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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
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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
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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( )
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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
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No. 3 Blake M. Andrews et al.: Assessment of buckling-restrained
braced frame reliability 377
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
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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
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No. 3 Blake M. Andrews et al.: Assessment of buckling-restrained
braced frame reliability 379
(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
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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).
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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
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382 EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION Vol.8
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
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braced frame reliability 383
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
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384 EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION Vol.8
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
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braced frame reliability 385
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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