Engineering Structures 46 (2013) 155–164

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Engineering Structures

journal homepage: www.elsevier .com/ locate /engstruct

On the quantification of seismic performance factors of Chevron Knee Bracings,in steel structures

Mojtaba Farahi, Massood Mofid ⇑Department of Civil Engineering, Sharif University of Technology, Tehran, Iran

a r t i c l e i n f o

Article history:Received 1 April 2011Revised 20 June 2012Accepted 22 June 2012Available online 11 September 2012

Keywords:Seismic performance factorChevron Knee BracingIncremental dynamic analysisCollapse margin ratio

0141-0296/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.engstruct.2012.06.026

⇑ Corresponding author. Tel./fax: +98 21 6601 4828E-mail addresses: mfarahi@mehr.sharif.edu (M. F

Mofid).

a b s t r a c t

As a matter of fact, it is necessary to have the values of Response Modification Factor R, Over-strengthFactor X0, and Deflection Amplification Factor Cd in order to design seismic-force-resisting systemsaccording to design and loading codes. This study is intended to evaluate these factors for a structurallateral bracing system called Chevron Knee Bracing (CKB). In this type of bracing, the knee elements assistthe system to dissipate energy through the formation of plastic flexural and/or shear hinges within thepresented bracing system. The approach utilized in this study is according to FEMA P695 based on lowprobability of structural collapse and involves nonlinear static and dynamic analyses. Over-strengthand ductility of this type of bracing is investigated through performing nonlinear static analyses. Con-ducting Incremental Dynamic Analyses (IDA), Collapse Margin Ratios (CMRs) of the defined archetypesmodel are achieved and modified to obtain an Adjusted Collapse Margin Ratio, ACMR for each archetype.The values of calculated ACMRs are compared with the accepted values proposed by FEMA P695 in whichthe total system collapse uncertainty is considered to prove the validity of presumed seismicperformance factors of CKB systems.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Having capacity to dissipate energy by tolerating reasonableplastic drifts is the main goal considered in designing seismic-load-resisting systems for a reduced seismic load. A reductionfactor, called Response Modification Factor, R, was introduced bydifferent codes for each routine approved seismic-load-resistingsystem. This study is intended to assess the seismic performancefactors of a structural lateral bracing system called Chevron KneeBracing (CKB) (Fig. 1). The main concept in application of this typeof structural lateral bracing system is dissipating more energythrough the formation of plastic hinges in the knee elements. Theformation of plastic hinges in knee elements can be in flexuralmode and/or shear mode depending on which mode they aredesigned for. Furthermore, each of these modes of yielding isinvestigated in this study.

Several studies are available on these systems, which utilizeknee elements as ductile fuses besides bracing elements to providemore ductility while the lateral stiffness of the frames is preserved.In two studies, more information was revealed on the behavior andcharacteristics of knee-bracing-frames in each yielding mode and anumber of experiments were conducted on these systems by

ll rights reserved.

.arahi), mofid@sharif.edu (M.

Balendra et al. [1,2]. They investigated frames, consisted of a diag-onal brace accompanied with a single knee element, in the righttop corner of the frames. According to the pseudo dynamic testsconducted by Balendra et al., the maximum displacement ductilityof 4 and 6 was achieved for frames with flexural and shear yieldingmodes, respectively. Frames with shear yielding mode dissipatedmore energy compared to the ones with flexural yielding modein these investigations. Mofid and Khosravi [3] presented anapproximate method for the determination of nonlinear behaviorof such diagonal knee bracing frames. They tried to characterizean appropriate geometry and diagonal element angle for this typeof bracing system.

Moreover, more details were illustrated about elastic andinelastic behaviors of CKB frames, by Mofid and Lotfollahi [4]. Intheir investigation, several parametric studies were carried outand the optimal shape and angle of the knee and the brace ele-ments had successfully been characterized. They further presentedstep-by-step algorithms in conjunction with some charts and ta-bles to model nonlinear behavior of these frame systems in eachyielding mode of the knee elements [5]. A procedure to designCKB frames was the final achievement of these studies.

This is a fact that all of aforementioned studies took only singlestory one-bay frames into consideration and no investigation havebeen conducted on real multistory CKB frames behavior. Therefore,further studies are quite necessary, regarding such frames andtheir performance under severe earthquake. In order to make

Fig. 1. General form of Chevron Knee Bracing frames under consideration in thisstudy. (a) A 2-stories frame with 2 bays; (b) a 6-stories frame with 4 bays.

156 M. Farahi, M. Mofid / Engineering Structures 46 (2013) 155–164

utilizing such frames possible in practical aspects, it is required tofind a technique to design CKB frames according to available designcodes; and to warrant the seismic behavior of the frames designedby this mean. Among different approaches proposed to computeResponse Modification Factor, R, ATC-19 has presented a simplifiedprocedure to estimate this factor by the product of three parame-ters: First, the over-strength factor; second, the ductility factor;third, the redundancy factor [6]. The Response Modification Factorsof several seismic-force-resisting systems have been assessedthrough this method [7–11]. FEMA P695 further proposed a meth-odology to quantify buildings seismic performance factors which isemployed in this study [12]. The objectives of this investigationare:

� To investigate multi-stories CKB frames nonlinear behavior andseismic response through conducting nonlinear dynamic andstatic analyses.� To introduce the seismic performance parameters needed based

on design codes for CKB frames in a manner that such frameshave a collapse margin ratios exceeding the acceptable collapsemargin ratios according to FEMA P695 [12].

2. General framework to evaluate seismic performance factors

The methodology proposed by FEMA P695 (2009) [12] and wasapplied in this study involves three main steps to establishingSeismic Performance Factors (SPFs). This includes characterizingsystem behavior through definition of appropriate index arche-types; further preparing nonlinear models of mentioned arche-types and conducting nonlinear static and dynamic analyses.Finally, SPFs are established utilizing results of analyses along withconsidering uncertainties explicitly in each step.

2.1. Development of archetypes and nonlinear models

As the first step, it is required to gather thorough data about theseismic-force-resisting system. These data includes type of con-struction materials, system possible configurations, inelasticenergy dissipation mechanisms, and intended range of application.Structural system archetypes are developed according to thesekinds of data in order to represent the bounds of proposedseismic-force-resisting system. Structural archetypes provide thebasis for preparing a finite number of designs, and then provide acorresponding number of idealized nonlinear models. These mod-els should appropriately represent nonlinear behavior of proposedseismic-force-resisting system.

2.2. Conducting nonlinear analyses

As the next step, utilizing the nonlinear models, nonlinear dy-namic and static analyses should be conducted. Pushover analysesare used to validate nonlinear behavior of models and to providethe necessary data on system over-strength and ductility. Nonlin-ear dynamic analyses are performed to assess Median CollapseCapacity (bSCT) and Collapse Margin Ratio (CMR) of each archetypemodel. CMR is defined as the ratio of ground motion intensity thatcauses median collapse to the Maximum Considered Earthquake(MCE) for each of the archetype models.

2.3. Performance evaluation

Finally, presumed R factor to design models is validated throughcomparison of CMRs, after adjusting, to acceptable values proposedby FEMA P695. These accepted values depend on total uncertaintyinvolved in the aforementioned steps. The following sources ofuncertainty are explicitly considered: (1) Record-to-record uncer-tainty, (2) Design requirement related uncertainty, (3) Test datarelated uncertainty and finally (4) Modeling uncertainty. Theover-strength factor can be evaluated from Pushover curves andthe deflection amplification factor will be calculated based on theaccepted value of the Response Modification Factor.

The flowchart in Fig. 2 represents the above procedure to eval-uate Seismic Performance Factors for a structural system.

3. Development of structural system archetypes

The process begins with gathering the required information andcharacteristics of the system such as design requirement underwhich the system is designed and information relating to compo-nent material properties, force–deformation behavior, and nonlin-ear response. The system possible behaviors are characterizedthrough establishment of the archetypes which are intended torepresent its typical applications. The following characteristicsare considered in the structural system archetypes and are definedin this study: (1) Different possible system height, and thereforedifferent fundamental periods are studied by defining 2-, 6-,10- and 14-stories frames; (2) Different number of bays in eachstory is considered through defining 2-bay and 4-bay systems;(3) The yielding mode of knee element is considered through defin-ing two sets of archetypes that knee elements yield in flexuralmode in one set and in shear mode in another set; (4) Due to theconsiderable effects of the ratio of seismic mass to gravity mass,two set of frames are defined including ‘‘space frames’’ and ‘‘perim-eter frames’’. By utilizing this method, structural system arche-types are assembled into bins called performance groupsaccording to major divisions.

In all the aforementioned 2-dimensional CKB frames, theheight of the stories is equal to 3 m and the bay length is as-sumed to be constant and equal to 4 m for all of the archetypes.In the case of space frames, the seismic mass is assumed to beequal to the mass conveyed to the frame. In perimeter cases,the ratio of seismic mass to gravity mass is assumed to be equalto 5. The dead load assumed to be 5.5 kN/m2 and the values of 2and 1.5 kN/m2 are considered as the live load for the stories andat the roof level, respectively.

The seismic performance factors are primarily assumed; next,the Equivalent Lateral Loading method (EQL) is employed accord-ing to part 12-8 of ASCE/SEI 7-05 in order to assessing seismicloads [13]. The values R = 7, X0 = 2, and Cd = 5 are presumed.According to Table 12-2-1 of ASCE/SEI 7-05, the R factor value forSCBF systems is proposed to be equal to 6; however, the assumedR factor is equal to 7 for CKB frames in this study, since they are

Fig. 2. The flowchart of the methodology used in this study.

M. Farahi, M. Mofid / Engineering Structures 46 (2013) 155–164 157

more ductile compared to general SCBF systems. MaximumConsiderable Earthquake (MCE) and Design Earthquake (DE) spec-tra have been defined in FEMA P695 methodology for structures inSeismic Design Categories (SDC) B, C, and D. In this study, these de-sign spectra are utilized to evaluate the equivalent static load ofearthquake in order to design archetypes. The performance

evaluation requires assessment of index archetypes which weredesigned for spectral intensities of the highest SDC in which thesystem is allowed. Therefore, the CKB index archetypes should bedesigned for maximum spectral intensities related to SDC-Daccording to Table 5-1 of FEMA P695. Occupancy category I andII is considered in this methodology.

Table 1Performance groups for evaluation of CKB frame archetypes.

GroupNo.

Grouping criteria Number ofarchetypes

Yielding mode inknee elements

Seismic mass togravity mass ratio

Perioddomain

PG1 Shear yielding mode Perimeter frames Short 2PG2 Long 6PG3 Space frames Short 0PG4 Long 2

PG5 Flexural yielding mode Perimeter frames Short 2PG6 Long 6PG7 Space frames Short 0PG8 Long 2

158 M. Farahi, M. Mofid / Engineering Structures 46 (2013) 155–164

The 2-dimensional CKB frames are considered as structural sys-tem archetypes developed under the design provisions of SpeciallyConcentrated Braced Frames (SCBF) [14,15]. IPB sections were usedfor columns as well as the beams over the bracing. Other beamssections were chosen among IPE sections. Double channel sectionswere used for brace elements. Furthermore, during the design pro-cess, the ratio of demand over elastic capacity for knee elementswas allowed to exceed one in order to foster dissipating energythrough knee elements before the buckling of the braces. The max-imum interaction in these elements was restricted to 130% of theirelastic capacity. To design the system with flexural yielding mode,box sections were used and the sections of knee elements in theframes with shear yielding mode were selected from I-shaped sec-tions [2].

The mode of yielding depends on the length of knee element. Byincreasing the length of knee element, the mode of yielding tendsto convert toward flexural mode from shear mode [2]. The ratio be-tween story height and the vertical distance between the ends ofthe knee element was taken equal to 0.25 for the frames with flex-ural yielding mode. Moreover, this ratio of 0.2 was selected for theframes with shear yielding mode [5]. In order to prevent knee ele-ments from yielding in unpredicted mode, the length of knee ele-ment should be greater than 4Mp/Vp for flexural yielding modeand should be less than 4M�

P=Vp for shear yielding mode. Mp, M�p

and Vp are the plastic moment of the knee element, section reducedplastic moment contributed by only flanges of the knee sectionsand plastic shear force of the knee sections respectively whichare defined by Eqs. (1)–(4) [5].

Mp ¼ Z � Fy ð1Þ

M�p ¼ tf � b � ðd� tf Þ � Fy ð2Þ

Vp ¼ tw � ðd� 2tf Þ � Fy=ffiffiffi3p

for I-shaped sections: ð3Þ

Vp ¼ 2tw � ðd� 2tf Þ � Fy=ffiffiffi3p

for Boxed sections: ð4Þ

Z is plastic modulus of the knee section and tf, tw, b and d are itsflange thickness, web thickness, width and depth of the section inthe above equations (see Table 1).

Table 2Summary of design results of two different archetypes.

Space/perimeter frame Story Beams over b

6-Stories shear yielding mode S 1,2 IPB4503,4 IPB4505,6 IPB400

6-Stories flexural yielding mode P 1,2 IPB6003,4 IPB6005,6 IPB450

Finally, designed frames were checked for maximum allowablestory-drift and for stability according to ASCE/SEI 7-05 [13]. Table 2shows design results of two cases from 6-stories archetypes.

4. Nonlinear analyses

4.1. Nonlinear modeling of the archetypes

After designing previously stated archetypes, nonlinear staticand dynamic analyses should be performed to investigate systembehavior in each case and in every performance group. It is re-quired to prepare nonlinear models of the mentioned archetypesin order to carry out such analyses. This modeling was carriedout with the aim of nonlinear modeling features of the OpenSeessoftware [16].

Columns are modeled with nonlinear beam-column elementsutilizing fiber sections. Nonlinear beam-column elements with asection consisting of an Elastic Uniaxial Material in order to con-sider axial stiffness and a nonlinear Material with 2% hardeningto consider flexural nonlinear behavior are used to model beamelements.

Nonlinear behavior has been assumed to be concentrated at theends of Knee elements. Therefore, these elements were composedof elastic beam-column elements and two zero-length elements atthe ends where these elements intersect with beams and columns.The zero-length elements represent shear nonlinear behavior aswell as nonlinear flexural behavior of the knee elements.

In order to model bracing elements in a way that make it pos-sible to capture post buckling behavior of these elements, nonlin-ear beam-column elements using fiber sections were employed.Moreover, an imperfection equal to 0.005 of the length of themembers was implemented in the middle of each brace length totrigger buckling in these members [17–20]. The behavior of a sin-gle bracing modeled in a similar way, under a cyclic loading hasbeen shown in Fig. 3.

4.2. Nonlinear static (pushover) analyses

Pushover analyses were conducted with the aid of OpenSeessoftware. Before these analyses, the models had been preloadedby the factored gravity combination being defined in Eq. (5) foundbelow:

1:05Dþ 0:25L ð5Þ

Fig. 4 shows the Pushover curve of a 6-stories model with 2 baysand with governed shear yielding mode in its knee elements. Thedrifts in which knee elements yielded in shear along with flexuralmode and compression braces buckled in each story have beenshown in Fig. 5. First story knee elements did not yield in flexuralmode and 6th story compression brace did not buckle in the men-tioned model. Furthermore, as the same result observed in all themodels, knee elements in all the stories reach the yielding pointin a short interval of roof drifts.

racing Beams Interior cols Exterior cols Braces Knee elements

IPE240 IPB240 IPB360 2UNP100 IPE200IPE240 IPB200 IPB260 2UNP100 IPE180,160IPE240 IPB160 IPB180 2UNP80 IPE160,140

IPE240 IPB240 IPB500 2UNP140 Box160IPE240 IPB200 IPB300 2UNP120 Box160IPE240 IPB160 IPB180 2UNP100 Box140,100

Fig. 3. (a) Schematic of a single brace; (b) cyclic displacement loading; (c) the response of the single brace under cycling loading.

Fig. 4. The Pushover curve of a 6-stories model with 2 bays.

Fig. 5. The roof drifts related to knees yielding and braces buckling of each story ofa 6-stories model with 2 bays.

M. Farahi, M. Mofid / Engineering Structures 46 (2013) 155–164 159

As it was expected, knee elements yielded first in all models inflexural or shear mode according to which mode of yielding theyhad been designed for. The results of nonlinear static analysesproved the validity of length limitations imposed on knee elementsto determine their yielding mode.

Fig. 6 shows the Pushover curves for 2-bays models with flex-ural yielding mode in their knee elements and Fig. 7 comparesPushover curves of two 6-stories models where one of them has2 bays and the other has 4 bays.

In this research, the nonlinear modeling which was utilized torepresent braces behavior indicated reasonable ability to capturethe buckling and to represent the post buckling behavior of braceelements. For instance, Fig. 8 shows the behavior of 3rd story com-pression brace of the 6-stories model, the Pushover curve of whichwas represented in Fig. 4. This brace has 2UNP120 section and hasa buckling force equal to 649 kN if the effective length coefficient isassumed to be one. The axial force of this brace drops after it

reaches the value equal to 642 kN, which is close to the bucklingforce of its section.

The maximum base shear capacity corresponding to ultimateroof displacement is obtained from these analyses. The ultimateroof drift (du) is taken as the roof drift in which plastic behaviorin one of the beams initiates for the first time [21]. The period-based ductility (lT) which will be used in the next steps for a givenarchetype model, is defined as the ratio of the ultimate roof dis-placement to the effective yield roof displacement, and is assessedin this step;

lT ¼du

dy;effð6Þ

dy;eff ¼ C0Vmax

Wg

4p2

h iðmaxðT1; TÞÞ2 ð7Þ

Fig. 6. The Pushover curves of 2-bays models with flexural mode of yielding in theirknee elements.

Fig. 7. The Pushover curves of 2-stories models with 2 bays and 4-bays and shearyielding mode.

Fig. 8. The axial force of 3rd story compression brace of a 6-stories model.

Table 3Summary of static nonlinear analyses.

Model ID Period (s) Drmax Vmax (kN) Dr y l X

2-Stories models2b2sm,P 0.38 0.020 663 0.002 9.2 2.72b2sv,P 0.37 0.021 582 0.002 11.1 2.34b2sm,P 0.36 0.024 1135 0.002 11.9 2.34b2sv,P 0.36 0.027 1074 0.002 12.9 2.1

6-Stories models2b6sm,P 0.99 0.055 1627 0.013 4.3 2.22b6sv,P 0.86 0.046 1622 0.009 4.9 2.24b6sm,P 0.76 0.043 3453 0.008 5.6 2.34b6sv,P 0.75 0.055 3326 0.007 7.7 2.22b6sm,S 0.63 0.037 1164 0.009 4.1 3.92b6sv,S 0.65 0.045 1139 0.010 4.7 3.810-Stories models2b10sm,P 1.39 0.023 2024 0.008 2.9 2.42b10sv,P 1.43 0.021 1763 0.007 3.1 2.34b10sm,P 1.25 0.026 4392 0.008 3.1 2.64b10sv,P 1.21 0.028 4403 0.008 3.4 2.62b10sm,S 1.13 0.037 1404 0.013 2.8 42b10sv,S 1.13 0.034 1287 0.012 2.8 3.7

14-Stories models2b14sm,P 2.15 0.026 2075 0.010 2.7 2.52b14sv,P 2.14 0.027 2012 0.009 2.9 2.54b14sm,P 1.72 0.044 5941 0.014 3.2 3.24b14sv,P 1.72 0.031 5475 0.010 3.0 2.8

160 M. Farahi, M. Mofid / Engineering Structures 46 (2013) 155–164

where C0 relates fundamental-mode (SDOF) displacement to roofdisplacement, T is the fundamental period (Cu � Ta according to ASCE7-05), T1 is the fundamental period calculated using eigenvalueanalysis, W represents the weight of the archetype and g is gravityconstant.

C0 ¼ u1;r

PN1 mxu1;xPN1 mxu2

1;x

ð8Þ

where u1,x(u1,r) is the ordinate of the fundamental mode at level x(roof), and N is the number of levels. The ductility of different mod-els is presented in Table 3. As it is expected, the ductility of modelswith shear yielding mode in their knee elements is larger than themodels with flexural yielding mode. In addition, an increase in thenumber of bays from 2 to 4 increases the ductility of the model.Finally, increasing the height and number of stories will decreasethe ductility of the models with the same number of bays and the

same yielding mode in knee elements which is consistent withthe results of previous researches [7,22].

Moreover, over-strength factor can be assessed through staticnonlinear analyses by evaluating the following ratio for eacharchetype:

X ¼ The maximum base shear during the analysisThe base shear for which the frame was designed

ð9Þ

The over-strength factors of the archetypes with flexural yieldingmode in their knee elements is greater than those of the archetypeswith shear yielding mode in their knee elements. Furthermore,increasing number of frame bays from 2 to 4 increases the over-strength factor.

4.3. Nonlinear dynamic analyses

Nonlinear dynamic analyses are conducted under a gravity loadcombination of Eq. (5) and input ground motions which are se-lected from the Far-Field record set proposed by FEMA P695. Thisset consists of 22 pairs of earthquake records. These analyses areutilized to establish the Median Collapse Capacity, bSCT, and Col-lapse Margin Ratio, CMR, for each index archetype model. MedianCollapse Capacity is the ground motion intensity in which half ofthe records in the set cause collapse of an index archetype model.Such intensity can be established through conducting IncrementalDynamic Analyses (IDA) [23] or Modal Incremental Dynamic Anal-yses (MIDA) [24–26]. However, in this study, bSCT has been evalu-ated through the concept of Incremental Dynamic Analyses (IDA).In these analyses, individual ground motions are scaled in orderto increase intensities until the structure reaches a collapse pointwhich is considered as a story-drift ratio equal to 10% accordingto Vamvatsikos and Cornell [23]. These analyses were conductedwith the aid of OpenSees software.

The record set had been scaled in two steps prior to its applica-tion in this study. First, the records were normalized by their peakground velocity in the purpose of removing unwarranted variabil-ity between records while preserving the variation between theirfrequency contents. Second, these normalized records are collec-tively anchored to a specific ground motion intensity such that

Fig. 9. (a) MCE design spectra for SDC-B, C and D. (b) Far field record set response spectra [12].

Fig. 10. (a) Incremental dynamic analysis response curves of the 6-stories modelwith 2 bays and shear yielding mode in its knees.

Fig. 11. Collapse Fragility Curve.

M. Farahi, M. Mofid / Engineering Structures 46 (2013) 155–164 161

the median spectral acceleration of the record set matches thedesign spectral acceleration at the fundamental period of thearchetype model under consideration.

Four design spectra are proposed by FEMA P695 that doesreflect the upper and lower bound hazard levels associated withSeismic Design Categories (SDC)-B through D of ASCE/SEI 7-05[13]. Fig. 9a demonstrates these MCE design spectra. The spectraof far field record set which are proposed and normalized accord-ing to their PGV by FEMA P695 are illustrated in Fig. 9b, includingthe statistical average spectrum.

Fig. 10 shows the results of IDA for a 6-stories model with 2bays and flexural yielding mode in its knee elements analyzed un-der each of 44 ground motion records with increasing intensities.Each point in this figure represents one nonlinear dynamic analysisand each curve represents the response of the frame to an individ-ual record where its intensity is increased in each step. As it isindicated in Fig. 10 the median collapse capacity, bSCT, was evalu-ated for this model and was equal to 2.95 g. MCE intensity for thismodel has been further shown in this figure.

The scatter in IDA response curves is caused by both the largedeviation of spectral intensities of the records pertaining to assessthe Median Collapse Capacity along with other specific character-istics of each ground motion such as duration and frequencycontent which are not fully reflected in spectral accelerationintensity [27].

As evidently illustrated in Fig. 10, in some of curves the rate ofincrease of inter-story drifts along with the increasing ground mo-tion record intensity is not always uniform. In these situations,there can be a pulse in a ground motion record which causes theframe to yield in a certain direction. When such a record is im-posed with greater intensities, an earlier pulse can be dominant

to cause the frame to substantially yield in an opposite direction,and this yielding prevent the frame from the effect of later pulse[28].

Using collapse data from IDA results, a collapse fragility curvecan be defined through a cumulative distribution function, whichrelates the ground motion intensity to the collapse probability.This curve for each model would be a lognormal curve which is de-fined by two parameters: (1) The median collapse intensity, bSCT,and (2) The standard deviation of the natural logarithm, bRTR,which represents the slope of logarithm distribution and reflectsdispersion in results due to record-to-record variability. In thisstudy, the value of bRTR is assumed to be constant and equal to0.4 according to what has been proposed by FEMA P695. Fig. 11shows the IDA results in the form of a fragility curve. However, itis an interim one, and other uncertainties has been not consideredin this fragility curve.

The ratio between the median collapse intensity and the MCEintensity for each model is the Collapse Margin Ratio (CMR).MCE intensity (SMT) is obtained from the design spectrum of MCEground motions for the seismic design category Dmax at the funda-mental period of the model.

CMR ¼bSCT

SMTð10Þ

For the aforementioned archetype model with Median Collapseintensity and MCE intensity equal to 2.95 g and 1.5 g respectively,the CMR is equal to 1.97. The values of CMR, bSCT and SMT for othermodels are summarized in Table 4. Index models with shorter peri-ods generally had greater collapse margins and 4-bays archetypemodels provided greater collapse margins in comparison with the

Table 4Summary of nonlinear dynamic analyses.

Model ID bSCTSMT CMR

2-Stories2b6sm,P 3.6 1.5 2.402b6sv,P 4 1.5 2.674b2sm,P 3.6 1.5 2.404b2sv,P 4.1 1.5 2.73

6-Stories2b6sm,P 2.95 1.5 1.972b6sv,P 2.62 1.5 1.754b6sm,P 3.98 1.5 2.654b6sv,P 3.75 1.5 2.502b6sm,S 6.14 1.5 4.092b6sv,S 5.28 1.5 3.50

10-Stories2b10sm,P 2.07 1.02 2.032b10sv,P 2.09 1.02 2.054b10sm,P 3.11 1.02 3.054b10sv,P 2.93 1.02 2.872b10sm,S 3.5 1.02 3.432b10sv,S 2.96 1.02 2.90

14-Stories2b14sm,P 1.12 0.8 1.42b14sv,P 1.08 0.8 1.354b14sm,P 1.18 0.8 1.484b14sv,P 1.14 0.8 1.43

Fig. 12. Collapse fragility curves considering: (a) record to record uncertainty, (b)total collapse uncertainty.

162 M. Farahi, M. Mofid / Engineering Structures 46 (2013) 155–164

archetype models with 2 bays. Moreover, the models with flexuralyielding mode in their knee elements had larger collapse margin ra-tios in comparison with the models having shear yielding mode intheir knee elements.

CMR value is modified to obtain an Adjusted Collapse MarginRatio, ACMR, for each archetype, i, through the application of a cor-rection factor called Spectral Shape Factor in FEMA P695 [12]. Thisadjustment accounts for the effects of spectral shape of the recordswhich are used to assess CMR value.

ACMRi ¼ SSFi � CMRi ð11Þ

Spectral Shape Factors, SSF, are a function of the fundamental per-iod, T, the period-based ductility, lT, and the applicable seismic de-sign category. The SSFs can be obtained from the tables in FEMAP695, which provide the value of SSF for different values of ductilityand period along with different Seismic Design Categories [12].

The effects of other significant sources of uncertainty should beconsidered on the grounds that the Record To Record uncertainty(RTR) is not the only source of uncertainty. The other sources ofuncertainty taken into account in this study are Design Require-ment Uncertainty (DR), Test Data Uncertainty (TD) and ModelingUncertainty (MDL) according to FEMA P695 [12]. DR uncertaintyreflects the robustness of the design requirements which are em-ployed to design the archetypes. TD uncertainty is related to thecompleteness and robustness of the quality of the test data whichis used to prepare final models of archetypes. Finally, MDL uncer-tainty deals with the ability of the index archetype models to rep-resent the full range of structural response characteristics and to

Table 5Quality rating of index archetype models (Table 5-3 of FEMA P695).

Representation of collapse characteristics

High. Index models capture the full range of the archetype design space and structurbehavioraleffects that contribute to collapse

Medium. Index models are generally comprehensive and representative of the designand behavioral effects that contribute to collapse

Low. Significant aspects of the design space and/or collapse behavior are not capturein the index models

capture structural collapse behavior. The total uncertainty isobtained by combining RTR, DR, TD and MDL uncertainties.

The effect of each of the aforementioned uncertainties is repre-sented by a lognormal standard deviation parameter. These param-eters are evaluated according to FEMA P695 [12] separated tables,which does relate them to the quantitative levels of the uncertain-ties stated above. The tables illustrated four levels of uncertaintydistinguished with the terms of superior, good, fair and poor withlognormal standard deviation parameters equal to 0.1, 0.2, 0.35and 0.5 respectively. Table 5 is one of these tables used in thisstudy to evaluate the lognormal standard deviation parameterwhich represents the Modeling Uncertainty (MDL).

bRTR is assumed to be constant and equal to 0.4. Other lognor-mal standard deviation factors, bDR, bTD and bMDL are obtained fromtables similar to Table 5 and are respectively equal to 0.1, 0.35 and0.2.

Considering all aforementioned sources of uncertainty, the col-lapse fragility of each index archetypes can be defined by a randomvariable, SCT, which is obtained from the production of, bSCT calcu-lated by nonlinear dynamic analyses, and a random variable, kTOT.Thus, kTOT is assumed to be log-normally distributed with the med-ian value of unity and a lognormal standard deviation of bTOT

which is defined by Eq. (13).

kTOT ¼ kRTRkDRkTDkMDL ð12Þ

kRTR, kDR, kTD and kMDL are assumed to have a median value of unityand lognormal distribution with standard deviation parameters,bRTR, bDR, bTD and bMDL, respectively. Assuming the four stated ran-dom variable to be independent, bTOT, reflects total collapse uncer-tainty can be given by Eq. (13).

bTOT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2

RTR þ b2DR þ b2

TD þ b2MDL

qð13Þ

Through these assumptions the lognormal standard deviationparameter, bTOT, which is calculated in this study is equal to0.577. The collapse fragility curve of the 6-stories model with 2 bays

Accuracy and robustness of models

High Medium Low

al (A) SuperiorbMDL = 0.10

(B) GoodbMDL = 0.20

(C) Fair bMDL = 0.35

space (B) Good bMDL = 0.20 (C) Fair bMDL = 0.35 (D) PoorbMDL = 0.50

d (C) Fair bMDL = 0.35 (D) Poor bMDL = 0.50 –

M. Farahi, M. Mofid / Engineering Structures 46 (2013) 155–164 163

and flexural yielding mode in its knee elements is revisedconsidering total collapse uncertainty and is plotted in Fig. 12 withprevious interim curve of Fig. 11.

5. Quantification of seismic performance factors

5.1. Evaluation of the response modification factor, R

To verify the seismic behavior of the lateral-force-resistingsystem which is under consideration, it is required for the ACMRvalues of each index archetypes and the average of these valuesin each performance group to be greater than the acceptable valuesproposed by FEMA P695. These acceptable values of AdjustedCollapse Margin Ratio are based on the total system collapseuncertainty and the values of acceptable collapse probabilities.The lesser probability of collapse accepted, the larger collapsemargin ratio is required to validate the seismic behavior of a sys-tem. The presumed Response Modification Factor, R, is acceptedif the performance of the archetypes designed with this value ofR is accepted which is achieved when the following two criteriaare fulfilled [12]:

� The average value of Adjusted Collapse Margin Ratio for eachperformance group exceeds the acceptable values of AdjustedCollapse Margin Ratio considering 10% acceptable collapseprobability which is equal to 2.1 in this study.� The individual values of Adjusted Collapse Margin Ratio for each

index archetype exceeds the acceptable values of Adjusted Col-lapse Margin Ratio considering 20% acceptable collapse proba-bility which is equal to 1.62 in this study.

Table 6Over-strength factors and summary of collapse margin ratios and comparison of them wi

Archetype ID Configuration and design Compute

No. of stories Framing Yielding mode No. of bays X

Performance group PG12b2sv,P 2 P Shear 2 2.34b2sv,P 2 P Shear 4 2.1Mean of performance group 2.2

Performance group PG22b6sv,P 6 P Shear 2 2.24b6sv,P 6 P Shear 4 2.22b10sv,P 10 P Shear 2 2.34b10sv,P 10 P Shear 4 2.62b14sv,P 14 P Shear 2 2.54b14sv,P 14 P Shear 4 2.8Mean of performance group 2.43

Performance group PG42b6sv,S 6 S Shear 2 3.82b10sv,S 10 S Shear 2 3.7Mean of performance group 3.75

Performance group PG52b2sm,P 2 P Flexural 2 2.74b2sm,P 2 P Flexural 4 2.3Mean of performance group 2.50

Performance group PG62b6sm,P 6 P Flexural 2 2.24b6sm,P 6 P Flexural 4 2.32b10sm,P 10 P Flexural 2 2.44b10sm,P 10 P Flexural 4 2.62b14sm,P 14 P Flexural 2 2.54b14sm,P 14 P Flexural 4 3.2Mean of performance group 2.53

Performance group PG82b6sm,S 6 S Flexural 2 3.92b10sm,S 10 S Flexural 2 4Mean of performance group 3.95

Collapse margin ratios and adjusted values of these ratios forindividual archetypes and different performance groups havebeen summarized in Table 6. Acceptance criteria for each arche-type and performance group have been further illustrated in thistable in order to be compared with the obtained results. As it isevident from Table 6, acceptable performance was achieved byall of index archetypes and performance groups, which are inves-tigated. Therefore, it can be inferred from this table that the pre-sumed R factored which is equal to 7, is appropriate to designChevron Knee Bracing systems.

5.2. Evaluation of the over-strength factor, X0

In order to quantify the system over-strength factor, X0, it is re-quired to primarily calculate the average value of archetypes over-strength factors for each performance group. Table 6 presentsthese average values.

The largest average over-strength factor among differentperformance groups is considered as the system over-strengthfactor, X0, according to FEMA P695. The largest average over-strength factor was equal to 3.95 which can be rounded to 4.On the other hand, the largest possible value for this factor is re-stricted by FEMA P695 to the largest practical values proposed inTable 12.2-1 of ASCE/SEI 7-05 [13] for all current approved Seis-mic Force Resisting systems. In this table, system over-strengthfactor is limited to X0 = 3.0 according to practical design consid-erations. Considering this upper bound value, the system over-strength factor equal to X0 = 3.0 is proposed for Chevron KneeBracing systems.

th acceptance criteria for different archetypes and different performance groups.

d over-strength and collapse margin parameters Acceptance check

CMR SSF ACMR Accepted ACMR Pass/fail

2.67 1.33 3.55 1.62 Pass2.73 1.33 3.64 1.62 Pass2.70 1.33 3.59 2.1 Pass

1.75 1.33 2.32 1.62 Pass2.50 1.4 3.50 1.62 Pass2.05 1.31 2.68 1.62 Pass2.87 1.3 3.73 1.62 Pass1.35 1.32 1.78 1.62 Pass1.43 1.32 1.88 1.62 Pass1.99 1.33 2.65 2.1 Pass

3.50 1.31 4.60 1.62 Pass2.90 1.25 3.63 1.62 Pass3.20 1.28 4.12 2.1 Pass

2.40 1.33 3.19 1.62 Pass2.40 1.33 3.19 1.62 Pass2.40 1.33 3.19 2.1 Pass

1.97 1.31 2.58 1.62 Pass2.65 1.33 3.53 1.62 Pass2.03 1.31 2.66 1.62 Pass3.05 1.29 3.93 1.62 Pass1.40 1.3 1.82 1.62 Pass1.48 1.34 1.98 1.62 Pass2.10 1.31 2.75 2.1 Pass

4.09 1.24 5.08 1.62 Pass3.43 1.25 4.29 1.62 Pass3.76 1.25 4.68 2.1 Pass

164 M. Farahi, M. Mofid / Engineering Structures 46 (2013) 155–164

5.3. Evaluation of the over-strength factor, Cd

The value of deflection amplification factor, Cd, is in directrelation with the value of Response Modification Factor, R. Thevalue of Cd is calculated by reducing the value of ResponseModification Factor of that system by the damping factor, BI, cor-responding to inherent damping of the system under theinvestigation.

Cd ¼RBI

ð14Þ

where BI refers to effective damping of the structure due to theinherent dissipating of energy. The value of the inherent dampingproposed is not to be taken greater than 5% of critical for all modeof vibration by section 18.6.2.1 of ASCE/SEI 7-05 [13]. The deflectionamplification factor can be considered equal to Response Modifica-tion Factor under such situation.

6. Conclusion

� The values of Adjusted Collapse Margin Ratio for all of indexarchetypes which were calculated according to the results ofnonlinear dynamic analyses exceeded acceptable collapse mar-gins as well as the average values of these ratios for all of per-formance groups. Hence, the presumed Response ModificationFactor equals to R = 7 is appropriate for Chevron Knee Bracingsystems (CKB). Furthermore, the over-strength factor equal toX0 = 3.0 was established for such seismic-force-resistingsystems.� As the height of the systems of interest increased, the collapse

margins decreased; therefore, long period systems are morevulnerable to collapse under severe ground motions comparedto short period ones according to the results found in thisinvestigation.� CKB frames with more braced spans provide more ductility as

well as more stiffness as the 4-bays frames showed more duc-tility and stiffness compared to 2-bays frames.� The performance groups governed the value of over-strength

factor are different from the critical performance groups indetermination of R factor. Space frames had greater over-strength factor in comparison with perimeter frames.� The sequence of elements yielding in nonlinear static analyses

is consistent with what was expected. Knee elements yieldedbefore the buckling of the braces in each story in all the inves-tigated models. Hence, it can be inferred that designing kneeelements according to the 130% of their elastic capacity is aproper strategy to ensure that knee elements are the first ele-ments which help the structure to dissipate absorbed energythrough formation of plastic hinges. Moreover, the sooner theknee elements yield, the later the braces buckle in a story.Therefore, further investigation is needed to find the optimumallowed ratio of demand in knee elements to their elastic capac-ity in order to design them.

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