seismic reliability of v-braced frames: influence of design methodologies

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2009; 38:1587–1608Published online 20 April 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.919

Seismic reliability of V-braced frames: Influence of designmethodologies

Alessandra Longo∗,†, Rosario Montuori and Vincenzo Piluso

Department of Civil Engineering, University of Salerno, via Ponte Don Melillo, Fisciano 84084, Italy

SUMMARY

According to the most modern trend, performance-based seismic design is aimed at the evaluation ofthe seismic structural reliability defined as the mean annual frequency (MAF) of exceeding a thresholdlevel of damage, i.e. a limit state. The methodology for the evaluation of the MAF of exceeding a limitstate is herein applied with reference to concentrically ‘V’-braced steel frames designed according todifferent criteria. In particular, two design approaches are examined. The first approach corresponds tothe provisions suggested by Eurocode 8 (prEN 1998—Eurocode 8: design of structures for earthquakeresistance. Part 1: general rules, seismic actions and rules for buildings), while the second approach isbased on a rigorous application of capacity design criteria aiming at the control of the failure mode (J.Earthquake Eng. 2008; 12:1246–1266; J. Earthquake Eng. 2008; 12:728–759). The aim of the presentedwork is to focus on the seismic reliability obtained through these design methodologies. The probabilisticperformance evaluation is based on an appropriate combination of probabilistic seismic hazard analysis,probabilistic seismic demand analysis (PSDA) and probabilistic seismic capacity analysis. RegardingPSDA, nonlinear dynamic analyses have been carried out in order to obtain the parameters describing theprobability distribution laws of demand, conditioned to given values of the earthquake intensity measure.Copyright q 2009 John Wiley & Sons, Ltd.

Received 2 May 2008; Revised 27 October 2008; Accepted 27 October 2008

KEY WORDS: cyclic behaviour; V-braced frames; global mechanism; seismic reliability

1. INTRODUCTION

In recent years, after Loma Prieta (1989) and Northridge (1994) earthquakes, building owners,managers and designers realized that buildings designed according to the minimum code require-ments for life safety protection could be subjected to extensive and costly damage under moderateearthquakes. Such awareness promoted the development of a new conceptual framework forseismic design called performance-based seismic design (PBSD). Even though buildings continue

∗Correspondence to: Alessandra Longo, Department of Civil Engineering, University of Salerno, via Ponte DonMelillo, Fisciano 84084, Italy.

†E-mail: [email protected]

Copyright q 2009 John Wiley & Sons, Ltd.

1588 A. LONGO, R. MONTUORI AND V. PILUSO

to be designed for the minimum levels of life safety protection offered by the code-based designapproach, PBSD permits a building owner/operator to select an acceptable mean annual frequency(MAF) of exceeding a threshold level of business interruption, economic losses and other conse-quences. Design objectives in PBSD concern the attainment of performance levels by relatingseveral structural performances with several seismic intensity levels. From a probabilistic pointof view, the design objective is defined by the MAF of exceeding a limit state. The probabilisticformulation of PBSD, in the most general form, takes into account both aleatory uncertainty andepistemic uncertainty. The first issue identifies the natural variability such as variability of masses,applied loads and material properties, the number and the magnitude of future earthquakes in aregion and record-to-record variability in acceleration time-history amplitudes and phases. Thesecond issue identifies the limit in available knowledge and data such as the modelling of struc-tural systems in highly nonlinear range. However, in this study only aleatory uncertainty has beenconsidered and, in particular, just the prominent one arising from the seismic action consideringboth aleatory in the number and magnitude of future earthquakes, by means of the hazard curve,and the record-to-record variability.

Aiming to account for all the sources of uncertainty, which influence the seismic structuralbehaviour, the need to adopt a probabilistic approach is recognized. An extensive effort has beencarried out by the Pacific Earthquake Engineering Research (PEER) [1] leading to a consistentapproach to performance assessment that is analytically rigorous and suitable to owner-specifiedperformance targets [2]. In this contest, the methodology proposed by Jalayer and Cornell [3]provides a simple formulation for evaluating the seismic reliability of structures accounting forthe record-to-record variability only [4].

The procedures for evaluating the seismic reliability of structures require an appropriate combi-nation of the results coming from probabilistic seismic demand analysis (PSDA), probabilisticseismic capacity analysis (PSCA) and probabilistic seismic hazard analysis (PSHA).

Two different approaches can be applied: the direct approach where the limit state is expressedby using a structural response parameter and the indirect approach where the limit state is expressedby using a seismic intensity measure. In this study the direct approach has been preferred because itis displacement based so that capacity is defined by means of parameters more commonly adoptedin engineering practice. On the basis of some simplifying assumptions [3], the direct methodleads to a closed-form analytic expression allowing a new framework for probabilistic seismicassessment of structures, namely demand and capacity factor design.

In the following, within a European framework, Jalayer and Cornell’s approach is applied aimingat the assessment of the seismic performance of V-braced frames (VBFs) designed according toEurocode 8 [5]. In addition, the probabilistic performance assessment is also carried out withreference to a new design approach recently proposed by the authors, based on a rigorous applicationof capacity design principles aiming at the development of a collapse mechanism of global type[6, 7]. The above design approaches are herein briefly summarized, while a detailed examinationis developed in a previous work [6, 7] where the outcomes of preliminary deterministic dynamicnonlinear analyses are also presented showing encouraging results concerning the superior seismicperformances, which can be obtained by means of the proposed design procedure, when comparedwith Eurocode 8 design provisions.

Probabilistic methods have been recognized as superior tools for the seismic performanceassessment of structures, whose outcome is the MAF of exceeding pre-defined limit states. Thecomparison between different structural solutions of the same design problem is immediatelyunderstandable when it is made in terms of seismic reliability. For this reason the comparison

Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:1587–1608DOI: 10.1002/eqe

SEISMIC RELIABILITY OF V-BRACED FRAMES 1589

between Eurocode 8 provisions and the proposed design approach for VBFs is herein carriedout by evaluating the MAF of exceeding pre-defined limit states. Even though the methodologyprovides the designer with the theoretical basis to account for all the sources of uncertainty, in thepresent study only the aleatory uncertainty, due to number and magnitude of future earthquakesand record-to-record variability, is considered.

Finally, it is recognized that when the structural weight is increased, it is likely to have animproved performance. Therefore, a cost versus benefit comparison between the different designsolutions is also presented.

2. EXAMINED DESIGN APPROACHES

In the case of concentrically braced frames, dissipative zones are constituted by the brace diagonalmembers. Conversely, beams, columns and connections to foundations are non-dissipative zones.Regarding the connections between braces and primary structure, the traditional design approach isbased on the use of non-dissipative connections. According to the traditional philosophy of capacitydesign, dissipative zones have to be designed considering the internal actions occurring under theseismic load combination; conversely, non-dissipative zones have to be designed considering themaximum internal actions that dissipative zones, yielded and strain-hardened, are able to transmit.

The structural scheme commonly adopted for evaluating the internal actions in beams, columnsand diagonals of VBFs subjected to seismic actions considers as active both the tensile and thecompressed diagonals. In addition, brace members are assumed to be pin-jointed to the primaryframe members.

Eurocode 8 [5] requires that the normalized slenderness � of braces has to be less than 2.0:

�= �

�y, �y=�

√E

fy(1)

where � is the brace slenderness, fy is the yield stress and E is the elastic modulus. The aim ofthis limitation is the reduction of the plastic out-of-plane deformation of the gusset plates due tobrace buckling, which otherwise are prone to failure due to low cycle fatigue.

Regarding the capacity design criterion for beams and columns, an overstrength coefficient �iof bracing elements is preliminarily defined for each storey:

�i = Nbr,Rdi

Nbr,Sdi(2)

where Nbr,Sdi is the design value of the brace internal axial force and Nbr,Rdi is the correspondingdesign resistance. Eurocode 8 requires the fulfilment of the following relationships with referenceto beams and columns, respectively:

Nb,Rd(MSd)�(NbSd,G+1.1 ·�ov ·�·NbSd,E), Nc,Rd�(NcSd,G+1.1 ·�ov ·� ·NcSd,E) (3)

where NRd is the buckling resistance of the element (Nb,Rd for beams and Nc,Rd for columns)reduced due to the contemporary action of the bending moment MSd deriving from the seismicload combination, NSd,G is the axial force due to the non-seismic loads included in the seismic



load combination, NSd,E is the axial force due to the seismic loads, 1.10 is an amplification coef-ficient accounting for strain-hardening effects, �ov is an overstrength factor accounting for randomvariability of material properties and � is the minimum value of the overstrength coefficients �i :

�= nsmini=1

�i (4)

where ns is the number of storeys.According to Eurocode 8 the overstrength factor can be taken as �ov=1.0 provided that the

lower value of yield strength fy,min and the upper value of yield strength fy,max are specifiedand, in addition, the nominal value of the yield strength of dissipative zones exceeds the uppervalue of yield strength fy,max of dissipative zones. If the above condition is not satisfied, the value�ov=1.25 is suggested.

It is evident that the overstrength factor is aimed to include all random effects of materialproperties. This study, conversely, is aimed at the evaluation of the nonlinear seismic responsefrom a probabilistic point of view, but only the uncertainty due to number and magnitude offuture earthquakes and record-to-record variability is accounted for. Random material variability isneglected and, therefore, a �ov factor equal to 1.00 has been considered in the following analyses.

In addition, as the study is aimed at the comparison between different design methodologies, itis important to underline that the effects of possible overstrength of dissipative members, due torandom material variability, are accounted for neither in the application of Eurocode 8 provisionsnor in the design approach proposed by the authors [6, 7]. In other words, within the aim of thestudy, the use of �ov factor greater than 1.0 would lead to a misleading comparison because randommaterial variability is faced in Eurocode 8, while it is neglected in the proposed design procedure.The aim of the study is the comparison of capacity design procedures independently of the effectsof random material variability.

In addition, the beams have to be verified accounting for the vertical action resulting from theunbalanced brace axial forces, due to the fact that the compressed diagonal is buckled when thetensile one yields. In Eurocode 8, this force is approximately evaluated according to the followingrelationship:

V =(Nbr,Rd,t ·sin�1−�pb ·Nbr,Rd,t ·sin�2) (5)

where �1 and �2 are the angles between the diagonal axes and the beam axis (typically �1=�2), Nbr,Rd,t is the design resistance of the diagonal in tension and �pb is a factor used for theevaluation of the postbuckling resistance of brace members. The suggested value is 0.3 accordingto Eurocode 8.

It is easy to recognize that Equation (3) provides the beam and the column axial forces, respec-tively, occurring when the first brace member is yielded and strain-hardened. Conversely, the aimof the proposed design methodology is the evaluation of axial forces in non-dissipative members(beams and columns) occurring when all the dissipative members (diagonals) are completelyyielded. Therefore, the evaluation of these forces is made by focusing the attention on the distri-bution of internal actions occurring when a collapse mechanism of global type is developed. Theconsidered structural scheme is characterized by column continuity as shown in Figure 1(a) wherethe structure is considered in a deformed configuration, showing that the collapse mechanismis governed by only one parameter: i.e. the base rotation � of the structure where real hingesare located. The � value corresponds to the yielding of all the diagonals in tension while thecompressed ones are buckled.



Figure 1. (a) V-braced frame in the deformed configuration corresponding to the global failure mode and(b) evaluation of compression and tension axial force.

Regarding the design procedure, it is assumed that cross-sections of bracing members areknown as they are designed to resist internal axial forces due to the seismic load combination.For this typology, braces work both in tension and in compression. Therefore, as the first step,the postbuckling behaviour of bracing members has to be predicted. This step, in this study, isperformed as suggested by Georgescu et al. [8]. The unknowns of the design problem are thebeam and the column sections required to guarantee the yielding of all the bracing members, i.e.the participation of all the storeys to the dissipation of the earthquake input energy.

The axial deformation corresponding to the yielding of the bracing member of the i th storey isgiven by

�pi = Py,i ·Li

E ·Ai= fy ·Li

E(6)

where Py,i is the yield load of the i th diagonal, fy is the yield stress, E is the elastic modulus andLi and Ai are the length and the cross-sectional area of the i th diagonal, respectively.

For each storey, it is possible to define the value of the interstorey drift angle �i correspondingto the yielding of the i th bracing member by means of the following yielding condition:

�i =�i ·di ·cos�i =�pi for i=1 . . .ns (7)

where di is the corresponding interstorey height and �i is the angle between the bracing memberand the beam axis:

�i =�pi

di ·cos�i for i=1 . . .ns (8)

The value of the base rotation � corresponding to the yielding of all the diagonals can beexpressed as

�m = nsmaxi=1

(�i ) (9)



Therefore, for each storey, the value of the axial deformation occurring in the diagonal memberswhen the collapse mechanism is completely developed can be easily computed as

�i =�m ·di ·cos�i (10)

The above value provides both the elongation of the diagonal in tension and the shortening of thecompressed diagonal subjected to buckling. Therefore, in the collapse condition, the axial force inthe tensile diagonal of the i th storey is equal to Py,i , while the axial force in the correspondingbuckled compressed diagonal can be derived according to Figure 1(b).

The net downward force resulting from the combination of the forces acting in the diagonalsof the i th storey is equal to

Vi = Py,i sin�i −Pc(�i ) ·sin�i (11)

The horizontal force transmitted by the braces to the beams is equal to

Hi = Py,i cos�i +Pc(�i )cos�i (12)

The unbalanced action (Vi ) applied to the beam, due to the postbuckling behaviour of braces, isused both to design the beam and to evaluate the axial forces in the columns, while the horizontalforce Hi , applied in the midspan of the beam, is used only for the beam design.

Therefore, by focusing the attention on the distribution of internal actions, in the kinematicmechanism condition, according to capacity design, the design value of the axial force in thenon-dissipative members is computed. In particular, with reference to columns, the axial force tobe considered in the design is given by

Nc,Sd,E,i =ns∑

j=i+1Pc(� j )·sin� j + 1

2

ns∑j=i

[Py, j −Pc(� j )]·sin� j +Nc,Sd,G,i for i<ns

Nc,Sd,E,n = Py,n ·sin�n−Pc(�n) ·sin�n2

+Nc,Sd,G,n for i=ns

(13)

where Py, j and Pc(� j ) are the tension and compression forces in the j th brace, respectively,while Nc,Sd,G,i is the axial force in the column due to gravity loads acting in the seismic loadcombination.

3. SEISMIC RELIABILITY OF STRUCTURES

In order to evaluate the seismic reliability of VBFs designed according to Eurocode 8 provisions [5]and according to the proposed design methodology [6, 7], a probability-based seismic performanceassessment has been carried out leading to the evaluation of the seismic reliability expressed interms of MAF of exceeding a limit state.

Using the general design approach of separately considering demand and capacity (displacement-based solution strategy), the MAF of exceeding a limit state can be determined in two steps.

The first step couples the demand measure DM and the spectral acceleration Sa to produce astructure-specific demand hazard curve HDM(dm), which provides the MAF that the demand Dexceeds a specific value d . The second and last step of procedure, aiming to determine the MAFof exceeding a limit state, combines the demand hazard curve with the capacity variables.



The demand hazard curve, HDM(dm), is obtained by means of the total probability theorem(TPT) and is expressed as

HDM(dm)=∫ ∞

0GDM(dm|Sa= x)·

∣∣∣∣ dHSa(x)

dx

∣∣∣∣ dx (14)

where GDM(dm|Sa= x) is the complementary cumulative distribution function (CDF) of demandconditioned to a given value Sa= x of the spectral acceleration, obtained by means of PSDA, andHSa(x) is the MAF of exceeding a spectral acceleration Sa= x , obtained by PSHA.Using again the TPT, the MAF of exceeding a limit state is determined:

HLS=∫ ∞

0Fc(dm)·

∣∣∣∣dHDM(dm)

ddm

∣∣∣∣ ddm (15)

where Fc(dm) is the CDF of structural capacity measure.Equations (14) and (15) can be numerically solved for any assumption regarding the probability

distribution functions of the parameter involved (demand, capacity and hazard curve).However, a closed-form solution of integrals (14) and (15) has been obtained by Jalayer and

Cornell based on three simplifying assumptions. The first assumption states that the site hazardcurve can be approximated, in the proximity of the limit state probability, through a linear relationin the logarithmic paper:

HSa(xi )= P[Sa�xi ]=k0 ·x−ki (16)

The second assumption regards the median value of the damage measure �DM/Sa , obtained fromnonlinear dynamic analyses, given the seismic intensity measure Sa. In particular, it can be expressedthrough the following relationship:

�DM|Sa =a ·sba · (17)

where a and b are linear regression parameters obtained in the logarithmic paper and is alognormal variable with median value equal to one and dispersion equal to

ln() =�DM|Sa (18)

The value of �DM/Sa will, in general, depend to some degree on the level of Sa; however, it isassumed to be constant for analytical tractability (homoscedasticity hypothesis). As suggested byJalayer and Cornell [3], the value of �DM/Sa should be chosen in the range of primary interest.

Finally, the third assumption regards PSCA stating that the distribution law of structural capacityis lognormal with median value equal to �C and dispersion equal to �C . In addition, it is assumedthat C is independent on the information about the demand level itself.

On the basis of the first two assumptions, the closed-form solution of the integral (14) is givenby [3]

HDM(dm)=HSa(sdma ) ·exp

[1

2· k

2

b2·�2DM|Sa

](19)

where

HSa(sdma )=k0 ·(sdma )−k =k0 ·

(dm

a

)−k/b

(20)



Equation (20) combines the aleatory uncertainty of structural demand due to record-to-recordvariability (PSDA) with the seismic hazard of the site (PSHA). By combining the probabilisticstructural capacity analysis (PSCA) with the PSDA, using the third simplifying hypothesis, aclosed-form solution is obtained also for the integral (15) providing the MAF of exceeding a limitstate HLS as

HLS=HSa(s�Ca ) ·exp

[1

2· k

2

b2·(�2DM|Sa +�2C )

](21)

where

HSa(s�Ca )=k0 ·(s�C

a )−k =k0 ·(�Ca

)−k/b(22)

Equation (22) shows that the MAF of exceeding a limit state is proportional both to the seismic

hazard HSa(s�Ca ), calculated for the spectral acceleration value s

�Ca corresponding to the median

value of the structural capacity, and to another factor accounting for the dispersion of structuraldemand and of structural capacity. The formulation herein briefly summarized accounts for theuncertainty implicit in the hazard curve from PSHA, the uncertainty due to seismic input (record-to-record variability) and uncertainty due to structural capacity. However, Jalayer and Cornell [3]developed a formulation able to also account other kinds of uncertainties such as the epistemicuncertainty, which is neglected in this study.

4. PROBABILISTIC SEISMIC HAZARD ANALYSIS (PSHA)

In view of the variability of future seismic events, the seismic risk needs to be described inprobabilistic terms. The seismic intensity can be expressed in terms of peak ground acceleration(PGA), spectral velocity (Sv), spectral displacement (Sd) or another parameter, which that containssufficient information about the ground motion to serve as an accurate and efficient predictor ofstructural response, and it should preferably be a variable for which the PSHA results are readilyobtainable. This problem has been studied by Shome et al. [9] and Luco and Cornell [10]. For shortand moderate-period structures, they found that the spectral acceleration at a period equal to thatof the fundamental mode of the structure satisfies the criteria mentioned above. For this reason,in this study the spectral acceleration corresponding to the first mode of vibration of analysedconsidered structures is adopted as the seismic intensity measure.

The hazard corresponding to a specific value of the ground motion intensity measure is definedas the MAF that the intensity measure for future ground motion events be greater than or equalto this specific value. Spectral acceleration hazard curves are normally provided by seismologistsfor a given site. Each curve provides the MAF of exceeding a particular spectral acceleration fora given period and damping ratio. It is advantageous to approximate such a curve in the regionof interest by a power law relationship [11] provided by Equation (16) in which k and k0 areparameters defining the shape of the hazard curve.

In this study, a curve representing high seismicity zones of Europe has been used. In particular,the formulation suggested in Eurocode 8 [12] for evaluating the PGA corresponding to seismicevents having a return period different from the reference one (475 years) has been used. Thedesign value ag,475 of PGA corresponding to the reference return period tr0 is related to the regional



Figure 2. (a) Peak ground acceleration hazard curve and (b) spectral acceleration hazard curve.

seismicity. Typical values of ag,475 are 0.35g, 0.25g and 0.15g for seismic zones of high, mediumand low seismicity, respectively. Eurocode 8 [12] provides a formula, which represents, with agood approximation, the relation between ag,475 and ag , i.e. the PGA corresponding to a returnperiod tr different from 475 years:

agag,475

=(trtr0

)1/k′

(23)

The value of the exponent 1/k′ is strictly related to the seismicity of the considered zone. Accordingto Eurocode 8 this value ranges between 0.30 and 0.40. Aiming at the evaluation of the coefficientsk and k0, the above formulation can be rearranged in the form given by Equation (16); it is easyto show that, given the shape of the elastic design response spectrum, Equation (23) leads to [13]

k0= ak′

g,475

tr0· f k′

, k= 1

k′ (24)

where f is the ratio between the spectral acceleration corresponding to the fundamental period ofthe analysed structure and the PGA value.

Equation (24) provides the k0 and k coefficients, which define the shape of the hazard curvein terms of spectral acceleration starting from the Eurocode 8 formulation [12]. It is important tounderline that the aim of this paper is the comparison of the seismic performances of concentricallybraced frames designed according to the two approaches summarized in Section 2, without referenceto a specific site. Therefore, the use of the hazard curve provided by Eurocode 8 has been preferredjust to represent high seismicity zones of Europe. The curves obtained with reference to theconsidered structures both in terms of PGA and spectral acceleration are depicted in Figure 2(a)and (b), respectively.

5. PROBABILISTIC SEISMIC DEMAND ANALYSIS (PSDA)

The aim of the study is the evaluation of seismic reliability of VBFs designed according to bothEurocode 8 provisions [5] and the methodology proposed by the authors [6, 7], as briefly presentedin Section 2.



Figure 3. The analysed structure.

Table I. Results for the two design criteria.

VBF-E VBF-P

Storey Fi (kN) Diagonals Beams Columns Beams Columns

1 159 HEA 280 HEM 550 HEB 500 HEM 550 HEB 4502 319 HEA 260 HEM 500 HEB 300 HEM 500 HEB 3003 479 HEA 240 HEM 450 HEB 200 HEM 450 HEB 2204 611 HEA 200 HEM 320 HEB 100 HEM 340 HEB 160

The study case is constituted by the four-storey structure depicted in Figure 3. Dead load andlive load are equal to 4 and 2kN/m2, respectively. The behaviour factor has been assumed to beequal to 2.5 according to Eurocode 8. The design value of PGA is equal to 0.35g with responsespectrum for stiff soil conditions. S235 steel grade has been adopted. The dynamic nonlinearstructural analysis has been carried out considering the design value of yield strength of steel.

Regarding the structural model, it accounts for column continuity while the beams are pinnedat their ends. In addition, beam continuity is assured at the intersection with the diagonals. All thebeam-to-column connections are pinned; therefore, all the seismic horizontal forces are withstoodby the concentrically braced frames, which are located along the perimeter of the structure. Themodelling of the brace behaviour has been performed by means of the Maison model [14], whichhas been calibrated in order to provide the same energy dissipation as the more refined Georgescumodel [7, 8].

The seismic horizontal forces adopted for the design of each brace are depicted in Figure 3and also given in Table I where the adopted profiles are also given both for the structure designedaccording to Eurocode 8 provisions (VBF-E) and for the structure designed according to theproposed methodology (VBF-P). For the same structures, in Table II the parameters of the hazardcurve are reported.



Table II. Hazard curve definition.

Structure T0 (s) k k0

VBF-E 0.505 2.5 0.2538VBF-P 0.495 2.5 0.2667

Figure 4. (a) Considered spectra and (b) ground motion spectra scaled to the samespectral acceleration at T =0.50s.

It is important to underline that the fundamental period of vibration of the two designed structuresis almost the same, being very close to 0.5 s. In addition, the structural weight has a variationof about 1.26%, looking just at the single VBF, while looking at the total structural weight thevariation is about 0.57%, with the proposed methodology (VBF-P) leading to the highest value.

The probabilistic analysis of structural response has been obtained by performing a set ofnonlinear dynamic analyses [15], using 30 natural seismic motions provided by PEER database[1]. The records are selected so that the mean spectrum shape is compatible with the designspectrum of Eurocode 8 for rigid soil (Figure 4(a)). Table III reports the name and other information



Table III. Selected records.

No. Record Date Component Sa(T =0.50)/g amax/g Length (s)

1 Irpinia, Italy (Bagnoli) 23/11/1980 N-S 0.433 0.133 72.612 Capte Mendocina 25/04/1992 E-W 0.542 1.039 30.003 Elcentro 19/05/1940 E-W 0.652 0.226 29.784 Montenegro (Ercegov) 15/04/1979 E-W 0.257 0.230 25.005 Ferndale 1 19/09/1939 S-E 0.136 0.091 20.526 Ferndale 3 03/10/1941 N-E 0.148 0.112 17.057 Gazli USSR (Karakyr) 17/05/1976 N-S 0.852 0.608 16.258 Tokaki-Kenoki (Hachinoe) 17/05/1968 N-S 0.374 0.229 36.009 Helena Montana (Carrol College) 31/10/1935 E-W 0.253 0.153 9.6710 Hollister (City Hall) 09/03/1949 S-W 0.287 0.117 17.0311 Kern County (Taft Lincoln School) 21/07/1952 S-E 0.349 0.178 54.1312 Kobe (Takarazuka) 16/01/1980 N-S 1.084 0.629 35.0013 Livermore (Morgan Terr Park) 27/01/1980 N-W 0.582 0.252 30.0014 Los Angeles 10/03/1933 N-E 0.144 0.065 25.7815 Mammoth Lakes (Convict Creek) 25/05/1980 E-W 0.496 0.416 29.9516 Ken-Oki (Miyagi) 1978 N-S 0.347 0.140 58.0017 Northridge (Sylmar—Olive View) 17/01/1994 N-W 1.776 0.842 15.5518 Olympia 13/04/1949 N-E 0.681 0.325 30.2819 Parkfield (Templor) 28/06/1966 S-W 0.489 0.357 30.3120 Montenegro (Petrovac) 15/04/1979 N-S 1.455 0.438 19.6221 San Fernando (Hollywood Stor Lot) 09/02/1971 E-W 0.301 0.210 27.9822 Friuli (San Rocco) 15/09/1976 N-S 0.246 0.035 16.9223 Seattle 13/04/1949 N-W 0.153 0.076 20.1024 Spitak, Armenia (Gukasian) 07/12/1988 N-S 0.359 0.199 19.9025 Superstitn Hills (A) 24/11/1971 E-W 0.221 0.132 29.81

(Wildlife Liquef. Array)26 Taft 1 21/07/1952 N-W 0.301 0.157 30.0027 Taft 3 21/07/1952 S-E 0.346 0.179 54.9028 Friuli (Tarcento) 15/09/1976 N-S 0.100 0.139 20.1029 Tokyo 1956 N-S 0.142 0.075 11.4030 Vernon 10/03/1933 S-E 0.220 0.192 42.17

regarding records selected from PEER database. In the same table, the values of spectral ordinatecorresponding to the fundamental period of the structure are reported (equal to 0.5 s).

The PC-ANSR computer program [14] has been used for incremental dynamic nonlinear analyses(IDA) [15]. IDA have been performed according to the multiple-stripe method. Multiple-stripeanalysis (MSA) is a collection of single-stripe analyses performed at multiple levels of spectralacceleration, where each stripe represents a set of results obtained by scaling all the earthquakerecords to provide the same spectral acceleration value corresponding to the fundamental period ofthe structure (Figure 4(b)). The results of MSA provide statistical information about the demandover a wide range of spectral acceleration values. Therefore, for each stripe it is possible to evaluatethe dispersion �DM|Sa and the median value �DM|Sa .

Several damage parameters have been used for PSDA:

• the peak interstorey drift ratio (PIDR) for each storey;• the ratio �/�max between the cyclic ductility demand of a brace element and the correspondingmaximum allowable value [16];



Figure 5. PIDR for the structure designed according Eurocode 8 methodology.

• the ratio NSd/Nb,Rd between the axial demand achieved in the generic column and thecorresponding value leading to out-of-plane buckling, computed according to Eurocode 3 [17].

IDA analyses have been performed starting from Sa value equal to 0.05g and increasing the Savalue, with 0.05g steps, until the occurrence of dynamic instability. This phenomenon is due tothe fact that, under load reversals, the previously buckled member could not return to its originalalignment and the member that was previously in tension could exceed its capacity in compression.As a consequence, both diagonal members could be in a buckled condition. The dynamic instabilityphenomenon occurs when, at any storey, both diagonals are in a buckled configuration so that asoft-storey mechanism develops. In Figure 5, as an example, with reference to the structure designedaccording to the proposed methodology, the results of IDA analyses are depicted in terms of PIDRoccurring at the first storey. IDA curves have a horizontal asymptote when dynamic instabilityarises. In addition, it can be observed that for a PIDR exceeding 0.002 rad significant inelasticresponse occurs due to the buckling of braces. In addition, in Figure 5 the points corresponding tothe Sa values in IDA curves leading to the out-of-plane buckling of columns or to the fracture ofbraces have been depicted. It can be noted that, in the case of braced frames designed according toEurocode 8, interstorey drift limitation is not a governing limit state, because out-of-plane bucklingof columns or fracture of braces occurs before the drift limit is reached.

As already noted, PSDA requires the estimation of the statistical parameters of the distribu-tion law of structural demand (DM). The great variability of nonlinear dynamic analyses results,combined with the dynamic instability phenomenon (absence of numerical convergence), requiresa more robust method than the moment method for evaluating the median �DM|Sa and the disper-sion �DM|Sa . In the study case herein presented, the so-called ‘three-parameter model’ has beenadopted [18].

According to this method, fractiles can be calculated by means of the following relationships:

dmp(sa) = a′ ·sb′a ·exp(�DM|NC ·�−1

DM|NC(p)) if sa�sa0 (25)

dmp(sa) = a′ ·sb′a ·exp

(�DM|NC ·�−1

DM|NC ·(

p

(sa/sa0)−B

))if sa>sa0 (26)



Figure 6. (a) Collapse probability versus spectral acceleration and (b) regression in the log–log plane ofthe demand median value (PIDR storey 1 of VBF-E) versus spectral acceleration with reference to the

subset of data corresponding to non-collapse situation.

where a′ and b′ can be obtained by means of a power regression of median values of the considereddemand parameter, calculated with the ordered statistic method with reference to the subset of datacorresponding to records for which dynamic instability does not occur (non-collapse). Therefore,�DM|NC=a′ ·sb′

a is the median value of DM conditioned to Sa and conditioned to the above subset,�DM|NC is the corresponding dispersion and � is the standard normal CDF. The coefficients sa0and B are evaluated by modelling the non-collapse probability (function of spectral acceleration)by means of the distribution provided by the following expressions:

pNC|Sa(sa) = 1 if sa�sa0 (27)

pNC|Sa(sa) =(sasa0

)−B

if sa>sa0 (28)

where sa0 is the highest IM value for which the non-collapse probability is equal to 1 (i.e. thehighest Sa value not leading to dynamic instability). The variation of collapse probability versusspectral acceleration is depicted in Figure 6(a). For a given value of Sa, median value and fractilescorresponding to 16 and 84% can be obtained with Equations (25) and (26) by imposing p equal,respectively, to 0.50, 0.16 and 0.84. For each DM parameter, using the fractiles and the medianvalue, the dispersion �DM|Sa can be evaluated as

�DM|Sa=1

2ln

dm84%

dm16%for pNC|Sa>0.84 and �DM|Sa= ln

dm50%

dm16%for 0.84>pNC|Sa�0.50 (29)

However, it has to be underlined that, as testified by Equation (29), �DM|Sa can be evaluated onlywhen the non-collapse probability (due to dynamic instability) is greater than or equal to 0.50.In Figure 7, with reference to the considered concentrically braced frames (VBF-E and VBF-P)and for all the damage parameters taken into account, the IDA curves, referred to the first-storeydamage parameters, and the 16th, 50th and 84th fractiles are depicted. The dispersion measure,�DM|Sa , has been computed (as a function of the spectral acceleration) starting from the knowledgeof 16th and 84th fractiles evaluated by means of Equation (29). Starting from the knowledge ofthe median value of the demand parameter, given Sa, for each considered Sa for which dynamicanalyses are performed, the a and b coefficients provided by Equation (17) can be obtained for eachstructure, for each storey and for each demand parameter by a linear regression in a log–log plane.



Figure 7. IDA results for the considered damage parameters (first storey) both for VBF-E and VBF-P.

Table IV summarizes the statistical parameters of demand distribution for each considereddamage parameter, for each storey and for each structure. In the case of chevron-braced frames,conversely to the case of X-braced frames [19], with reference to all the considered demandparameters computed for the different storeys, the assumption corresponding to Equation (17) isnot satisfied so that a single linear regression of median demand versus spectral acceleration cannotbe adopted in all the range of Sa. This result can be easily explained in the case of NSd/Nb,Rddamage parameter. In fact, in this case, the column axial force increases as far as the spectralacceleration increases, but this trend is obviously limited by the internal actions that dissipativezones are able to transmit. As a consequence, the column axial force attains a maximum valuewhen all the diagonal braces, above the considered storey, are yielded. As a result, both in the caseof VBF-E and in the case of VBF-P structures, for high values of the spectral acceleration, the



Table IV. Values of a and b coefficients.

VBF-E VBF-P

DM Sa(T0) (m/s2) a b a b

PIDR 1 �6.5 0.0004 0.9279 0.0004 0.9306>6.5 1.0E−08 5.9277 1.00E−08 6.0943

PIDR 2 �6.5 0.0005 1.0116 0.0005 0.9878>6.5 6.0E−06 3.1898 7.00E−09 6.4299

PIDR 3 �6.5 0.0007 0.7573 0.0006 0.9407>6.5 0.0003 1.2299 2.00E−05 2.6873

PIDR 4 �6.5 0.0006 1.2237 0.0006 1.1657>6.5 3.00E−05 2.588 2.00E−06 3.9332

�1/�1,max �9.0 0.0324 0.8557 0.0401 0.7068>9.0 1.00E−08 7.6329 5.00E−07 5.7404

�2/�2,max �9.0 0.0299 0.927 0.0288 1.017>9.0 2.00E−05 4.2909 6.00E−09 8.5307

�3/�3,max �9.0 0.0278 0.9068 0.0274 0.9313>9.0 2.00E−04 3.2558 3.00E−04 3.1929

�4/�4,max �9.0 0.0229 1.3538 0.0281 1.0463>9.0 1.00E−06 5.8237 9.00E−09 8.1488

N1/Nb,Rd,1 �4.0 0.2172 0.6775 0.2357 0.6846>4.0 0.4920 0.071 0.4546 0.2085

N2/Nb,Rd,2 �4.0 0.2354 0.6414 0.228 0.6185>4.0 0.4431 0.1671 0.3716 0.2694

N3/Nb,Rd,3 �4.0 0.2311 0.6016 0.214 0.5249>4.0 0.4416 0.1577 0.3813 0.1257

N4/Nb,Rd,4 �3.5 0.4212 0.0001 0.109 0.01133.5–4.5 0.001 4.8742 5.00E−08 10.7224.5–8.0 0.4979 0.7293 0.3965 0.0088>8.0 0.0008 3.8528 0.0092 1.8622

curve relating the median value of the considered demand parameter (NSd/Nb,Rd) to the spectralacceleration shows a vertical asymptote (Figure 7), because in the analyses the strain-hardeningeffects are neglected.

For the above reason, the log–log representation of the median demand versus spectral accelera-tion curve shows two or more ranges (Figure 8) where a linear regression, leading to a relationshipin the form of Equation (17), can be applied. The Sa value separating the two or more ranges hasbeen evaluated by means of a procedure aiming at the maximization of the correlation coefficientfor each regression curve.

In Table IV the values of a and b and the range of Sa for the applicability of Equation (17),which have been evaluated by means of a power regression of the median values obtained withthe three-parameter model, are reported.

6. PROBABILISTIC SEISMIC CAPACITY ANALYSIS (PSCA)

The structural capacity defines the limit or the threshold value for the demand variable to identifyacceptable structural behaviour. In other words, on the one hand the ‘demand’ represents the



Figure 8. Regression in the log–log plane of demand median value versus spectral acceleration.

seismic nonlinear behaviour of the structure and, on the other hand, the ‘capacity’ represents the‘demand’ limit value. In the general formulation of Jalayer and Cornell [3] also the ‘capacity’is random according to a lognormal distribution with a median value �C = C and dispersion �C .As three damage parameters have been considered, there are three capacity values correspondingto the collapse prevention limit state: the limit value of PIDR, the limit value of cyclic ductilityof brace elements (�,max) and the limit value of the out-of-plane buckling resistance of columns(Nb,Rd) provided by Eurocode 3 [17] formulation.

The limit value of PIDR is not simple to establish. Different suggestions can be found in FEMA273 [20] and FEMA 350 [21]. In the following, reference is made, for the collapse preventionlimit state, to a median value of capacity �C equal to 2%, while the dispersion value �C has beenassumed to be equal to 0.20.

Regarding the limit value of cyclic ductility of bracing members, it has been evaluated accordingto the following expression suggested by Tremblay [16] on the basis of available experimentalresults:

�max=2.4+8.3 · � (30)

As shown by Equation (30), the cyclic ductility limit is a function of the normalized slenderness� and this limit increases as far as the normalized slenderness increases, i.e. the limit is morerestrictive for stocky braces rather than for intermediate or slender braces. However, by consideringthe ratio between the brace ductility demand and the corresponding ductility supply as the damageparameter, the median value of the normalized capacity becomes independent of the consideredstorey, being equal to 1.0. The dispersion of capacity associated with �/�,max damage parametercan be derived from the available experimental data. In particular, according to the experimentaldata collected by Tremblay [16], the ratio between the experimental value of the cyclic ductilitysupply �exp and the theoretical value provided by Equation (30) has a mean value equal to 1.0,while the coefficient of variation is equal to 0.25.

The third considered damage parameter is related to the column axial load and it is equal toNSd/Nb,Rd, where, NSd is the maximum axial load obtained by the nonlinear analysis and Nb,Rdis the theoretical limit value corresponding to the out-of-plane buckling computed according to



Eurocode 3 [17]. The theoretical limit value of the above ratio is equal to 1.0, because the columncan be considered in a buckled configuration when NSd exceeds the value corresponding to theout-of-plane buckling resistance Nb,Rd. Regarding the dispersion, a coefficient of variation equalto 0.10 has been assumed according to the general suggestions given in EN 1990 [22].

7. MAF OF EXCEEDING A LIMIT STATE

The application of Jalayer and Cornell [3] approach allows one to determine the MAF of exceedinga specific limit state as soon as the seismic hazard curve, the statistical distribution law of structuraldemand and the capacity are known. This approach is based on the hypotheses provided byEquations (16)–(18) and, in particular, on the assumption that the dispersion of demand �DM|Sa isindependent of the spectral acceleration (i.e. homoscedasticity hypothesis). This assumption doesnot comply with the physical interpretation of structural response whose dispersion is expectedto increase under increasing earthquake intensity measures (Figure 9), because the structure ismore and more strongly engaged in plastic range. Aiming at the application of Jalayer and Cornellprocedure, an appropriate evaluation of �DM|Sa value is required. In the original report of Jalayerand Cornell [3], the authors suggest that the value of �DM|Sa to be used in the formulation has tobe chosen for Sa values in the ‘range of primary interest’. Therefore, in this study, the constantvalue of �DM|Sa required for the application of Equation (21) is assumed to be equal to the meanvalue evaluated with reference to the range of Sa where Equation (17) holds. In the case of VBFstructures, it is necessary to define two or more ranges of Sa for the application of the assumptionprovided by Equation (17); therefore, the mean value of �DM|Sa is evaluated with reference to therange of Sa in which �DM|Sa =�C . Finally, MAF of exceeding the limit state is evaluated.

The results obtained by the application of Jalayer and Cornell approach are compared with thoseobtained by means of numerical integration of Equations (14)–(15), for the examined structures, inTable V for all the considered damage parameters and for all the storeys. In addition, in this table,also the coefficients a, b and �DM|Sa used for the application of Jalayer and Cornell formulationare reported. Furthermore, the return periods TR,LS calculated as the inverse of MAF of exceedinga limit state are also provided.

Figure 9. Demand dispersion versus spectral acceleration curve.



TableV.Meanannual

frequenciesof

exceedingalim

itstateandcorrespondingreturn

periods.

JalayerandCornellapproach

Num

erical

integrationof

Equ

ation(18)

DM

ab

� DM

|S aHLS

T R,LS(years)

HLS

T R,LS(years)

VBF-E

PIDR

11.50

0E−0

85.92

770.52

136.97

9E−0

414

336.85

6E−0

414

59PIDR

26.00

0E−0

63.18

980.41

994.81

7E−0

420

764.91

9E−0

420

33PIDR

32.74

0E−0

41.22

990.31

815.67

8E−0

517

611

3.48

6E−0

528

684

PIDR

43.37

0E−0

52.58

800.37

925.93

1E−0

416

865.86

3E−0

417

06� 1

/� 1

,max

1.00

0E−0

87.63

290.70

496.12

3E−0

416

336.22

2E−0

416

07� 2

/� 2

,max

1.78

0E−0

54.29

090.58

714.45

6E−0

422

445.14

6E−0

419

43� 3

/� 3

,max

1.87

1E−0

43.25

580.42

263.54

9E−0

428

183.65

6E−0

427

35� 4

/� 4

,max

1.50

0E−0

65.82

370.45

528.00

6E−0

412

498.62

9E−0

411

59N1/Nb,Rd,1

4.69

2E−0

10.07

100.04

351.09

0E−0

9>10

000

07.36

8E−0

9>10

000

0N2/Nb,Rd,2

4.43

1E−0

10.16

710.04

525.03

9E−0

6>10

000

05.04

6E−0

6>10

000

0N3/Nb,Rd,3

4.41

6E−0

10.15

770.04

522.72

9E−0

6>10

000

02.89

4E−0

6>10

000

0N4/Nb,Rd,4

1.02

5E−0

34.87

420.02

087.62

7E−0

313

17.49

9E−0

313

3

VBF-P

PIDR

11.00

0E−0

86.09

430.50

707.11

3E−0

414

066.82

2E−0

414

66PIDR

27.00

0E−0

96.40

000.40

128.14

4E−0

412

287.63

6E−0

413

10PIDR

32.00

0E−0

53.00

000.30

758.61

5E−0

411

617.03

8E−0

414

21PIDR

48.00

0E−0

63.28

420.38

527.29

7E−0

413

706.59

0E−0

415

17� 1

/� 1

,max

5.00

0E−0

76.00

000.72

096.28

6E−0

415

915.48

0E−0

418

25� 2

/� 2

,max

5.00

0E−0

98.53

070.50

109.77

6E−0

410

238.44

5E−0

411

84� 3

/� 3

,max

3.00

0E−0

43.19

290.46

564.68

3E−0

421

355.06

2E−0

419

76� 4

/� 4

,max

9.00

0E−0

98.14

880.53

748.81

7E−0

411

348.47

8E−0

411

80N1/Nb,Rd,1

4.88

4E−0

10.10

250.03

521.87

7E−0

7>10

000

01.87

4E−0

7>10

000

0N2/Nb,Rd,2

3.45

4E−0

10.25

860.02

681.51

4E−0

566

045

1.64

6E−0

560

767

N3/Nb,Rd,3

3.41

2E−0

10.13

340.05

024.14

9E−0

9>10

000

01.71

9E−0

7>10

000

0N4/Nb,Rd,4

8.35

7E−0

31.86

210.37

474.95

6E−0

414

066.37

5E−0

415

69



The first consideration about the obtained results concerns the difference between the HLS valueobtained by means of Jalayer and Cornell approach and the value obtained by numerical integrationof PEER equation. It can be observed that, in the case of VBF-P structure, the smallest value of HLSis generally (but not always) obtained when the numerical integration is performed. It means that,for this structure, the value of dispersion, used in the Jalayer and Cornell formulation, representsan overestimation of the actual value for a relatively small value of Sa (i.e. Sa values having highprobability to be exceeded). Conversely, in the case of VBF-E structure, a general trend cannotbe observed. However, in both cases (VBF-P and VBF-E), the values of HLS and TR,LS computedby means of Jalayer and Cornell approach and by means of the numerical integration of PEERequation are sufficiently close for practical application, with reference to the governing failuremode leading to the minimum value of TR,LS.

The second consideration regards the comparison between the seismic performance obtainedby means of the two investigated design approaches (Eurocode 8 and the design methodologyproposed by the authors). For sake of simplicity, in the following the discussed results are thoseobtained by means of numerical integration.

Both the structure designed according to Eurocode 8 provisions (VBF-E) and the structuredesigned according to the proposed procedure (VBF-P) provide similar return periods regardingthe limit states of excessive storey damage (PIDR) and fracture of braces (�/�max). Conversely,regarding the damage parameter corresponding to column buckling, Eurocode 8 [5] provisions leadto the worst behaviour. In fact, the probability of failure of VBF-E structure is very significant andthe return period corresponding to the considered limit state is very low (133 years). Regardingthis issue, it is useful to note that, according to the new Italian seismic code [23], with referenceto the collapse prevention limit state the probability of failure has to be less than 5% in a numberof years equal to the mean life of the structure (50 years for residential buildings). In the caseof residential buildings, this corresponds to a return period of 975 years. On the other hand,the structure designed by means of the procedure proposed by the authors [6, 7] exhibits a goodperformance with reference to all the considered damage parameters, leading to a return period ofthe collapse prevention limit state exceeding 975 years. Comparing the maximum probability offailure occurring with reference to the designed structures (maximum value between all the damageparameters and for all the storeys), it results that the seismic reliability of VBF-P is about 9.0times better than that of VBF-E (TR,LS=1180 years versus TR,LS=133 years). Regarding the ratioNSd/NRd, it is easy to observe that the structure designed according to Eurocode 8 is highly proneto out-of-plane buckling leading prematurely to collapse. In fact, the return period associated withthis phenomenon is very small (133 years). On the contrary, for the structure designed accordingto the proposed methodology, the out-of-plane buckling of columns is not the governing principalfailure mode. This result represents the main design goal of the proposed methodology [6, 7].

In terms of seismic reliability, the structure designed by means of the proposed method-ology (VBF-P) provides a significant performance improvement when compared with concen-trically braced frames designed according to European provisions (VBF-E). Therefore, aimingat a preliminary evaluation of the economic meaning related to the adoption of more stringentdesign procedures, the seismic performances of the considered structures, expressed in terms ofMAF of exceeding a limit state, have been compared with the cost needed to realize the wholebuilding.

In Table VI, the weight of structural steel required by the designed buildings (VBF-E andVBF-P) is given. In particular, the proposed design criteria lead to a very small increase of thestructural building weight (only 0.57%) and, as a consequence, to a not significant increase of the



Table VI. Comparison in terms of weight.

VBF-E VBF-P

Weight of horizontal force resisting systems (ton) 47.60 48.20Weight of vertical loads resisting system (ton) 56.80 56.80Total weight of the structure (ton) 104.40 105.0Percentage variation — +0.57%

cost to realize the whole building when compared with the structure dimensioned according toEurocode 8.

An appropriate seismic performance assessment of different design strategies has to be basedon benefits versus costs comparison. In this case, the benefit provided by the proposed designprocedure can be measured as the ratio between the values of the return period corresponding tothe exceeding of the collapse prevention limit state. As already stated, in the case of the examinedstructures, the above ratio is about 9, so that the structural performance benefits due to the proposeddesign procedure are very large, while the increase of the whole building cost is negligible.

8. CONCLUSIONS

In this study some procedures for evaluating the seismic reliability of structures have been presentedand applied with reference to concentrical VBFs. In particular, the seismic reliability of a four-storey building, designed according to two different criteria, has been expressed in terms of MAFof exceeding the collapse prevention limit state.

The obtained results have pointed out a good seismic behaviour of the structure designedaccording to the proposed methodology for all the considered damage parameters. In particular,by comparing these results with those obtained for the structure designed according to Eurocode 8provisions, a big difference arises due to the out-of-plane buckling of columns. The MAF ofexceeding this limit state, HLS, with reference to the structure designed according to the proposedmethodology is low and the corresponding return period TR,LS is clearly greater than the returnperiod assumed for defining the collapse prevention limit state. On the contrary, the collapse returnperiod TR,LS corresponding to the out-of-plane buckling of columns for the structure designedaccording to Eurocode 8 provisions is only 133 years. In addition, it has to be stressed that thestructural weight is practically the same for the two considered structures, but it is differentlydistributed among beams and columns (diagonals are exactly the same). In particular, the applicationof Eurocode 8 rules leads to greater columns at the base of the structure and smaller columns at thetop. In conclusion, VBFs designed according to Eurocode 8 provisions exhibit poor performanceswith reference to the ultimate limit state due to the premature out-of-plane buckling of columns.Conversely, VBFs designed according to the proposed methodology exhibit good performancesfor all the considered damage parameters.

REFERENCES

1. Pacific Earthquake Engineering Research Center, PEER, 2000. http://peer.berkeley.edu/smcat.2. Cornell CA, Krawinkler H. Progress and Challenges in Seismic Performance Assessment, 2000.



3. Jalayer F, Cornell A. A technical framework for probability-based demand and capacity factor design (DCFD).Seismic Formats. PEER Report 2003/06, 2003.

4. Cornell CA, Jalayar F, Hamburger RO, Foutch DA. Probabilistic basis for 2000 SAC Federal EmergencyManagement Agency steel moment frame guidelines. Journal of Structural Engineering (ASCE) 2002; 128(4):526–533.

5. prEN 1998—Eurocode 8: design of structures for earthquake resistance. Part 1: general rules, seismic actionsand rules for buildings, 1998.

6. Longo A, Montuori R, Piluso V. Plastic design of V-braced frames. Journal of Earthquake Engineering 2008;12:1246–1266.

7. Longo A, Montuori R, Piluso V. Failure mode control of X-braced frames under seismic actions. Journal ofEarthquake Engineering 2008; 12:728–759.

8. Georgescu D, Toma C, Gosa O. Post-critical behaviour of ‘K’ braced frames. Journal of Constructional SteelResearch 1992; 21:115–133.

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17. CEN. EN 1993-1-1—Eurocode 3: design of steel structures. Part 1: general rules and rules for buildings. ComiteEuropeen de Normalisation, 2005.

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