[Computational Methods in Applied Sciences] Computational Methods in Earthquake Engineering Volume 30 || Improving Static Pushover Analysis by Optimal Bilinear Fitting of Capacity

Improving Static Pushover Analysis by OptimalBilinear Fitting of Capacity Curves

Flavia De Luca, Dimitrios Vamvatsikos, and Iunio Iervolino

Abstract An improvement of codes’ bilinear fit for static pushover (SPO) curves isput forward aimed at decreasing the error introduced in the conventional SPO anal-ysis by the piecewise linear fitting of the capacity curve. In the approach proposedherein, the error introduced by the bilinear fit of the force-deformation relationshipis quantified by studying it at the single degree of freedom (SDOF) system level,away from any interference from multiple degree of freedom (MDOF) effects. In-cremental Dynamic Analysis (IDA) is employed to enable a direct comparison ofthe actual curved backbones versus their piecewise linear approximations in termsof the spectral acceleration capacity for a continuum of limit-states, allowing an ac-curate interpretation of the results in terms of performance. A near-optimal elastic-plastic bilinear fit can be an enhanced solution to decrease systematically the errorintroduced in the SPO analysis if compared to the fit approaches provided by mostcodes. The main differences are (a) closely fitting the initial stiffness of the capac-ity curve and (b) matching the maximum strength value, rather than disregardingthem in favor of balancing areas or energies. Employed together with selective areadiscrepancy minimization, this approach reduces the conservative bias observed forsystems with highly curved force-deformation backbones.

Keywords Pushover curve · Incremental dynamic analysis ·Single-degree-of-freedom · Piecewise linear fit

F. De Luca (B) · I. IervolinoDipartimento di Strutture per l’Ingegneria e l’Architettura, Università degli Studi di NapoliFederico II, Via Claudio 21, 80125 Naples, Italye-mail: [email protected]

I. Iervolinoe-mail: [email protected]

D. VamvatsikosSchool of Civil Engineering, National Technical University of Athens, 9 Heroon Polytechneiou,15780 Athens, Greecee-mail: [email protected]

M. Papadrakakis et al. (eds.), Computational Methods in Earthquake Engineering,Computational Methods in Applied Sciences 30, DOI 10.1007/978-94-007-6573-3_14,© Springer Science+Business Media Dordrecht 2013

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http://dx.doi.org/10.1007/978-94-007-6573-3_14

274 F. De Luca et al.

1 Introduction

In the last decades, improvements in the computational capabilities of personalcomputers have allowed the employment of nonlinear analysis methods in manyearthquake engineering problems. In this framework, nonlinear static analysis is be-coming the routine approach for the assessment of the seismic capacity of existingbuildings. Consequently, nonlinear static procedures (NSPs) for the evaluation ofseismic performance, based on static pushover analysis (SPO), have been codifiedfor use in practice. All such approaches consist of the same five basic steps: (a) toperform static pushover analysis of the multi-degree-of-freedom (MDOF) systemto determine the base shear versus displacement (e.g., roof) response curve; (b) tofit a piecewise linear function (often bilinear) to define the period and backbone ofan equivalent single degree of freedom system (SDOF); (c) to use a pre-calibratedR-μ-T (reduction factor-ductility-period) relationship for the extracted piecewiselinear backbone to obtain SDOF seismic demand for a given spectrum; (d) to usethe static pushover curve to extract MDOF response demands; (e) finally, to comparedemands to capacities; see [1] for example.

In fact, NSP is a conventional method without a rigorous theoretical foundationfor application on MDOF structures [2], as several approximations are involved ineach of the above steps. On the other hand, its main strength is its ability to providenonlinear structural capacity in a simple and straightforward way.

Although several improvements and enhancements have been proposed since itsintroduction, any increase in the accuracy of the method is worth only if the cor-responding computational effort does not increase disproportionately. Extensivelyinvestigated issues are the choice of the pattern considered to progressively loadthe structure and the implication of switching from the nonlinear dynamic analysisof a multiple degree of freedom (MDOF) system to the analysis of the equivalentSDOF sharing the same (or similar) capacity curve. Regarding the shape of the forcedistributions, it was observed that an adaptive load pattern could account for the dif-ferences between the initial elastic modal shape and the shape at the collapse mecha-nism [3–5]. Contemporarily, other enhanced analysis methodologies were proposedto account for higher mode effects and to improve the original MDOF-to-SDOF ap-proximation (e.g., [6]). Regarding the demand side, efforts have been put to provideimproved relationships between strength reduction factor, ductility, and period (R-μ-T relationships), to better evaluate the inelastic seismic performance at the SDOFlevel [7, 8].

One of the issues that have not yet been systemically investigated is the approxi-mation introduced by the piecewise linear fit of the capacity curve for the equivalentSDOF (for some preliminary suggestions see the recent NIST GCR 10-917-9 [9]report).

The necessity to employ a multilinear fit (an inexact, yet common, expression todescribe a piecewise linear function) arises due to the use of pre-determined R-μ-Trelationships that have been obtained for idealized systems with piecewise linearbackbones. This has become even more important recently since nonlinear mod-eling practice has progressed towards realistic multi-member models, which often

Improving Static Pushover Analysis by Optimal Bilinear Fitting 275

Fig. 1 (a) Example of exact capacity curve versus its elastoplastic bilinear fit according to EC8,and (b) the corresponding median IDA curves showing the negative (conservative) bias due tofitting for T = 0.5 s (from [18])

accurately capture the initial stiffness using uncracked section properties. The grad-ual plasticization of such realistic elements and models introduces a high curvatureinto the SPO curve that cannot be easily represented by one or two linear segments.It is an important issue whose true effect is often blurred, being lumped within thewider implications of using an equivalent SDOF approximation.

The investigation presented deals only with the bilinear approximation of thecapacity curves. Although R-μ-T relationships that can capture far more com-plex backbones have recently appeared [10], the bilinear approach is by far themost widely employed in guidelines and literature [11–16]. The approach presentedherein will be based on the accurate assessment of the effect of the equivalent SDOFfit on the nonlinear static procedure results. This can be achieved by quantificationof the bias introduced into the estimate of the seismic response at the level of theSDOF itself.

Incremental dynamic analysis (IDA) [17] will be used as the benchmark methodto quantify the error introduced by a bilinear fit with respect to the exact capac-ity curve of the SDOF. Figure 1a shows a typical example, where an elastoplasticbackbone fit is used according to Eurocode 8 or EC8 [11]. While this fit approachis meant to result to an unbiased approximation in terms of seismic performance,the median IDA results of Fig. 1b show the actual error that is introduced by suchcode-mandated fitting rules. In most cases, they lead to an unintended and hiddenbias that is generally conservative. On the other hand, this bias can become unrea-sonably high in many situations.

Therefore, three issues emerge for addressing: first, to develop a methodologyaimed at quantifying the bias introduced by the fitting of a capacity curve; second,to assess the error introduced by the fitting rules already employed in codes and lit-erature; third, to perform a systematical investigation aimed at providing alternativefits that can reduce this discrepancy to almost a minimum. The comparison withexisting approaches will function as the benchmark to evaluate the improvementintroduced by the alternative fitting rules proposed.


A near optimal solution in terms of the bilinear fit will, thus, be examined forboth hardening and softening behavior. A more comprehensive solution based on apiecewise linear fit with three or four linear segments is studied in [18].

2 Methodology

The first main target is the quantification of the error introduced in the NSP-basedseismic performance assessment by the replacement of the original capacity curveof the system, termed the “exact” or “curved” backbone, with a piecewise linearapproximation, referred to as the “fitted” or “approximate” curve (e.g., Fig. 1a). Thiswill enable a reliable comparison between different fitting schemes in an attemptto minimize the observed discrepancy between actual and estimated performance.In all cases, to achieve an accurate and focused comparison of just the effect offitting, it is necessary to disaggregate the error generated by the fit from the effect ofapproximating an MDOF structure via an SDOF system. Thus, all the investigationsare carried out entirely at the SDOF level.

An ensemble of SDOF oscillators is considered with varying curved shapes offorce-deformation backbones. They are all fitted accordingly with bilinear elastic-plastic shapes. For each considered curved backbone shape, 5 % viscous damp-ing was used and appropriate masses were employed to obtain periods of 0.2 sec,0.5 sec, 1 sec, and 2 sec. In all cases, both the exact and the approximate systemshare the same mass, thus replicating the approach followed in the conventionalNSP methodology (e.g., [1]).

When comparing an original system with its approximation, having a piecewiselinear backbone, the same hysteretic rules are always employed, so that both systemsdisplay the same characteristics when unloading and reloading in time-history anal-yses. In other words, all differences observed in the comparison can be attributed tothe fitted shape of the approximate backbone, obviously also capturing any corre-sponding differences in the oscillator period.

For each exact shape of the SDOF’s capacity curve and for each period value,several piecewise linear fit approximations have been considered according to dif-ferent fitting rules. To enable a precise comparison that will allow distinguishingamong relatively similar backbones in consistent performance terms, as it was pre-viously stated, IDA is employed. Median IDA curves of the exact capacity curvesand their backbones are compared according to the intensity measure given engi-neering demand parameters (IM|EDP) approach, see [19] and [18] for details.

To perform IDA for each exact and approximate oscillator considered, a suite ofsixty ground motion records was used, comprised of both horizontal componentsfrom thirty ordinary records from the PEER NGA database. They represent a largemagnitude, short distance bin having no near-source directivity or soft-soil effects.Using the hunt and fill algorithm [20], 34 runs were performed per record to captureeach IDA curve with excellent accuracy. The IM of choice was the 5 %-dampedspectral acceleration at the period T of the oscillator, Sa(T ), while the oscillator


displacement δ was used as the corresponding EDP, being the only SDOF responseof interest when applying the NSP.

Once the IM and EDP are decided, interpolation techniques allow the generationof a continuous IDA curve from the discrete points obtained by the 34 dynamicanalyses for each ground motion record.

The resulting sixty IDA curves can then be employed to estimate the summa-rized IDA curves for each exact and approximate pair of systems considered. Still,in order to be able to compare an exact system with a period T with its approxi-mation, having an equivalent period Teq, it was necessary to have their summarizedIDA curves expressed in the same IM. In this case it is chosen to be Sa(T ); i.e.,the spectral ordinate at the period of the exact backbone oscillator. Thus, while theapproximate system IDA curves are first estimated as curves in the Sa(Teq)-δ plane,they are then transformed to appear on Sa(T )-δ axes. This is achieved on a record-by-record basis by multiplying all the Sa(Teq) values, from the runs that comprisethe i-th IDA curve, by the constant spectral ratio (Sa(T )/Sa(Teq)) that characterizesthe i-th record [21].

The error due to the fitting is evaluated for every value of displacement, δ, ornormalized displacement, δn, in terms of the relative difference between the twosystem median Sa-capacities, both evaluated at the period T of the exact system, asit is shown in Eq. (1).

e50(δn) = Sfita,50(δn) − Sexact

a,50 (δn)

Sexacta,50 (δn)

(1)

3 Overview of Code and Guidelines Fits

The piecewise linear fit made of two branches (bilinear) is the most common approx-imation adopted by codes and guidelines; unsurprisingly most R-μ-T relationshipshave been calibrated for such approximate backbones. Eurocode 8 [11] suggests abilinear fit based on the equivalence of the area discrepancy above and below the fit,assuming an elastic-plastic idealized backbone for the equivalent SDOF, see Fig. 2a.This approach is similar to the original N2 method [1]. As a consequence, EC8 pre-scribes an R-μ-T relationship [7] based on the elastic-perfectly-plastic fit.

The EC8 approximation is fitted up to the point of attainment of a plastic mecha-nism (point A in Fig. 2a). The only remaining parameter to be evaluated is the yielddisplacement (δy ), determined by the equivalence of the areas (or the correspond-ing energy). At its basis, the EC8 fitting process is not iterative. Still, an optionaliterative adjustment of the fit for a given target displacement is also possible [11].

FEMA documents (FEMA 273 [12]; FEMA 356 [13]; FEMA 440 [14]) gradu-ally upgraded the proposed piecewise linear fit by integrating rules to account forsoftening behavior. According to FEMA 356 [13] (see Fig. 2b), the idealized rela-tionship is bilinear with an initial slope and a post yield slope (positive or negative)evaluated by balancing the area above and below the capacity curve up to the tar-get displacement (δt ) and calculating the initial effective slope at a base shear force


Fig. 2 Fitting procedure (a) according to EC8 [11], (b) according to FEMA 356 [13]

Fig. 3 Fitting procedure (a) according to FEMA 440 [14] and ASCE/SEI 41-06 [15], (b) accord-ing to Italian seismic code (CS.LL.PP. [16, 22])

equal to 60 % of the nominal yield strength. The proposed graphical procedure is it-erative. All FEMA documents are based on the coefficient method for the evaluationof demand.

FEMA 440 [14], and consequently ASCE/SEI 41-06 [15] that provides exactlythe same fitting method, assume the basic approach provided by FEMA 356 [13]with additional rules regarding the softening behavior, see Fig. 3a. The softeningslope αe is employed only in the evaluation of the limiting value of strength reduc-tion factor to avoid dynamic instability, and not in determining the target displace-ment.

It is worth to note that αe is evaluated according to a simplified expression, seeEq. (2), that accounts for both softening caused by P -� effect (αP -�) and for thenegative stiffness (α2) caused by cyclic and in-cycle degradation. The ad hoc λ

coefficient in Eq. (2) is meant to account for near-field effects.

αe = αP -� + λ(α2 − αP -�) (2)

In FEMA 440 [14] it is clearly emphasized that the whole fitting procedure; inparticular the definition of intersection at 60 % of Vy and the suggested values forthe λ coefficient are based purely on judgment. The latter observation emphasizes


the need of a more specific investigation of the fit approach adopted. Some prelimi-nary suggestions on this subject are provided in the recent NIST GCR 10-917-9 [9]report, while a more systematical validation on this specific issue is provided in thestudy herein.

Italian guidelines (CS.LL.PP. [16, 22]) suggest an elastoplastic fit that may alsoaccount for a limited softening behavior up to the point of a 15 % degradation of themaximum base shear in the capacity curve. The fit is based on the 60 % rule for theinitial stiffness, in analogy with all FEMA documents. Then, an equal area criterionis applied to derive the plateau of the bilinear fit; the latter can be extended up to thepoint in which a 15 % degradation of the maximum base shear is reached (Fig. 3b).

The result of such an approach is that the plateau of the elastic-plastic fit is al-ways lower than the maximum shear strength of the exact backbone. Obviously,if the structural model adopted cannot display any negative stiffness (e.g., elasticplastic behavior of the element without consideration of P -� effects), this fittingcriterion simply becomes equivalent to FEMA provisions. The obtained bilinear fitaccording to the Italian code provisions is then accompanied by the classical [7]R-μ-T relationship, also employed in Eurocode 8.

4 Investigation of Bilinear Fits

Bilinear elastic-plastic or elastic-hardening fits are the fundamental force-defor-mation approximations employed in all NSP guidelines, as shown in the previoussection. The simplicity of the bilinear shape means that the only need is to estimatethe position of the nominal “yield point” and select a value for the constant post-elastic stiffness. According to codified prescriptions the general criteria followedare: (a) using the intersection at a specific point of the “exact” capacity curve, and/or(b) area balancing approaches that in turn conform to an equal energy criterion.

The definition of a near-optimal bilinear fit is attempted by a two steps process:first the definition of the proper initial slope is investigated considering generalizedelastic plastic exact backbones, and then the post-elastic part is investigated consid-ering generalized hardening and softening in the exact backbone studied. The scopeof the investigation is always limited to bilinear non-softening fits.

For the first sample of generalized elastic-plastic systems studied, the stiffnessbecomes zero beyond a displacement of 0.10 m and the target displacement is as-sumed to be well into the fully plastic region. Thus, the post-elastic segment is fixedto have zero stiffness while the initial “elastic” part can be fitted at will.

In both the FEMA and EC8 fitting rules, the fitted “elastic” stiffness can be afunction of the target displacement due to the area balancing used. This can makethe bilinear fitting yield different results depending on the limit-state of interest andmay initially put the code-mandated fits (as implemented herein) at a slight dis-advantage. It will be remedied in the second part of the study where generalizedelastic-hardening systems will be tackled. They have varying post-yield stiffnessand lower target displacements, allowing more fitting flexibility and providing a


Fig. 4 Comparison of generalized elastic-plastic capacity curves and their corresponding fits hav-ing (a) insignificant versus (b) significant changes in initial stiffness (from [18])

more rigorous comparison of code fits against any proposal for a near-optimal bi-linear fit in the case of non-softening “exact” capacity curves. Finally, in analogywith the approach followed by the recent Italian seismic code, also “exact” capacitycurves characterized by degrading behavior will be investigated. In this latter casean elastoplastic fit is still employed and the general aim is to check up to whichpoint in the softening range it remains accurate.

Generally, the target, herein, is the development of a simple fitting rule that per-forms well for a continuum of limit states. Therefore, the performance of all ruleswill be carefully examined even outside the immediate region of interest defined bya target displacement.

4.1 Bilinear Fits of Generalized Elastic-Plastic Systems

First a family of generalized elastic-plastic capacity curves is considered that ex-hibit a stiffness gradually decreasing with deformation, starting from the initial elas-tic and reaching zero slope. The shapes are characterized by the magnitude of thechanges in stiffness. Figures 4a and 4b give an example of the shapes employed inthe investigation of this family of backbones and emphasize two opposing cases. Thefirst (Fig. 4a) is not characterized by significant curvature, while the second (Fig. 4b)shows a significant change in slope that can be representative of the behavior of amodel that accounts for uncracked stiffness. The kinematic strain hardening and amoderately pinching hysteresis rule [23] are considered.

Three basic fitting rules are compared: (a) the FEMA fit (60 % rule) assuming atarget displacement high enough so that the slope of the second branch of the bilin-ear can be assumed to be zero; (b) the EC8 fit using a simple equal area criterion;(c) the 10 % fit, defined so that the intersection between the capacity curve and the


Fig. 5 Median relative error comparison between the 10 %, FEMA and equal area fits forT = 0.2 sec, when applied to the capacity curves of Fig. 4: (a) insignificant versus (b) signifi-cant changes in initial stiffness (from [18])

Fig. 6 Median relative error comparison between the 10 %, FEMA and equal area fits forT = 0.5 sec, when applied to the capacity curves of Fig. 4: (a) insignificant versus (b) signifi-cant changes in initial stiffness

fitted elastic segment is at 10 % of the maximum base shear. The latter is the pro-posal for a simple rule that can better capture the initial stiffness. Figure 4a showsthat in the case in which the capacity curves are not characterized by significantstiffness changes at the beginning of the backbone, the three fits are very similarto each other. They differ significantly though when the initial backbone stiffnessdiminishes rapidly, as in Fig. 4b. In all cases the fitted post-elastic stiffness is set tozero.

As described in the methodology section, IDA is performed for each of the SDOFsystems presented in Fig. 4 and for their fitted approximations for a range of peri-ods. Figures 5 and 6 show the comparison in the median Sa-capacity for T equal to0.2 sec and 0.5 sec, respectively, as functions of normalized displacement δn. The


Fig. 7 (a) Backbones and (b) hysteretic behavior according to standard kinematic strain hardeningrule of the generalized elastic-plastic systems considered (adapted from [18])

normalizing value is in this case equal to 0.1 m, the point in which all the backbonesturn to zero stiffness. Obviously, the shape of the backbones has a significant im-pact. In all cases, the error becomes significant for the shape with non-trivial curvedsegments, the maximum error appearing at the earlier backbone segments. The 10 %fit leads to a significant decrease in the error for any deformation level, even for thehighly curved shape of Fig. 4b where it clearly violates any notion of equal area (orequal energy) that seems to be prevalent in current guidelines.

The new fit introduced leads to a slightly non-conservative estimation of thecapacity for displacements before the full plasticization (for δn up to 1) and onlyfor short-period systems, T = 0.2 sec (Fig. 5). On the other hand, even in case ofhighly-curved backbones (Fig. 5b), only a 10 % underestimation appears at most.Conversely, it has to be noted that code approaches are always conservative for allthe displacement levels and all the shapes considered, but at a cost of almost 20–40 % underestimation of capacity. The trends identified are generally confirmed forall other periods considered (T equal to 1.0 sec and 2.0 sec).

Beyond the two curved shapes shown in Fig. 4, a number of different gener-alized elastic-plastic shapes were also investigated. A sample of ten backbones isconsidered, aimed at drawing out general conclusions regarding the proposed fit.The sample consists of five different shapes (see Fig. 7a) times the two hystereticrules. In Fig. 7b the standard kinematic strain hardening rule is shown for the fiveshapes considered.

Figures 8, 9, 10 and 11 show the relative error on the median Sa-capacity eval-uated at each period for the 10 % fit proposed and the conventional FEMA fit. Thebias is evaluated up to δn equal to 2, where most of the significant differences ap-pear. This is a more focused view, compared to the two shapes in Figs. 4a and 4bwhere it was evaluated up to δn equal to 10. Figures 8 to 11 show gray dotted linesrepresenting the error related to a specific shape of the backbone and to a specifichysteretic rule. The effect of the hysteretic rule was found to be insignificant, as


Fig. 8 The median relative error for (a) the 10 % fit and (b) the 60 % FEMA fit, T = 0.2 sec, incase of elastic-plastic SDOF system family (gray dotted lines) (adapted from [18])

Fig. 9 The median relative error for (a) the 10 % fit and (b) the 60 % FEMA fit, T = 0.5 sec, incase of elastic-plastic SDOF system family (gray dotted lines)

the error depends primarily on the shape of the fit; thus the two different hystereticbehaviors have been considered together to build up a heterogeneous family of gen-eralized elastic-plastic behavior.

The 10 % fit enjoys an insignificant bias on average for all the periods consideredand the error introduced by the fit never exceeds 20 %. FEMA fit shows a strictlynegative (i.e., conservative, bias of 20 % or even 60 %) depending on the shape of theoriginal backbone. Most of the bias is concentrated at the beginning of the backboneas it was already emphasized by the examples in Figs. 5 and 6. Error comparisonsfor the Sa-capacity dispersion (record to record) are not shown as all fits roughlyachieve similar performance. Some differences may appear in the region precedingthe nominal yield point of the approximation.


Fig. 10 The median relative error for (a) the 10 % fit and (b) the 60 % FEMA fit, T = 1.0 sec, incase of elastic-plastic SDOF system family (gray dotted lines) (adapted from [18])

Fig. 11 The median relative error for (a) the 10 % fit and (b) the 60 % FEMA fit, T = 2.0 sec, incase of elastic-plastic SDOF system family (gray dotted lines)

Therein the fitted system will predict no dispersion whereas the actual one showssome small variability. Still, this is to be expected, and it is not important enough tofavor one fit with respect to another.

Summarizing, it can be stated that the fit should capture as close as possible theinitial stiffness of the backbone while the generally low secant stiffness assumed inmost of the guidelines and codes tends to be overly-conservative. The only possibleexception to this rule appears only for initially ultra-stiff systems that very quicklylose their initial properties. This is the reason why fitting the “elastic” secant at 5 %or 10 % of the maximum base shear, as opposed to 0.5 % or 1 % is considered amore robust strategy.


Fig. 12 Comparison of generalized elastic-hardening capacity curves and their corresponding fitshaving (a) insignificant versus (b) significant changes in initial stiffness (from [18])

4.2 Bilinear Fits of Generalized Elastic-Hardening Systems

The second family of shapes investigated is characterized by a generalized elastic-hardening behavior. Only the pinching hysteretic rule was considered for this familyof backbones, given the insignificant differences observed earlier when compared tothe kinematic hardening.

Each backbone considered was characterized by different curvatures and finalhardening stiffness, allowing a wide coverage of the typical shapes that can be ob-tained considering different structural behaviors and modeling options.

In analogy with the two examples showed in the previous subsection, two dif-ferent backbones will be presented in detail from the family considered. The first(Fig. 12a) is not characterized by significant changes in the stiffness, in contrast tothe second (Fig. 12b). The target displacement is assumed to be equal to 0.2 m. TheEC8 fit is not applied as it is restricted to elastic-plastic approximations which areclearly inferior for the shapes shown in Fig. 12. On the other hand, the FEMA fit rulecan be applied without problems, although it might call for different approximationsdepending on the value of the target displacement (FEMA-356 fit). Still, the resultsand the corresponding conclusions remain the same in all cases. The alternative fitproposed, termed the H-10 % fit rule, determines the initial stiffness at 10 % (in-stead of 60 %) of the nominal yield shear defined in accordance with FEMA, whilethe post-elastic stiffness is determined by minimizing the absolute area discrepancybetween the capacity curve and the fitted line.

The proposed H-10 % procedure came out as the simplest rule with a near-minimum error for this family of backbones. While many alternatives were consid-ered, they are not showed herein for the sake of brevity. It has to be noted that elastic-plastic fits according to the proposal of the previous subsection have been also con-sidered in the investigation of this family of backbones, assuming the plateau of thefit at the force corresponding to the target displacement (e.g., at 0.2 m here); even ifless performing than the H-10 % fit rule, considered in the following, the still low


Fig. 13 Median relative error comparison between the H-10 % and FEMA fits for T = 0.2 sec,when applied to the capacity curves of Fig. 12: (a) insignificant versus (b) significant changes ininitial stiffness (from [18])

Fig. 14 Median relative error comparison between the H-10 % and FEMA fits for T = 0.5 sec,when applied to the capacity curves of Fig. 12: (a) insignificant versus (b) significant changes ininitial stiffness

bias of those trials compared to the FEMA-356 fitting rule, enforced the generalconclusion that capturing the initial stiffness is the dominant criterion that leads toa significant bias reduction with respect to actual code fit prescriptions.

The results of the two fitting procedures (H-10 % fit and FEMA-356 fit) appliedto the example shapes appear in Fig. 12. Obviously, when the stiffness of the back-bone is not characterized by abrupt changes in the curvature (Fig. 12a) both fits tendto be practically the same. Figures 13 and 14 show the error introduced by each fit,for both backbone shapes considered in Fig. 12, in the cases of T = 0.2 and 0.5 sec,respectively. In analogy with the results presented for the elastic-plastic case, mostof the error is concentrated at the beginning of the backbone. In the case of the back-bone with low changes in the stiffness (low curvature), it can be observed that the


Fig. 15 Backbones (a) and hysteretic behavior according to pinching hysteresis rule (b) of thegeneralized elastic-hardening systems considered (adapted from [18])

error is very small and very similar for both fits. A higher curvature of the backboneincreases the error introduced by the fit and emphasizes that although at the targetdisplacement the backbones and their fits in both cases are coincident, this is notenough to guarantee the same error. The earlier fitted segments and especially thedefined equivalent period can make a large difference. As the H-10 % rule man-ages to capture the initial stiffness better, it provides better predictive capability forhigher displacements.

The evidence, again, confirms the general trend already shown in the previoussubsection, that it is important to capture the initial stiffness of the capacity curveto have an unbiased fit. In this case, as well, the code approach results in a conser-vative error that can be over 50 % in the case of non-trivial shapes and at the lowerdisplacement values (Fig. 12b). In the same range, the H-10 % fit leads to a slightlynon-conservative solution only for the short periods (T = 0.2 sec), an effect thatwas also observed in the elastic-plastic backbone family. Otherwise, the fit remainsconservative and overall it can be considered to be relatively unbiased.

In this case a sample of only four shapes is considered, the different shapesshown in Fig. 15a and the pinching hysteretic rule assumed for these backbonesappearing in Fig. 15b. In Figs. 16, 17, 18 and 19 the relative errors in the medianSa-capacity, are compared for the four different periods and the different SDOFbackbones. The importance of capturing the initial stiffness is highlighted by theresults at each period, and the H-10 % fit leads to a small and relatively unbiasederror, which seldom exceeds 10 %. In this case the sample of backbones consideredfor the elastic-hardening case was smaller than the elastic-plastic case, but robust-ness of the general results, showing the same trends in both cases, supports previousremarks. It should be noted that the results of the FEMA approximation will im-prove somewhat at low displacements if we refit for a lower target displacement.Still, the corresponding improvement by a similar refit of H-10 % at the target dis-placement nullifies this advantage, thus the general fitting hierarchy observed ismaintained.


Fig. 16 The median relative error for (a) the H-10 % fit and (b) the 60 % FEMA fit, T = 0.2 sec,in case of elastic-hardening SDOF systems family (gray dotted lines) (adapted from [18])

Fig. 17 The median relative error for (a) the H-10 % fit and (b) the 60 % FEMA fit, T = 0.5 sec,in case of elastic-hardening SDOF systems family (gray dotted lines)

4.3 Bilinear Fits of Generalized Elastic-Hardening-SofteningSystems

The Italian hybrid fitting rule (CS.LL.PP. [16, 22]) is worth to be investigated, espe-cially to check whether considerations regarding the best approach for non-softeningbehaviors are still reliable to capture early negative response. In other words, thegeneral principle of getting the initial stiffness (10 % rule) is going to be checkedfor a backbone family of softening curves and consequently compared with the 60 %rule. In addition, the merits of balancing the mismatch areas above and below thecapacity curve will also be discussed. While the Italian Code, similar to other guide-lines, is concerned with providing a specialized fit for a given target point on thecapacity curve. Following the methodology described in Sect. 2, herein, it will be


Fig. 18 The median relative error for (a) the H-10 % fit and (b) the 60 % FEMA fit, T = 1.0 sec,in case of elastic-hardening SDOF systems family (gray dotted lines) (adapted from [18])

Fig. 19 The median relative error for (a) the H-10 % fit and (b) the 60 % FEMA fit, T = 2.0 sec,in case of elastic-hardening SDOF systems family (gray dotted lines)

sought to provide near-optimal fits for a continuum of limit states. Thus, any depen-dence on any specific target point on the backbone is intentionally avoided, that is,checking all possible targets at the same time. This is achieved employing a simpledirect search approach, where the height of the plastic plateau for the fit is graduallymoved from 80 % to 100 % of the peak shear in the exact capacity curve. Conse-quently, while checking for a broadly optimal fit, all possible fits that could arisefrom applying the Italian Code approach to different target points will be efficientlycaptured.

Two example backbones from the numerous tests conducted are shown in Fig. 20.They were chosen to emphasize two different softening trends, combined withdifferent curvature changes in the initial part of the backbones: insignificant (seeFig. 20a) versus significant (see Fig. 20b). Four different fits are displayed out of


Fig. 20 Comparison of two capacity curves with softening branch and their corresponding fitshaving (a) insignificant versus (b) significant changes in initial stiffness (Fig. 20(b) from [18])

Fig. 21 Median relative error comparison between the 10%L, 10%P, 60%L and 60%P fits forT = 0.2 sec, when applied to the capacity curves of Fig. 20: (a) insignificant versus (b) significantchanges in initial stiffness and different softening slopes (Fig. 21(b) from [18])

the large number that has been checked: the initial stiffness is fixed at 10 % or 60 %of the maximum shear combined with two plateau levels at 80 % (L) and 100 % (P)of peak shear, thus obtaining four fitting approaches named 10%L, 10%P, 60%L and60%P, respectively. The performances of each of the four fits considered are shownin Figs. 21 and 22 for T = 0.2 sec and 0.5 sec, respectively.

Results show how the changes in curvature in the initial part still play an im-portant role for fitting performances. The 10 % fit improves results as long as thereare significant changes in the curvature of the exact backbone. Furthermore, catch-ing the peak point (P versus L fits) seems to offer better performance than a lowerplateau value, regardless of any area-balancing rules. The latter results are partiallyconfirmed by the fits suggested in other studies where the maximum shear value isselected as one of the criteria for fitting softening backbones (e.g., [24]).


Fig. 22 Median relative error comparison between the 10%L, 10%P, 60%L and 60%P fits forT = 0.5 sec, when applied to the capacity curves of Fig. 20: (a) insignificant versus (b) significantchanges in initial stiffness and different softening slopes

Fig. 23 Backbones (a) and example hysteretic behavior according to pinching hysteresis rule (b)of the family of capacity curves considered

For low frequencies and significant changes in the initial stiffness, the 10 % fit,in both its versions showed herein, P and L, can lead to slightly non-conservativeresults at the beginning of the backbone. The same effect was observed also in thecase of non-softening backbones. Still, for conventional limit states of interest andfor most practical applications for which a static pushover is used, the target pointswill not be located in this early near-elastic part of the backbone.

The fit approaches showed have been tested for a sample family of exact back-bones in which curvature at the beginning and the slope of the softening have beenvaried. The sample family of backbones considered and their hysteresis loops arepresented in Fig. 23. The median error trends of the 60%L and the 100%P fits arecompared in Figs. 24, 25, 26 and 27, respectively, for the four periods considered(0.2, 0.5, 1.0 and 2.0 seconds). The error is mapped according to the characteris-tic points of the exact backbone; the peak point “p”, where the maximum shear


Fig. 24 The median relative error at T = 0.2 sec if (a) the 60%L and (b) 10%P fits are employed,respectively, for the family of capacity curves in Fig. 23 (gray dotted lines) (adapted from [18])

Fig. 25 The median relative error at T = 0.5 sec if (a) the 60%L and (b) 10%P fits are employed,respectively, for the family of capacity curves in Fig. 23 (gray dotted lines)

is attained, and the ultimate point “u”, where the exact capacity curve reaches theresidual branch.

Figures 24 to 27 are meant to represent the median error to be expected whenfitting a generic capacity curve with a specific rule (in this case 60%L and 100%Prules respectively). The 10%P fit is found to be an unbiased fit approach that canbe extended to the mildly softening part of the backbone at each of the periodsinvestigated, showing robust performances for this portion of backbone in a widerange of frequencies.

5 Conclusions

Structural seismic assessment based on the nonlinear static procedure is a methodthat, in its coded variants, has become common in the last decades among re-


Fig. 26 The median relative error at T = 1.0 sec if (a) the 60%L and (b) 10%P fits are employed,respectively, for the family of capacity curves in Fig. 23 (gray dotted lines) (adapted from [18])

Fig. 27 The median relative error at T = 2.0 sec if (a) the 60%L and (b) 10%P fits are employed,respectively, for the family of capacity curves in Fig. 23 (gray dotted lines)

searchers and practitioners. It is based on the fundamental approximation that thebehavior of an MDOF system can be interpreted by the response of an equivalentSDOF. This necessitates a number of approximations at various stages of the pro-cedure. Herein, the problem of the optimal bilinear fit to be chosen as the idealizedbase shear-displacement curve is systematically investigated.

Assessment of different fits is achieved via incremental dynamic analysis on anintensity-measure capacity basis that allows a straightforward comparison of theperformance of structural systems characterized by different periods; in this case,the period of the capacity curve and its bilinear approximation. Therefore, the ap-proach followed allows the investigation of a continuum of limit states at the SDOFlevel, excluding other sources of error already investigated in detail in other studies.

The initial focus is on the non-softening behavior of the exact capacity curve to befitted. Code fits (Eurocode 8, FEMA documents and Italian code) were found to bealways characterized by a mostly conservative bias due to the enlarged equivalent


period assumed. Italian code recommendations to capture early softening behav-ior by means of extended elastoplastic fits are shown to result to acceptable errorswhen matching the maximum strength and not exceeding the displacement that cor-responds to 85 % of the maximum shear.

Finally, the investigations yielded to suggest the rule of setting the elastic seg-ment to match the initial stiffness of the exact capacity curve (e.g., fitting at 10 %of the peak base shear). A subsequent hardening segment should be fitted by min-imizing the area discrepancy up to the target point, while a plastic segment shouldinstead be set at the peak base shear value. The low bias introduced by these en-hanced fit solutions is found to be an improvement with respect to existing codeapproaches. Essentially, such fitting rules can be assumed to be a near-optimal so-lution especially in the case of capacity curves with significant changes in stiffness,representative of modern modeling approaches that account for the uncracked sec-tion properties. In fact, the error introduced by a near-optimal bilinear fits does notexceed 20 %.

Furthermore, the latter results represent a step towards the definition of a fullyoptimized fit rule that can represent a further enhancement to be considered as anupgrade in current seismic provisions. While any enhanced three-segment piecewiselinear fit that incorporates a softening branch would further improve the accuracyin the equivalent SDOF backbone, it would also require changes in the R-μ-T rela-tionships. On the other hand, the procedures presented herein can fit seamlessly incurrent seismic codes without requiring any further changes.

Acknowledgements The analyses presented in the paper have been developed in cooperationwith Rete dei Laboratori Universitari di Ingegneria Sismica—ReLUIS for the research programfunded by the Dipartimento della Protezione Civile—Executive Project 2010–2013.

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[Computational Methods in Applied Sciences] Computational Methods in Earthquake Engineering Volume 30 || Improving Static Pushover Analysis by Optimal Bilinear Fitting of Capacity