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Journal of Constructional Steel Research 90 (2013) 209–220
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Journal of Constructional Steel Research
Drift-based and dual-parameter fragility curves for concentrically bracedframes in seismic regions
Dimitrios G. Lignos a,⁎, Emre Karamanci b
a McGill University, 817 Sherbrooke West, Macdonald Engineering Building, Room 278C, Montreal, Quebec H3A 2K6, Canadab McGill University, 817 Sherbrooke West, Macdonald Engineering Building, Room 165, Montreal, Quebec H3A 2K6, Canada
⁎ Corresponding author. Tel.: +1 514 398 2198.E-mail addresses: [email protected] (D.G. Lign
[email protected] (E. Karamanci).
0143-974X/$ – see front matter © 2013 Elsevier Ltd. All rihttp://dx.doi.org/10.1016/j.jcsr.2013.07.034
a b s t r a c t
a r t i c l e i n f oArticle history:Received 7 March 2013Accepted 30 July 2013Available online xxxx
Keywords:Steel bracesDrift-based fragility curvesDamage statesRapid earthquake damage assessmentDual-parameter fragilitiesFracture ductility
This paper discusses the development of drift-based and dual-parameter fragility curves for steel braces as part ofconcentrically braced frames designed in seismic regions. The experimental results from 24 different researchprograms are compiled into a database for this effort. Drift-based fragility curves are developed for three damagestates of steel braces subjected to cyclic loading associated with brace flexural buckling, local buckling and bracestrength loss due to fracture. The effects of material variability, brace cross sectional shape and loading protocolon the drift-based fragility curves are investigated. The effect of global and local slenderness ratios on the fractureductility of various shapes of steel braces is examined through dual-parameter fragility curves that relate thesegeometric ratios with the expected story drift ratios that each of the three pre-described damage states occur.The proposed fragility curves can be employed for rapid assessment of the seismic vulnerability of concentricallybraced frames.
© 2013 Elsevier Ltd. All rights reserved.
1. Introduction
The next generation of performance-based earthquake designand evaluation procedures for new and existing buildings necessi-tate the development of efficient tools for rapid earthquake damageassessment of buildings with various lateral seismic resisting sys-tems. These procedures should evaluate the earthquake perfor-mance of individual buildings based on their unique site, structural,non-structural and occupancy characteristics [1,2]. As part of theseprocedures, fragility curves are employed in order to estimate thedamage of various structural and/or non-structural components ofa building after various intensities of ground motion shaking [3,4].Fragility curves are cumulative distribution functions that provide thevariation of increasing probability of reaching or exceeding a damagestate as an engineering demand parameter such as story drift ratio(SDR) increases. A number of past research studies summarize drift-based fragility curves for structural failures associated with reinforcedconcrete assemblies [5–7], and steel beam-to-column connections [8,9].Similar efforts have been devoted for drift- and acceleration-sensitivenon-structural components [3,5,10]. Aslani and Miranda [5] employedsuch curves for the development of a probabilistic loss estimationmeth-odology to provide quantitative measures of seismic performance ofbuildings in terms of economic losses. More recently, the Applied Tech-nology Council (ATC) has undertaken to transfer this methodology toprofessional practice [1,2]. This has led to the development of drift-
os),
ghts reserved.
based fragility curves that classify earthquake damage for steel plateshear walls and steel braced frames [11,12].
In the aforementioned studies the emphasis was placed on thedevelopment of singe variable fragility curves that relate peak oraverage story drift ratios (SDRs) with pre-defined damage statesthat have been observed through experimental research and fieldobservations in various structural and non-structural components.However, for an accurate representation of earthquake damage ofstructural and non-structural components as part of a building,one should also relate other geometric and material parameters ofthese components with their corresponding damage states. It is gen-erally accepted that global and local slenderness ratios of steelbraces are among the primary geometric parameters that affecttheir pre- and post-buckling behavior during an earthquake [13].Prior work indicated the importance of brace connection detailing(e.g., gusset plate configuration, beam-to-column connection) onthe drift-based fragility curves for CBFs [12]. To the best of ourknowledge, there is no reported study that relates engineering de-mand parameters such as SDRwith geometric parameters or materi-al properties of steel braces in order to quantify the probability ofbeing or exceeding a pre-described damage state after an earth-quake. Furthermore, the effects of aleatory and epistemic uncer-tainties on these fragilities have not been studied in the case ofsteel braced frames. The magnitude of these uncertainties in struc-tural component fragility curves is significant and should be consid-ered for their quality, implementation and updating [14].
This paper first discusses the development of drift-based fragilitycurves that characterize three discrete damage states for differentsteel braces that are commonly used in steel construction practice in
210 D.G. Lignos, E. Karamanci / Journal of Constructional Steel Research 90 (2013) 209–220
North America. These involve square or rectangular Hollow StructuralSteel (HSS) braces, round HSS (pipes), wide flange (W-) and angle (L)braces. Furthermore, a statistical framework has been developed thatincorporates various sources of epistemic uncertainty in the developedfragility curves of various steel brace types. Dual-parameter fragilitycurves are also developed that associate the three discrete damagestates of a steel brace with its global and local slenderness ratios for agiven SDR. These fragilities demonstrate the effectiveness of presentlyemployed compactness limits for highly ductile braces at target perfor-mance levels associated with SDRs. The effect of the loading protocol onthe fracture ductility of round HSS braces is also investigated.
2. Definition of damage states for steel brace components
Major earthquakes in the United States (Northridge, 1994) andJapan [Hyogoken-Nanbu (Kobe), 1995] and extensive experimentalresearch related to the seismic performance of CBFs have demon-strated that their seismic behavior and subsequently their observeddamage states should describe failure modes that depend on vari-ous parameters including (1) the steel brace cross section [13];(2) the global and local slenderness ratios of the steel brace [15]and (3) the gusset plate connections [16–18]. The investigationpresented in this paper is only concerned with three key damagestates associated with the steel brace member. These damage stateswere selected based on damage observations from large-scale ex-periments (see Section 3) and from earthquake reconnaissance re-ports [19–22]. The three damage states are associated with braceflexural buckling, brace local buckling at its mid-length and finallybrace stress loss, i.e., loss of axial force capacity due to fracture atthe mid-length of the brace. These failure modes are shown inFig. 1a to c and are defined in detail in the subsequent sections.Steel braces that fail in a brittle manner at their ends due to prema-ture fracture are not part of this investigation. These failure modesare typically observed in older CBFs designed prior to the AISCcode provisions adopted in the early 2000, ordinary CBFs and CBFswith a seismic reduction factor R = 3 [23]. In addition, damagestates associated with weld tearing of gusset plates and damage to
(a) DS1 (global buckling) (b) DS2 (loca
(d) hysteretic behaviour
Brace Elongation δ
Axi
al L
oad
F
DS3
DS2
DS1
Fig. 1. Definition of damage states for
framing members of the CBFs are not part of the subsequent discus-sion. A reader is referred to [12] in this case.
2.1. DS1 brace flexural buckling
Brace flexural buckling corresponds to the first occurrence of axialforce strength deterioration of a steel brace subjected to compressivestresses. These stresses are typically less than the yield compressivestresses that the steel material is capable of withstanding. During thisdamage state, the out-of-plane brace deformation is typically less thanthe brace depth [13]. The SDR that corresponds to this damage state iseither based on values reported by the investigator conducting the ex-periment or obtained from the measured axial load, F, — brace elonga-tion, δ, hysteretic diagram as shown in Fig. 1d. Fig. 1a illustrates anexample of DS1 based on a recent experiment of rectangular HSS braces[15].
2.2. DS2 brace local buckling
This damage state corresponds to a case that a plastic hinge formsat the midpoint of a steel brace when it is subjected to compressivestresses. The steel brace experiences local buckling at this location(see Fig. 1b). Fig. 1d illustrates the associated definition of DS2 onan axial force, F — brace elongation, δ, hysteretic diagram. In casethat the associated SDR that local buckling occurred during theexperiment was not reported by the investigator, this point wasassumed to be at a corresponding SDR associated with 40% axialforce strength reduction after global buckling occurred in the com-ponent. This value was determined based on experimental reportsthat explicitly reported the SDR that local buckling occurred suchas [15].
2.3. DS3 brace strength loss — loss of force capacity
This damage state corresponds to the case that considerable loss ofthe tensile force capacity of a steel brace has occurred due to ductilefracture at its midpoint. Fig. 1d shows the definition of this damage
l buckling) (c) DS3 (brace fracture)
(e) distribution of SDRs corresponding to DS1 to DS3
0 0.02 0.04 0.06
DS1
DS2
DS3
SDR [rad]
Dam
age
Sta
te, D
Si
steel braces (photos from [15]).
211D.G. Lignos, E. Karamanci / Journal of Constructional Steel Research 90 (2013) 209–220
state on a hysteretic diagram of a steel brace subjected to cyclic loading.This failure mode is triggered by cyclic loading at the point of plastichinge formation of the steel brace as discussed in DS2. The fracture initi-ates due to low cycle fatigue and then propagates by ductile tearingthrough the brace cross-section. Fig. 1c illustrates this phenomenonthat occurred during cyclic testing of an HSS steel brace [15].
3. Experimental database
The experimental results from 281 tests documented in twenty-four different experimental programs were compiled into a consis-tent format in order to express the probability that a steel brace isbeing or exceeding one of the three damage states summarizedabove given a SDR. An extensive summary of these tests includingthe associated SDR that each damage state occurred, their measuredmaterial properties and their global and local slenderness ratios isincluded in [24,25]. In summary, the global slenderness ratio (KL/r)in the plane of buckling of the steel braces involved in the evaluationvaried from 22.4 to 200. The brace length of the specimens L rangedfrom 410 mm to 6230 mm. Their tensile strength AgFy ranged from50 kN to 6430 kN (Ag: gross area, Fy is the expected yield stress of thesteel material). A wide range of steel sections is included in the experi-mental database. These include: 143 rectangular hollow structuralsections (HSS); 51 round HSS (pipes); 50 W- and 37 L-shapes.
Based on the local slenderness b/t (width-to-thickness ratio) of thesteel brace cross sections contained in the experimental dataset, mostof them are classified as Class 1 according to the section classification
(a) square or rectangular HSS
(c) W-shape
0 0.005 0.01 0.0150
0.2
0.4
0.6
0.8
1
SDRDS1
[rad]
Pro
babi
lity
of E
xcee
denc
e
SDRDS1
values
Fitted Lognorm., =0.44
0 0.005 0.01 0.0150
0.2
0.4
0.6
0.8
1
SDRDS1
[rad]
Pro
babi
lity
of E
xcee
denc
e
SDRDS1
values
Fitted Lognorm., =0.59
Fig. 2. Drift-based fragility curves for damage state DS1 for square
limits per CISC [26] and CSA-S16 [27]. Thirty braces are classified asClass 2 based on the same requirements. Following the AISC 360-10[28] requirements, the steel brace classification is slightly different be-cause thirty-three braces are classified as moderate ductility members.These are mostly angle (L) braces. The rest are highly ductile members.
In many cases, there was no information to establish the SDR atwhich all three damage states took place either because the informationfrom the experimental reports did not include sufficient details for iden-tifying the SDR at which the damage states occurred or because thedamage state did not occur (e.g., not all the specimens were testedthrough fracture). Fig. 1e shows the distribution of SDRs correspondingto the three damage states discussed in Section 2 together with theircounted medians. The large variability in the reported SDR levels withthe progression of damage from DS1 to DS3 necessitates the use of aprobabilistic approach for estimating the probability of reaching orexceeding the three discrete damage states discussed above.
4. Fragility curves for steel braces
The steel database thatwas briefly discussed in the previous section isutilized to first develop drift-based fragility curves for performance-based seismic evaluationof CBFs. Theprocess discussedherein to developfragility curves for steel braces is general and has been successfully ap-plied to other structural and non-structural components [5,8,9,29]. Theempirical cumulative frequency distribution function for each damagestate is obtained by sorting in ascending order the corresponding abso-lute peak SDRs at which DS1, DS2 and DS3 are observed from the
(b) round HSS
(d) L-shape
0 0.005 0.01 0.0150
0.2
0.4
0.6
0.8
1
SDRDS1
[rad]
Pro
babi
lity
of E
xcee
denc
e
SDRDS1
values
Fitted Lognorm., =0.51
0 0.005 0.01 0.0150
0.2
0.4
0.6
0.8
1
SDRDS1
[rad]
Pro
babi
lity
of E
xcee
denc
e
SDRDS1
values
Fitted Lognorm., =0.51
or rectangular HSS, round HSS, W- and L-shape steel braces.
(a) square or rectangular HSS (b) round HSS
(c) W-shape (d) L-shape
0 0.01 0.02 0.030
0.2
0.4
0.6
0.8
1
SDRDS2
[rad]
Pro
babi
lity
of E
xcee
denc
e
SDR valuesDS2
Fitted Lognorm., =0.44
0 0.01 0.02 0.030
0.2
0.4
0.6
0.8
1
SDRDS2
[rad]
Pro
babi
lity
of E
xcee
denc
e
SDRDS2
values
Fitted Lognorm., =0.45
0 0.01 0.02 0.030
0.2
0.4
0.6
0.8
1
SDRDS2
[rad]
Pro
babi
lity
of E
xcee
denc
e
SDRDS2
values
Fitted Lognorm., =0.54
0 0.01 0.02 0.030
0.2
0.4
0.6
0.8
1
SDRDS2
[rad]
Pro
babi
lity
of E
xcee
denc
e
SDRDS2
values
Fitted Lognorm., =0.65
Fig. 3. Drift-based fragility curves for damage state DS2 for square or rectangular HSS, round HSS, W- and L-shape steel braces.
212 D.G. Lignos, E. Karamanci / Journal of Constructional Steel Research 90 (2013) 209–220
experiments. The absolute peak SDR values are plotted against the com-puted probability equal to (i − 0.5)/n, where i is the position of the peakSDR and n is the number of the steel brace specimens included in the datasubsets used for each damage state. Figs. 2, 3 and 4 show the empiricalcumulative distributions for rectangular HSS braces, round HSS, W- andL-shape braces for flexural buckling (DS1), local buckling (DS2) and frac-ture (DS3), respectively.
In order to characterize the cumulative empirical distribution foreach damage state, three different probability distributions wereemployed to fit the empirical data including the Wiebull, Gumbeland lognormal distribution. These distributions were determinedbased on the maximum likelihood method [30]. The best fit wasfound by using a lognormal distribution that is fully defined by twostatistical parameters that incorporate the central tendency (μSDR)and dispersion (βlnSDR) of each one of the datasets used as follows,
P DS≥dsijSDR ¼ sdrð Þ ¼ Φln sdrð Þ− ln μSDRð Þ
β lnSDR
� �: ð1Þ
In Eq. (1), P(DS ≥ dsi|SDR = sdr) is the conditional probability ofreaching or exceeding damage state i, at a story drift ratio (sdr) and Φis the standard normal cumulative distribution. In order to verify if thelognormal cumulative distribution adequately characterizes each oneof the empirical distributions shown in Figs. 2 to 4, a Kolmogorov–Smirnov (K–S) goodness-of-fit test was employed [31]. The same fig-ures show a graphical representation of the K–S tests at 5% significancelevel for the fragility curves developed as part of this research. Becauseall the data points from the empirical cumulative distributions lie
between the two dashed lines for all the cases presented in Figs. 2 to4, the hypothesis that the assumed lognormal distribution is acceptableholds true for all three damage states regardless of the steel brace shape.
The statistical parameters that fully define the lognormal cumula-tive distributions for each subset of steel braces and for the three dis-crete damage states are summarized in Table 1. From this table,aleatory and epistemic uncertainties are reflected in the dispersionterm βlnSDR of the fragility curves. Note that aleatory uncertaintiescan be reduced by the analyst but epistemic cannot [32]. From thecorresponding fairly low values of dispersion that are reported inTable 1, the developed fragility curves shown in Figs. 2 to 4 providea rationale way to estimate the likelihood of buckling (global andlocal) and fracture at the midpoint of steel braces if only the peakSDR is used as a single engineering demand parameter (EDP) to char-acterize damage in CBFs. However, as it will be shown later, the useof single EDP parameter fragility curves may be misleading for ulti-mate damage states such as DS3. Note that L-shape braces tend tobuckle globally at smaller SDR values compared to other types ofbraces because they are more susceptible to lateral torsional buck-ling. In general, round HSS and W-shape braces tend to fracture atlarger SDRs compared to square or rectangular HSS braces. Thereason is that the latter exhibit severe local buckling and crimpingat their corners. These phenomena tend to amplify local strains atthe plastic hinge location of the steel brace. On the other hand,local buckling is more gradual at the midpoint of round HSS andW-shape braces and therefore the associated strain gradient at thesame location is less severe compared to square or rectangular HSSbraces. However, for local slenderness ratios equal to the seismic
(a) square or rectangular HSS (b) round HSS
(c) W-shape (d) L-shape
0 0.02 0.04 0.06 0.080
0.2
0.4
0.6
0.8
1
SDRDS3
[rad]
Pro
babi
lity
of E
xcee
denc
e
SDRDS3
values
Fitted Lognorm., =0.48
0 0.02 0.04 0.06 0.080
0.2
0.4
0.6
0.8
1
SDRDS3
[rad]
Pro
babi
lity
of E
xcee
denc
e
SDRDS3
values
Fitted Lognorm., =0.51
0 0.02 0.04 0.06 0.080
0.2
0.4
0.6
0.8
1
SDRDS3
[rad]
Pro
babi
lity
of E
xcee
denc
e
SDRDS3
values
Fitted Lognorm., =0.41
0 0.02 0.04 0.06 0.080
0.2
0.4
0.6
0.8
1
SDRDS3
[rad]
Pro
babi
lity
of E
xcee
denc
e
SDRDS3
values
Fitted Lognorm., =0.52
Fig. 4. Drift-based fragility curves for damage state DS3 for square or rectangular HSS, round HSS, W- and L-shape steel braces.
Table 1Statistical parameters for drift-based fragility curves for steel braces.
Brace shape Parameter Damage state
DS1 DS2 DS3
Square or rectangular HSS μSDR% 0.40 1.02 1.60βlnSDR 0.43 0.44 0.48μSDR% | 10% CI 0.43 1.09 1.73βlnSDR | 10% CI 0.39 0.40 0.43μSDR% | 90% CI 0.37 0.95 1.48βlnSDR | 90% CI 0.48 0.50 0.54Number of specimens (n) 116 112 104
Round HSS μSDR% 0.41 0.96 2.75βlnSDR 0.51 0.45 0.51μSDR% | 10% CI 0.46 1.08 3.25βlnSDR | 10% CI 0.44 0.38 0.41μSDR% | 90% CI 0.36 0.85 2.33βlnSDR | 90% CI 0.62 0.56 0.67Number of specimens (n) 48 37 25
W-shape μSDR% 0.30 1.01 3.10βlnSDR 0.58 0.54 0.41μSDR% | 10% CI 0.32 0.99 3.63βlnSDR | 10% CI 0.50 0.46 0.32μSDR% | 90% CI 0.25 0.76 2.64βlnSDR | 90% CI 0.69 0.66 0.57Number of specimens (n) 56 44 18
Angle (L) shape μSDR% 0.27 0.70 1.43βlnSDR 0.51 0.65 0.52μSDR% | 10% CI 0.32 0.88 2.10βlnSDR | 10% CI 0.41 0.52 0.34μSDR% | 90% CI 0.23 0.56 0.98βlnSDR | 90% CI 0.68 0.87 1.23Number of specimens (n) 23 22 5
213D.G. Lignos, E. Karamanci / Journal of Constructional Steel Research 90 (2013) 209–220
compactness requirements for high ductility members by AISC-341-10 [28], square or rectangular HSS braces have about 40% probabilityof being in DS3 at a 2% SDR (see Section 4.3).
4.1. Incorporating other sources of uncertainty into the fragility curves
The drift-based fragility curves discussed earlier include thespecimen-to-specimen variability only. However, other sources ofepistemic uncertainty should be taken into consideration for theupdating of the fragilities discussed in the previous section. Thefirst source of epistemic uncertainty is related to the finite samplesize n data. The accuracy of any mean estimate derived from a finitesample relies on the central limit theorem of probability. For an infi-nite sample size and for uncorrelated data, the distribution of samplemeans will tend to a normal distribution independent of the form ofthe parent distribution [33,34]. For the case of uncorrelated finitesample size n data the rate at which the sample mean converges tothe parent distribution mean is fast. The sample mean will convergeslower than 1=
ffiffiffin
pif correlations exist in the data.
The second source of uncertainty is related to the fact that differentgeometric testing configurations and brace connection detailinghave been used to compile the steel brace database including differ-ent boundary conditions. The third source of uncertainty is related tothe fact that the corresponding SDRs are associated with peak valuesof a pre-defined loading [5,8]. A steel brace when subjected to aloading protocol can reach any of the three discrete damage statesdiscussed previously during a loading cycle and not necessarily at apre-defined peak. The last source of epistemic uncertainty is associatedwithmaterial variability. The finite sample epistemic uncertainty can be
(a) square or rectangular HSS (b) round HSS
(c) W-shape (d) L-shape
0 0.02 0.04 0.06 0.080
0.2
0.4
0.6
0.8
1
SDRDS3
[rad]
Pro
babi
lity
of E
xcee
denc
e
SDR3 values
Original FragilityEpistemic Uncert. Envel.
0 0.02 0.04 0.06 0.080
0.2
0.4
0.6
0.8
1
SDRDS3
[rad]
Pro
babi
lity
of E
xcee
denc
e
SDR3 values
Original FragilityEpistemic Uncert. Envel.
0 0.02 0.04 0.06 0.080
0.2
0.4
0.6
0.8
1
SDRDS3
[rad]
Pro
babi
lity
of E
xcee
denc
e
SDR3 values
Original FragilityEpistemic Uncert. Envel.
0 0.02 0.04 0.06 0.080
0.2
0.4
0.6
0.8
1
SDRDS3
[rad]
Pro
babi
lity
of E
xcee
denc
e
SDR3 values
Original FragilityEpistemic Uncert. Envel.
Fig. 5.Drift-based fragility curves for damage state DS3 for square or rectangular HSS, round HSS, W- and L-shape steel braces including the envelope of epistemic uncertaintieson the fragility curves.
214 D.G. Lignos, E. Karamanci / Journal of Constructional Steel Research 90 (2013) 209–220
considered for each damage state and steel brace structural shape bycomputing confidence intervals for the standard deviation βln SDR
based on [35],
n−1ð Þ � β2lnSDR
X2a=2;n−1
" #1=2
andn−1ð Þ � β2
lnSDR
X21−a=2;n−1
" #1=2
: ð2Þ
In Eq. (2), Xa/2,n−1 and X1−a/2,n−1 are the inverses of the X2 distribu-tion having n−1 degrees of freedom and a probability of occurrence ofa/2 and (1−a)/2, respectively. All sources of epistemic uncertaintydiscussed in this section were taken into consideration in the centraltendency μSDR of the SDR associated to each damage state by approxi-mating μSDR as discussed in [35],
μSDR−SDRμSDR
2
� �� exp �za=2 �
β lnSDRffiffiffin
p� �
: ð3Þ
In Eq. (3), za/2 is the value in the standard normal distribution suchthat the probability of a random deviation numerically greater thanza/2 is a and n is the data sample size. Table 1 summarizes the effectsof epistemic uncertainty on the mean μ and standard deviations β ofthe lognormal cumulative distributions for each one of the threediscrete damage states and the structural shape used per brace fora 10% and 90% Confidence Interval (CI). Fig. 5 also demonstratesthe influence of the additional sources of epistemic uncertainty on
the fragility curves for DS3 for the four steel brace shapes under in-vestigation. This figure essentially illustrates the envelope of shifted fra-gilities for DS3 that are summarized in Table 1. The shifted fragilities areparticularly useful for computing the probability of being or exceedingDS1,DS2 andDS3 if SDR is used as a single EDP to characterize earthquakedamage in CBFs. For instance, from Fig. 5b there is a 50% probability thatroundHSS braces fracture at theirmidpoint at a SDR between 2.33% and3.25%. Moreover, the probability of exceedence of a damage state suchas fracture can be shifted. From the same figure, the probability of ex-ceeding DS3 varies from 11.8% to 41% when a round HSS steel brace issubjected to a 2% SDR, while including only specimen-to-specimen var-iability indicates that there is only a probability of 26.5% for the sameSDR. This simple example illustrates the value of incorporating episte-mic uncertainties in the development of drift-based fragility curves forsteel braces. Note that the proposed drift-based fragility curves do notconsider separately the effect of beam-column connections, net sectionreinforcement and weld sizes on the cyclic performance of steel bracesas part of CBFs. Roeder et al. [12] demonstrated that when the steelbraces meet the balanced design requirements [18] the average defor-mation capacity associated with DS2 (local buckling) and DS3 (fracture)typically increases compared to steel braces that meet the minimumSCBF requirements per AISC [28]. The reason is that more rotationallyflexible gusset plates reduce the end restraint on steel braces, decreas-ing the curvature at the point of maximum out-of-plane deformationand thus reducing the plastic strain at the critical location and elongat-ing brace fracture life. Thus the upper bound of the shifted fragilities
215D.G. Lignos, E. Karamanci / Journal of Constructional Steel Research 90 (2013) 209–220
presented herein should be considered in case that the connection de-tailing is known to be beneficial on the fracture life of steel bracessubjected to cyclic loading.
4.2. Influence of the material type on the fragility curves
This section discusses the influence of the material type of a steelbrace on the fragility curves for DS1, DS2 and DS3. Angle (L-shape)braces are excluded because the collected ones have been fabricatedby either A36 or SS400 (Japanese Industrial Standard JIS G 3101)structural steel. These two steel types have the same nominalyield stress [fy,nominal = 235 MPa (36 ksi)] and the influence of thematerial type on L-shape braces is negligible. The same processdiscussed in Section 4.1 is employed to develop the drift-based fra-gility curves for different material types for rectangular HSS, roundHSS and W-shape steel braces. Table 2 summarizes the statisticalparameters for steel braces depending on their steel material typein terms of mean μSDR and logarithmic standard deviation βlnSDR.Fig. 6 illustrates the fragility curves for these three steel braceshapes for DS1, DS3, respectively. From Fig. 6a, c and e, it is observedthat in average, the deformation associated with global buckling of asteel brace, is typically increased with the increase of the yield stressof the same brace. For instance, from Fig. 6a, a rectangular HSS bracefabricated by A500 Gr.B steel (fy,nominal = 345 MPa) in average bucklesglobally at a μSDR = 0.39% compared to a rectangular HSS brace that isfabricated by S235 JRH steel (fy,nominal = 235 MPa). The latter buckles
Table 2Statistical parameters for steel braces depending on their steel material type.
Brace shape Material type Parameter Damage state
DS1 DS2 DS3
Square or rectangular HSS A500 Gr.B μSDR% 0.39 0.93 1.52βlnSDR 0.44 0.54 0.55n 54 46 53
CSA-G40.21-350W μSDR% 0.40 1.34 1.40βlnSDR 0.19 0.16 0.20n 19 18 16
A500 Gr.C μSDR% 0.42 0.72 1.72βlnSDR 0.11 0.22 0.31n 7 7 7
S235 JRH μSDR% 0.32 1.02 1.89βlnSDR 0.54 0.29 0.51n 13 13 9
HSS EN1 μSDR% 0.68 1.27 2.07βlnSDR 0.19 0.26 0.20n 9 9 9
Round HSS A500 Gr.B μSDR% 0.56 1.42 2.72βlnSDR 0.18 0.13 0.21n 7 6 4
A500 Gr.C μSDR% 0.28 1.00 naβlnSDR 0.44 0.20 nan 8 6 2
SS400 (SS41) μSDR% 0.43 na naβlnSDR 0.53 na nan 9 1 1
STK400 μSDR% 0.69 1.40 3.08βlnSDR 0.25 0.39 0.32n 9 9 9
A53/A53M μSDR% 0.28 1.20 3.65βlnSDR 0.26 0.44 0.23n 10 10 8
W-shape A992 Gr.50 μSDR% 0.27 1.23 2.79βlnSDR 0.26 0.62 0.33n 11 11 8
A36 μSDR% 0.23 1.40 2.77βlnSDR 0.43 0.66 0.08n 13 13 4
SS400 (SS41) μSDR% 0.30 1.16 2.71βlnSDR 0.64 0.28 0.31n 28 16 3
globally in average at a μSDR = 0.32%. Similar findings hold true forround HSS (see Fig. 6c) andW-shape braces (see Fig. 6e). From the cor-responding mean values for DS2 summarized in Table 2, it is observedthat in average a steel bracewith higher nominal yield stress buckles lo-cally at its midpoint at a larger SDR compared to a steel brace with alower nominal yield stress.
The influence of the steel material on the fragilities for DS3 is theopposite compared to the one observed for DS1 and DS2 regardlessof the steel brace shape. This can be seen in Fig. 6b, d and f. Forinstance, rectangular HSS steel braces (see Fig. 6b) fabricated byA500 Gr.B (fy,nominal = 345 MPa) fracture in average at aboutμSDR = 1.5% compared to rectangular HSS braces fabricated bySS235 JRH steel (fy,nominal = 235 MPa) that fracture in average atabout μSDR = 2.3%. This issue is in part attributed to the fabricationprocess that is necessary to achieve higher nominal yield stresses.The fabrication process of high strength steel typically involvesalloying or quenching and tempering [36]. These processes normal-ly reduce the maximum elongation at fracture and the length of theplastic plateau of a stress strain curve of a steel material. FromFig. 6f, the material influence on the fragility curves for DS3 dimin-ishes for W-shape steel braces. However, it is believed that thisoccurs due to the small sample of available W-shape braces thathave been tested through fracture.
4.3. Influence of the global and local slenderness ratios on the fragility curves
This section summarizes the influence of the global slendernessratio KL/r and local slenderness ratio on the fragility curves for thefour main brace shapes discussed in this paper. Dual-parameterfragility curves have been developed for DS1, DS2 and DS3 to betterexplain the influence of global slenderness ratio on different dam-age states for steel braces. To develop these distributions, a jointlognormal cumulative distribution is employed. The SDR associatedwith each one of the three damage states is assumed to be jointlylognormal with KL/r. In order to construct a joint lognormal cumu-lative distribution, the 2 × 2 covariance matrix needs to be deter-mined. The cross correlation of SDR with respect to KL/r is almostzero based on the data sample that is available regardless of thesteel brace shape [24]. This indicates that SDR and KL/r can be treat-ed as statistically independent variables. Therefore, the joint log-normal distribution can be written as follows based on the totalprobability theorem,
P DS≥dsijSDR ¼ sdr&KL=r ¼ kl=rð Þ
¼ Φln sdrð Þ− ln μSDRð Þ
β ln SDR
� �Φ
ln kl=rð Þ− ln μKL=r
� �β lnKL=r
24
35: ð4Þ
The mean μ and standard deviation β of the two distributions isestimated with the maximum likelihood method. Table 3 summa-rizes the statistical parameters for global slenderness ratios KL/r ofsteel braces depending on their steel shape. In average, all the spec-imens included in the evaluation have similar KL/r ratios regardlessof the steel brace shape. Figs. 7 and 8 show the dual-parameterfragility curves for DS1 and DS3, respectively, for the four basicsteel brace shapes. From these figures and Table 3, the followingfindings are summarized:
• From Fig. 7, in average L-shape braces typically buckle globally at alower SDR for a given KL/r ratio compared to the other three steelbrace shapes. This is attributed to the fact that L-shape braces aresusceptible to lateral torsional buckling [27].
• The effect of KL/r on DS1 (global buckling) is important regardlessof the steel brace shape because a steel brace will typically bucklein flexure rapidly for KL/r ratios larger than 100. Therefore, thelimit of KL/r b 100, which is currently used in design code seismic
(a) square or rectangular HSS, DS1 (b) square of rectangular HSS, DS3
(c) round HSS, DS1 (d) round HSS, DS3
(e) W-shape, DS1 (f) W-shape, DS3
0 0.002 0.004 0.006 0.008 0.010
0.2
0.4
0.6
0.8
1
SDRDS1
[rad]
Pro
babi
lity
of E
xcee
denc
e
A500 Gr.BA500 Gr.CCSA G40.21 350WS235 JRHEN1
0 0.01 0.02 0.03 0.04 0.050
0.2
0.4
0.6
0.8
1
SDRDS3
[rad]
Pro
babi
lity
of E
xcee
denc
e
A500 Gr.BA500 Gr.CCSA G40.21 350WS235 JRHEN1
0 0.005 0.01 0.0150
0.2
0.4
0.6
0.8
1
SDRDS1
[rad]
Pro
babi
lity
of E
xcee
denc
e
STK400A53MA500 Gr.BA500 Gr.CSS400
0 0.01 0.02 0.03 0.04 0.050
0.2
0.4
0.6
0.8
1
SDRDS3
[rad]
Pro
babi
lity
of E
xcee
denc
e
STK 400A53MA500 Gr.B
0 0.005 0.01 0.0150
0.2
0.4
0.6
0.8
1
SDRDS1
[rad]
Pro
babi
lity
of E
xcee
denc
e
SS400A992 Gr.50A36
0 0.01 0.02 0.03 0.04 0.050
0.2
0.4
0.6
0.8
1
SDRDS3
[rad]
Pro
babi
lity
of E
xcee
denc
e
SS400A992 Gr.50A36
Fig. 6. Material influence on the drift-based fragility curves for damage states DS1 and DS3 for square or rectangular HSS, round HSS and W- shape steel braces.
216 D.G. Lignos, E. Karamanci / Journal of Constructional Steel Research 90 (2013) 209–220
provisions [27,28] for highly seismic zones, is satisfactory basedon DS1 fragility curves developed herein (see Fig. 7).
• From Fig. 8, the effect of KL/r on DS3 (fracture) diminishes regard-less of the steel brace shape. This is to be expected because DS3takes place after local buckling occurs at the midpoint of a steelbrace. However, it should be noted that steel braces designedwith large KL/r ratios tend to have increased fracture life [13].The large KL/r ratio typically leads to slight relaxation of localstrains at the cross section. To further investigate the effect of glob-al and local slenderness on the fracture ductility of braces, the axialdisplacement at which the individual steel braces experiencedfracture (noted as Δ) is normalized with the length of the brace(L). The term Δ/L represents an “equivalent” strain at fracture
assuming that the cross section remains uniform. Fig. 9a and b dis-play the effect of global (KL/r) and local slenderness ratios (w/t) onthe normalized axial displacement at fracture (Δ/L) for rectangularHSS braces, respectively. In these figures, the straight line illus-trates the tendency of Δ/Lwith respect to the global and local slen-derness ratios for rectangular HSS braces. The figure suggests thatthe fracture ductility of braces defined as, Δ/L, does not stronglydepend on the global slenderness term, ΚL/r, and that it stronglydepends on the local slendernessw/t. This agrees with recent find-ings from full-scale brace tests [15].
• From Fig. 8c, for a given loading history and a KL/r ratio, W-shapebraces would typically fracture at a larger SDR followed by roundHSS, followed by rectangular HSS and finally followed by L-shape
Table 3Statistical parameters for global and normalized local slenderness ratios of steel bracesdepending on their steel shape.
Brace shape Parameter Damage state
DS1 DS2 DS3
Square or rectangular HSS μKL/r 62.8 62.5 62.3βln(KL/r) 0.33 0.33 0.35μ(w/t)/λHD
a 1.03 1.07 1.07βln(w/t)/λHD
a 0.42 0.41 0.35Round HSS μKL/r 63.6 66.1 68.9
βln(KL/r) 0.46 0.45 0.40μ(D/t)/λHDa 0.97 1.02 1.11βln(D/t)/λHD
a 0.47 0.42 0.33W-shape μKL/r 62.9 64.5 72.3
βln(KL/r) 0.47 0.45 0.37μ(bf/2tf)/λHDa 0.71 0.78 0.95βln(bf/2tf)/λHD
a 0.35 0.35 0.27Angle (L) shape μKL/r 76.8 81.5 97.3
βln(KL/r) 0.45 0.44 0.39μ(b/t)/λHDa 1.41 1.47 1.71βln(b/t)/λHD
a 0.43 0.43 0.18
a λHD is the corresponding seismic compactness limit per AISC-341-10 [28] for highlyductile members in compression.
217D.G. Lignos, E. Karamanci / Journal of Constructional Steel Research 90 (2013) 209–220
braces. This indicates that in average, the fracture life of W-shapesteel braces is larger compared to the steel braces with othershapes. However, W-shape braces typically increase the seismicdemands on the gusset plate welds, which could subsequently re-sult to weld tearing at smaller SDRs. This agrees with observationsfrom prior experimental programs [12,13,15]. In addition, fromTable 3, the average KL/r ratio of W-shape braces is slightly largerthan the one of round and rectangular HSS braces. This would typ-ically lead to slight relaxation of local strains at the cross sectionand therefore increase the fracture life of a steel brace [13].
(a) square or rectangular HSS
(c) W-shape
00.005
0.010.015
050
100150
0
0.5
1
SDRDS1
[rad]KL/r
Pro
babi
lity
of E
xcee
denc
e
00.005
0.010.015
050
100150
0
0.5
1
SDRDS1
[rad]KL/r
Pro
babi
lity
of E
xcee
denc
e
Fig. 7. Dual parameter fragility curves for damage state DS1 for squa
To investigate the influence of local slenderness ratios on thefracture life of various types of steel braces, dual-parameter fragilitycurves are developed for these braces. This time the local slendernessof the corresponding cross section as defined per AISC-341-10 [28] istaken into account together with SDR. Fig. 10 shows joint lognormaldistributions for steel brace shapes that express the probability ofbeing or exceeding DS3 given their local slenderness ratio and SDR.Because the compactness limits per AISC-341-10 [28] depend onthe material properties of the individual steel braces, in this figurethe local slenderness ratio is normalized with respect to the corre-sponding limit λHD for highly ductile members per [28]. Table 3 sum-marizes the normalized local slenderness ratios of steel bracesdepending on their steel shape. The main findings from Fig. 10 andTable 3 are summarized as follows:
• From Fig. 10a, for square or rectangular HSS shapes withw=tNλHD ¼0:55 � ffiffiffiffiffiffiffiffiffiffiffi
E=Fyp
(highly ductile members), the influence of local slen-derness w/t on the probability of fracture of square or rectangularHSS braces becomes significant given any SDR. From Fig. 10a, asquare or rectangular HSS brace with w/t = λHD (AISC-341-10limit) for a SDR = 2.0% has at least 40% chance to fracture at itsmid-length due to low cycle fatigue. From the same figure, a reduc-tion to about three-fourths of the current compactness limit forhighly ductile HSS braces would achieve a probability of fractureless than 10% for a 2% SDR. This implies that single EDP parameterfragility curves may be over conservative for DS3 for square andrectangular HSS braces if we compare the corresponding meanfrom Table 1 (μSDR = 1.6%).
• There is a 10% probability of being or exceeding DS3 for round HSSbraces (see Fig. 10b) designed with a local slenderness ratio D/t =λHD = 0.038 ⋅ E/Fy (AISC-341-10 limit) indicating that the current seis-mic compactness criteria for highly ductile round HSS braces are satis-factory.
(b) round HSS
(d) L-shape
00.005
0.010.015
050
100150
0
0.5
1
SDRDR1
[rad]KL/r
Pro
babi
lity
of E
xcee
denc
e
00.005
0.010.015
050
100150
0
0.5
1
SDRDS1
[rad]KL/r
Pro
babi
lity
of E
xcee
denc
e
re or rectangular HSS, round HSS, W- and L-shape steel braces.
(a) square or rectangular HSS (b) round HSS
(c) W-shape (d) L-shape
0
0.05
050
100150
0
0.5
1
SDRDS3
[rad]KL/r
Pro
babi
lity
of E
xcee
denc
e
0
0.05
050
100150
0
0.5
1
SDRDS3
[rad]KL/r
Pro
babi
lity
of E
xcee
denc
e
0
0.05
050
100150
0
0.5
1
SDRDS3
[rad]KL/r
Pro
babi
lity
of E
xcee
denc
e
0
0.05
050
100150
0
0.5
1
SDRDS3
[rad]KL/r
Pro
babi
lity
of E
xcee
denc
e
Fig. 8. Dual parameter (global slenderness: KL/r and story drift ratio: SDR) fragility curves for damage state DS3 for square or rectangular HSS, round HSS, W- and L-shape steel braces.
218 D.G. Lignos, E. Karamanci / Journal of Constructional Steel Research 90 (2013) 209–220
• The W-shape braces have a 10% probability of being or exceedingDS3 when bf =2t f ¼ λHD ¼ 0:30 � ffiffiffiffiffiffiffiffiffiffiffi
E=Fyp
(AISC-341-10 limit) giventhat the SDR is 2% (see Fig. 10c). For the same local slenderness,the probability of being or exceeding DS3 becomes more than 50%for SDRs N 4%.
• From Fig. 10d, angle (L-shape) braces fracture at much lower SDRscompared to the other three brace shapes because their local slen-derness b=tNλHD ¼ 0:30 � ffiffiffiffiffiffiffiffiffiffiffi
E=Fyp
, which classifies them as moder-ate and low ductility braces per AISC-341-10 [28]. Note also thatL-shape braces are more susceptible to lateral torsional bucklingcompared to the other three types of steel braces.
The dual-parameter fragility curves summarized in this sectionhave the potential to be a very effective tool for preliminary seismic
(a)
20 40 60 80 1000
0.01
0.02
0.03
0.04
0.05
0.06
KL/r
/L
/L= 1.95e 05*KL/r + 0.013
Fig. 9. Effect of geometric parameters on the normalized axial displacement at fractureΔ/L for sq
design of special concentrically braced frames (SCBFs) and ordinaryconcentrically braced frames (OCBFs) in a performance-basedearthquake-engineering context. Because these fragilities associatethe global and local slenderness ratios of steel braces with SDRs, astructural engineer may have a sense how efficient a brace shapeselection could be for a targeted performance level, which is nor-mally associated with an expected SDR range during a design levelor a maximum considered earthquake.
4.4. Influence of the loading protocol on the fragility curves
Themajority of the collected braces in the experimental databasewere subjected to a symmetric loading protocol that typically repre-sents ordinary or far field ground motions. However, few of the
(b)
0 10 20 30 400
0.01
0.02
0.03
0.04
0.05
0.06
w/t
/L
/L= 0.000579* w/t + 0.0203
uare or rectangular HSS steel braces; (a) global slenderness KL/r; (b) local slendernessw/t.
(a) square or rectangular HSS (b) round HSS
(c) W-shape (d) L-shape
0
0.05
0
1
20
0.5
1
SDRDS3
[rad](w/t)/HD
Pro
babi
lity
of E
xcee
denc
e
0
0.05
0
1
20
0.5
1
SDRDS3
[rad](D/t)/HD
Pro
babi
lity
of E
xcee
denc
e
0
0.05
0
1
20
0.5
1
SDRDS3
[rad](bf/2t
f)/
HD
Pro
babi
lity
of E
xcee
denc
e
0
0.05
0
1
20
0.5
1
SDRDS3
[rad](b/t)/HD
Pro
babi
lity
of E
xcee
denc
e
Fig. 10.Dual parameter fragility curves (normalized local slenderness λHD and story drift ratio: SDR) for damage state DS3 for square or rectangular HSS, round HSS,W- and L-shape steelbraces.
219D.G. Lignos, E. Karamanci / Journal of Constructional Steel Research 90 (2013) 209–220
collected roundHSS braceswere subjected to loadingprotocols that rep-resent pulse-like (near-field) ground motions [15]. Based on a muchsmaller data sample, themean μSDR atwhich a roundHSS brace fractures(DS3) when it is subjected to a pulse-like loading protocol is about twotimes larger compared to the corresponding μSDR for round HSS bracessubjected to symmetric type of loading protocols. This is illustrated inFig. 11 that compares the drift-based fragility curves for DS3 for roundHSS steel braces for symmetric and pulse-like loading protocols. Themain reason for such discrepancy is that during a pulse-like loadingprotocol, a steel brace is subjected to fewer reverse loading cycles com-pared to the case that a symmetric loading protocol is employed forcomponent experimentation. This demonstrates that particularly forDS3 the effect of loading history on the fracture performance of steel
0 0.02 0.04 0.06 0.080
0.2
0.4
0.6
0.8
1
SDRDS3
[rad]
Pro
babi
lity
of E
xcee
denc
e
Sym.ProtocolSDR
DS3 values
NF Protocol, =0.22K S bounds
Fig. 11. Effect of loading protocol on thedrift-based fragility curve for damage stateDS3 forround HSS steel braces.
braces is important and necessitates further investigation if fragilitycurves are employed as tools for rapid estimation of earthquake damagefor CBFs.
5. Conclusions
This paper presents single and dual-parameter fragility curvesthat permit the rapid estimation of damage in four different typesof steel braces as part of concentrically braced frames (CBFs).These types are square or rectangular HSS, round HSS, W-shapeand angle (L) braces. The fragility curves express the probability ofbeing or exceeding three discrete damage states observed in steelbraces namely, global buckling, local buckling and brace strengthloss due to ductile fracture at the brace's midpoint. For all threedamage states it was statistically verified that the cumulative distri-bution function to describe themmay be assumed to be lognormallydistributed regardless of the steel brace shape. For a given loadinghistory, a W-shape brace would fracture with a 50% chance atabout 3% SDR followed by round HSS (μSDR = 2.75%), followed byrectangular HSS (μSDR = 1.6%) and finally followed by angle braces(μSDR = 1.4%). Square or rectangular HSS braces exhibit severe localbuckling and crimping at their corners. These phenomena tend toamplify local strains at the plastic hinge location of an HSS steelbrace. However, when five different sources of epistemic uncer-tainties are incorporated in the single parameter fragilities, the cor-responding mean story drift ratio (SDR) that a square or rectangularHSS brace may fracture is 1.7% with a 90% confidence.
In case that the material type of a steel brace as part of CBFs isavailable, a more refined set of drift-based fragility curves was de-veloped for square or rectangular HSS, round HSS and W-shapebraces. In average, the SDR associated with global buckling of asteel brace, is typically increased with the increase of the yieldstress of the same brace. However, the fracture life of steel braces
220 D.G. Lignos, E. Karamanci / Journal of Constructional Steel Research 90 (2013) 209–220
fabricated with high strength steel is smaller than the correspond-ing one of low yield ratio steel braces. This is in part attributed tothe fabrication process of high strength steel because it involvesalloying or quenching and tempering.
Based on dual-parameter fragility curves for steel braces thatassociate global and local slenderness ratios of steel braces withSDRs to express the probability of being or exceeding in a discretedamage state, it was found that:
• The effect of KL/r on global buckling of steel braces is important re-gardless of the steel brace shape. A steel brace will buckle in flexurerapidly for KL/r N 100; therefore, the corresponding limit employedby current seismic requirements for special concentrically bracedframes is satisfactory based on DS1 fragility curves developed herein.
• The effect of the global slenderness ratio KL/r on fracture ductilityof a steel brace diminishes regardless of the steel brace shape.The local slenderness ratio of the steel brace seems to primary con-trol the damage state associated with brace strength loss of thetensile carrying capacity. However, slender braces tend to relaxlocal strains at the brace cross section and therefore, their fracturelife tends to increase.
• Currently employed seismic compactness requirements per AISC-341-10 for highly ductile round HSS and W-shape steel braces aresatisfactory because at a targeted 2% SDR, which represents theseismic demand in terms of SDRs for a CBF during a design levelearthquake, the probability of exceeding the fracture damagestate is less than 10%. The corresponding value for square or rect-angular HSS braces is about 40%. A reduction to about three-fourths of the current compactness limit for highly ductile rectan-gular HSS braces would achieve a probability of fracture less than10% for a 2% SDR. This implies that single EDP parameter fragilitycurves (drift-based only) may be over conservative for damagestates associated with fracture of steel braces.
Based on a small subset of round HSS steel braces subjected to anear-field loading protocol, the associated mean μSDR at which around HSS brace fractures is about two times larger compared tothe corresponding μSDR for round HSS braces subjected to a symmet-ric loading protocol. A steel brace subjected to a symmetric loadingprotocol experiences many more inelastic cycles prior to fracturecompared to the case that a near-field protocol is employed. There-fore, the effect of cumulative damage on the fracture ductility of asteel brace is detrimental in the first case. This finding requires fur-ther experimental testing of various steel brace shapes subjected todifferent types of loading protocols that better represent earthquakeshaking. It should also be clearly stated that connection detailing(e.g., gusset plate details, beam-to-column connections, weld de-tails) plays a significant role on the steel brace performance, as partof CBFs. This effect is not discussed as part of this paper. The readeris referred to [12] in this case.
Acknowledgments
This study is based on work supported by the National Science andEngineering Research Council of Canada (NSERC). The McGill SummerUndergraduate Research in Engineering (S.U.R.E) program providedpartial funding for Mr. Gabriel Martin, who contributed to the steelbrace database development. This support is gratefully acknowledged.The authors would like to sincerely thank Prof. Ben Fell (SacramentoState), Dr. Patzi Uriz (Exponent Failure Analysis) and Prof. StevenMahin (University of California at Berkeley) for their willingness toshare their experimental data in order to assemble part of the steelbrace database discussed in the paper. Any opinions, findings, and con-clusions or recommendations expressed in this paper are those of theauthors and do not necessarily reflect the views of sponsors.
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