1 INTRODUCTION

Recent advancements in the evaluation of frame buildings subjected to earthquake loading have fo-cused on Performance-Based Earthquake Engineer-ing (PBEE) as discussed in (FEMA, 2012a, b). This probabilistic framework necessitates the use of reli-able computer models that are able to predict the seismic behavior of a building from the onset of damage through collapse. It is recognized that the effect of the associated modeling uncertainty on the structural response needs to be quantified.

As part of the same PBEE methodology, fragility curves are used to estimate the probability of a building and its structural components sustaining a certain level of damage as a function of engineering demand parameters, such as story drift ratios and ab-solute floor accelerations. It is understood that the development of structural component databases is essential for (1) the reliable representation of struc-tural component models that simulate the hysteretic response of various building structural components from the onset of damage through their complete loss of resistance; (2) the quantification of modeling uncertainties and their effect on structural perfor-mance and (3) the development of fragility curves for structural components. These curves allow a rap-id damage assessment of frame buildings subjected to earthquake loading. The same curves may also be used to facilitate loss estimation methodologies of buildings or urban areas after an earthquake (FEMA, 2012a, b, Ramirez et al., 2012a, Ramirez and Miranda, 2012).

Aiming to achieve such an outlook, this paper discusses the development of structural component

databases that include extensive statistical infor-mation for various properties of different types of structural components including (a) steel W-shape beams and (b) steel braces. The first author has de-veloped reinforced concrete (RC) beam and tubular steel column databases (Lignos and Krawinkler 2012); however, these two databases are not dis-cussed here in due to brevity. The structural compo-nent databases serve as a source for the develop-ment, further refinement and extensive calibration of state-of-the-art inelastic cyclic models, which are able to simulate phenomena associated with strength and stiffness deterioration of various structural com-ponents as part of lateral resisting structural systems subjected to earthquake loading. The same databases are employed for the development of drift-based and dual-parameter cumulative distribution functions that describe well-known discrete damage states that these components may undergo through during an earthquake.

The utilization of these databases is demonstrated through an example of the potential use of drift-based fragility curves for rapid damage assessment of instrumented steel buildings in line with the new generation of Performance-Based Earthquake De-sign and evaluation techniques for new and existing buildings. In addition, structural performance data-base are employed to assess the seismic behavior of concentrically braced frames by performing multiple nonlinear response history analyses for a set of ground motion records, the latter scaled at increasing intensity levels. Emphasis is placed on the collapse quantification of the reserved capacity of such lat-eral resisting systems.

Structural Component Databases for Performance-Based Earthquake Engineering

D.G. Lignos, E. Karamanci, & N. Al-Shawwa Department of Civil Engineering & Applied Mechanics, McGill University, Montreal, Canada

ABSTRACT: This paper discusses the development of structural performance databases that can serve for the development of modeling recommendations of component models that simulate the complex hysteretic re-sponse of structural members. These databases serve for the development of drift-based and dual-parameter fragility curves for structural components. Emphasis in placed on a steel W-shape beams and steel brace data-bases that contain data from more than 300 experiments each. The utilization of the two databases is illustrat-ed through two case studies. The first one demonstrates the concept of rapid earthquake damage assessment of instrumented steel buildings. The second one illustrates how sidesway collapse due to seismic excitations can be traced in the case of special concentrically braced frames.

2 STRUCTURAL COMPONENT DATABASES Four structural component databases have been de-veloped during the past six years for modeling of structural components as part of frame buildings: These databases include detailed information in the form of metadata and digitized results for (a) W-section steel beams; (b) reinforced concrete (RC) beams; (c) tubular hollow square columns; and (d) steel braces. The first three databases have been ex-tensively documented in (Lignos and Krawinkler, 2011, 2012); The steel brace database was recently developed. This paper summarizes information for the W-section steel beam and steel brace databases.

2.1 Steel W-sections Database The steel W-sections database includes 304 tests from a wide range of fully restrained beam-to-column connections, including beams with reduced sections (RBS); welded unreinforced flange bolted web and welded web (WUF-B, WUF-W) connec-tions; hunches and cover plate connections. Eighty specimens have beams with RBS. For individual connection types (other-than-RBS) the number of tests is not sufficient to establish trends between crit-ical geometric and material parameters that control flexural plastic hinging of steel beams and analytical model parameters that can be used to assess the seismic performance of steel frame buildings. For this reason, these connections are lumped together in a group called beams “other-than-RBS”. Lignos and Krawinkler (2011) calibrated extensively a refined analytical model that simulates the steel component deterioration in strength and stiffness (Ibarra et al., 2005, Lignos and Krawinkler, 2011). They proposed empirical equations to model component deteriora-tion in support of collapse assessment of steel mo-ment frames. These equations have been adopted by PEER/ATC (2010) for modeling and acceptance cri-teria of tall buildings in United States.

More recently, (Elkady and Lignos, 2013) pro-posed modeling recommendations to simulate the assymetric deterioration in strength and stiffness for composite beams with RBS. Based on these rec-ommendations, they assessed the current strong col-umn/weak beam requirements by (AISC, 2010a) for steel special moment frames (SMFs). Figure 1 shows an example of the hysteretic response of the modified Ibarra-Medina-Krawinkler (IMK) deterio-ration model, calibrated with available experimental data from full-scale tests on composite beam-to-column connections that have been conducted by Ricles et al. (2004a). A user needs to define a back-bone curve of the composite steel beam based on its elastic stiffness, Ke; the effective yield flexural strength, My*, that inherently accounts for cyclic hardening; the capping-to-yield flexural ratio, Mc/My*; the pre- and post-capping plastic rotation capacities, θp and θpc, respectively; the steel beam

residual flexural strength Mr and the ultimate rota-tion capacity θu associated with ductile tearing. Cy-clic deterioration in strength and stiffness is simulat-ed with the reference energy dissipation capacity, Λ, of the steel beam. Note that in the case of composite steel beams, the aforementioned parameters become assymetric due to the presence of slab as discussed in Elkady and Lignos (2013).

−0.05 0 0.05−6

−4

−2

0

2

4

6 x 104

Chord Rotation θ [rad]M

omen

t [kip

-in]

θp

M*y+M*c+

M*c - M*y -

Ke

+

Figure 1. Example of calibration of deterioration parameters of the modified IMK model to simulate the hysteretic response of composite steel beams with RBS (data from Ricles et al. 2004).

The deterioration model parameters for steel

beams can be explicitly described by probabilistic distributions and correlated to factors derived from experimental data. In particular, Table 1 summarizes the median (exponent of the average of the natural logarithms of the data) and standard deviation of the natural logarithms of the data for beams with RBS and “other-than-RBS”.

Table 1. Statistics of random variables for beams with RBS and other-than-RBS.

Connection Type µ θp β θp µ θpc β θpc µ Λ β Λ

RBS 0.025 0.32 0.230 0.32 1.140 0.34 Other-than-

RBS 0.025 0.43 0.160 0.41 1.000 0.43

Table 2 summarizes the correlation coefficients

between the deterioration model parameters that are typically used to fully define a moment plastic hinge of a steel moment resisting frame. Kazantzi et al. (2013) employed these values in order to investigate the model parameter uncertainty effects on the seis-mic performance of a steel SMF. They concluded that for code-compliant low-rise steel buildings that employ steel SMFs as their primary lateral load re-sisting system, modeling uncertainties can be safely ignored. The reader is referred to Lignos and Krawinkler (2012) for more information related to statistical information for modeling parameters af-fecting cyclic deterioration in strength and stiffness

of steel and reinforced concrete beams and steel hol-low square columns.

Table 2. Random variables correlation coefficients for beams with RBS.

Beams θp θpc Λ My/My,p Mc/My 1 0.54 0.65 0 0 0.54 1 0.63 0 0 0.65 0.63 1 0 0 0 0 0 1 0 0 0 0 0 1

Another aspect of the steel W-sections database is the development of fragility curves that describe the probability of reaching or exceeding a pre-described damage state for beams with RBS and pre-Northridge fully-restrained beam-to-column connec-tions, respectively (Lignos et al., 2010, Ramirez et al., 2012b). Figures 2a and 2b show the probability of reaching fracture given the story drift ratio for typical RBS beam-to-column connections and typi-cal pre-Northridge WUF-B beam-to-column connec-tions. A comparison of the two figures indicates that a typical pre-Northridge beam-to-column connection would fracture in average at about 1.5% story drift ratio (SDR), which is much lower than the equiva-lent average SDR at fracture for a typical beam with RBS (see Figure 2a). The main reasons for this are attributed to:

the steel connection detailing practice that of-ten resulted in stress concentrations in typical pre-Northridge beam-to-column connections,

the deposition of low toughness weld metal and large defects in critical beam flange-to-column joints,

excessive panel zone shear inelastic distor-tions that caused fracture to occur at the bot-tom flanges of steel beams.

The aforementioned reasons agree with earlier findings from the 1994 Northridge earthquake (Deierlein, 1998, Mahin et al., 1998) and recent RBS connection testing (Ricles et al., 2004b).

Al-Shawwa and Lignos (2013) utilized the fra-gility curves of typical pre-Northridge beam-to-column connections and develop a methodology for rapid earthquake damage assessment of instrument-ed steel buildings that computes the probability of structural connection damage along the height of a steel building within minutes after an earthquake. Findings from this study are summarized in Section 3.1 of the present paper.

2.2 Steel Brace Database

This section discusses the development of a steel brace database for modeling the hysteretic response of rectangular hollow square sections (HSS), round HSS and W-shape braces. A total of 317 steel braces of various geometries were collected for this pur-

pose. These braces have been fabricated from 14 dif-ferent material grades that are commonly used in the U.S., Canada, Japan and Europe (Lignos et al. 2012). More details regarding the steel brace data-base are summarized in Lignos et al. (2012) and Karamanci and Lignos (2013).

0.00

0.20

0.40

0.60

0.80

1.00

0.0 3.0 6.0 9.0 12.0SDR(%)

P[DS|SDR]

Lognormal fit

K-S test, 5%Significance

0.00

0.20

0.40

0.60

0.80

1.00

0.0 1.0 2.0 3.0 4.0

Lognormal fit

90% Confid. Intervals

P[DS|SDR]

(b)

(a)

SDR(%) Figure 2. Drift-based fragility curves for fracture of typical ful-ly restrained beam-to-column connections; (a) RBS; (b) pre-Northridge WUF-B.

The full hysteretic response of the collected steel

braces in terms of axial load – axial displacement was digitized with the program called digitizer (Lignos and Krawinkler, 2011, 2012). The digitized data for various types of steel braces was employed to calibrate the hysteretic response of a state-of-the-art fiber-based steel brace model developed by Uriz et al. (2008). An example of such calibrations is shown in Figure 3, which shows a comparison of the predicted versus experimental axial load versus brace elongation of a round HSS brace tested by Fell et al. (2009). Fracture due to low cycle fatigue is simulated with a fatigue material model, which is based on a linear strain accumulation rule based on the Coffin-Manson relationship that relates a materi-al parameter εo that represents the strain amplitude at which one complete cycle of an undamaged material will cause fracture.

Based on extensive calibrations of the steel brace component model discussed earlier, multivariate re-gression equations were developed to simulate cy-clic buckling and fracture of rectangular HSS, round HSS and W-shape steel braces as discussed in Lignos and Karamanci (2013). These predictive equations relate the material parameter εo with geo-metric and material properties of a steel brace.

Brace Elongation [in]-4.0 -2.0 0.0 2.0 4.0

Axi

al L

oad

[kip

s]

0.0

110

220

440

-220

-440

SimulationExperimental Data

Figure 3. Comparison of predicted versus experimental hyster-etic response of a round HSS brace (data from Fell et al. 2009).

The predictive equation that best describes the dataset of rectangular HSS braces in terms of εo is,

!0 = 0.291kLr

!

"#

$

%&'0.484 w

t!

"#

$

%&'0.613 E

Fy

!

"##

$

%&&

0.3

(1)

in Equation (1), kL/r is the global slenderness ratio of the rectangular HSS brace in the direction that global buckling occurs; w/t is the width to thickness ratio of the steel brace; Fy and E are the yield stress and Young’s modulus of the steel material, respec-tively. Equation (1) is valid for the following range of parameters, 27 ! kL/r ! 85; 4.20 ! w/t ! 30.40; 223 ! Fy ! 532 MPa.

The predictive equation that best describes the dataset of round HSS braces in terms of εo is,

!0 = 0.748kLr

!

"#

$

%&'0.399 D

t!

"#

$

%&'0.628 E

Fy

!

"##

$

%&&

0.2

(2)

in Equation (2), D/t is the depth-to-thickness ratio of the round HSS steel brace. This equation is valid for the following range of predictive parameters, 29 ! kL/r ! 128; 12.75 ! D/t ! 39.91; 326 ! Fy ! 521 MPa.

The predictive equation that best describes the dataset of W-shape braces in terms of εo is,

!0 = 0.0391kLr

!

"#

$

%&'0.234 bf

2t f

!

"##

$

%&&

'0.169htw

!

"#

$

%&

'0.065EFy

!

"##

$

%&&

0.351

(3)

in Equation (3), the local cross section slenderness ratios, bf/2tf and h/tw are defined based on the AISC-360-10 (AISC, 2010b) seismic provisions. The same equation is valid for the following range of parame-ters, 39 ! kL/r ! 153; 4.19 ! bf/2tf ! 10.20; 7.99 ! h/tw ! 49.40; 284 ! Fy ! 414 MPa. Further de-tails regarding the multivariate regression equations are summarized in Karamanci and Lignos (2013) in-cluding (1) modeling recommendations for the fiber cross section per steel brace shape; (2) the number of integration points used within the brace element; and (3) the initial camber imperfections to trace flexural buckling of the steel brace member. In aver-age, round HSS and W-shape steel braces tend to fracture at larger SDRs compared to square or rec-tangular HSS steel braces since the latter exhibit se-vere local buckling and crimpling at their corners.

The steel brace database was utilized to develop drift-based fragility curves for steel braces that ex-press the probability of reaching or exceeding global buckling, local buckling and fracture during low cy-cle fatigue at their mid-length. Figures 4a and 4b show examples of such curves for rectangular HSS steel braces for global buckling and fracture, respec-tively.

In addition, dual-parameter fragility curves were developed for the same braces. These curves relate the influence of local slenderness ratios on the frac-ture life of various types of steel braces given a SDR. Figure 4c shows an example of these curves for rectangular HSS steel braces. From this figure, the local slenderness of rectangular HSS braces is normalized with the corresponding compactness limit λHD for highly ductile members per AISC-341-10 (AISC, 2010a). This figure indicates that for square or rectangular HSS shapes with w/t > 0.55 (E/Fy)1/2 the influence of local slenderness on the probability of fracture of rectangular HSS braces be-comes significant at a given SDR. From the same figure, a rectangular HSS brace with w/t = 0.55 (E/Fy)1/2 has at least 40% chance to fracture at its mid-length due to low cycle fatigue for SDR = 2.0%. Comparing these values with Figure 4b (drift-based fragility curve for fracture) we observe that it may be over conservative for ultimate damage states such as fracture to ignore the geometric parameters of steel braces in order to compute the probability of reaching of exceeding the associated fracture dam-age state. Similar findings were obtained for other steel brace types such as round HSS and W-shape braces. Karamanci and Lignos (2013) also devel-oped drift-based fragility curves for various steel brace shapes and they quantified the effect of mate-rial type on these fragilities.

0 0.005 0.01 0.0150

0.2

0.4

0.6

0.8

1

SDRGlobal Buckling [rad]

SDRDS1 valuesFitted Lognorm., β=0.44

P[DS|SDR]

0 0.02 0.04 0.06 0.080

0.2

0.4

0.6

0.8

1

SDRFracture [rad]

SDRDS3 valuesFitted Lognorm., β=0.48

P[DS|SDR]

0

0.05

01

20

0.5

1

SDRFracture [rad](w/t)/λHD

P[DS|SDR,(w/t)/λ ]HD

(a)

(b)

(c) Figure 4. Drift-based and dual-parameter fragility curves for square or rectangular HSS steel braces; (a) global buckling; (b) fracture; (c) dual-parameter fragility curve for fracture.

3 UTILIZATION OF STRUCTURAL PERFORMANCE DATABASES IN PBEE

3.1 Rapid Estimation of Earthquake Damage in Instrumented Steel Buildings

The fragility curves that were developed for pre-described damage states for typical fully-restrained beam-to-column connections are utilized for rapid estimation of earthquake damage in instrumented

steel buildings. This section demonstrates this concept by using a 17-story steel building, which was instrumented and experienced the 1994 Northridge earthquake in California. Further information can be retrieved from (Chi et al., 1998). Figure 5 shows the elevation view of the steel building, which consists of perimeter steel moment resisting frames (MRFs) in the North-South (NS) and East-West (EW) loading directions.

Figure 5. Geometry and member sizes of the 17-story steel building in Woodland Hills, Canoga Avenue (figure from Chi et al. 1998). Based on a recently developed system identification approach, we are able to predict the expect story drift ratio profile of the steel building in its loading directions. This is achieved by using a continuous model (Miranda and Akkar, 2006), which is calibrated with the recorded absolute acceleration and relative displacement histories at the instrumented floors of the same bulding. This is achieved with an optimization approach developed by Al-Shawwa and Lignos (2013) that within minutes is able to conduct the system identification of the continuous model. Figure 6a shows the computed peak SDR profile along the height of the steel building in the NS loading direction. This profile matches well with the computed profile based on Chi et al. (1998). The peak SDR profile is used to compute the expected probability of reaching or exceeding connection fracture along the height of the same building. This is shown in Figure 6b. In this figure, the probability of fracture is computed based on drift-based fragility curves for pre-Northridge beam-to-column connections developed by Ramirez et al. (2012b). From Figure 6b it is evident that there is a high chance to have connection fractures between stories 7 to 14 of the 17-story steel building. This agrees with damage inspection reports that summarize the extent of structural damage along the height of the 17-story steel building during the 1994 Northridge

earthquake (Chi et al., 1998). Similarly, the estimated structural damage in the EW loading direction was minimal based on the tool for rapid estimation of earthquake damage for instrumented steel buildings. This also agrees with damage reports for the same building.

0 0.01 0.02 0.03 0.040.00.10.20.30.40.50.60.70.80.91.0x=z/H

SDR [rad]0 0.25 0.5 0.75 10.0

0.10.20.30.40.50.60.70.80.91.0

Prob. of Fracture

x=z/H

(a) (b) Figure 6. Distribution of peak SDRs and probability of connec-tion fracture along the height of the 17-story steel building in Woodland, Canoga Avenue.

3.2 Collapse Assessment of Steel Structures The steel component databases that were summa-rized in Section 2 can be utilized to facilitate nonlin-ear response history analysis of steel frame struc-tures through collapse. Lignos et al. (2011) employed the modeling recommendations from the steel beam-to-column connection database (see Sec-tion 2.1) for collapse assessment of steel Special Moment Frames (SMFs). They successfully predict-ed numerically the dynamic response of a scale model of a 4-story SMF that was tested through col-lapse at the NEES facility at the University at Buffa-lo (SUNY) in 2007. More recently, Lignos et al. (2013) successfully reproduced numerically the dy-namic response through collapse of a 4-story steel building with SMFs designed in accordance with the Japanese practice. In other salient analytical studies (NIST, 2010), the collapse potential of a range of ar-chetype steel buildings with SMFs was quantified with the FEMA P695 methodology (FEMA, 2009).

The modeling recommendations for steel braces summarized in Section 2.2 can be employed to trace the dynamic stability of special concentrically braced frames (SCBFs) designed in urban areas with high seismicity. In this section, the predictive equa-tions for inelastic buckling and fracture of rectangu-

lar HSS steel braces are utilized to simulate the dy-namic stability of a 2-story SCBF. Uriz (2005) test-ed quasi-statically the same frame through complete failure at the University of California, Berkeley. Figure 7 shows the geometry and member sizes of the 2-story SCBF.

ColumnW10x45

BeamW24x117

2,74

3mm

2,74

3mm

6,096mm

ReactionBeam

PL 22(A572 Gr.50)

PL 22(A572 Gr.50)

Lateral support

Loading beam

305mm

Figure 7. Geometry and member sizes of the 2-story SCBF.

A 2-dimensional model of the 2-story SCBF was developed in OpenSees (Mckenna, 1997). Strength and stiffness deterioration of various structural com-ponents of the SCBF are simulated with the modi-fied IMK deterioration model. The parameters of this model are defined based on regression equations by Lignos and Krawinkler (2011). The out-of-plane flexibility and finite yield flexural strength of the gusset plates was simulated with a model proposed by Hsiao et al. (2012). The hysteretic response of the gusset plate beam-to-column shear connections was simulated with the pinching version of the modified IMK model (Lignos and Krawinkler, 2012). The modeling parameters of this model were derived based on past experiments by (Liu and Astaneh-Asl, 2000) and Stoakes and Fahnestock (2012).

The same frame is subjected to a sweep of 40 ground motions compiled by Medina and Krawinkler (2003) that characterize the seismic risk of urban California. The 5% spectral acceleration at the first mode period of the 2-story SCBF is used as the intensity measure (IM) to scale the set of 40 ground motions. Incremental dynamic analysis (IDA) is employed for this purpose (Vamvatsikos and Cornell, 2002) till sidesway collapse instability is traced for each one of the ground motions. Col-lapse is traced explicitly as defined from measured experimental data from small and full-scale shake table collapse tests of steel frame buildings (Suita et al., 2008, Lignos et al., 2011, Lignos et al., 2013).

Figure 8a shows the IDA curves for the set of 40 ground motions in terms of IM versus maximum sto-ry drift ratios (SDR) for the 2-story SCBF. This data is utilized to compute the collapse fragility curve of the same frame. This curve is shown in Figure 8b. The 2-story SCBF has a median collapse capacity of

1.17g and a logarithmic standard deviation of 0.43 for the set of 40 ground motions that were used.

0 0.02 0.04 0.06 0.080

0.5

1

1.5

2

2.5

3

Maximum Story Drift Ratio, SDR [rad]

S a(T1,5

%) [

g]

(a)

0 1 2 3 40

0.2

0.4

0.6

0.8

1

Sa(T1,5%) [g]

Prob

abili

ty o

f Col

laps

e

(b) Figure 8. IDA curves and collapse fragility curve of the 2-story SCBF.

4 CONCLUSIONS

This paper discusses the development and utilization of structural component databases in the context of Performance-Based Earthquake Engineering (PBEE). Emphasis is placed on the steel W-shape beams and steel brace databases. In both cases, modeling recommendations have been developed for steel beams and various types of steel braces that are commonly used in steel construction of special mo-ment frames (SMFs) and special concentrically braced frames (SCBFs) designed in seismic urban areas. Furthermore, both databases have been uti-lized for the development of drift-based and dual pa-rameter fragility curves. These curves describe the probability of reaching or exceeding discrete dam-age states observed in steel components. Both data-bases have been used to compute important statisti-cal parameters that can be used for quantification of the effects of modeling uncertainties on the seismic

performance of steel frame buildings in addition to ground motion uncertainties.

A case study was presented in which drift-based fragility curves of standard pre-Northridge beam-to-column connections are employed in order to rapidly assess the earthquake damage along the height of in-strumented steel buildings. This methodology may be easily used after an earthquake in order to man-age seismic risk in urban areas and allocate re-sources more efficiently.

A second case study was presented in which the dynamic response of a 2-story SCBF was predicted through collapse. A set of 40 ground motions was used for this purpose. This particular example is very important since it demonstrates the use of mod-eling recommendations for inelastic buckling and fracture of various steel braces and the quantifica-tion of the collapse margin of SCBFs beyond brace fracture.

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