic and inelastic spectra.Therefore, Eq. (17) may be written as:qD 5 (SA)elc /(SA)eld qcode (18)

By assuming that the response acceleration spectra atcollapse and at yield have the same dynamic amplifi-cation, the behaviour factors can be expressed as a func-tion of peak ground acceleration as below:

q9 5ag(design)ag(yield)

(19)


qD 5ag(collapse)ag(design)

qcode

(20)

Eq. (19) is the analytical translation of the definitionof behaviour factor, but does not account for the effect ofoverstrength which is significant for the structures underconsideration. The proposed behaviour factor of EC8 forthe considered structural type is 5, while the minimumbehaviour factor calculated by Salvitti and Elanshai [27]using Eq. (20) is 12.5. The 7-story building analysed bythe ATC project team [28] was designed for a behaviourfactor equal to 3.4. In order to define the key componentsof the behaviour factor as formulated in Eq. (3), thedefinition of ductility-dependent behaviour factor pro-posed by Miranda and Bertero [24] is invoked. The duc-tility-dependent key component is 2.73. The building hasan excellent redundancy in the horizontal direction ofloading, hence a redundancy factor of 1 is used. Abehaviour factor of 8.14 was calculated using Eq. (3).The main conclusion of the aforementioned work wasthat the behaviour factor values proposed by the codesare too conservative. The overstrength of structuresdesigned according to this code lead to high values ofbehaviour factor capacity, or supply.

For both buildings considered, the ductility corre-sponding to the proposed behaviour factors is obtainedby means of a comparison with the behaviour factorspectra (Fig. 12). The ductility supply, evaluated bynonlinear static analysis, and the ductility demand cor-responding to the behaviour factor proposed by the codesare reported in Table 8.

The above observations lead to the conclusion thatboth the European and US standards are too conserva-tive, due to the overstrength shown by the structures. Onthe other hand, the ductility demand corresponding tothe behaviour factor proposed by codes is higher thanthe ductility supply measured in the nonlinear static

Fig. 12. Definition of the ductility demand corresponding to the behaviour factors given by seismic codes design for the structure analysed by(a) Salvitti and Elnashai [27], and (b) ATC project team.

Table 8Ductility supply and ductility demand

msupply mdemand Difference

Salvitti and Elnashai [27] 2.95 3.79 22%ATC [28] 2.36 2.61 10%

analyses. As a consequence, the shear resistanceassumed for the design of both the buildings is inad-equate for the required ductility capacity. The abovetreatment shows that if the latter buildings did not exhi-bit rather high overstrength, the demand could haveexceed the supply in terms of their ability to dissipateenergy in the inelastic range thus leading to an unsafesituation in terms of deformation capacity. In otherwords, had the buildings not been of a much higherstrength than their seismic design base shear wouldimply, their seismic performance would have beendemonstrably inadequate.

7. Conclusions

Regression coefficients for inelastic acceleration spec-tra have been calculated using the elastic perfectly-plas-tic (EPP) and the hysteretic hardening-softening (HHS)structural response models. The data-set employed iscarefully selected to exhibit consistent distribution in themagnitude, distance and site condition spaces. A total of364 records were used. The results obtained with theEPP model were employed to highlight the influence ofinput motion parameters on inelastic acceleration spec-tra, while some different sets of parameters for the HHSmodel were used to evaluate the influence of hystereticbehaviour. In the light of the obtained results it isobserved that:

O the influence of input motion parameters on inelasticspectra is similar to that for the elastic spectra;


O the hysteretic models herein assumed only mildlyinfluence the inelastic acceleration spectra. Thereforethe level of force imposed on structures is not heavilyinfluenced by their global hysteretic behaviour.

A ductility-based behaviour factor (q or R) is definedas the ratio between elastic and inelastic spectral ordi-nates. This is considered as a demand value, whichgives an indication of the minimum level of ductility,and energy dissipation capacity that structures have topossess. Coupled with this is the response acceleration,which is proportional to the required strength of the sys-tem. Utilising a well controlled and evenly distributeddata-set, period-dependent force reduction factors havebeen calculated. These are recommended for assessmentand design and are associated with the ductility factorsindicated. A trilinear representation was derived in afully parametric fashion for use as a benchmark for code-recommended response modification factors. The appli-cability of these functions to practice is underlined bythe fact that they provide uniform reliability designs, e.g.structures designed with these factors across the wholeperiod range will be subjected to the same probabilityof their capacity being exceeded. As such, the proposedformulation is consistent with elastic uniform hazardspectra employed in codes of practice.

Two RC structures, studied in detail elsewhere, rep-resenting typical European and US seismic design prac-tice were investigated, as an application exercise. Theirsupply and demand force reduction factors were studied.It was concluded that had it not been for their high over-strength (ratio of calculated-to-design horizontal forceresistance), both structures would not have possessedsufficient deformational supply to meet the imposeddemand, the latter estimated from the proposed spectra.

Acknowledgements

The writers would like to express their gratitude toDr. J.J Bommer and Mr. G.O. Chlimintzas for their helpwith the strong-motion data-set. The regression programwas kindly provided by Dr. S.K. Sarma, whilst supportwas given by Mr. D. Lee during the implementation ofthe hysteretic model. Valuable advice on statistical errormeasures was given by Dr. M. K. Chryssanthopoulosand Ms. C. Dymiotis. All the above are from ImperialCollege. Thanks are also due to Professor E. Faccioli(Politecnico di Milano) and Professor G.M. Calvi(Universita degli Studi di Pavia) who supervised part ofthe work. Funding for the primary author whilst atImperial College was provided by the EU network pro-grammes ICONS and NODISASTR.

References

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