Seismic reliability implied by behavior-factor-based design

Force-based seismic design involves the reduction of elastic spectra by introducing a behavior factor, q . This approach is widespread in engineering practice; however, recent studies have shown that structures consistently designed at different sites will not share the same level of seismic risk, which can be defined as the annual rate of the structure failing to meet a seismic performance objective, despite seismic actions having the same exceedance return period at all sites. This paper investigates whether the definition of site-specific q factors can lead to uniform risk across sites characterized by varying levels of seismic hazard, based on the pushover curves of bare frame reinforced concrete buildings. These pushover curves are used to establish the backbones of equivalent single degree of freedom systems with varying lateral resistance. These systems are fictitiously placed at several Italian sites and their seismic failure risk is computed by integrating their fragility, assessed by means of incremental dynamic analysis, with each site’s hazard curve. By assuming an arbitrary risk threshold, the same for all sites, the corresponding lateral strength leading to said threshold is determined and the corresponding behavior factor is back calculated. As expected, risk-targeted q factors tend to increase with decreasing seismic hazard and are highly sensitive to the shape of the hazard curve beyond the design return period. Coupled with the fact that at low hazard sites lateral strength is determined by detailing for gravity-load design and minimum code requirements, rather than seismic design actions, the results suggest that q factor-based design is unsuitable for warranting territorially uniform seismic safety, yet it may be suitable for setting an upper-bound to the annual failure probability.

F I G U R E 1 (A) Reduction of a horizontal elastic spectrum via q factor to obtain horizontal design spectrum (the design spectrum was constructed according to the provisions of the Italian code NTC18; the EC8 design spectrum would differ in the  <   segment, but this is not relevant for the present study, as all structural vibration periods considered fall beyond   , that is in the zone where the definitions of NTC18 and EC8 design spectra coincide); (B) graphical representation of overstrenght on a static pushover curve same exceedance return period (  ) at the site of interest; i.e., by uniform-hazard spectra.Design of ordinary building structures covered by the codes is often driven by a force-based verification against the life-safety limit state, which is typically performed considering seismic actions with   = 475 years.The performance objective for the structure against that level of ground shaking is to dissipate seismic energy via inelastic excursions of structural members, while retaining its structural integrity and sufficient bearing capacity against gravity loads.To this aim, the force-based design philosophy assigns strength to the various structural elements according to force demands calculated via linear-elastic analysis, for example modal response analysis, using spectral ordinates that have been reduced (  ) by means of a so-called behavior factor, or q factor, by a margin that depends on the vibration period , as shown in Figure 1A.
The reasoning behind reducing the design actions by means of the q factor, is to allow structural elements to yield under the design ground motion to trigger hysteretic dissipation.Current European standards, such as EC8 and NTC18, prescribe a maximum allowable behavior factor,  max , to be applied to all structures of a specific building typology or class, that is to buildings sharing the same primary building material(s) and structural system adopted for resisting horizontal actions.This maximum allowable value of the behavior factor is known to depend on the deformation and dissipative capacity of the lateral-load-resisting system and on the, so-called, overstrength that said system exhibits.These concepts are illustrated in Figure 1B, which plots the static pushover (SPO) curve of a generic building structure.In the figure, the ordinate shows the resultant of monotonically increasing lateral forces acting on the building, that is base shear , and the abscissa registers the roof-level lateral displacement  caused by these forces.The SPO curve is indicatively shown to end at an ultimate displacement   , representing an admissible deformation that can be reached without compromising gravity-load bearing capacity.The figure also highlights that the base shear causing the first cross-section to reach yield,   , and the maximum base shear the building can sustain prior to the onset of strength deterioration under monotonic loading,  max , exceed the demand resulting from elastic analysis,   .In other words, the actions that will actually force the structure to exhibit inelastic deformations are higher than the design actions, for reasons related to section-level design, reinforcement detailing and the system's capacity for redistribution of internal forces. 3ssuming, for the time being, that overstrength and deformation capacity can be considered invariant among buildings of the same typology, it can be postulated that having the different buildings belonging to that typology share a common code-prescribed  max , stems from the implied expectation that this should lead to uniformity of seismic reliability (i.e., seismic risk) of these structures.On the other hand, the Italian national-level research program Rischio Implicito -Norme Tecniche per le Costruzioni (RINTC; 4 the risk implicit in the technical code for construction-translated from Italian) sought to analytically quantify seismic risk of code-conforming building structures by computing the annual failure rate, that is the annual occurrence rate of earthquakes that cause failure of the structure to meet a predefined seismic performance objective.6][7][8] In fact, it was observed that the seismic risk for newly-designed buildings increases with increasing seismic hazard at the site, even though the design actions increase commensurately with any increase in hazard.
The results of the RINTC project constitute the foundations of the present study, which investigates the possibility of calculating site-dependent risk-targeted q factors for some typical building structures.][10] The present investigation is based on the pushover curves of selected reinforced concrete (RC) frame buildings that were designed, modelled, and analyzed within the activities of the aforementioned research project RINTC.The static pushovers of these RC frames, which had been designed against different seismic hazard levels at various sites, are used to establish the backbone curves and dynamic characteristics of equivalent single degree of freedom (ESDoF) systems. 11Furthermore, by interpolating among these backbones, which exhibit varying levels of lateral resistance according to the hazard at the site for which they were designed, additional SDoF oscillators are defined, to obtain a family of simple (idealized) structures with an almost continuous variation of lateral strength (e.g., in terms of  max ), and whose characteristics can be held to be representative of the building typology of the original designs.These ESDoF systems are then subjected to incremental dynamic analysis (IDA), 12 in order to analytically procure their fragility curves for a selected performance objective.Subsequently, each set of SDoF oscillators is fictitiously placed at various Italian sites, each characterized by a different hazard curve, and their annual failure rates are evaluated.By assigning an (arbitrary) seismic risk target, the lateral strength level, necessary to meet that risk criterion, is established per site.Finally, the contribution of overstrength is considered in order to back-calculate the necessary q factor that would have led to that specific design.This study, although applying the Italian code, could attract international interest given the similarities with other international codes, most notably EC8.
This study is discussed in the remainder of this article, which is structured as follows: first, an overview of the methodology used for obtaining the necessary computational tools is provided, including a description of the case study buildings, the use of pushover analyses to define equivalent SDoF models, fragility analysis and probabilistic seismic hazard analysis for each site considered.Then, the assumptions and method for determining risk-targeted behavior factors for each case are presented, followed by a presentation and discussion of the results.The conclusions of this study are summarized in the final section.

Reliability assessment
For the purposes of the present investigation, the seismic safety of structures is quantified in terms of their annual failure rate,   .According to the performance-based earthquake engineering paradigm (PBEE), 13   can be computed via Equation (1): where [| = ] denotes the probability of failure of a specific performance objective, given that some ground motion intensity measure () takes a specific value, , and   is a measure of seismic hazard at the construction site, defined as the annual rate of earthquakes exceeding that value of shaking intensity.The terms typically used in PBEE for [| = ] and   are the structure's fragility function and the site's hazard curve.At the state-of-art of seismic risk studies, analytical estimation of fragility functions is performed by conducting a series of nonlinear dynamic analyses on a numerical model of the structure, while   is obtained by means of a probabilistic seismic hazard analysis (PSHA). 14,15

Case-study structures
The case-study structures used in this investigation were chosen among the buildings designed, modelled and analyzed within the activities of the RINTC-project.More specifically, six RC bare frames are considered, that is three-, and sixstorey RC moment-resisting frame structures, regular in plan and elevation, that were designed according to the Italian NTC18 code at three Italian sites, namely L'Aquila, Naples, and Milan.Each of these sites is representative of a different design level of seismic hazard, with L'Aquila being the highest-hazard site of the three and Milan the lowest.All buildings are destined for residential use and share the same basic layout in plan, shown in Figure 2, with first-storey height of 3.40 m and 3.05 m for the remaining storeys.Each structure was designed against gravity and seismic loads, for the latter via modal response spectrum analysis using a behavior factor q equal to 3.90 and assuming soil class C, according to NTC18 classification (the same as EC8).As a consequence, buildings with the same height but designed against the seismic actions of different sites, only differ in the dimensions and reinforcement detailing of structural elements, while their overall configuration is the same.Details on the design and modelling of these buildings can be found in dedicated literature. 16

Structural models and ESDoF systems
As part of the activities of the RINTC project, two separate numerical models were developed for each one of these buildings: an elastic model used for modal response analysis during code-based design and a more refined lumped-plasticity model. 17The reason behind dual modelling was that the former was intended to replicate standard engineering design practice, while the latter followed the state-of-the-art in modelling the post-yield hysteretic behavior of RC members and was used for conducting static non-linear analyses in the two principal directions of each building, arbitrarily designated X and Y.Some of the six buildings' SPOs can be seen in Figure 3, where the previously mentioned characteristic points, defined by   ,   ,  max and nominal elastic base shear demand   , are marked against each curve along with their corresponding roof displacements.Each pushover curve is plotted up to the point corresponding to a fifty percent drop in lateral resistance, which was used within the RINTC project to define the collapse threshold in terms of displacement response.The pushover curves of these buildings form the starting point of the analyses described in the remainder of the article.
Design base shear per principal direction   is determined via spectral modal response analysis of the linear-elastic model used for design. 18Strictly speaking, this modal analysis entails multiple-mode contribution to base shear; however, given that all structures considered are first-mode dominated in both directions, with primarily translational modes, for the purposes of the present study the simplification is made to only consider the first mode contribution, according to Equation (2): where  1 is the fundamental period of vibration resulting from the numerical model used for design purposes,   ( 1 ) is the design spectral acceleration at that period and m1 is the associated participating modal mass, also calculated from the elastic model used for design (see Figure 1A).According to the Italian code NTC18, design spectral acceleration   () is in turn given, as a function of the q factor, by Equation (3): where () are the elastic horizontal pseudo-acceleration spectral ordinates of the code at vibration period ,   is the period marking the beginning of the constant-acceleration branch of the code spectrum and (0), (  ) are the elastic spectral ordinates at  = 0s (i.e., the peak ground acceleration) and  =   , respectively.Analogously, the nominal elastic base shear demand,   , can be obtained via Equation (4): These design-related parameters for the six case-study structures considered, specifically  1 , m1 ,   and   , are provided in Table 1.

Reference ESDoF systems
For the purposes of the present study, twelve reference equivalent SDoF oscillators were defined using these SPO curves, one for each structure's two principal directions. 11Examination of each direction separately is justified by the buildings' regularity in plan, which precludes torsional effects dominating seismic response.The transition from a multiple DoF (n-story) building to an ESDoF system is schematically presented in Figure 4.Each MDoF structure's SPO curve was converted to a so-called capacity curve of the ESDoF system by dividing both base shear and roof displacement with the first-mode modal participation factor Γ. 19 The resisting force and displacement of the ESDoF oscillator are denoted as  * = ∕Γ and  * = ∕Γ, respectively.Subsequently, each ESDoF's backbone curve is approximated via a tri-linear function, defined by five parameters, as shown in Figure 4C: the nominal yield force and displacement,  *  and  *  respectively, the capping-point force and displacement,  *  and  *  , and the ultimate displacement,  *  , which corresponds to complete loss of strength.The tri-linear approximation to the capacity curve follows the procedure implemented

[t] [s] [-] [kN] [kN] [m] [m] [m] [t] [s] [-] [kN] [kN] [m] [m] [m]
L'Aquila 1177 in previous applications, 20 based on a dedicated study, 21 with the exception that  *  is deliberately set at the point corresponding to the formation of a plastic mechanism for each frame, which results in the ESDoF systems having nominal yield points that correspond to similar roof drifts.
The equivalent mass of each ESDoF is calculated 19 as  * = ∑  =1   ⋅   , where   and   with  = {1, 2, … , } are, respectively, the i-th floor mass and corresponding component of the first modal vector.In this context, the modal vector is normalized to have unit value at roof level and the vibration period of the ESDoF oscillator can be calculated as per Equation ( 5): These parameters are reported for each of the twelve reference ESDoF systems in Table 2.

Generated ESDoF systems
The reference ESDoF oscillators presented in the preceding paragraph correspond to RC frames of the same configuration, but designed for sites in Milan, Naples and L'Aquila, listed here in ascending order of seismic design actions for the considered (life-safety) limit state.In the abscissa the roof displacement has been normalized with respect to each building's height, , thus reporting roof drift ratio  = Γ ⋅  * ∕.It can be observed that nominal yield points for all structures are around  = 1%, which is a drift around which a global plastic mechanism is expected to form for RC structures. 22,23The equivalent masses assigned to each of these generated oscillators are also the result of linear interpolation between the  * values of the reference models and thus the vibration period  * of each generated SDoF oscillator can be calculated via Equation (5).It is convenient to express the lateral resisting capacity of these systems using a dimensionless parameter termed the yield strength coefficient,   , defined as the ratio of spectral acceleration at the structure's vibration period causing yield,   ( * ), divided by the acceleration of gravity, according to Equation ( 6): Figure 5 also shows that, for each building's principal direction, a number of backbones were also obtained from extrapolation beyond the lowest and highest lateral resistance cases (i.e., Milan and L'Aquila, respectively).However, for the two backbones generated above the ones corresponding to the L'Aquila designs, that is the reference systems with the highest lateral strength,  *  was not extrapolated but held constant, to avoid adjudicating response in a region that will have no impact in the final results, as will become evident later on.
An overall summary of the yield strength coefficient   , the yield and capping-point displacements  *  ,  *  and the vibration period  * of these SDoF systems is provided in Table 3.The result of this exercise is four sets of SDoF systems with varying levels of lateral strength, which are considered to be approximately representative of variants of the original reference structures, as they would have ostensibly resulted from design against seismic actions of varying intensity.While the same premise cannot be made for the SDoF systems obtained via extrapolation to   values beyond the bounds of the

Fragility functions
Within this study, each structure is (arbitrarily) considered to have failed if  * max ≥  *  , with  * max being the maximum (absolute value) transient displacement of each ESDoF oscillator undergoing base-acceleration.In other words, nominal failure is taken to occur whenever seismic displacement demand exceeds a threshold value coinciding with each backbone's capping-point displacement; note that this is not intended to be representative of collapse, but simply a possible performance objective for seismic design of a nonlinear structure.This corresponds to a displacement demand of around four-and-a-half times the nominal yield displacement, for all ESDoF systems concerned.Although more than one performance objective could have been considered for the quantification of seismic safety, the choice was made to contain the investigation within the limits of this single performance.
To calculate the associated seismic risk in terms of   , fragility functions are needed; i.e., [| = ] = [ * max ≥  *  | = ].These fragilities were evaluated analytically, using the, so-called, IM-based approach, 24 based on IDA of the SDoF oscillators.The IM-based approach uses the IDA curves and introduces a random variable that is the shaking intensity value causing failure, designated as   .By assuming that   is lognormally distributed, fragility can be expressed as per Equation ( 7): where  and  are, respectively, the mean and standard deviation of the natural logarithm of   , both estimated from a sample of structural responses provided by dynamic analysis, Φ(⋅) denotes the standard Gaussian function,  is the number of acceleration records used in the IDA, and  , with  = 1, 2, … ,  is the   realization obtained from dynamic analysis with the j-th record.The IM adopted in this study was an average spectral acceleration, 25  avg , defined as the geometric mean of 5% damped spectral accelerations at vibration periods   ,  = 1, 2, … , , according to Equation ( 8): In this case the following vibration periods   were selected for the definition of   : {  } = {0.70,0.75, 0.80, 0.85, 0.90, 0.95, 1.00, 1.10, 1.20, 1.30, 1.40, 1.50, 1.60, 1.70, 1.80, 1.90, 2.00}, with all values being in seconds.Past research has shown that such IMs are more efficient in predicting inelastic displacement response than single-spectral-ordinate IMs, by virtue of accounting for spectral shape beyond the structure's fundamental period. 26his efficiency is quantified in terms of the dispersion of structural responses given various levels of the conditioning IM and has the practical result of limiting the amount of dynamic analyses required to obtain risk estimates within a target standard error. 27,28In fact, this period interval contains almost all  * of the generated ESDoF oscillators, bar a few with   < 0.12, and therefore   could be a common IM to express the fragilities of all structures considered. 29Finally, the so-called sufficiency of   with respect to the seismological parameters affecting hazard at each site, 30 implies that the same fragility functions can be exported to different sites.
IDA was conducted for each ESDoF oscillator using the DYANAS 31 interface for the OpenSees platform. 32The modified Ibarra-Medina-Krawinkler peak-oriented hysteretic rule, 33 without cyclic degradation, was assigned to the oscillators for these analyses, along with five-percent-of-critical viscous damping.The forty-four acceleration records constituting the far-field FEMA-P695 set 34 were used in these analyses, which comprises accelerograms recorded on firm soil, compatible with the assumed ground type C, predominantly recorded during events in California, of magnitude range from 6.5 to 7.6 and distances between the site and the horizontal projection of the fault within the range from 11 to 27 kilometers.
The procedure used for defining each ESDoF's fragility function from the IDA results 35 is illustrated in Figure 6.The intersection of the j-th record's IDA curve with the vertical line  * max =  *  , provides one realization of the random variable   , denoted as  , with  = 1, 2, … , 44.The resulting fragility parameters for each ESDoF system, which are reported in Table 4, are then estimated based on this sample of   realizations, according to Equation (7).The structure-specific fragility functions thus derived only account for one source of uncertainty, which is the so-called, record-to-record variability of structural response, 36 while uncertainty in the structural model is neglected, as previous RINTC-specific studies 37 supported to do so (it is worth observing that these fragility parameters are subject to estimation uncertainty, by virtue of being derived from samples of non-linear responses, 38 and this estimation uncertainty carries over to the calculated failure rates; even though this aspect is not treated here for the sake of brevity, it does not alter the results of the comparisons).
The fragility parameters are plotted in Figure 7 against the corresponding ESDoF oscillators' yield strength coefficients   , for one of the principal directions of the three-storey and six-storey configurations.The figure displays continuous functions of   that are fitted to the fragility parameters {, } of each structural configuration (as will be discussed further on, this operation allows to calculate failure rates for any level of lateral resistance within a given range of site's design hazard covered by the continuous functions).

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0 The figure shows that median capacity   increases with increased yield strength of the structure, as expected.On the other hand, the dispersion appears to be higher for the weaker variants of each building type with respect to the more resistant ones; in other words,  is higher at yield strength coefficients.This can be explained by the fact that  * increases with decreasing   .While it is convenient to maintain the same IM for all structures considered, at the lower end of   the period  * tends to approach the upper bound of the period interval behind the definition of   ; it is therefore to be expected that the IM exhibit lower efficiency for those structures.

Benchmark sites and hazard curves
For the purposes of this study, ten benchmark Italian sites were considered and seismic hazard curves were computed via PSHA, with the aid of the software REASSESS. 39These hazard curves provide    for each site, that is the annual rate of earthquakes exceeding any threshold of average spectral acceleration   , as defined in Equation (8).
It is important to highlight that the seismic source model 40 used in PSHA is the same as the one underlying the code's seismic design actions.This means that the failure rates from the risk analysis that will follow, and the exceedance rates of design actions are comparable, because they stem from the same model.Furthermore, the seventeen vibration periods F I G U R E 8 (A) Hazard curves in terms of   for the case-study sites; (B) 475-year-return-period horizontal elastic (uniform-hazard) spectra according to NTC18 used to define   were chosen so that model parameters were directly obtainable from the ground motion prediction equation that was used in the PSHA, in combination with a correlation model for the spectral ordinates. 41he location of the considered sites and the corresponding hazard curves are shown in Figure 8, along with their peak ground acceleration (PGA) value of the design spectra for   = 475 years, denoted as  475 (hereafter roughly considered as a proxy of each site's design-level hazard), and with the respective site-specific 475 year return period elastic design spectrum for ground type C, according to EC8 classification (which is the same as the Italian code's).Note that the sites are numbered from one to ten in descending order of  475 ; however, it is evident from the figure that this order is not preserved by   at longer return periods, as the corresponding hazard curves can be seen to intersect.In addition to the numbering labels, the figure also reports the name of the corresponding administrative municipality for the sake of context.

Assigning behavior factors to the ESDoF systems
The set of ESDoF oscillators, representing incarnations of the reference structural configurations, differing only in their levels of lateral strength, can be ideally placed in any of the benchmark sites under consideration, that is assuming that each one's yield strength coefficient is the result of a different behavior factor that has been applied to that site's elastic design spectrum.In other words, it is assumed that each ESDoF system of given   , could have been the outcome of a seismic design at any of the benchmark sites using a free choice of q factor, disregarding for the moment the code-mandated  max .Essentially, this exercise entails assigning a different q factor per reference structural configuration,   and site.This operation must account for overstrength, which can be quantified via two factors that are respectively denoted in EC8 as  1 and   , and defined by Equation ( 9): Factor  1 is larger than unity because, at section-level design, material partial safety factors are used and nominal strength demand can be exceeded due to practical detailing considerations, while   mainly depends on the redundancy of the structural system and preferential mobilization of ductile over brittle local failure mechanisms due to capacity design.

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L Before proceeding any further, two considerations must be made.The first emerges directly from observing Figure 3C, showing pushover curves of the structures designed for Milan, which is characterized by the lowest seismic hazard among the three sites of the original (reference) designs (the other two being, Naples, and L'Aquila).From the figure it can be surmised that, due to the design base shear   for that site being so low, the actual lateral resistance  max is no longer determined by   and typical overstrength due to redundancy and material partial safety factors.In fact,  max is determined by other design requirements, such as the intrinsic lateral resistance of frames proportioned by gravity-load design and minimum design provisions for the geometry and reinforcement of structural members.This can be seen in Table 5, which shows the  1 and   overstrength ratios and the ratio of unmitigated elastic demand to lateral strength,   ∕ max , for all reference structures.In fact, the latter drops just about below unity for the Milan buildings, meaning that no arbitrary reduction of the q factor would have led  max to be governed by seismic design.
Based on this observation, the reference buildings designed for the site in Milan are considered to exhibit a minimum value of the yield strength coefficient,  min  , which cannot be undercut by reducing elastic seismic design forces.In other words, structures with yield strength coefficient lower than  min  , whose values are reported in Table 6, would fail to meet various non-seismic functionality and safety requirements.Therefore, values of   <  min  are only considered in the following for completeness of the investigation, all the while being aware that structures weaker than that are unlikely to result from engineer design.
The second consideration emanates from Equation ( 4), which reflects the fact that the elastic spectral acceleration demand ( 1 ) is read off the code spectrum at the first-mode period from modal analysis of the numerical model assembled for design purposes.As mentioned previously, this period is different than  * , which corresponds to the secant stiffness at nominal yield assigned to the ESDoF oscillators and, while  1 is known for the reference structures that were actually the object of design, this is not so for the ESDoF systems generated by interpolating between the reference backbones.For this reason, a design-model period  1 was assigned to each ESDoF oscillator, by interpolating between the ratios  1 ∕ * of the reference structures.These values are shown in Table 7 for each of the two structural configurations and principal directions.
With these considerations in mind, the behavior factor to be allotted to each ESDoF system placed at each one of the benchmark sites, is given by Equation ( 10): where ( 1 ) is taken from each benchmark site's elastic spectrum in Figure 8, and there is the implicit assumption that  max ≈ Γ ⋅  *  , in other words, that the maximum base shear corresponding to each backbone can be obtained from the capping-point resisting force multiplied by Γ.For this calculation, the modal participation factor value Γ = 1.3 is assumed for all ESDoF systems, a simplification that can be justified by the fact that Γ values obtained for the structures with available numerical models varied from 1.26 to 1.33.Furthermore, a overstrength ratio   had to be assigned to each ESDoF oscillator's backbone.The overstrength factors are assigned by assuming a constant value for  1 = 1.3, which is rounded down from 1.36, that being the average of the  1 ratios obtained from the SPOs of the reference structures designed for L'Aquila and Naples, and by interpolating the ratio   ∕ 1 between reference structures.This is intended to reflect the fact that the overstrength ratio  1 depends on material partial safety factors applied at section-level flexural design and on provided reinforcement overshooting the nominal requirement; it can therefore be considered invariant among the different-strength incarnations of a structure.The q factors assigned to each ESDoF oscillator at each benchmark site, are plotted in Figure 9, where continuous q -  curves have been fitted to the single values.Details on the analytical form of the curves and corresponding least-squares fitting can be found in an electronic supplement available at http://wpage.unina.it/iuniervo/papers/q-factor_supplement.pdf.The figure also delineates  min  considered for each structural configuration, so that the part of the curve to the left of the dashed line represents cases of low lateral strength that are only theoretical and are not expected to correspond to modern code-conforming structures.

Risk-targeted q factors
Having obtained fragility parameters as a function of   and hazard curves at each benchmark site, the failure rate   can be calculated for any combination of an ESDoF system's presumed design location and lateral strength, according to Equation (1).As a first step,   was calculated for the structures having a yield strength coefficient that corresponds to design under  = 3.9, denoted as  =3.9 , this being the behavior factor used for design of the reference structures.The results of this exercise are provided in Table 8.
The failure rates listed in the table confirm a fact that was already highlighted by past research, 4-8 that structures designed with the same behavior factor, applied to design spectra of the same   at different sites, exhibit different levels of seismic risk.One worthwhile observation is that, while the sites are ordered in terms of descending  475 , and the yield strength coefficients duly follow suit, the failure rates do not follow a strictly descending order.As will be discussed later, this is also consequence of the different shapes of each site's hazard curve beyond the design return period intensity, which are evident in Figure 8A.Even so, the difference in   , between the site with the highest risk in the list (no.4 Brienza) and the lowest (no.10 Parma), is almost two orders of magnitude.
The same procedure as above was implemented for all four structural configurations, values of   and benchmark sites.The results of this operation are provided in Figure 10, which plots failure rate against yield strength coefficient as a continuous curve for each benchmark site and structural configuration.Also in this plot, each panel is divided in two by a vertical line corresponding to  min  , so that portions of the curves to the left of the dashed lines are assumed non-applicable for modern code-conforming structures.
According to the discussion of the previous paragraph, a q factor that would have ostensibly led to each   value via design has been assigned for all ESDoF systems and benchmark sites.Thus, by replacing all   values in the plots of Figure 10 with their counterpart q factors shown in Figure 9, a new set of curves was obtained, shown in Figure 11, providing  failure rates and associated behavior factors for each benchmark site and structural configuration.In this new mapping of failure rate against q factor, the delimitation becomes slanted but remains approximately linear, dividing each plot of Figure 11 into two regions shown with different shading; in this case, the shaded regions to the right correspond to the domain where lateral strength is likely determined by design requirements unrelated to the behavior factor, according to the assumptions presented in the previous paragraphs.The figure gives a broader picture of what was already observed in the failure rate values of Table 8: seismic risk as a function of behavior factor differs from one site to another and, while risk is generally higher at sites under higher seismic hazard, the ranking order of the risk curves deviates from that of  elastic demand in terms of force.This can be attributed to the fact that risk depends on the shape of the hazard curve well beyond the design intensity as will be discussed in the following paragraphs.At this point, an arbitrary risk target is assumed for   , equal to 10 −4 events/year, 8,42 uniform across all benchmark sites.Thus, for each benchmark site and structural configuration, a risk-targeted behavior factor,   , can be estimated by finding the intersection of the   = 10 −4 horizontal line with each curve in Figure 11.The risk-targeted behavior factors corresponding to the assumed risk target, are given in Table 9, where the   values are shown in parentheses whenever the intersection of the target risk line and the curve falls within the shaded area, indicating that the lateral strength level required would likely be overridden by non-seismic design requirements.In the table, dashes appear whenever no   value could be obtained due to lack of intersection of the curves in Figure 11 with the risk target line.Obtaining an intersection for those cases, which comprise all structures at sites no. 9 and 10, would entail considering SDoF systems with   < 0.05, which was ruled out.The   values are also given in Figure 12, where they are plotted against the elastic spectral acceleration demand ( 1 ) of each benchmark site, marked with a grey circle when the corresponding yield strength coefficient is below  min  and a black circle otherwise.The figure shows the general trend of decreasing risktargeted behavior factors with increasing hazard, the latter expressed in terms of the code's nominal 475 year   spectral ordinate, which is nevertheless not perfectly monotonic.The yield strength coefficients corresponding to the risk-targeted behavior factor at each site,    , as well as the corresponding fragility parameters,   and   are reported in Table 10.Yield strength coefficients that fall below the assumed  min  are reported in parentheses.

Evaluation of the results
As a general preamble to the discussion of these results, it should be noted that, due to both the performance threshold and target risk being arbitrary, the   values shown in the Table 9 are intended for comparison with one another, and not with the  max value adopted for the design of the original reference structures.In this light, there are two observations to make.The first is that there is some variation of the risk-targeted behavior factor even for structures that are located at the same site and share the same type of lateral-load resisting system but differ in actual layout of structural elements in plan or in the number of storeys.The second observation is that harmonization of risk across different sites, via calibration of the q factor, only appears to be possible among sites characterized by higher seismic hazard, since the lateral resistance levels

[-] [g] [-] [-] [g] [-] [-] [g] [-] [-] [g] [-]
4 -Brienza 0.36 required to render risk at lower-hazard sites equal to that of the higher ones, appears to be unrealistically low.In fact, as can be seen in Figure 11, even if the arbitrary risk target assumed for the example were to be lowered, the curves of some of the benchmark sites would continue to be intersected by the horizontal threshold line within the shaded area, indicating that it was not possible to achieve the uniform risk objective set forth for that benchmark site, without resorting to yield strength coefficients that are unacceptably low for code-conforming engineered structures.This lower bound of lateral strength is attributed to two factors: first, cross section dimensions and flexural reinforcement are designed in compliance with minimum code-mandated values, independently of the severity of design seismic actions.Second, the redundancy that is typical of cast-in-situ moment-resisting RC frames, means that even if the latter were designed exclusively for gravity loads, they would still exhibit some level of lateral force resistance that is independent of any behavior factor applied to seismic actions.
A tempting way to sum-up this fact, could be to say that it is not possible to achieve the same level of seismic risk between high-and low-hazard sites via q factors.However, this qualitative distinction between high-and low-hazard sites is hard to make based, for example, on the PGA of the code's elastic spectrum at typical design return periods.A comparison between Figure 8 and Figure 11 reveals that the order of the   −  curves also depends on the shape of the hazard curves, mainly at return periods higher than the ones considered for design, which is not unexpected according to previous studies. 43erein, disaggregation of structure-specific seismic risk is used to take a closer look at this effect.Disaggregation of the failure rate is calculated according to Equation (11): where  | () is the conditional density of the shaking intensity given that structural failure has occurred.This operation was performed for two ESDoF systems, corresponding to the Y direction of the three-storey building and the Xdirection of the six-storey building, each under two assumptions for the behavior factor, namely the risk-targeted   from Table 9 or  = 3.9, and at three benchmark sites, namely Brienza, Caorera and Pedara.The resulting plots are given in Figure 13, where risk disaggregation of the  = 3.9 structures is shown in black and that of the   structures in grey (small irregularities visible in the disaggregation curves are due to numerical issues).On the same plots, the median capacities corresponding to the two choices of q factor,   and  =3.9 , are indicated by vertical dashed lines and so is the average spectral acceleration value with   = 475 years,  ,475 .
Broadly speaking, the risk disaggregation shown in the figure can be interpreted as indicative of the shaking intensity that will be most likely causative of an observed failure of the structure at the given site.Figure 13A shows the case of the three-storey building's Y direction at Brienza, where passing from   = 1.14 ⋅ 10 −4 events/year for  = 3.9 to the (arbitrary) target of   = 10 −4 events/year required an adjustment to  ≈ 3.5.In all other cases, achieving this specific level of uniform risk entailed shifting median capacity to the left of that corresponding to  = 3.9 by increasing the behavior factors to achieve lower lateral strength.Therefore, the transition to uniform risk was achieved by altering structural capacity, so that different segments of the site-specific hazard curve become relevant in determining the failure rate.
F I G U R E 1 3 Risk disaggregation for ESDoF oscillators corresponding to uniform  and uniform risk, corresponding to the three-storey building in the Y direction at (A) Brienza, (B) Caorera, (C) Pedara, and the six-storey building in the X direction at (D) Brienza, (E) Caorera, (F) Pedara Additionally, it is important to highlight that, in all cases, failure to meet this specific performance objective will most likely be caused by intensities larger than the one corresponding to the return period of the design seismic actions.This explains the fact that, when the same behavior factor is used across all sites, the failure rates do not always rank under an order that can be explained by the ranking of design spectral ordinates at   = 475 years: as can be seen from Figure 8, the hazard curves beyond the design return period differ in shape and may even intersect, but it is that part of the curves that mostly influences precedence in terms of risk.As a result, the trend of   against said spectral ordinates shown in Figure 12 is neither regular, nor monotonic.A clear example of this comes from comparing the results for sites no.4 (Brienza) and no.5 (Caorera): passing from a site with a  475 of 0.24 g to one of 0.23 g, the q factor needed for maintaining the arbitrary common risk level shoots from approximately 3.5 to 7.3.This is because risk associated with large displacement demands is more likely to be determined by seismic actions of larger return periods and the PGA value of the design spectrum cannot account for those.
For the cases examined here, it was shown that there is leeway for obtaining uniform levels of seismic safety by adjusting the q factor only between sites characterized by higher hazard levels.Figure 11 shows that, for each structure and site, the point where the failure rate-behavior factor curve breaks into the shaded area corresponds to a maximum value of   , that cannot be superseded by further increase of the q factor, for the reasons described above.This implies that this point corresponds to a minimum level of seismic safety that the structure will exhibit when designed under any q factor to the right of the demarcation line, since lateral resistance past that point should become insensitive to the reduction of seismic design actions.Thus, although it appears that uniform seismic safety cannot always be achieved between all sites by adjusting the behavior factor, one could acquiesce to q factors that allow structures to exhibit the same risk across high-hazard sites while remaining below a certain risk threshold at low-hazard sites, which can be possibly defined as acceptable according to some criterion. 44However, as already discussed, what constitutes a high-hazard or a low-hazard site, in this context, is by no means obvious from the elastic design spectrum alone.

CONCLUSIONS
The present study investigated the question of whether it is practical to achieve uniform levels of seismic safety for codeconforming structures designed at different sites, using force-based design and behavior factors.To address this issue, seismic risk calculations for several case-study reinforced concrete structures were developed.In these risk calculations a single performance objective was considered, one that entails entry of the structures in the inelastic range, and the only source of uncertainty considered was that due to record-to-record variability of structural response.Another simplification lies in the fact that only bare frames were considered and the potential influence of infill walls in the response was neglected.
Two main results emerged from this investigation.The first result was that, at some of the sites considered, the behavior factors that would have supposedly led to uniform risk among all sites, are too large to be applicable in practice.The reason for this is that code-mandated detailing requirements and gravity-load design should provide a minimum level of lateral strength that is insensitive to risk-targeted fluctuations of the behavior factor.Although this issue has also emerged in past research, the present investigation additionally highlighted the fact that the dividing line between high-hazard sites, where uniformity of seismic risk via appropriately calibrated q factors is applicable, and low-hazard sites, where it is not, cannot be defined based on the code's elastic design spectrum.The reason behind this, is that the distinction is decided by the shape of the hazard curve beyond the return period of the design actions.This was highlighted by one case where a five percent decrease in the design spectrum's peak ground acceleration when passing from one Italian site to another, was accompanied by more than doubling of the behavior factor required to maintain the same failure rate.Nevertheless, it should be pointed out that the non-applicability of uniform seismic safety via risk-targeted behavior factors at lowerhazard sites, stems from the fact that, at those sites, a minimum level of safety is implicitly guaranteed by design against other actions that cannot be reduced further by increasing q factors.
The second main result is that a risk-targeted behavior factor is structure-and site-specific to the point that a-priori definition in a normative document appears unfeasible.Although the present study considered a limited number of structures, the ones that it did consider stemmed from design of the same layout-in-plan of structural elements for different levels of seismic design actions.In other words, only the size of structural members and amount of reinforcing steel were allowed to vary between frames with the same number of storeys.Thus, not only were risk-targeted behavior factors of a given structure observed to vary from site to site, as expected from similar past research, but the same was true for the behavior factors of similar structures at a given site.This implies that the definition of such risk-targeted q factors can only be undertaken when the analyst is armed with information about the structure that is only available after design, which hints at an iterative procedure being more realistic than a-priori definition by the code.

A C K N O W L E D G E M E N T S
This study was developed within the activities of the ReLUIS-DPC 2019-2021 research program, funded by Presidenza del Consiglio dei Ministri -Dipartimento della Protezione Civile, with the disclaimer that the opinions and conclusions presented herein are those of the authors and are not necessarily shared by the funding organization.The authors wish to express their gratitude to the RINTC project research group, whose results were instrumental in the development of the present study.

D ATA AVA I L A B I L I T Y S TAT E M E N T
The data that supports the findings of this study are available in the supplementary material of this article.

F I G U R E 2 F I G U R E 3
View of the structural elements of the case-study RC frames and designation of the two principal directions X and Y: (A) three-storey frame; (B) six-storey frame; (C) layout of structural elements in plan, common to both RC frames (L X , L Y represent the lenghts of each side of the building) Pushover curves of the case-study structures; (A) 3-storey building in L'Aquila, Y direction; (B) 3-storey building in Naples, Y direction; (C) three-storey building in Milan, Y direction; (D) six-storey building in L'Aquila, X direction; (E) six-storey building in Naples, X direction; (F) six-storey building in Milan, X direction

F I G U R E 4
Schematic representation of pushover-based conversion of the building into an ESDoF oscillator; (A) pushover analysis of the lumped mass MDoF structural model per direction; (B) mass, displacement and restoring force of the ESDoF system; (C) tri-linear backbone approximation of the capacity curve in the Y direction of the three-storey building designed for L'Aquila TA B L E 2 Parameters of the reference ESDoF oscillators three-storey, X direction three-storey,

F I G U R E 5
Definition of additional backbone curves via interpolation of the backbones of (A) the three-storey building, in the X direction; (B) the three-storey building, in the Y direction; (C) the six-storey building, in the X direction; (D) the six-storey building, in the Y direction backbones are shown in Figure5, where the backbone curves of the ESDoF oscillators corresponding to structures that resulted from actual design are plotted in black lines.

F
I G U R E 6 (A) IDA curves and   sample at the failure threshold and (B) lognormal fragility curve fitted to the  , sample corresponding to a three-storey building's Y direction, with   = 0.48

F I G U R E 9
Continuous fitted curves of q factor against yield strength coefficient   for each site and type of structure: (A) three-storey building, X direction; (B) three-storey building, Y direction; (C) three-storey building, Y direction; (D) six-storey building, Y direction TA B L E 8 Failure rates of the SDoF oscillators corresponding to q factor equal to 3.90 at each site

F I G U R E 1 0
Failure rate against yield strength coefficient curves for (A) three-storey building, X direction; (B) three-storey building, Y direction; (C) six-storey building, X direction; (d) six-storey building, Y direction TA B L E 9 Risk-targeted q factor for the case-study sites with different seismic hazard level

F G U R E 1 1
Curves providing the variation of   as a function of q factor per benchmark site for (A) the three-storey building in the X direction; (B) the three-storey building in the Y direction; (C) the six-storey building in the X direction; (D) the six-storey building in the Y direction F I G U R E 1 2 Scatter plots of the risk-targeted behavior factors against the code's elastic spectral ordinate for (A) the three-storey building, X direction; (B) the three-storey building, Y direction; (C) the six-storey building, X direction; (D) the six-storey building, Y direction

TA B L E 1
Design parameters for the case-study structures used in this investigation

[kN] [kN] [s] [t] [kN] [kN] [s] [t] [kN] [kN]
Overstrength factors from the SPOs of the case-study structures TA B L E 5 Design model first-mode periods  1 assigned to the generated ESDoF systems TA B L E 7