2010 Structures Congress © 2010 ASCE
A Steel Component Database for Deterioration Modeling of Steel Beams with RBS under Cyclic Loading D.G. Lignos1, F. Zareian2, and H. Krawinkler3 1
Disaster Prevention Research Institute (DPRI), Kyoto University; #S301D Gokasho, Uji, Kyoto 611-0011, JAPAN, PH +81-77- 438-4085; email:
[email protected] 2 The Henry Samueli School of Engineering, University of California, Irvine, 92697, USA; PH +1-(949) 824-9866; email:
[email protected] 3 Department of Civil and Environmental Engineering, Stanford University, Stanford, CA, 94305, USA; PH +1-(650) 723-4129; email:
[email protected] ABSTRACT In recent years much effort has been devoted to the development of reliable analytical tools that model component deterioration and can be used to predict global collapse of structural systems under seismic excitations. This paper focuses on development of a steel component database that can serve as the basis for validation and improvement of analytical models that explicitly model deterioration in structural steel components and can be used in collapse assessment of steel moment resisting frames. Relationships that associate deterioration model parameters with geometric and material properties that control deterioration of steel beams with reduced beam sections (RBS) are proposed. The relationships are based on calibration and evaluation of steel beams with RBS subjected to monotonic and cyclic bending moments. The use and importance of deterioration modeling based on the proposed relationships for collapse prediction is demonstrated on a case study of a 20–story steel building designed based on current seismic provisions and evaluated based on FEMA P695 (ATC-63) performance methodology that requires explicit modeling of structural collapse. INTRODUCTION Recent methodologies for quantification of building system performance and response parameters for use in seismic design (FEMA P695) necessitate the collapse assessment of building structures under earthquakes. Many analytical studies (Ibarra et al. 2002, Sivaselvan and Reinhorn, 2002, Ibarra and Krawinkler, 2005, Haselton and Deierlein, 2007, Zareian and Krawinkler, 2009, Christovasilis et al. 2009) that have focused on the evaluation and prediction of the collapse capacity of structural systems have emphasized on the importance of sophisticated analytical models that simulate component deterioration phenomena such as strength and stiffness deterioration. A detailed description of various analytical models can be found in Lignos and Krawinkler (2009a). However, reliable analytical modeling to predict global collapse of structural systems under extreme earthquake events requires
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experimental data for model validation and improvement. Even though there is available experimental data through tests that have been conducted worldwide on steel components, there is no systematic way to gather this data in an organized way for better assessment of the hysteretic response of structural components. The missing information in the available steel component databases (SAC, http://www.sacsteel.org/connections/) is the digitized load - displacement (deduced moment - rotation) needed for calibration of the simulated response of the component. This paper summarizes the development of a steel database (Lignos and Krawinkler, 2007, 2009a, 2009b) for deterioration modeling of beams with reduced beam sections (RBS), one of the most common connection types in U.S design practice. The main advantage of this database is that provides complete digitized load – displacement (deduced moment - rotation) diagrams of each component. Empirical relationships are proposed that associate deterioration model parameters with geometric and material properties of steel beams with RBS. Such information can be used for more reliable collapse assessment of steel moment resting frames with RBS. This paper shows such an exercise by adopting the recently developed methodology for quantification of building system performance and response parameters (FEMAP695) on a 20 story building with RBS beams. This building is part of a research project (ATC-76-1) on quantification of seismic performance parameters of a group of special moment resisting frames summarized in Zareian et al. (2010). DETERIORATION MODELING A recently developed analytical model (Ibarra et al. 2005, Lignos and Krawinkler, 2009a), denoted as IK deterioration model, is used in this research to simulate deterioration of steel beams with RBS. The modified IK deterioration model (see Figure 1) is based on a reference backbone curve that is defined by the elastic stiffness Ke, pre-capping plastic rotation θp, post-capping plastic rotation θpc, residual strength ratio κ and ultimate rotation capacity θu of the component associated with ductile tearing of steel components. The modified IK model with bilinear hysteretic response is able to simulate strength, post-capping strength and unloading stiffness deterioration.
Moment (k-in)
6000 M
y
M
K = 840000 e M+ = 5340 y + θ = 0.045 p + θpc = 0.220 M /M+ = 1.06
c
5000 4000 3000
c
θ
θ
p
y
pc
2000
Strength Det.
6000
Post Cap. Strength Det.
4000 2000 Unload. Stiff. Det.
0 -2000 -4000
1000 0 0
8000
Moment (k-in)
7000
-6000 0.05
0.1 0.15 0.2 Chord Rotation (rad)
0.25
-8000
-0.05
Ref. BackBone Curve 0 0.05 Chord Rotation (rad)
(a) monotonic response (b) cyclic response Figure 1. Modified IK deterioration model (data from Tremblay et al. 1997)
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COMPONENT DATABASE FOR STEEL BEAMS WITH RBS In order to assess the deterioration characteristics of steel beams under cyclic and monotonic loading a steel component database of more than 300 beams was developed with experimental data that has been conducted worldwide. The database is organized in metadata, experimental results and deduced data. The metadata includes information related to test configuration and beam-to-column connection types, geometric and material properties and a report excerpt that provides a summary of each test. Experimental results include observations related to the test as reported by the experimentalist. Valuable information that is also included in this category is the complete digitized load - displacement diagram for each test. Deduced data include the deduced moment - rotation diagram of the steel beam based on the individual test configuration. The steel database includes 73 beams with RBS. The main configurations of these tests are standard single cantilever beam, standard single beam with column pinned at top and bottom and standard two beams with column pinned at top and bottom. Sixteen specimens include slab. The beam sizes range from W21x62 to W36x150. Fourteen specimens include additional lateral bracing near the RBS region (Uang et al. 2000, Tremblay et al. 1997). Calibration of Component Deterioration Parameters for beams with RBS. The information stored in the steel database is utilized to calibrate the cyclic and monotonic moment-rotation behavior of plastic hinge regions in steel beams with RBS. The objective of the calibration process is to provide complete moment-rotation relationships that incorporate experimental evidence to quantify cyclic deterioration that is insufficiently described by presently available models of engineering mechanics. The calibration process is based on a Matlab-based (Mathworks, 2006, http://www.mathworks.com/access/helpdesk/help) interactive software named Calibrator (Lignos and Krawinkler, 2009a) that was developed to facilitate the calibration effort. This software is an interface between hysteretic moment-rotation relationships stored in the steel database and a single-degree-of-freedom (SDOF) inelastic analysis program that computes the hysteretic response based on modeling parameters input into the interface. The initial selections of deterioration parameters of computed moment rotation response is based on a combination of visual observations and concepts of mechanics. These parameters are altered as deemed necessary to improve the match between computed and experimental momentrotation diagram for each test (see Figure 1b). More information about the calibration process of the modified IK model parameters is summarized in Lignos and Krawinkler (2009a). Trends for deformation modeling parameters of beams with RBS. This section summarizes trends that show the dependence of modeling parameters, such as precapping plastic rotation θp, post-capping plastic rotation θpc and cumulative rotation capacity Λ, with selected geometric properties of steel beams with RBS such as beam depth (h) to thickness (tw) ratio of the beam web h/tw, width (bf) to thickness (tf) ratio of the flange bf/2tf, and lateral torsional buckling controlled by Lb/ry ratio. The parameter Lb is defined here as the distance from the column face to the nearest brace
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and ry is the radius of gyration about the y-axis of the beam. The main findings for beams with RBS and beams other-than-RBS are summarized in detail in Lignos and Krawinkler (2009a, 2009b). The geometric parameter that is found to be the most influential for all three modeling parameters for beams with RBS is the h/tw ratio. The primary reason is that a beam with large h/tw ratio is more susceptible to web local buckling. Since web local buckling is coupled with flange local buckling and lateral torsional buckling (Lay and Galambos, 1966) the initiation of web local buckling triggers the other two buckling phenomena. The dependence of θp and θpc on h/tw ratio is shown in Figures 2a and 2b, respectively. The effect of bf/2tf ratio on θp and θpc is small when the bf/2tf ratio is viewed in isolation. A section with large fillet to fillet web depth over web thickness ratio (i.e. relatively small θp, θpc and Λ) will generally have a large bf/2tf ratio, which implies that the beam is more susceptible to flange local buckling. However, this effect is smaller for beams with RBS compared to beams other than RBS due to the flange width reduction at the RBS region. Beams with RBS: θ p versus h/tw ratio
Beams with RBS: θ pc versus h/tw ratio
0.6
0.08
0.5 0.06 (rad)
0.3
pc
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θ
θ (rad) p
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0.2 0.02 0.1 0 0
20
40 h/tw
60
80
0 0
20
40 h/tw
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(a) (b) Figure 2. Dependence of θp and θpc on beam depth to thickness h/tw ratio Per AISC (2005) design provisions the Lb/ry ratio (Lb is defined here as the distance from the column face to the nearest brace and ry is the radius of gyration about the y-axis of the beam) is required to be less than 2500/Fy (Fy expected yield strength of the steel beam section). In almost all the tests with RBS beams the AISC requirement was satisfied. It was found that the Lb/ry ratio does not greatly affect θp, θpc provided that the Lb/ry ratio is kept less equal to 2500/Fy. Additional bracing near the RBS region does not greatly improve θp and θpc values of beams with RBS but decreases the rate of cyclic deterioration, since twisting of the RBS region is delayed. The same findings are confirmed experimentally by Uang et al. (2000). Dependence of modeling parameters on beam depth d, shear span L to depth ratio L/d and expected yield strength Fy are summarized in Lignos and Krawinkler (2009a, 2009b).
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RELATIONSHIPS FOR COMPONENT DETERIORATION MODELING OF BEAMS WITH RBS AND WITHOUT SLAB Based on the available experimental data for beams with RBS empirical relationships are proposed that associate parameters of the modified IK deterioration model with geometric and material properties that control deterioration in steel beams with RBS. For this purpose, stepwise multivariate regression analysis is used (Chatterjee et al. 2000). To consider the coupling between web and flange local buckling and lateral torsional buckling a general nonlinear model is used that predicts the response parameter (RP) of interest, 2 ⎛ h ⎞ ⎛ bf RP = a1 ⋅ ⎜ ⎟ ⋅ ⎜ ⎜ ⎝ tw ⎠ ⎝ 2 ⋅ t f
a
a3
a4
6 2 ⎞ ⎛ Lb ⎞ ⎛ L ⎞ a5 ⎛ ⋅ Fy ⎞ ⎛ cunit d ⎟⎟ ⋅ ⎜⎜ ⎟⎟ ⋅ ⎜ ⎟ ⋅ ⎜ 1 ⎟ ⋅ ⎜⎜ ⎠ ⎝ ry ⎠ ⎝ d ⎠ ⎝ cunit ⋅ 21" ⎠ ⎝ 50
a
⎞ ⎟⎟ ⎠
a7
(1)
In which α1, α2,..., α7 are constants known as regression coefficients and c1unit and c unit are coefficients for units conversion. They both are 1.0 if inches and ksi are used, and they are c1unit = 25.4 and c2unit = 0.145 if d is in mm and Fy is in MPa, 2
respectively. For the regression equations discussed in the subsequent sections only variables that are statistically significant at the 95% level using a standard t-test are kept in Equation (1). Predictive equations discussed below are valid within the following range of geometric and material properties of beams with RBS, 21 ≤ h t w ≤ 55; 20 ≤ Lb ry ≤ 65, 4.5 ≤ b f 2t f ≤ 7.5, 2.3 ≤ L d ≤ 6.3, 21” ≤ d ≤ 36” and 38ksi ≤ Fy ≤ 63ksi. The effect of composite action is not considered in any of the empirical
equations since the experimental data is limited to W36x150 sections only. Pre-capping plastic rotation θp. The proposed relationship for predicting precapping plastic rotation θp is based on 55 beams with RBS and is given by, ⎛h⎞ θ p = 0.19 ⋅ ⎜ ⎟ ⎝ tw ⎠
−0.314
⎛ b ⋅⎜ f ⎜ ⎝ 2⋅t f
⎞ ⎟⎟ ⎠
−0.10
⎛L ⎞ ⋅⎜ b ⎟ ⎜ ⎟ ⎝ ry ⎠
−0.1185
⎛L⎞ ⋅⎜ ⎟ ⎝d⎠
0.113
⎛ ⎞ d ⋅⎜ 1 ⎟ ⎝ cunit ⋅ 21" ⎠
−0.76
⎛ c2 ⋅ F ⋅ ⎜ unit y ⎜ ⎝ 50
⎞ ⎟⎟ ⎠
−0.07
(2)
R 2 = 0.56 , σ ln = 0.24
The coefficient of determination R2 and the lognormal standard deviation σ ln of the equation illustrate that the data set of beams with RBS exhibits considerable scatter in the predicted versus calibrated θp values but the fit is acceptable. The same observation is illustrated in Figure 3 that shows the predicted versus calibrated θp values of this dataset. Post-capping plastic rotation θpc. In order to create an empirical equation that predicts post capping plastic rotation θpc for beams with RBS, only specimens with clear indication of post capping behavior in their cyclic or monotonic response are
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considered in the multivariate regression (see Figure 1). The predictive equation for θpc is given by, −0.513
⎛ b ⎛h⎞ ⋅⎜ f θ pc = 9.62 ⋅ ⎜ ⎟ ⎜ ⎝ tw ⎠ ⎝ 2⋅tf R 2 = 0.48 , σ ln = 0.26
⎞ ⎟⎟ ⎠
−0.863
⎛L ⎞ ⋅⎜ b ⎟ ⎜ ry ⎟ ⎝ ⎠
−0.108
⎛ c2 ⋅ F ⋅ ⎜ unit y ⎜ 50 ⎝
⎞ ⎟⎟ ⎠
−0.36
(3)
Equation (3) confirms the observations from deterioration parameter trends summarized earlier in this paper, where the geometric parameters h/tw, bf/2tf and Lb/ry were treated as independent random variables ignoring their correlations. 0.08
0.04
θ
p,pred.
(rad)
0.06
0.02
0 0
0.02 0.04 0.06 Calibrated θ p (rad)
0.08
Figure 3. Predicted versus calibrated pre-capping plastic rotation θp values for beams with RBS Cumulative rotation capacity Λ. To determine an empirical relationship for predicting cumulative rotation capacity Λ of steel beams with RBS only specimens that deteriorate gradually (absence of brittle fracture) have been considered in the data set to conduct the multivariate regression analysis. For simplicity in the calibration process the parameters that define strength (Λs), post-capping strength (Λc) and unloading stiffness (Λk) deterioration of the steel beam with RBS under cyclic loading are set to be equal. The proposed relationship for predicting Λ is given by, ⎛h⎞ Ε Λ = t = 592 ⋅ ⎜ ⎟ My ⎝ tw ⎠
−1.138
⎛ b ⋅⎜ f ⎜ ⎝ 2⋅tf
⎞ ⎟⎟ ⎠
−0.632
⎛L ⎞ ⋅⎜ b ⎟ ⎜ ⎟ ⎝ ry ⎠
−0.205
⎛ c2 ⋅ F ⋅ ⎜ unit y ⎜ ⎝ 50
⎞ ⎟⎟ ⎠
−0.391
(4)
R 2 = 0.486 , σ ln = 0.35
As seen from Equation (4) the effect of Lb/ry ratio on Λ is somewhat more important compared to θp and θpc because additional Lb/ry ratio delays twisting of the RBS region (see Uang et al. 2000) as was previously mentioned. Residual strength ratio κ. Steel components whose hysteretic behavior deteriorates due to local instabilities (no brittle behavior) typically approach stabilization of the
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hysteretic response at large inelastic deformations. The modified IK deterioration model is able to capture this stabilization with the residual strength ratio κ, which is a fraction of the component yield strength. Since within the data set of beams with RBS only few specimens attained stabilization (due to testing equipment limitation or stopping condition of the test), there is no clear trend of κ with respect to geometric or material parameters of the steel beams. A value of κ=0.40 is assumed to be a reasonable estimate of the yield strength ratio for beams with RBS. Ultimate rotation capacity θu. At large inelastic deformations the steel base material, at the apex of the most severe local buckle, may develop cracks that will then propagate, followed by ductile tearing and essentially complete loss of strength. For beams with RBS the ultimate rotation capacity θu associated with ductile tearing is about 0.06 radians using a symmetric cyclic loading protocol (AISC, 2005) as reported by Fry et al. (1997) and Ricles et al. (2004). However, it was recently shown (Uang et al. 2000, Lignos and Krawinkler, 2009a) that the ultimate rotation θu depends on the loading history that the steel beam experiences and may be extremely large for cases in which only a few very large cycles are executed (e.g., near-fault loading history). CASE STUDY The applicability of the steel component database summarized in this paper for modeling of deterioration phenomena of steel beams with RBS is illustrated through a case study of a 20-story office building with perimeter 3-bay special moment resisting frames. The case study addresses the application of the FEMA P695 methodology to steel moment-resisting frames as part of the ATC-76-1 study. The FEMA P695 methodology requires explicit modeling of structural collapse (i.e. reliable deterioration modeling of steel beams and columns). The special momentresisting frames of the 20-story office building have been designed in accordance with ASCE 7-05 and AISC 341-05 provisions. It is assumed that the seismic design category is Dmax and the deflection amplification factor Cd is equal with the reduction factor R = 8. Response Spectrum Analysis (RSA) was employed for the design process. The plan view of the office building can be seen in Figure 4. The first story height of the building is 15ft and all the other stories have a height of 13ft. A detailed summary of all the design decisions for the building is included in ATC-76-1 and in Zareian et al. (2010). The 20-story-RSA-Dmax building is modeled with a modified version of the DRAIN-2DX (Prakash et al. 1993) analysis program that includes the modified IK deterioration model discussed in this paper. The analysis program has been verified that can reliably predict seismic response of structural systems at large deformations near collapse with recent earthquake simulator collapse experiments (Suita et al. 2008, Lignos and Krawinkler, 2009a). Panel zone shear distortions are included in the analytical model. The predominant period based on analysis of the East West (EW) steel moment-resisting frame (see Figure 4) is 4.47sec. Deterioration parameters of steel beams with RBS are based on Equations (2) to (4). Deterioration parameters of steel columns are based on empirical relatioships that are similar with Equations (2)
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to (4) but are based on experimental data of steel beams other than RBS (see Lignos and Krawinkler, 2009b).
100’
3@20’
140’
3@20’
Figure 4. Plan view of archetype 20-story office building (ATC-76-1)
0.2
0.2
0.15
0.15 Vy/W
Vy/W
Performance evaluation based on nonlinear static procedure. In order to compute the system overstrength factor (Ω0) and period-based ductility (µT), nonlinear static (pushover) analysis is performed in accordance with Section 6.3 of FEMA P695. The lateral load pattern distribution, Fx, at each story level, x, is assumed to be in proportion to the fundamental mode of the 20-story building. Figure 5a shows the normalized base shear of the 20-story-RSA-Dmax building versus the roof drift ratio (roof displacement normalized with respect to total height H=262ft of the building). From this figure, global yielding occurs around 1% roof drift. Using the procedure outlined in Section 6.3 of FEMA P695 the period based ductility µT = 2.05 and the overstrength factor Ω0 = 2.23 of the 20-story-RSA-Dmax building. As seen from Figure 5a the building is not as ductile as expected. The main reason why the 20story-RSA-Dmax building is not ductile is the increase in P-Delta effects due to partial mechanism that forms in the first 4-stories, i.e. the story drift ratios at these stories get greatly amplified compared to the roof drift. This can be seen in Figure 5b that we plot the normalized base shear of the 20-story-RSA-Dmax building with respect to the first story drift ratio (relative displacement of the first story over its height h1 = 15ft).
0.1
0.05
0.05 0 0
0.1
0.01 Roof Drift Δ/H
0.02
0 0
0.01 SDR1
0.02
(a)Norm. base shear versus roof drift (b) Norm. base shear versus SDR1 Figure 5. Nonlinear static curve for 20-story-RSA-Dmax design
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Performance evaluation based on nonlinear response history analysis. To conduct nonlinear response history analysis 2.5% Rayleigh damping is assigned at the first more period T1=4.47sec of the structure and at 20% of T1. Based on the FEMA P695 methodology the use of complete incremental dynamic analysis (IDA) is not necessary. Instead, target scale factors for a set of 44ground motions are selected and are varied until 22 collapses are obtained. The objective is to obtain the Collapse Margin Ratio (CMR), which is obtained as the ratio of median collapse intensity SˆCT to the Maximum Considered Earthquake (MCE) ground motion demand (SMT) at the code period CuTa. The median collapse intensity SˆCT is defined as the scale factor that causes 22 collapses multiplied by the median Sa at the code period CuTa of the unscaled records. For the 20-story-RSA-Dmax building discussed in this paper the CMR is equal to 1.19. In order to account for the effect of spectral shape on the median collapse capacity of the 20-story-RSA-Dmax building we calculate the adjacent collapse marginal ratio (ACMR) of this structure to be 1.47 (see Section 7.2 of FEMA P695). If we compare this value with the Acceptable Collapse Marginal Ratio (ACMR = 1.43) for a conditional collapse probability less equal to 20% at the MCE level we can see that the 20-story-RSA-Dmax building barely passes the performance check per FEMA P695. The primary issue for the low collapse capacity of the 20-story-RSA-Dmax steel moment-resisting frame is P-Delta effects since they dominate response in the highly inelastic range and lead to a mechanism that involves only the bottom stories of the steel moment-resisting frame as was pointed out using the nonlinear static analysis procedure. This can also be seen in Figures 6 and 7 that show the roof drift and first story drift ratio (SDR1) histories, respectively, of the 20-story building using one ground motion (record 120721) from the FEMA P695 set. In both figures two lines are included. The first one (solid line) is based on a ground motion scale factor of 4.5. The dashed line is based on a ground motion scale factor of 4.85. The latter scale factor leads the structure to collapse (last stable point). For a relatively low increase in ground motion intensity the increase in deformation demands in the first story is sufficient to cause collapse.
Roof Drift (rad)
0.03 0.02
S.Factor=4.5 S.Factor=4.85
0.01 0 -0.01 0
10
20 Time (sec)
30
Figure 6. Roof drift histories of 20-story-RSA-Dmax for 4.5 and 4.85 ground motion intensity scale factors
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SDR1 (rad)
0.1 0.075
S.Factor=4.5 S.Factor=4.85
0.05 0.025 0 -0.01 0
10
20 Time (sec)
30
Figure 7. First story drift histories of 20-story-RSA-Dmax for 4.5 and 4.85 ground motion intensity scale factors CONCLUSION A recently developed database for modeling of deterioration of steel beams under cyclic and monotonic bending moments is summarized in this paper. The focus is on deterioration modeling of steel beams with reduced beam sections (RBS) for collapse assessment of steel moment resisting frames under extreme seismic events. The steel component database, which is available through the Network for Earthquake Engineering Simulation (NEES) central repository (https://central.nees.org/?action=DisplayExperimentMain&projid=84&expid=159 0) is organized in metadata, experimental results and deduced data and provides complete digitized load - displacement (deduced moment - rotation) diagrams of the steel beam components. These diagrams serve for analytical modeling validation and improvement for reliable collapse assessment of steel frame structures. Using a hysteretic model that can simulate adequately component deterioration of beams with RBS, it is concluded that the effect of web local buckling on deterioration parameters (pre-capping plastic rotation θp, post-capping plastic rotation θpc and cumulative plastic rotation Λ) of beams with RBS is critical since it triggers flange local buckling and lateral torsional buckling. Additional bracing near the RBS region does not lead to a significant improvement of θp and θpc values but slows down the rate of cyclic deterioration since delays twisting of the RBS region. A residual strength ratio of 40% as a fraction of yield strength is recommended for beams with RBS. The ultimate rotation θu is about 6 % based on symmetric loading protocols. The steel database is utilized to provide empirical relationships that predict θp, θpc and Λ values of beams with RBS as a function of critical geometric and material properties of the steel sections that control component deterioration. These relationships that are based on actual experimental data are used for reliable seismic performance assessment of special steel moment-resisting frames since recent performance evaluation recommendations require explicit modeling of structural collapse (i.e. incorporation of component deterioration in the analytical models of steel frame structures). After applying the FEMA P695 performance methodology on a case study of a 20-story steel building designed based on current seismic provisions, tall steel moment resisting frames seem to have low collapse capacities due to
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amplification of deformations concentrated at lower stories due to large P-Delta effects. AKNOWLEDGEMENTS This study is based on work supported by the United States National Science Foundation (NSF) under Grant No. CMS-0421551 within the George E. Brown, Jr. Network for Earthquake Engineering Simulation Consortium Operations, by a grant from the CUREE-Kajima Phase VI joint research program and ATC-76-1 Project “Quantification of Building System Performance and Response Parameters,” funded by the NEHRP Consultants Joint Venture (a partnership of the Applied Technology Council and Consortium of Universities for Research in Earthquake Engineering), under Contract SB134107CQ0019, Earthquake Structural and Engineering Research, issued by the National Institute of Standards and Technology. This financial support is gratefully acknowledged. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of sponsors. REFERENCES AISC (2005). “Seismic provisions for structural steel buildings, including supplement No. 1”, American Institute of Steel Construction, Inc. Chicago, Illinois. ASCE-7-05 (2006), “Minimum design loads for buildings and other structures,” American Society of Civil Engineers, 424pages. Chatterjee, S., Hadi, A.S., and Price, B. (2000), “Regression Analysis by Example”, 3rd Edition, John Wiley and Sons Inc., New York. Christovasilis, I. P., Filiatrault, A., Constantinou, M. C., and Wanitkorkul, A. (2009). “Incremental dynamic analysis of woodframe buildings”, Earthquake Engineering and Structural Dynamics, 38(12), 477-496. FEMA P695 (2009). “Quantification of building seismic performance factors”, Rep. FEMA P695, Federal Emergency Management Agency, Washington, D.C. Fry, G. T., Jones, S. L., and Holliday., S. D. (1997). “Protocol for fabrication, inspection, testing, and documentation of beam-column connection tests and other experimental specimens,” Rep. No. SAC/BD-97/02, SAC Joint Venture, Sacramento, CA. Ibarra, L. F., Medina, R., and Krawinkler, H. (2002). “Collapse assessment of deteriorating SDOF systems”, Proc. 12th European Conference on Earthquake Engineering, London, UK, Paper 665, Elsevier Science Ltd. Ibarra L. F., and Krawinkler, H. (2005). “Global collapse of frame structures under seismic excitations”, Rep. No. TB 152, The John A. Blume Earthquake Engineering Center, Stanford University, Stanford, CA. Ibarra L. F., Medina R. A., and Krawinkler H. (2005). “Hysteretic models that incorporate strength and stiffness deterioration”, Earthquake Engineering and Structural Dynamics, 34(12), 1489-1511.
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