VERIFICATION OF SPECTRAL REDUCTION FACTORS FOR SEISMIC ASSESSMENT OF BRIDGES

Within a nonlinear static analysis procedure perspective for the assessment of structures, one of the key issues is the employment of a demand spectrum that takes also into account, through an adequate reduction of its spectral ordinates, the hysteretic energy dissipation capacity of the structure being assessed. There are certainly a relatively large number of past parametric studies dedicated to the validation of different approaches to translate such structural energy dissipation capacity into spectral reduction factors, however such studies have focused mainly, if not exclusively, on single-degree-offreedom (SDOF) systems. It seems, therefore, that verification on full structural systems, such as complete bridges, is conspicuously needed in order to verify the adequacy of using existing SDOFderived relationships in the assessment of multiple-degree-of-freedom (MDOF) systems. In this work, eleven different spectral reduction proposals, involving diverse combinations of previously proposed equivalent damping and spectral reduction equations, are evaluated, for various intensity levels, using a preliminarily validated nonlinear static procedure. A wide set of bridges, covering regular and irregular configurations as well as distinct support conditions, is used. The accuracy of the results is checked by direct comparison with Time-History Analyses performed with ten real ground motion records. Overall conclusions are then presented with the purpose of providing practitioners and researchers with indications on the most adequate spectral reduction schemes to be employed in nonlinear static analysis of bridges.


INTRODUCTION
For what seismic assessment of structures is concerned, the use of simplified procedures, such as pushover-based ones, may be considered as a useful alternative to the more rigorous nonlinear time-history analyses, if and when all relevant variables and effects are suitably taken into account.Those variables include, amongst others, the hysteretic damping associated to the energy dissipation capacity that a structure inherently presents during an earthquake.One of the main concerns, when applying any Nonlinear Static Procedure (NSP), is thus the definition of a demand spectrum that features ordinates appropriately scaled-down to take due account of the aforementioned capacity of structures for dissipating seismic energy through hysteresis.
Reduction of spectral ordinates may be carried out through the use of either over-damped elastic or constant-ductility inelastic spectra.The former make use of equations that estimate, as a function of ductility, values of the so-called equivalent viscous damping which is then used as input into another set of expressions that provide the spectral scaling factor.In alternative, the use of constant-ductility inelastic spectra, although perhaps less commonly, has also been proposed as a means to estimate seismic demand within the scope of nonlinear static assessment of structures [e.g.1,2].
There are a relatively large number of past parametric studies dedicated to the derivation and/or validation of different approaches to estimate spectral reduction factor values [e.g.3,4], however such studies seem to have focused mainly, if not exclusively, on single-degree-of-freedom (SDOF) systems.It seems, therefore, that verification on full structural systems is conspicuously needed in order to verify the adequacy of using existing SDOF-derived relationships in the assessment of multiple-degree-of-freedom (MDOF) systems.
In the present work, the case of bridges is considered (building frames are being currently investigated through a separate endeavour), and eleven different approaches for taking into account, through spectral scaling, the energy dissipation capacity of such systems are considered.In order to do so, the Adaptive Capacity Spectrum Method (ACSM), recently proposed by Casarotti and Pinho [5] is employed as the reference nonlinear static procedure.The method starts by assuming an initial start-up target damping/ductility value, with which the scaled response spectrum is then computed, after which the intersection between the latter and the capacity curve is calculated, and the corresponding ductility/damping determined, carrying out a new iteration in case the latter differs from the initial assumption.The rationale behind such choice is the fact that it has been shown [6,7] that ACSM provides bridge seismic performance predictions that are superior (or equivalent in worst cases) to those obtained from other NSPs commonly used in practice.
The accuracy of bridge behaviour predictions, in terms of displacements, shear forces or bending moments obtained at the performance point using each of the selected spectral reduction possibilities, is evaluated throughout direct comparison with results from Nonlinear Time-History Analyses, carried out for a sufficiently large number of ground motion records.The study is performed for a wide range of bridge configurations as well as different support conditions.

SPECTRAL REDUCTION FACTORS
As previously discussed, spectral Reduction Factors (RF) can be roughly divided in two groups: damping-based and ductility-based.Roughly, the first family consists of all those methods which, through the application of a reduction factor B based on the equivalent viscous damping (elastic viscous plus hysteretic), reduce by the same amount both displacement and acceleration spectral ordinates (see equation (1) and Figure 1 left).To the second category belong all those approaches which make use of a 5%-damping elastic response spectrum and then reduce the spectral acceleration ordinates by a factor defined as a function of ductility (equation ( 2) and Figure 1 right).The spectral reduction within ductility-based methods is not exactly vertical, given that displacements are modified as well, however, for the range of periods considered in this work, R ≈ µ.A few hybrid methodologies have also been proposed, as discussed subsequently.

Damping-based methods
With regard to damping-based spectrum scaling methods, a combination of different ways of calculating the (i) equivalent viscous damping ξ eq and (ii) corresponding spectral reduction factor B are available.Some of the most commonly used ones are presented and reviewed in what follows.

Equivalent viscous damping models
In the following equations, ξ 0 stands for the so-called elastic viscous damping and µ for ductility (which was a variable of the work, given that it varied with the intensity level and the characteristics of the response of each bridge).
A) ATC40, based on the modified Rosenblueth and Herrera model (herein termed ATC40) This proposal by Rosenblueth and Herrera [8], and subsequently adopted by ATC40 [9], was the first equivalent linear method to suggest the use of secant stiffness at maximum deformation as the basis for considering inelastic response.In such approach, if one considers a bilinear system with a post-yield stiffness ratio α, the viscous damping for the equivalent linear elastic system is given by equation ( 3), where κ is an empirical parameter that takes account the degree to which the hysteresis response cycle resembles a parallelogram or not; three possibilities are defined (A, B or C), depending on structural system configuration and duration of ground shaking.
B) Kowalsky, based on the Takeda hysteretic model with post-yield hardening (herein termed TakKow) Kowalsky [10] derived an equation for equivalent viscous damping ratio that was based on the Takeda hysteretic model.For a thin response mode (empirical parameter b = 0) with unloading stiffness factor of 0.5 and a post-yield stiffness ratio α, the equivalent damping ratio is given by equation ( 4).
( ) C) Gulkan and Sozen, based on the Takeda model without hardening (herein termed TakGS) Gulkan and Sozen [11] used the Takeda hysteretic model and experimental shaking table results of small-scale reinforced concrete frames to develop empirical equation (5) to compute equivalent viscous damping ratio values.
D) Iwan (herein termed Iwan) Iwan [12] derived empirically equations to estimate the equivalent viscous damping ratio, equation (6), using a hysteretic model derived from a combination of elastic and Coulomb slip elements together with results from time-history analyses using 12 earthquake ground motion records.[13] recently developed new equivalent viscous damping relations for four structural systems, defined as a function of ductility and effective period of vibration.With the latter, the authors claim to having managed to significantly reduce the error in predicting inelastic displacements and minimize the scatter of results.Considering the structural system that best fits the case of continuous deck bridges, the equivalent viscous damping relation is that shown in equation (7), where C ST is a parameter dependent on the effective period, ranging from a minimum of 0.3 to a maximum value of 0.65.The approach proposed by Priestley et al. [14] can, in a somewhat simplified manner, be represented by equation (8).Actually, the equation should be applied to each individual pier, and then a weighted average based on shear forces and response displacements would be used to estimate the overall damping of SDOF system.Herein, for reasons of simplicity and congruency with the employed NSP, the simplification of applying equation (8) directly to the full system is carried out.
. 0 0 Pr iestley eq (8) Figure 2 plots the six previously listed approaches for equivalent viscous damping estimation.It is noted that ATC40 results are computed for a structural type B (the most appropriate for the structures considered), no post-yield hardening was considered for ATC40 and TakKow approaches, and Dwairi's estimates are plotted for two representative values of 0.4 and 0.5.It is readily observed that whilst ATC40 and TakKow models distinguish themselves from the rest by providing significantly higher equivalent viscous damping estimates, the remaining four approaches (TakGS, Iwan, Dwairi, Priestley) yield results that are very close.With the latter in mind, and considering its relative contemporariness with the work of Gulkan and Sozen [11], the expression proposed by Iwan [12] will not be considered on the subsequent parametric study, also because, in opposite to the other relationships, it does not feature an upper bound limit.Such rationale could also have led to the exclusion of one the two proposals from Dwairi et al. [13] and Priestley et al. [14], however in this case both were kept in the parametric study given their diverse nature in terms of application.

Damping-based reduction factors
As previously discussed, the computation of ξ eq is then followed by the calculation of the corresponding spectral reduction factor B. Different approaches may again be considered: A) Newmark-Hall and ATC40 (herein termed NH-ATC40) In the well known method proposed by Newmark and Hall [15], the damping reduction factors B NH for median estimates of response (i.e.50% probability of exceedance) are given by: The data of Newmark and Hall were limited to viscous damping ratios of 20% and are obtained from a limited number of earthquakes prior to 1973.In addition they were derived from the displacement response spectrum or pseudoacceleration response spectrum.It is noted that, for damping ratios higher than 5%, B NH (acc) < B NH (vel) < B NH (disp) .
The method has been adapted by most of the American design codes and guidelines, such as the ATC-40 document [9], among others, where B ATC40 is defined by SR A and SR V , corresponding to constant acceleration and velocity regions, respectively: In both equations ( 9) and ( 10) a period T c' given by equation (11) should be applied in order to guarantee continuity conditions with respect to the corner period.
The constraints imposed by equation ( 10), referring to Table 1, depend on the aforementioned ATC40 structural typologies (A, B or C), and imply maximum admitted damping ratios of 37-40% for type A, 28-29% for type B and 19-20% for type C. Analogously, SR A < SR V .In the present study, an approach blending the proposals of Newmark and Hall [15] and ATC-40 [9] is taken into account.
The original NH formulation is therefore complemented by considering the lower limits introduced by ATC40 for the velocity and acceleration zones, and introducing a factor SR D (see Table 1) with a limitation similar to SR A and SR V , i.e., maximum admitted damping ratios of 40%, 29% and 20%, for types A, B and C respectively.

B) Eurocode 8 (herein termed EC8)
Eurocode 8 [16] recommends the use of the spectral reduction factors given in equation ( 12), with a minimum of 0.55.
Ramirez et al. [17] proposed a bilinear relationship -equation ( 13) -between the reduction factor B short and equivalent damping ratio ξ eq , valid up to damping ratios of 50%.Exceeding that value the relation becomes trilinear and dependent on B long -equation (14).T b and T c are the first and third spectral characteristic/corner periods, whilst B short and B long can be found in Table 2.
) In a recent study, Lin and Chang [18] proposed a perioddependent reduction factor, equation ( 15), based on an extensive time-history analysis parametric study of linear elastic SDOF systems, using real USA records.Such RF has the advantage that, whilst still following the general trend of the others with respect to the structural period T, it is inherently continuous, hence not requiring the addition of any other type of continuity constraints or conditions.Recently, Priestley et al. [14] proposed the application of equation ( 16), which was included in earlier versions of Eurocode 8, but was then subsequently abandoned.Figure 3 presents results obtained with the above-listed spectral reduction equations for values of response period from three representative spectral regions; constant acceleration zone (T= 2T b ), constant velocity zone (T= 2T c ), constant displacement zone (T= 1.5T d ).It is observed that those approaches that feature a lower bound limit (i.e.NH-ATC40, EC8 and Priestley) tend to yield the higher spectral reductions for low damping ratios (Priestley across the entire period range, EC8 in the displacement zone and NH-ATC40 in the acceleration zone).On the other hand, for higher values of damping (larger than 0.25-0.36)Ramirez and Lin-Chang equations provide the higher spectral reductions.The dispersion among the different proposals also increases with damping ratio.
Finally, it is perhaps also noteworthy to see that the NH-ATC40 formulae is highly sensitive to period values; if one considers a constant high value of damping (e.g.ξ eq = 0.3), the spectral reduction factor may vary from 0.44 to 0.67, depending on the response period of the structure.To shed further insight into this latter issue (i.e.period dependency of spectral reductions), Figure 4 has also been produced.It is  observed that the dispersion among the several RF approaches, indicated in the plots as a percentage of the maximum RF, increases with the damping ratio.It is also noted that the NH-ATC40 variant generally yields the highest reduction factor, especially for periods higher than 3 or 4 seconds.
In the parametric study described subsequently, three out of the above-listed five possible reduction factor approaches have been selected: (i) Lin-Chang (equation 15), (ii) Priestley (equation 16), (iii) a combination of EC8 and Ramirez (equation 17).This last hybrid approach, herein termed EC8Ram, aimed especially at creating a more simplified version of EC8 and Ramirez proposals that would nonetheless envelop the general period-dependent trend of RF; EC8 is employed for low periods where it yields lower RF estimates and Ramirez is considered for the high period range where it provides high RFs.
The somewhat cumbersome trilinear NH-ATCH40 relationship was not considered further, whilst the equation by Priestley et al. [14] was employed only in tandem with the equivalent viscous damping equations proposed by the same authors (equation 8, above).

Ductility-based methods
As mentioned before, the reduction of the spectral ordinates may also be achieved by an alternative type of approach, whereby a reduction factor, directly dependent on ductility, is used.Amongst the different approaches present in literature, two have been considered in the current work: A) Miranda (2000) for firm soils (herein termed Mir2000) Miranda [4] observed that, for sites with average shear-wave velocities higher than 180 m/s in the upper 30 m of the soil profile (typically soil types A-B-C-D in ATC and FEMA  guideline documents), inelastic displacement ratios are not significantly affected by local site conditions, nor by changes in earthquake magnitude, nor by changes in epicentral distance (with the exception of very near-field sites that may be influenced by forward directivity effects).As a result, the following displacement modification factor expression was proposed: ( ) [19] suggested a relationship for a ductility-based reduction factor, with a corner period T C' dependent of the characteristic spectral period T C , equation ( 19): In the parametric study that follows, a number of diverse spectrum scaling approaches will be employed, considering combinations of different equivalent viscous damping models and damping-based scaling factors, together with ductilitybased scaling equations.With reference to the nomenclature introduced above, the eleven cases considered are hence: ATC40 -EC8Ram, ATC40 -LinChang, TakKow -EC8Ram, TakKow -LinChang, TakGS -EC8Ram, TakGS -LinChang, DwaKowNau -EC8Ram, DwaKowNau -LinChang, Priestley, Mir2000, VidFajFish.

PARAMETRIC STUDY -DESCRIPTION
The parametric study considered two bridge lengths (viaducts with four and eight 50 m spans), with regular, irregular and semi-regular layout of the piers' height and with two types of abutments; (i) continuous deck-abutment connections supported on piles, with a bilinear behaviour, and (ii) deck extremities supported on linear pot bearings.The total number of bridges is therefore fourteen, as implied by Figure 5, where the label numbers 1, 2 and 3 stand for pier heights of 7, 14 and 21 metres, respectively.The employed set of seismic excitation, referred to as LA, is defined by an ensemble of ten records selected from a suite of historical earthquakes scaled to match the 10% probability of exceedance in 50 years (475 years return period) uniform hazard spectrum for Los Angeles [21], which corresponds, in the current endeavour, to the intensity level 1.0.Additional intensity levels, linearly proportional by a factor of 0.5, 0.75, 1.5, 3.0 and 3.5, were also considered.The ground motions were obtained from California earthquakes with a magnitude range of 6-7.3 recorded on firm ground at distances of 13-30 km.The characteristics of the records are summarized in Table 3, where the significant duration is defined as the interval between the build up of 5% and 95% of the Total Arias Intensity [22].Further details on the input motion can be found in [20].
The demand spectrum employed with the NSP was defined as the median response spectrum of the ten accelerograms, since individual accelerogram spectra contain singularities that are not only of limited significance from the statistical point of view, which stands at the base of RF relationships, but may constitute a source of convergence problems in the application of code-geared nonlinear static assessment methods, as shown by Miranda and Ruiz-García [3], for instance.
The seismic demand on the bridge models is evaluated by means of nonlinear time-history analyses (THA), with admitted equivalent viscous damping of 2%, assumed to constitute the most accurate tool to estimate the 'true' earthquake response of the structures, using the fibre-based finite elements program SeismoStruct [23], whose accuracy in predicting the seismic response of bridge structures has been demonstrated through comparisons with experimental results derived from pseudo-dynamic tests carried out on large-scale models [24].The same software package was employed in the running of the displacement-based adaptive pushover analyses [25,26] that are required by the employed nonlinear static procedure, ACSM.Within this sort of procedure, when computing elastic/inelastic response spectra, it is typical for 5% equivalent viscous damping to be included so as to account, amongst other things, for the pre-yield energy dissipation capacity of the structure (due to the fact that structural elements are not all cracked right from the start).This could sound inconsistent with the 2% assumed in THA.However, when carrying out fibre-modelling analysis, on the other hand, such pre-yield energy dissipation capacity is already implicitly accounted for in the element formulation, and thus needs not to be introduced by means of an equivalent viscous damping.The results are presented in terms of different response parameters: the estimated displacement pattern (D) and flexural moments (M) of the bridge deck at the nodes above the piers, and the shear forces at the base of the piers (V) and abutments (ABT).
Then, in order to appraise the accuracy of the NSP results obtained with the different spectral reduction schemes, these are normalised with respect to the median of the corresponding response quantities obtained through the incremental THAs; this provides an immediate indication of the bias for each of the eleven spectral scaling procedures.Equation (20) shows, for a generalised parameter ∆ at a given location i, how the results from the incremental dynamic analyses (IDA), run for each of the ten records considered, are first processed.
[ ] The aforementioned results' normalisation consists thus in computing, for each of the parameters and for each of the considered locations, the ratio between the result coming from ACSM application and the median result coming from THA, as illustrated in Figure 6 and numerically translated into equation ( 21); ideally the ratio should be one.This normalisation renders also somewhat "comparable" all deck displacements, moments and shear forces, since all normalised quantities have the same unitary target value, thus allowing in turn the definition of a so-called bridge index [20].

ˆ, ,
The bridge index (BI) is computed as the median of normalised results for the considered parameter over the m deck locations; deck displacements (BI D ), deck moments (BI M ) or shear forces at the piers and abutments (BI V and BI ABT ), as shown in equation (22).The standard deviation STD measures, on the other hand, the dispersion with respect to the median, for each of the eleven damping modelsequation (23).

PARAMETRIC STUDY -RESULTS
In the following, results are analysed to evaluate the validity of the different spectrum scaling approaches, starting from a somewhat global perspective, where the entire set of results (for all bridges and for all intensity levels) are first considered together, and then sub-structured in terms of seismic input and bridge model, in order to detect trends and dependencies, if possible.

Global results
This global results overview consists in the computation of the bridge index for each spectrum scaling approach over all the 14 bridges and 6 intensity levels.In other words, the median bridge index over all bridge configurations and intensity levels represents the median of the single BI of every considered bridge configuration, at every intensity level.Analogously, median standard deviation is also computed.4 features the corresponding STDs, both referring to each spectrum scaling modality.
Considering the results shown, the first conclusion is that it is the employed damping model, rather than the chosen damping-dependent spectral reduction equation, that conditions the results; there is essentially no difference between LinChang and EC8Ram results.On the other hand, the two ductility-based approaches (Mir2000 and VidFajFish) and the damping-based ATC40 method show the worst results when compared to nonlinear dynamic analyses.This trend seems to contradict opposite results, found in previous studies [1][2][27][28][29][30], which state that inelastic spectra-based methods yield better results with respect to their elastic highly-damped counterparts.However, such findings come from the use of a different NSP from the used in the current framework, the Adaptive Capacity Spectrum Method (ACSM), which, within the elastic highlydamped methods, actually yields more consistent results in the assessment of the seismic response of bridges [6,7] when compared to other NSPs, and this may explain the difference.
The underpredicting trend of ATC40 is probably due to the well known overprediction that such approach carries out within the equivalent damping estimation in structures which will not exactly exhibit elastoplastic behaviour.
Similar tendency is found when ductility based reduction factors are used.Mir2000 approach has been primarily developed for elastoplastic behaviour but has also been widely verified for other models such as Takeda or Clough, proving to work well for periods longer than 1.2 seconds, which is not the case of the considered bridge structures with maximum periods of about 0.8 seconds.One would therefore expect overprediction, as found by Miranda and Ruiz Garcia [3] for short periods.However, the low ductility levels achieved for the considered bridge structures have lead to barely unitary displacement modification factors.VidFajFish approach, on the other hand, is known to be highly dependent on period and ductility for the short period region, which may apply to the selected bridge structures; for this region, the R factor increases linearly with increasing period, remaining constant for the rest of the periods, which may cause some excessive reduction, and, hence, under prediction, given that periods tend to be low, as early stated, for this sort of structures.
The Gulkan and Sozen approach, based on the Takeda hysteretic model, is over predicting displacements.According to equation (5) and Figure 2, it is the one with lowest equivalent damping prediction, among all the approaches, for ductility levels not higher than approximately 3. Adding to this the fact that most cases of application correspond to ductility levels equally distributed between 1 and 3, one may explain the overprediction by the underestimation of dampingbased energy dissipation.
The superiority found when using TakKow, DwaKowNau or Priestley approaches is firstly, in opposite to the others, due to the intermediate nature on estimating the equivalent viscous damping.Additionally, the latter two, which are the most recent, have been particularly developed and therefore, more suitable, to the structural behaviour where bridges fit in.DwaKowNau's equation, for instance, depends on the effective period which is another advantage, whereas Priestley's expression refers exactly to bridge piers.

Intensity level results
Herein, at each intensity level, the median Bridge Index and Standard Deviation over the 14 bridge configurations is computed for all possible spectrum scaling modalities.For reasons of succinctness, only displacements and piers shear forces results are presented in Figure 8, the former being then repeated in Table 5, to assist in the interpretation of the results.
Observing this new set of results, it turns out apparent that amongst the three approaches previously mentioned to be performing superiorly, the one proposed by Priestley et al. [14] seems to be slightly better, even if only marginally, very closely followed by those of Kowalsky [10] and Dwari et al. [13].The latter approach seems then to introduce a not relevant dependency of the coefficient C ST on the effective period.In addition, it is also noted that there is a general tendency for the displacements' predictions to improve with increasing intensity level, probably due to the fact that for low levels of nonlinearity, thus dissipation, current damping and RF relationships overestimate the reduction, whilst for the case of piers shear forces, the opposite is instead verified, since higher intensity levels are associated to a decrease in the accuracy of the NSP predictions.No noteworthy influence of the intensity level was found in deck moments and abutment forces results.
As expected (see Figure 2), ATC40 damping model seems to overestimate damping, leading to underestimated displacements at each intensity level.
As for the standard deviation, this did increase with growing intensity in the case of deck displacements (e.g. from 0.18 to 0.33 with the Priestley approach), but was instead constant for all other response parameters.Again, generally, the worst modalities predicting Bridge Index have lower, but not considerably different, dispersion levels, as can be confirmed by the displacement-only results given in Table 6.

Bridge configuration results
Herein, for each bridge configuration, the median Bridge Index and Standard Deviation across the 6 intensity levels is plotted considering each of the eleven possible modalities for energy dissipation (i.e.spectrum scaling) modelling.A bridge configuration detailed level of results enables the analysis of the influence of symmetry, regularity, length, abutments type, among other variables, on the global results previously presented and discussed.
Figures 9 and 10 show the median BI and STD across all intensity levels, for each bridge configuration.Again, for the sake of brevity, only displacement results are shown.As expected, the response predictions do appear to be very bridge-dependent, even if the observations/conclusions previously drawn still hold for the majority of configurations; e.g. the Priestley approach consistently leads to the closest-tounity BIs.It was also observed, though it is not shown herein, that variations in the accuracy of deck moments and shear forces predictions were not significant from one bridge model to the other.Evidently, BI values are better in the case of regular bridge configurations, featuring also lower STDs.An overestimating trend for short bridges and underestimating for longer ones can also be observed, with the latter cases leading also to larger scatter.These observations stand for all response parameters considered.

Figure 3 :
Figure 3:Reduction factor variation with damping in the three spectral regions.

Figure 4 :
Figure 4:Reduction factor variation with period for different damping ratios.

Figure 7
Figure 7 represents graphically the global median BI values and Table4features the corresponding STDs, both referring to each spectrum scaling modality.

Figure 8 :
Figure 8:Median Bridge Index per intensity level.

Table 2 . Reduction factors for Ramirez et al. approach.
D) Lin and Chang (herein termedLin-Chang)

Table 6 . Median Standard Deviation per intensity level -Displacements.
DisplacementsFigure 10:Median Standard Deviation per bridge configuration.