buckling restrained structures seismic simulation

5/27/2018 Buckling Restrained Structures Seismic Simulation

Vol.8, No.3 EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION September, 2009

Earthq Eng & Eng Vib(2009) 8: 373-385 DOI: 10.1007/s11803-009-9013-8

Assessment of buckling-restrained braced frame reliability using anexperimental limit-state model and stochastic dynamic analysis

Blake M. Andrews1, Junho Song2 and Larry A. Fahnestock2

1. Wiss, Janney, Elstner Associates, Inc., Northbrook, Illinois, USA

2.Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

Abstract: Buckling-restrained braces (BRBs) have recently become popular in the United States for use as primarymembers of seismic lateral-force-resisting systems. A BRB is a steel brace that does not buckle in compression but instead

yields in both tension and compression. Although design guidelines for BRB applications have been developed, systematicprocedures for assessing performance and quantifying reliability are still needed. This paper presents an analytical framework

for assessing buckling-restrained braced frame (BRBF) reliability when subjected to seismic loads. This framework

efficiently quantifies the risk of BRB failure due to low-cycle fatigue fracture of the BRB core. The procedure includes a

series of components that: (1) quantify BRB demand in terms of BRB core deformation histories generated through stochastic

dynamic analyses; (2) quantify the limit-state of a BRB in terms of its remaining cumulative plastic ductility capacity based

on an experimental database; and (3) evaluate the probability of BRB failure, given the quantified demand and capacity,

through structural reliability analyses. Parametric studies were conducted to investigate the effects of the seismic load,

and characteristics of the BRB and BRBF on the probability of brace failure. In addition, fragility curves (i.e., conditional

probabilities of brace failure given ground shaking intensity parameters) were created by the proposed framework. While the

framework presented in this paper is applied to the assessment of BRBFs, the modular nature of the framework components

allows for application to other structural components and systems.

Keywords: risk and reliability analysis; buckling-restrained brace; stochastic dynamic analysis; first-order reliabilitymethod; cumulative plastic ductility capacity

Correspondence to: Larry A. Fahnestock, Department of Civil

and Environmental Engineering, University of Illinois at

Urbana-Champaign, 205 North Mathews Avenue, Urbana, IL,

61801, USA

Tel: (217) 265-0211; Fax (217) 265-8040E-mail: [email protected]; Assistant Professor

Supported by: Federal Highway Administration Under Grant No.

DDEGRD-06-X-00408

ReceivedFebruary 5, 2009; AcceptedMarch 22, 2009

1 Introduction

1.1 Description of buckling-restrained braces

Buckling-restrained braced frames (BRBFs), whichare concentrically-braced frames that incorporatebuckling-restrained braces (BRBs), are beingimplemented as the primary lateral-force-resisting

system in a rapidly increasing number of structureslocated in high seismic regions of the United States. Thebasic concept of the BRB, which was first developed inJapan several decades ago as a seismic damper (Uangand Nakashima, 2004; Xie, 2005), is illustrated in Fig.1. Unlike a typical steel brace, a BRB does not buckle incompression but yields in both tension and compression.Although design guidelines for BRB applications

have been developed (e.g., AISC, 2005), systematicprocedures for assessing performance and reliability ofBRBFs are not well established.

1.2 Seismic performance assessment

Extensive research over the past several decadeshas established a broad performance-based earthquakeengineering (PBEE) framework (e.g., Krawinkler andMiranda, 2004). The most common PBEE framework,which has both design and assessment components(Krawinkler et al., 2004), uses the total probabilitytheorem to link intensity measures, engineering demandparameters, damage measures, and decision variables.Although this framework is comprehensive and rigorous,alternate procedures may provide valuable insight intosystem behavior and performance by using limit-statemodels developed based on experimental databases anddemand determined by stochastic dynamic analysis.In an effort towards developing such an alternativeframework, this paper presents a new procedure for

quantifying BRBF system reliability based on potentialBRB core fracture. Stochastic dynamic analysis isperformed as the starting point for quantifying uncertainseismic demands. A probabilistic BRB remainingcapacity model is developed to determine its limit state


374 EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION Vol.8

based on a database of experimental research. Using thedeveloped limit-state model and the stochastic demands,structural reliability analysis is performed to assess thereliability of the system. The overall architecture and

analysis flow of the framework is presented in Fig. 2.The components of the framework can be dividedinto three categories: Modules, Analyses, and Results.Modules were mathematical constructs used to modelthe physical reality, Analyses were mathematicalsimulations performed using the modules, and Resultswere the outputs from the analyses. The components ofthe framework include: stochastic modeling of seismicloads, dynamic analyses of the BRBF, cumulative plasticductility (CPD) capacity models for BRBs, structuralreliability analyses, parametric studies of how BRBand BRBF properties affect performance, and fragility

modeling.Using the Input Module, input ground accelerationrecords were randomly generated from power spectrummodels and then modulated with envelope functions toaccount for non-stationarity of the stochastic processes.The generated acceleration records were used as inputexcitations to the BRBF System Model, which was asingle-degree-of-freedom lumped-mass system for thisresearch. Within the BRBF System Model, the BRB

behavior was modeled using a Bouc-Wen hysteresismodel (Wen, 1976; Song and Der Kiureghian, 2006).Nonlinear Dynamic Simulations were performed toobtain BRB core deformation time history records.

A BRB CPD Limit State model developed by Andrewset al. (2009) was used in the assessment frameworkto describe the fatigue fracture limit state of a BRB interms of CPD. Given BRB core deformation historyparameters as inputs, the CPD limit state models predictthe remaining CPD capacity of the brace, where valuesless than zero indicate failure. The epistemic uncertaintyin the model was taken into account explicitly by anoverall error term identified by a maximum likelihoodestimation (MLE) method (Devore, 2000). Given BRBdemand (i.e., core deformation histories generated fromthe dynamic analyses) and limit state (i.e., remaining

capacity predicted by the CPD models), ReliabilityAnalyses were performed to evaluate the probability ofbrace failure and to quantify BRBF System Reliability.The analyses were conducted using the first orderreliability method (FORM) (see Der Kiureghian, 2005for a review) and facilitated by the Matlab open-source code, FERUM (Der Kiureghian et al., 2006). Inthe reliability analyses, the epistemic uncertainty in thelimit state identification was accounted for explicitly,

Fig. 1 Typical BRB configuration

Fig. 2 Assessment framework architecture

Modules Analyses Results

Nonlinear dynamic

simulation (Section 3) BRBF system reliability

(Section 4)

Parametric studies and

fragility analyses(Section 5)Reliability analysis

(Section 4)

Input module

(Section 2)

BRBF system model

(Section 3)

BRB CPD limit state

(Section 4)

Core yielding region

Steel plate core

Core elastic region

Concrete-filledsteel tube (CFT)

Top view

Side view

Core is debonded from concrete

Core


No. 3 Blake M. Andrews et al.: Assessment of buckling-restrained braced frame reliability 375

and, as a result, the probabilities of brace failure werecalculated in terms of mean probability, 90% confidencelevel probability, and 95% confidence level probability(Der Kiureghian, 2001).

Using the tools described above, a ParametricStudy was conducted to explore the effects of theseismic loading, BRB, and BRBF characteristics on theprobability of brace failure. For given seismic loadings,surfaces of reliability indices were constructed in orderto determine the probability of brace failure directlyfrom BRB and BRBF properties. Also, for a given setof BRB and BRBF properties, Fragility Analyses wereperformed to provide conditional probability of bracefailure given ground motion intensity parameters. Thepaper concludes with the lessons learned from thedevelopment of the assessment framework and outcomes

of the parametric study and fragility analyses.

2 Input module

The origin of the data flow in the proposedassessment framework is the Input Module, whichcreates randomly-generated ground acceleration timehistories that are applied as base acceleration recordsto the BRBF system model. Acceleration time historiesare generated based on an input power spectral density(PSD) function (Clough and Penzien, 1993; Newland,2005), where the input PSD function is calibrated suchthat the acceleration time histories produced result ina 5%-damped elastic spectrum that matches a targetspectrum. Resulting acceleration time history values aremultiplied by a modulation function to account for non-stationarity inherent in seismic loadings. Details of theformulation are presented in the following sections.

2.1 Analytical flow of input module

The analytical flow of the Input Module is illustratedin Fig. 3. The steps of input generation are as follows:

(a) A random acceleration time history a tbase ( )

is generated using the spectral representation method

(Shinozuka and Deodatis, 1991), which gives astationary zero-mean Gaussian random time historiesthat are compatible with a given input PSD function

( ) . The random acceleration time history is given

by:

a t tii

n

i ibase ( ) ( ) cos( )= +2 (1)

where ( )i is the value of the input accelerationPSD function at = i , = +i i1 , andi U= ( , )0 2 is a random phase angle that conformsto a uniform distribution bounded by 0 and 2. Thevector of frequencies is chosen to cover the range offrequencies of interest.

(b) a tbase

( ) is multiplied in time by a modulationfunction ( )t to produce a non-stationary accelerationtime history &&z tg ( ) , which is used as input to the BRBFsystem model.

(c) The 5%-damping elastic acceleration responsespectrum S Ta ( ) is calculated for &&z tg ( ) .

(d) S Ta ( ) is compared to the target spectrumS Ta ( )( )target .

(e) The input PSD ( ) is calibrated by varyingits ordinate values systematically such that S Ta ( ) matches S Ta ( )( )target within a tolerance.

(f) Following calibration of the input PSD, it isused repeatedly to generate acceleration time historiesfor input to the BRBF system model.

The target elastic spectra, modulation function, andcalibration of the input PSD function are described indetail below.

2.2 Target elastic spectra

The basic shape of the target elastic spectra wasdefined per Minimum Design Loads for Buildings andOther Structures: ASCE/SEI 7-05 (ASCE, 2005) asshown in Fig. 4. To define the parameters of the spectrum,a high-seismic location in Southern California was

chosen, and the location was assumed to be categorized

Fig. 3 Analytical flow of input module

Input PSD Modulation BRBF system

model&&z tg ( )

Acceleration spectrumCompareTarget spectrum

Calibrate to match

Further analysesa tbase( )



as site class B per ASCE/SEI 7-05 (i.e., rock with soilshear wave velocity between 762 and 1524 m/s). Twotarget spectra were used in this research: (1) a spectrumthat conformed to a design basis earthquake (DBE), and

(2) a spectrum that conformed to a maximum consideredearthquake (MCE), where DBE and MCE levels aredefined per ASCE/SEI 7-05. For the MCE level, S

a(T) is

1.5 times that of the DBE level. The parameters for theDBE and MCE target spectra used in this research aredefined in Table 1.

2.3 Modulation function

Earthquake ground acceleration processes aregenerally recognized as being non-stationary, that is,their stochastic properties change over time. To create

a non-stationary input process during the procedureto generate input time histories, each stationaryacceleration time history generated based on a PSDfunction, i.e., a tbase( ) , is modulated by a deterministicfunction of time, ( )t as follows:

&&z t a t tg base( ) ( ) ( )= (2)

where &&z tg ( ) is the resulting modulated acceleration timehistory. Clough and Penzien (1993) offer the followingmodulation function to represent the non-stationaritytypical in seismic ground acceleration processes:

( )t tt=

1

2e (3)

According to Clough and Penzien (1993), statisticalstudies of accelerograms during the San Fernando,California earthquake have shown that constants

1and

2can be assigned values of 0.45 and 1/6, respectively.

These values of 1and

2 were considered typical for this

research. In Fig. 5, the modulation function ( )t in Eq. 3is plotted for these values.

2.4 Calibration of the input PSD function

The input PSD ( ) was calibrated such thatthe actual 5%-damping elastic acceleration responsespectrum as determined using a standard single-degree-

of-freedom (SDOF) oscillator matched the targetspectrum. The algorithm below was used to accomplishthis. In the algorithm, the calculations in (a) to (c) areperformed at all selected discrete values of T

iover the

domain of the PSD function and are repeated over theindexj.

(a) Determine the actual elastic pseudo-accelerationresponse spectrum for iteration j, S Ta i j( )( ) , using astandard SDOF oscillator (5% damping) with input PSDj iT( ) .

(b) Compare the actual and target spectra foriterationjand calculate a multiplierM

jas follows:

M TS T

S Tj i

a i

a i

j

( )( )

( )=

( )

( )target

(4)

(c) Apply the multiplier to the input PSD for thecurrent iteration, j iT( ) , to generate a new input PSD foriterationj+1, j iT+1( ) , as follows:

j i j i j iT M T T + = 1( ) ( ) ( ) (5)

(d) Iterate over index j and repeat (a) to (c) untilthe change in the old PSD to the new PSD is less than atolerance value, which was set to 1% in this research.

(e) As the last step in this process, the value of

the input PSD at T= 0 may need to be altered, because

Table 1 Target elastic spectrum parameters

Parameter DBE spectrum MCE spectrum

aPG

(g) 0.38 0.58

SDS

(g) 1.0 1.5

SD1

(g) 0.40 0.60

T1(s) 0.08 0.08

TS(s) 0.40 0.40

TL

(s) 8.0 8.0

Note: aPG

is peak ground acceleration, (PGA).

1.0

0.8

0.6

0.4

0.2

0

(t)

0 5 10 15 20 25 30

Time (s)

Fig. 5 Modulation function

Fig. 4 Target elastic spectrum

aPG

T (s)

T1 TS 1.0 TL

SD1

SDS

Sa( ) targetS

S

Ta

D( ) =target1

SS T

Ta

D( ) =

target1

2L



the process tends to result in very high first values ofthe input PSD. To solve this, the first value of the inputPSD was always set equal to the second value. Thiseliminated the spike in the PSD function at T= 0 .

The calibrated input PSD was then used repeatedly togenerate time histories that conform to the target spectra.Calibration was performed only once for a given targetspectrum. In some cases, minor manual manipulationof the input PSD was required to ensure that the actualelastic spectrum matched the target, particularly nearT

S. A comparison between the actual and target elastic

spectra for the DBE level is presented in Fig. 6; thecalibrated PSD for the DBE level is presented in Fig. 7;and a sample modulated time history for the DBE levelis given in Fig. 8.

3 BRBF system model and numericalsimulation

3.1 Prototype structural system

A simple prototype structural system was selectedbased on average values from a BRB test database of76 specimens compiled from the literature (Andrewset al., 2009). The prototype system is shown in Fig.9. All connections in the system were assumed to bepinned and the beams and columns were assumed to berigid so that all of the lateral stiffness was provided bythe BRB. The structural system was assumed to have asingle lumped mass massociated with it. A simplifiedBRB structural model was used in this research, wherethe BRB consisted of three regions: the core yielding

region and two non-core regions (the non-core regionsare sections of the brace that do not yield since they havea larger cross-sectional area than the core region). Thepertinent BRB and BRBF properties are identified inTable 2.

3.2 Mathematical models

The mathematical model of the prototype system isillustrated in Fig. 10, and the parameters are defined inTable 3.

Fig. 6 Comparison between actual and target spectra for

DBE level (1,000 comparison points)

Fig. 7 Calibrated input PSD for DBE target spectrum

Fig. 8 Sample modulated time history for DBE target

spectrum

Table 2 BRB and BRBF properties

Parameter Variable

Core area Ac

Core length Lc

Non-core area Anc

Non-core length Lnc

Core yield force Py

Elastic modulus E

Structure mass m

Brace angle

Fig. 9 Prototype structural system

1.4

1.2

1.0

0.8

0.6

0.4

0.2

00 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Period (s)

ActualTarget

Spectralacceleration(g)

201816141210

86420

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Period (s)

PSDvalue(cm2/s3)

0.4

0.2

0

-0.2

-0.4

-0.60 5 10 15 20 25 30

Period (s)

Acceleration(g)

Work point (WP)

4.57 m

BeamPin

4.57m

Column

Work point (WP)

Non

-core

BRB

core

Non

-core

&&zg

Column



The equation of motion (EOM) for the system,assuming no coupling of the structure and the ground(i.e., the ground serves only as a filter) is:

&& & &&x xP

mz+ + = 2 0 0 cos g (6)

where0 is the modal damping ratio of the structure and

Pis the BRB axial force; and

0

2

=K

m

wp cos (7)

is the elastic natural frequency of the structure in which

Kwpis the elastic stiffness of the BRB between the workpoints (WPs) given by:

KL

A E

L

A E

WPc

c

nc

nc

=+

1

2 (8)

A Bouc-Wen (BW) model (Wen, 1976) was used tomodel the hysteric response of the BRB. Per Black et al.(2004), the BRB axial force is given by:

P K u K u z= + wp wp y

( )1 (9)

where is the ratio of post-yield to pre-yield stiffness,u is the relative displacement between the WPs in thedirection of the BRB (hereafter referred to as BRBWP deformation), u

y is the BRB WP deformation at

initial yielding of the BRB (i.e., uy= P

y/ K

wp), andzis

an auxiliary variable describing the inelastic response,evolved by the following nonlinear differential equation(Black et al., 2004):

u z u z z u z un n

y BW BW& & & &+ + =

1

0 (10)

where BW

and BW

are model parameters that affect

the shape of the hysteretic response, and n is a modelparameter that controls the smoothness of the transitionfrom pre-yield to post-yield behavior.

The BW model parameters for the BRB were takenfrom research by Black et al. (2004) that calibrated theBW model parameters such that the hysteretic behaviorpredicted by the BW model matched test specimenbehavior. Table 4 shows the standard BW modelparameter values selected for this research. Note that

in the BW model is increased artificially by 25%from the value calculated using BRB properties basedon the observation by Black et al. (2004) that this

change resulted in better agreement between predictedand measured behavior.

The final form of the EOM is determined bysubstituting Eq. (9) into Eq. (6) and noting that for smallanglesx = u / cosand setting n= 1:

&& & &&x xm

K x K u z z+ + + + =2 1 00 0

cos

[ cos ( ) ]wp wp y g

(11)

u z x z x z xy& & & &+ ( ) + ( ) ( )= BW BWcos cos cos 0

(12)

3.3 Nonlinear dynamic analysis

Nonlinear dynamic analysis was performed basedon Eqs. (11) and (12) using the Simulink toolboxof Matlab. The input base acceleration time history&&z tg ( ) generated from the Input Module was used asthe input to the BRBF system mathematical model. Thetotal length of the simulation was 30 s, and a Dormand-Prince ordinary differential equation solver algorithm(Dormand and Prince, 1980) was used to perform time-step integration of the governing EOM given by Eqs.

Fig. 10 BRBF mathematical model

Table 3 BRBF mathematical model parameter definitions

Parameter Variable

Ground acceleration &&zg (t)

Relative displacement of mass x(t)

Absolute acceleration of mass &&x ta ( )

System mass m

Mass of the ground mg

Damping coefficient of system c0

Table 4 BRBF mathematical model parameter definitions

BW

BW

n

0.025 0.45 0.55 1

xa

zg

x

m

mg

c0

BRB

Datu

m



(11) and (12). A variable time step was used for time-stepintegration where the time step was varied by Simulinkbased on the dynamical response of the system. While avariable time step was used for integration, a fixed time

step was used to sample values from the integration(such as input values and output values). This time stepwas selected to be 10 times smaller than the naturalfrequency of the system. Parametric studies showed thatthis ratio was adequate for the level of precision requiredto adequately calculate BRB demands.

3.4 BRB deformation calculations

The BRB CPD capacity model requires as input theimposed BRB core deformation history. This is calculatedas the difference between total BRB deformation and the

elastic deformation of the BRB non-core regions:

u u u uP

Kc nc

nc

= = 2 2 (13)

whereKnc

= AncE / L

ncis the elastic stiffness of a non-

core region. Substituting the expression for the braceaxial forcePgiven by Eq. (9) yields:

u uK

K u K u zcnc

wp wp y= + ( )2

1 ( ) (14)

where u is the BRB work-point deformation given

by u x= cos, and x(t) and z(t) are outputs from thenonlinear dynamic simulation.

4 Capacity models and reliability analyses

4.1 BRB CPD capacity models

The goal of the reliability analysis within theassessment framework (as shown in Fig. 2) is tocalculate the probability of brace failure based onBRB CPD capacity models and the structural demandsestimated by nonlinear dynamic simulation. The BRBCPD capacity models developed by Andrews et al.(2009) were used to accomplish this. Specifically,remaining capacity models were used, which predict theremaining CPD capacity of a BRB after it is subjected toan imposed deformation history, where values less thanzero indicate its limit state or failure. The general formof the remaining capacity models is given as (Andrewset al., 2009):

C C C h hi i j jR T U= = +

(15)

where CR

is the remaining (or unused) capacity (RC);C

Tis the total capacity (the capacity of the brace in an

undamaged state); and CU is the used capacity. In this

expression, all components are given in terms of CPD.

CRvaries with the applied deformation history, from a

value of CTat the beginning of the applied deformation

history to a value of 0 when the brace fractures. In theexpression, h = {h

i} are predictive parameters whose

values influence CPD capacity. The models developedby Andrews et al. (2009) include predictive parametersdescribing BRB material properties (yield strength,yield strain, ultimate strength, and ultimate strain), BRBgeometric properties (BRB core length and core area),and deformation history predictive parameters, whichact to describe the imposed deformation history (e.g.,maximum strains, cumulative plastic strains, etc.).= {

i}

are model parameters that are introduced to cause theRC model to most accurately match values from BRBtest results.

The parameters and characterize the epistemic

error in the model, making the model probabilistic innature. is the model parameter that represents thestandard deviation of the model prediction error, and is the standard normal random variable (zero meanand unit variance). Together, the quantity representsthe error in the model. Model parameters and thestandard deviation prediction error are calibrated so asto maximize the probability that the predicted C

Rfrom

the model matches values from test results. This MLEmethod was used to calibrate the parameters based onthe test results of 76 specimens. The following RC modeldeveloped by Andrews et al. (2009) was employed in the

assessment framework:

CA

A

L

LR=

2 21 200 425 0 044

3

.

max

.

max

.

( ) ( )

c

c

c

c

yc..

.

max. .451 46

152 9 1 12

F

F

u

y

c

ult ult

(16)

which uses the following predictive parameters: Ac

isthe cross-sectional area of the BRB core; (A

c)

max is the

largest core cross-sectional area of all BRBs in the test

database used to calibrate the RC models;Lcis the lengthof the yielding core region of the BRB; (L

c)

max is the

maximum core length of all BRBs in the test database;

ycis the yield strain of the BRB core ;F

uis the ultimate

tensile stress of the BRB core (from coupon tests);Fyis

the yield stress of the BRB core (from coupon tests);max

is the maximum absolute ductility demand (both tensileand compressive) throughout the imposed deformationhistory, where a ductility demandis defined as c yc/ ;c is the instantaneous deformation of the BRB core(measured acrossL

c); and yc is the core deformation

at incipient yielding of the core. cis the CPD demand,

which is the summation of all plastic core deformation

( p ) occurring up to a specific deformationincrement, normalized by the yield deformation (i.e.,

c = p / yc). Finally, ult is the ultimate ductility



capacity, which is assumed to be equal to the value ofductility demand at the ultimate tensile strain of thesteel core. This is given by

ult=

uc/

yc, where

uc is

the ultimate tensile strain of the core, assumed to be

35%. The value for the standard deviation of the modelprediction error is not stated explicitly in Eq. (16), butafter calibration was determined to be = 434.

4.2 Reliability analysis

The probability of brace failure (Pf) can be

calculated based on the probabilistic distribution ofrandom variablesxand a subset of their outcome spacethat defines a failure event. The probability of failure iscalculated using the following multi-fold integral:

P ff= ( )x x

d (17)

wheref(x) is the joint probability density function (PDF)ofx, and is the failure domain. The failure domain is defined by use of a limit state functiong(x), i.e.,g(x)0 indicates that xis in . In this research, the limitstate functiong(x) is given by the equation for remainingcapacity (Eq. (16)) in which the vector of randomvariables x is defined by the deformation descriptorparameters (

maxand

c). BRB property parameters (A

c,

Lc,

yc,F

u.F

y, and

ult) were considered deterministic and

not included inxsince the variability in the BRB propertyparameters was much smaller than the variability in thedeformation parameters. The randomness inherent in theexpression is dealt with explicitly as described above.When many random variables are present, the integralgiven by Eq. (17) can be difficult and time consuming tosolve. In this research, the first order reliability method(FORM) (see Der Kiureghian, 2005 for a review) wasused to obtain the integral efficiently with reasonableaccuracy. The steps to perform the reliability analysis,which are described in detail in the following section,are as follows:

(a) Extract deformation descriptor parameters c

and

maxfrom the nonlinear dynamic simulation.

(b) Fit analytical distribution functions of x ={

c,

max} to their empirical distributions and construct a

joint distribution model based on individual distributionsand the correlation coefficients.

(c) Construct the limit state function g(x) (i.e.,remaining capacity function) using Eq. (16).

(d) Given the joint distribution model ofxfrom (b)and g(x) from (c), perform a reliability analysis usingFORM to calculate the probability of brace failure.4.2.1 Deformation descriptor terms

For reliability analyses, the probabilistic distributions

of uncertain deformation descriptors in x (i.e., c and

max) must be known. These terms may be calculated

directly from the imposed BRB core deformation history

c(t) (Andrews et al., 2009) produced as an output from

the nonlinear dynamic simulation. A single value for

each term is produced from each complete simulation(in which the BRBF is subjected to a 30 s groundacceleration history). To find the distribution of eachterm, multiple simulations must be performed to produce

random samples generated according to the probabilisticdistribution of basic random variables. An analysis wasperformed to determine the number of simulations thatmust be run to adequately characterize the distributionsof

c and

max. It was determined that at least 50

simulations were required for the mean and standarddeviation of

cand

maxto converge approximately, and

so 50 simulations were performed for each reliabilityanalysis. These 50 simulations produced 50 parametervalues that composed the empirical distribution for eachterm, and these distributions were then fit with analyticalPDFs, as described below.

4.2.2 Distribution fittingAfter creation of empirical distributions (50

observations) of cand

max, in Step (a), analytical PDFs

were fitted to these distributions. This was accomplishedusing statistical tools in Matlab, which fit specifiedPDFs to the empirical distributions to minimizethe difference between the empirical and analyticaldistributions. After comparison of the empiricaldistributions with various analytical distributionforms (e.g., normal, lognormal, Weibull), it was foundthat lognormal distributions best fit

c and

max. The

correlation coefficients between random variables were

also estimated from the samples to construct a Natafjoint PDF (Liu and Der Kiureghian, 1986) of the randomvariables.4.2.3 Limit state function and first-order reliability

analysisThe limit state function g(x) was defined from the

remaining capacity model given by Eq. (16) such thatg Cx h( )= ( )R , , 0 defined failure of the brace.The FORM was used to calculate an approximatesolution to Eq. (17) to give the probability of bracefailure. The original random variablesxare transformedinto uncorrelated standard normal random variables uthrough a transformation u = T(x). Using a nonlinear

constrained optimization algorithm, a point on the limit-state surface G g( ) ( ( ))u x u= = 0 may be obtained that isnearest to the origin of the u-space. At this design point,the limit-state function is linearized, and the probabilityintegral in Eq. (17) is approximately evaluated by theprobability in the half-space determined by the linearizedlimit-state function:

Pf ( ) (18)

where is the standard normal cumulative distributionfunction, and is the distance from the origin to the

design point, typically referred to as the reliabilityindex. The reliability index is often used as a relativemeasure of the reliability of structural components orsystems (e.g., from Eq. (18), = 0 indicates a 50%probability of failure).



The limit state function defined by Eq. (16) containsepistemic uncertainty due to the capacity model errorin which determines the magnitude of the model errorquantified by the MLE method, and is the standard

normal random variable. Therefore, the uncertaintyin the random variable affects the confidence in thereliability index estimated by the FORM analysis.In order to provide the quantified confidence of theestimated reliability index based on this model error, thefirst order approximation of the mean and variance of

() are obtained as (Der Kiureghian, 2001):

( ) = ( )0 (19)

2 2

=

= =0

2

0

2

(20)

whereand , respectively, denote the mean and standarddeviation of the corresponding random variable; (0)means the reliability index evaluated by FORM with fixed as zero; and the sensitivity of the reliability indexwith respect to is obtained as:

=

1

G

g

( ; )

( ; )

*

*

u

x (21)

wherex* and u*are the design points in the original andstandard normal space, respectively. This sensitivity wasobtained by use of the open-source software FERUM(Der Kiureghian et al., 2006). Assuming () followsa normal distribution with mean () and standarddeviation

, thep-percentile (i.e., the value of a random

variable below whichp% of the observations fall) of thereliability index is obtained as:

p p= + k (22)

where kp is the p-percentile of the standard normalrandom variable. In particular, the following percentiles,which are explained in Table 5, were obtained in thisresearch:

50= (23)

10 1 28=

. (24)

5 1 65=

. (25)

For a more conservative estimate of system reliability,lower p-percentile values would be used, whereas for

a less conservative estimate, higher values would beused.

50 provides an average value of reliability. In

addition, the magnitude of the differences between thereliability indices above is proportional to the value of

. If = 0, then the value of is deterministic. Moreaccurate RC models (those with smaller ) provide theability to avoid unnecessary conservatism in evaluatingbrace reliability. The reliability analysis by FORM wasperformed using FERUM (Der Kiureghian et al., 2006).Three inputs were required to use FERUM:

(a) The limit state functiong(x).(b) Distribution types for the random variables inx

and their parameters.(c) Quantification of the uncertainty in g(x) (i.e.,

model error).The limit state functiong(x) was defined by the RC

model as shown in Eq.(16). Computation of the analyticaldistribution parameters for the two random variables inthe RC model (i.e.

c and

max) was described above. The

uncertainty ing(x) was characterized by the value of associated with the RC model. Using these inputs, theFERUM code calculates values of

50,

10, and

5.

5 Parametric studies and fragility analyses

5.1 Parametric studies

A wide variety of parametric study analyses were

performed to investigate the influences of BRB, BRBF,and seismic loading properties on system reliability.Tables 6 and 7 present parameter values used during theparametric studies. In each parametric study analysis,a large number of reliability analyses (typically 1,000)were performed to investigate the influence of certainparameters on system reliability. The investigated andvaried parameters includedL

c,A

c, andR, whereRis the

response modification coefficient (ASCE, 2005). Rwasdefined for the simple BRBF system studied as:

Rm S

Va= ( )targety

(26)

where Sa( )target is the value of the design responsespectrum at the natural period of the system and V

yis the

base shear at first yield of the BRB (i.e., when the BRBforce is equal to P

y).The ranges of variation for L

cand

Acwere set the same as those of the BRB test database

(Andrews et al., 2009), whereas the range for R wasset based on typical design values. Multiple suites ofparametric studies were created in terms of Sa( )target ,tsim

(simulation duration),Fy, andF

u. Table 8 summarizes

the suite of analyses performed and discussed in thispaper.

The analysis suites were configured to investigatethe effects of different seismic events or sequencesof seismic events and different steel grades (ASTM



A36 and ASTM A572 Grade 50). In Table 8, Hazardrepresents different seismic events and sequences ofseismic events, as follows:

DBE: design basis earthquake (30 s) MCE: maximum considered earthquake (30 s) DBE-MCE: DBE followed by MCE (30 s each for

a total of 60 s)

MCE-MCE: two MCEs in a row (30 s each for atotal of 60 s)

Figures 11 (a) through (f) present50

values versusLc

andAcfor different anlyses conditions withR = 8.

The effects of BRB, BRBF, and seismic loading

properties on the system reliability can be classified bythe behavior of the reliability index with respect to thefollowing parameters:

Core AreaAcand LengthL

cof BRB: In general,

the system reliability varies with bothAcand Lc. AsAcandL

cbecome smaller, the system reliability goes down.

This arises from the lower cumulative ductility capacityobserved for BRBs with smaller core regions (smallerA

cand L

c), which have less material volume to absorb

energy through yielding. Core Yield Strength, F

y: The yield stress has a

significant effect on system reliability, as observed bycomparing the values of

50 for Analysis 1 (A36) and

Analysis 2 (A572 Grade 50). The values from Analysis1 (Fig. 11(a)) are higher than values from Analysis 2(Fig.11(b)). This trend results from the influence of F

y

on the CTterm in the RC model:

CF

F

F

E

F

FT yc

u

y

u

y

y

=

3 451 46 3 45

.

. . 11 46

1 99 1 46

3 45

.

. .

.

=

F F

E

y u (27)

Thus, the RC model predicts that as the core yieldstrength increases, the predicted C

T decreases. This

prediction is in line with test results, which show thatBRBs with higher yield stress tend to have lower CPDcapacity.

Seismic Loading Sequence: The seismic loadingapplied to the BRB has a significant effect on theBRB system reliability. This effect can be observed bycomparing the

50 values for the various analyses. In

general, the system reliability decreases as the seismicloading severity increases. The system reliability,ordered from highest to lowest is approximately DBE >MCE > DBE-DBE > DBE-MCE = MCE-DBE > MCE-MCE, though the differences in the system reliabilitybetween the various seismic loadings vary over thedomain ofA

candL

c.

As defined in Table 5, various reliability indices50

,

10, and

5, can be calculated and used to assess system

performance. To demonstrate the variation in thesereliability indices, consider results for two exampleBRBF systems, which are labeled BRBF 1 and BRBF 2.The properties for these systems are described in Table 9,which shows that the systems were the same except thatthe yielding region of the core for BRBF 2 was half thatof BRBF 1. Values of

50,

10, and

5for various levels

of seismic demand (DBE, MCE, DBE-MCE) are shownin Table 10 for these systems. Logically, for a given

scenario, reliability is lower when the desired confidencelevel is higher. Also, the reliability decreases as thehazard level increases, as expected. Finally, the systemBRBF 1 is more reliable for all cases, as anticipated,since its core length is twice that of system BRBF 2.

Table 5 Definition of reliability indices

ParameterConfidence

level

Probability (%)

thatwill be

observed below

the parameter

Probability (%)

thatwill be

observed above

the parameter

50

50 50 50

10

90 10 90

5

95 5 95

Table 6 BRB parameters for study

Parameter RangeIncrements in

range

Specified

Ac(cm2) 4.06 to 185 10 -

Anc(cm2) - - 2Ac

Lc(cm) 43.8 to 472 10 -

Lnc

(cm) - - (646 Lc)/2

E(GPa) - - 200

Table 7 BRBF parameters for study

Parameter RangeIncrements in

rangeSpecified

R 0 to 10 10 -

0

- - 0.05

(t) - - 0.45te0.167t

(deg) - - 45

Table 8 Suite of analyses performed

Analysis

numberHazard

tsim

(s)

Fy

(MPa)

Fu

(MPa)

1 DBE 30 289 427

2 DBE 30 379 496

3 MCE 30 289 427

4 DBE-DBE 60 289 427

5 DBE-MCE 60 289 427

6 MCE-MCE 60 289 427



5.2 Fragility analysis

The assessment framework created in this researchfacilitates fragility analyses, which provide conditionalprobability of BRB fatigue fracture given ground

shaking intensity. A fragility curve can be createdfor any assumed structure and BRB. In this research,fragility analyses were performed for the two examplestructures described in Table 9. Structural properties ineach analysis were based on average values from the

180

160

140120

100

80

60

40

20

Ac

(cm2)

1 2 3 4

Lc(m)

(a) Analysis 1 (Grade 36, DBE)

180

160

140120

100

80

60

40

20

Ac

(cm2)

1 2 3 4

Lc(m)

(b) Analysis 2 (Grade 50, DBE)

180

160

140

120

100

80

60

40

20

Ac

(cm2)

1 2 3 4

Lc(m)

(c) Analysis 3 (Grade 36, MCE)

180

160

140

120

100

80

60

40

20

Ac

(cm2)

1 2 3 4

Lc(m)

(d) Analysis 4 (Grade 36, DBE-DBE)

180

160

140

120

100

8060

40

20

A

c(cm2)

1 2 3 4

Lc(m)

(e) Analysis 5 (Grade 36, DBE-MCE)

180

160

140

120

100

8060

40

20

A

c(cm2)

1 2 3 4

Lc(m)

(f) Analysis 6 (Grade 36, MCE-MCE)

Fig.11 50

values versusLcandA

cfor different analyses



BRB test database and are the same for each analysis,except that the core length for BRBF 2 was half that ofBRBF 1. To construct the fragility curves, a sequence ofreliability analyses was performed where the intensity ofthe base acceleration history (which may be modulatedby increasing the target elastic spectrum in the InputModule) was systematically increased.

Figure 12 shows the conditional probability ofbrace failure given the peak ground acceleration. Whilethe analyses produce a series of points, log-normalCDFs can easily be fitted to the data (e.g., Nielson andDesRoches, 2007). BRBF 2 (which has a yielding corelength half of analysis BRBF 1) is a more vulnerablesystem than BRBF 1 since the BRB with the shorteryielding region develops larger inelastic deformations

for the same overall brace deformation when comparedto the BRB with longer yielding region. Note that thisrepresents the probability of brace failure for a seismicloading duration of 30 s. Longer duration loadings wouldlikely increase the probability of brace failure due to theincrease in ductility demand. These fragility analyses areone simple example of the many analyses that could beperformed using the assessment framework presented inthis paper. For example, analyses could be conductedto investigate the effects of BRB core area, responsemodification coefficient or seismic loading duration.

6 Summary and conclusions

This research developed an analytical frameworkfor evaluating the risk of fatigue fracture of buckling-restrained braces (BRBs) subjected to seismic loadingsusing a limit state model derived from an experimentalresearch database and demands quantified by stochasticdynamic analysis (Fig. 2). The Input Module generatesrandom input ground acceleration records using powerspectrum density models and modulating envelopefunctions that account for non-stationarity such that therecords match a given target spectrum. The generatedtime records were used as input excitations to thebuckling-restrained brace frame (BRBF) System Model,which was a single-degree-of-freedom lumped-mass

system. Within the BRBF System Model, the BRBbehavior was modeled using a Bouc-Wen hysteresismodel. Nonlinear Dynamic Simulations were performedto obtain BRB core deformation time history records.This study used BRB Remaining Capacity Modelsdeveloped by Andrews et al. (2009) to predict theremaining cumulative plastic ductility (CPD) capacityof the brace based on the BRB core deformation history.Given BRB demand (i.e., core deformation historiesgenerated from the dynamic analyses) and supply(i.e., remaining capacity predicted by the remainingcapacity models), structural reliability analyses were

performed to evaluate the probability of brace failure.The analyses were conducted efficiently using thefirst order reliability method and were facilitatedusing the Matlab open-source code FERUM (DerKiureghian et al., 2006). During the reliability analyses,the epistemic uncertainty in the fatigue capacitypredictions was accounted for explicitly, and, as aresult, the probabilities of brace failure were calculatedin terms of mean probability, 90% confidence levelprobability, and 95% confidence level probability.

Using the tools described above, extensiveParametric Studies were conducted to explore the effectsof the seismic loading, BRB, and BRBF characteristics

on the probability of brace failure. Also, for a given setof BRB and BRBF properties (average values from theBRB test database), Fragility Analyses were performedto predict the conditional probability of brace failuregiven ground shaking intensity parameters. The primary

Table 9 System parameters for BRBF case studies

Parameter BRBF 1 BRBF 2

Ac(cm2) 53.9 53.9

Anc(cm2) 107.7 107.7L

c(m) 2.46 1.23

Lnc

(m) 2 2.62

E(GPa) 200 200

Fy(MPa) 289 289

Fu(MPa) 427 427

Table 10 System reliability indices

System Hazard 50

10

5

BRBF 1 DBE 6.8 3.2 2.2

BRBF 2 DBE 4.3 1.5 0.7

BRBF 1 MCE 4.1 1.4 0.7

BRBF 2 MCE 1.7 0.3 -0.1

BRBF 1 DBE-MCE 1.9 -0.3 -1.0

BRBF 2 DBE-MCE -0.9 -1.7 -1.9

Fig. 12 Fragility analysis results

1.0

0.8

0.6

0.4

0.2

0

Proabilityoffailure

BRBF 1

BRBF 2

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

PGA (g)

Fragility curve



results from this research include the development of theassessment framework and its components, and multipleparametric analyses performed using the frameworkand limit state models of BRBs. The parametric studies

resulted in plots which related the system reliability tothe BRB core area, BRB core length, and the responsemodification coefficient. They were constructed forgiven values of other BRB properties (e.g., steel grade,seismic hazard). These could be used directly in a designscenario to evaluate the reliability of a BRB and BRBFin reference to the potential for fatigue fracture givenBRB, BRBF, and seismic loading properties. In addition,fragility curves relating peak ground acceleration to theprobability of brace failure could be applied in individualor regional loss assessment studies.

While the analyses performed using the framework

are specific to a range of BRB, BRBF, and seismicloading properties, the framework itself is flexibleenough to allow for alterations to account for: (1)different seismic loadings, (2) various BRB and BRBFconfigurations (single or multi-degree-of-freedom), and(3) new CPD limit state models. Future work may usethis basic framework and its flexibility to develop better,more capable modules and run more advanced analyses.As design paradigms continue to shift away fromprescriptive procedures to adaptable performance-baseddesign frameworks, this research may be used as-is orin a more-developed form as a useful tool for reliability

assessment of BRBs and BRBF or other structuralcomponents and systems.

Acknowledgements

Funding for the first author was provided by theDwight David Eisenhower Transportation FellowshipProgram, administered by the National HighwayInstitute, an organization of the Federal HighwayAdministration (FHWA), under FHWA Grant No.DDEGRD-06-X-00408.

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