Seismic Collapse Safety of Reinforced ConcreteBuildings.II:Comparative Assessment of Nonductile and Ductile Moment FramesAbbie B.Liel,M.ASCE 1;Curt B.Haselton,M.ASCE 2;and Gregory G.Deierlein,F.ASCE 3Abstract:This study is the second of two companion papers to examine the seismic collapse safety of reinforced concrete frame buildings,and examines nonductile moment frames that are representative of those built before the mid-1970s in California.The probabilistic assessment relies on nonlinear dynamic simulation of structural response to calculate the collapse risk,accounting for uncertainties in ground-motion characteristics and structural modeling.The evaluation considers a set of archetypical nonductile RC frame structures of varying height that are designed according to the seismic provisions of the 1967Uniform Building Code.The results indicate that nonductile RC frame structures have a mean annual frequency of collapse ranging from 5to 14×10À3at a typical high-seismic California site,which is approximately 40times higher than corresponding results for modern code-conforming special RC moment frames.These metrics demonstrate the effectiveness of ductile detailing and capacity design requirements,which have been introduced over the past 30years to improve the safety of RC buildings.Data on comparative safety between nonductile and ductile frames may also inform the development of policies for appraising and mitigating seismic collapse risk of existing RC frame buildings.DOI:10.1061/(ASCE)ST.1943-541X .0000275.©2011American Society of Civil Engineers.CE Database subject headings:Structural failures;Earthquake engineering;Structural reliability;Reinforced concrete;Concrete structures;Seismic effects;Frames.Author keywords:Collapse;Earthquake engineering;Structural reliability;Reinforced concrete structures;Buildings;Commercial;Seismic effects.IntroductionReinforced concrete (RC)frame structures constructed in Califor-nia before the mid-1970s lack important features of good seismic design,such as strong columns and ductile detailing of reinforce-ment,making them potentially vulnerable to earthquake-induced collapse.These nonductile RC frame structures have incurred significant earthquake damage in the 1971San Fernando,1979Imperial Valley,1987Whittier Narrows,and 1994Northridge earthquakes in California,and many other earthquakes worldwide.These factors raise concerns that some of California ’s approxi-mately 40,000nonductile RC structures may present a significant hazard to life and safety in future earthquakes.However,data are lacking to gauge the significance of this risk,in relation to either the building population at large or to specific buildings.The collapse risk of an individual building depends not only on the building code provisions employed in its original design,but also structuralconfiguration,construction quality,building location,and site-spe-cific seismic hazard information.Apart from the challenges of ac-curately evaluating the collapse risk is the question of risk tolerance and the minimum level of safety that is appropriate for buildings.In this regard,comparative assessment of buildings designed accord-ing to old versus modern building codes provides a means of evalu-ating the level of acceptable risk implied by current design practice.Building code requirements for seismic design and detailing of reinforced concrete have changed significantly since the mid-1970s,in response to observed earthquake damage and an in-creased understanding of the importance of ductile detailing of reinforcement.In contrast to older nonductile RC frames,modern code-conforming special moment frames for high-seismic regions employ a variety of capacity design provisions that prevent or delay unfavorable failure modes such as column shear failure,beam-column joint failure,and soft-story mechanisms.Although there is general agreement that these changes to building code require-ments are appropriate,there is little data to quantify the associated improvements in seismic safety.Performance-based earthquake engineering methods are applied in this study to assess the likelihood of earthquake-induced collapse in archetypical nonductile RC frame structures.Performance-based earthquake engineering provides a probabilistic framework for re-lating ground-motion intensity to structural response and building performance through nonlinear time-history simulation (Deierlein 2004).The evaluation of nonductile RC frame structures is based on a set of archetypical structures designed according to the pro-visions of the 1967Uniform Building Code (UBC)(ICBO 1967).These archetype structures are representative of regular well-designed RC frame structures constructed in California between approximately 1950and 1975.Collapse is predicted through1Assistant Professor,Dept.of Civil,Environmental and Architectural Engineering,Univ.of Colorado,Boulder,CO 80309.E-mail:abbie .liel@ 2Assistant Professor,Dept.of Civil Engineering,California State Univ.,Chico,CA 95929(corresponding author).E-mail:chaselton@csuchico .edu 3Professor,Dept.of Civil and Environmental Engineering,Stanford Univ.,Stanford,CA 94305.Note.This manuscript was submitted on July 14,2009;approved on June 30,2010;published online on July 15,2010.Discussion period open until September 1,2011;separate discussions must be submitted for individual papers.This paper is part of the Journal of Structural Engineer-ing ,V ol.137,No.4,April 1,2011.©ASCE,ISSN 0733-9445/2011/4-492–502/$25.00.492/JOURNAL OF STRUCTURAL ENGINEERING ©ASCE /APRIL 2011D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y S u l t a n Q a b o o s U n i v e r s i t y o n 06/21/14. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .nonlinear dynamic analysis of the archetype nonductile RC frames,using simulation models capable of capturing the critical aspects of strength and stiffness deterioration as the structure collapses.The outcome of the collapse performance assessment is a set of measures of building safety and relating seismic collapse resistance to seismic hazard.These results are compared with the metrics for ductile RC frames reported in a companion paper (Haselton et al.2011b ).Archetypical Reinforced Concrete Frame StructuresThe archetype nonductile RC frame structures represent the expected range in design and performance in California ’s older RC frame buildings,considering variations in structural height,configuration and design details.The archetype configurations explore key design parameters for RC components and frames,which were identified through previous analytical and experimental studies reviewed by Haselton et al.(2008).The complete set of archetype nonductile RC frame buildings developed for this study includes 26designs (Liel and Deierlein 2008).This paper focuses primarily on 12of these designs,varying in height from two to 12stories,and including both perimeter (P )and space (S )frame lateral resisting systems with alternative design details.All archetype buildings are designed for office occupancies with an 8-in.(20-cm)flat-slab floor system and 25-ft (7.6-m)column spacing.The 2-and 4-story buildings have a footprint of 125ft by 175ft (38.1m by 53.3m),and the 8-and 12-story buildings measure 125ft (38.1m)square in plan.Story heights are 15ft (4.6m)in the first story and 13ft (4.0m)in all other stories.Origi-nal structural drawings for RC frame buildings constructed in California in the 1960s were used to establish typical structural configurations and geometry for archetype structures (Liel and Deierlein 2008).The archetypes are limited to RC moment frames without infill walls,and are regular in elevation and plan,without major strength or stiffness irregularities.The nonductile RC archetype structures are designed for the highest seismic zone in the 1967UBC,Zone 3,which at that time included most of California.Structural designs of two-dimensional frames are governed by the required strength and stiffness to satisfy gravity and seismic loading combinations.The designs also satisfy all relevant building code requirements,including maximum and minimum reinforcement ratios and maximum stirrup spacing.The 1967UBC permitted an optional reduction in the design base shear if ductile detailing requirements were employed,however,this reduction is not applied and only standard levels of detailing are considered in this study.Design details for each structure areTable 1.Design Characteristics of Archetype Nonductile and Ductile RC Frames Stucture Design base shear coefficient a,bColumn size c (in :×in.)Column reinforcementratio,ρColumn hoop spacing d,e (in.)Beam size f (in :×in.)Beam reinforcementratios ρ(ρ0)Beam hoop spacing (in.)Nonductile2S 0.08624×240.0101224×240.006(0.011)112P 0.08630×300.0151530×300.003(0.011)114S 0.06820×200.0281020×260.007(0.014)124P 0.06824×280.0331424×320.007(0.009)158S 0.05428×280.0141424×260.006(0.013)118P 0.05430×360.0331526×360.008(0.010)1712S 0.04732×320.025926×300.006(0.011)1712P 0.04732×400.032930×380.006(0.013)184S g 0.06820×200.028 6.720×260.007(0.014)84S h 0.06820×200.0281020×260.007(0.014)1212S g 0.04732×320.025626×300.006(0.011)1112S h 0.04732×320.025926×300.006(0.011)17Ductile2S 0.12522×220.017518×220.006(0.012) 3.52P 0.12528×300.018528×280.007(0.008)54S 0.09222×220.016522×240.004(0.008)54P 0.09232×380.016 3.524×320.011(0.012)58S 0.05022×220.011422×220.006(0.011) 4.58P 0.05026×340.018 3.526×300.007(0.008)512S 0.04422×220.016522×280.005(0.008)512P0.04428×320.0223.528×380.006(0.007)6aThe design base shear coefficient in the 1967UBC is given by C ¼0:05=T ð1=3Þ≤0:10.For moment resisting frames,T ¼0:1N ,where N is the number of stories (ICBO 1967).bThe design base shear coefficient for modern buildings depends on the response spectrum at the site of interest.The Los Angeles site has a design spectrumdefined by S DS ¼1:0g and S D1¼0:60g.The period used in calculation of the design base shear is derived from the code equation T ¼0:016h 0:9n ,where h n isthe height of the structure in feet,and uses the coefficient for upper limit of calculated period (C u ¼1:4)(ASCE 2002).cColumn properties vary over the height of the structure and are reported here for an interior first-story column.dConfiguration of transverse reinforcement in each member depends on the required shear strength.There are at least two No.3bars at every location.eConfiguration of transverse reinforcement in ductile RC frames depends on the required shear strength.All hooks have seismic detailing and use No.4bars (ACI 2005).fBeam properties vary over the height of the structure and are reported here are for a second-floor beam.gThese design variants have better-than-average beam and column detailing.hThese design variants have better-than-average joint detailing.JOURNAL OF STRUCTURAL ENGINEERING ©ASCE /APRIL 2011/493D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y S u l t a n Q a b o o s U n i v e r s i t y o n 06/21/14. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .summarized in Table 1,and complete documentation of the non-ductile RC archetypes is available in Liel and Deierlein (2008).Four of the 4-and 12-story designs have enhanced detailing,as described subsequently.The collapse performance of archetypical nonductile RC frame structures is compared to the set of ductile RC frame archetypes presented in the companion paper (Haselton et al.2011b ).As sum-marized in Table 2,these ductile frames are designed according to the provisions of the International Building Code (ICC 2003),ASCE 7(ASCE 2002),and ACI 318(ACI 2005);and meet all gov-erning code requirements for strength,stiffness,capacity design,and detailing for special moment frames.The structures benefit from the provisions that have been incorporated into seismic design codes for reinforced concrete since the 1970s,including an assort-ment of capacity design provisions [e.g.,strong column-weak beam (SCWB)ratios,beam-column and joint shear capacity design]and detailing improvements (e.g.,transverse confinement in beam-column hinge regions,increased lap splice requirements,closed hooks).The ductile RC frames are designed for a typical high-seismic Los Angeles site with soil class S d that is located in the transition region of the 2003IBC design maps (Haselton and Deierlein 2007).A comparison of the structures described in Table 1reflects four decades of changes to seismic design provisions for RC moment frames.Despite modifications to the period-based equation for design base shear,the resulting base shear coefficient is relatively similar for nonductile and ductile RC frames of the same height,except in the shortest structures.More significant differencesbetween the two sets of buildings are apparent in member design and detailing,especially in the quantity,distribution,and detailing of transverse reinforcement.Modern RC frames are subject to shear capacity design provisions and more stringent limitations on stirrup spacing,such that transverse reinforcement is spaced two to four times more closely in ductile RC beams and columns.The SCWB ratio enforces minimum column strengths to delay the formation of story mechanisms.As a result,the ratio of column to beam strength at each joint is approximately 30%higher (on average)in the duc-tile RC frames than the nonductile RC frames.Nonductile RC frames also have no special provision for design or reinforcement of the beam-column joint region,whereas columns in ductile RC frames are sized to meet joint shear demands with transverse reinforcement in the joints.Joint shear strength requirements in special moment frames tend to increase the column size,thereby reducing axial load ratios in columns.Nonlinear Simulation ModelsNonlinear analysis models for each archetype nonductile RC frame consist of a two-dimensional three-bay representation of the lateral resisting system,as shown in Fig.1.The analytical model repre-sents material nonlinearities in beams,columns,beam-column joints,and large deformation (P -Δ)effects that are important for simulating collapse of frames.Beam and column ends and the beam-column joint regions are modeled with member end hinges that are kinematically constrained to represent finite joint sizeTable 2.Representative Modeling Parameters in Archetype Nonductile and Ductile RC Frame Structures Structure Axial load a,b (P =A g f 0c )Initial stiffness c Plastic rotation capacity (θcap ;pl ,rad)Postcapping rotation capacity (θpc ,rad)Cyclicdeterioration d (λ)First mode period e (T 1,s)Nonductile2S 0.110:35EI g 0.0180.04041 1.12P 0.030:35EI g 0.0170.05157 1.04S 0.300:57EI g 0.0210.03333 2.04P 0.090:35EI g 0.0310.10043 2.08S 0.310:53EI g 0.0130.02832 2.28P 0.110:35EI g 0.0250.10051 2.412S 0.350:54EI g 0.0290.06353 2.312P 0.140:35EI g 0.0450.10082 2.84S f 0.300:57EI g 0.0320.04748 2.04S g 0.300:57EI g 0.0210.03333 2.012S f 0.350:54EI g 0.0430.09467 2.312S g 0.350:54EI g 0.0290.06353 2.3Ductile2S 0.060:35EI g 0.0650.100870.632P 0.010:35EI g 0.0750.1001110.664S 0.130:38EI g 0.0570.100800.944P 0.020:35EI g 0.0860.100133 1.18S 0.210:51EI g 0.0510.10080 1.88P 0.060:35EI g 0.0870.100122 1.712S 0.380:68EI g 0.0360.05857 2.112P0.070:35EI g0.0700.1001182.1a Properties reported for representative interior column in the first story.(Column model properties data from Haselton et al.2008.)bExpected axial loads include the unfactored dead load and 25%of the design live load.cEffective secant stiffness through 40%of yield strength.dλis defined such that the hysteretic energy dissipation capacity is given by Et ¼λM y θy (Haselton et al.2008).eObtained from eigenvalue analysis of frame model.fThese design variants have better-than-average beam and column detailing.gThese design variants have better-than-average joint detailing.494/JOURNAL OF STRUCTURAL ENGINEERING ©ASCE /APRIL 2011D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y S u l t a n Q a b o o s U n i v e r s i t y o n 06/21/14. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .effects and connected to a joint shear spring (Lowes and Altoontash 2003).The structural models do not include any contribution from nonstructural components or from gravity-load resisting structural elements that are not part of the lateral resisting system.The model is implemented in OpenSees with robust convergence algorithms (OpenSees 2009).As in the companion paper,inelastic beams,columns,and joints are modeled with concentrated springs idealized by a trilinear back-bone curve and associated hysteretic rules developed by Ibarra et al.(2005).Properties of the nonlinear springs representing beam and column elements are predicted from a series of empirical relation-ships relating column design characteristics to modeling parame-ters and calibrated to experimental data for RC columns (Haselton et al.2008).Tests used to develop empirical relationships include a large number of RC columns with nonductile detailing,and predicted model parameters reflect the observed differences in moment-rotation behavior between nonductile and ductile RC elements.As in the companion paper,calibration of model param-eters for RC beams is established on columns tested with low axial load levels because of the sparse available beam data.Fig.2(a)shows column monotonic backbone curve properties for a ductile and nonductile column (each from a 4-story building).The plastic rotation capacity θcap ;pl ,which is known to have an important influence on collapse prediction,is a function of the amount of column confinement reinforcement and axial load levels,and is approximately 2.7times greater for the ductile RC column.The ductile RC column also has a larger postcapping rotation capacity (θpc )that affects the rate of postpeak strength degradation.Fig.2(b)illustrates cyclic deterioration of column strength and stiffness under a typical loading protocol.Cyclic degradation of the initial backbone curve is controlled by the deterioration parameter λ,which is a measure of the energy dissipation capacity and is smaller in nonductile columns because of poor confinement and higher axial loads.Model parameters are calibrated to the expected level of axial compression in columns because of gravity loads and do not account for axial-flexure-shear interaction during the analysis,which may be significant in taller buildings.Modeling parameters for typical RC columns in nonductile and ductile archetypes are summarized in Table 2.Properties for RC beams are similar and reported elsewhere (Liel and Deierlein 2008;Haselton and Deierlein 2007).All element model properties are calibrated to median values of test data.Although the hysteretic beam and column spring parameters incorporate bond-slip at the member ends,they do not account for significant degradations that may occur because of anchorage or splice failure in nonductile frames.Unlike ductile RC frames,in which capacity design require-ments limit joint shear deformations,nonductile RC frames may experience significant joint shear damage contributing to collapse (Liel and Deierlein 2008).Joint shear behavior is modeled with an inelastic spring,as illustrated in Fig.1and defined by a monotonic backbone and hysteretic rules (similar to those shown in Fig.2for columns).The properties of the joint shear spring are on the basisofFig.1.Schematic of the RC frame structural analysismodel(a)(b)Fig.2.Properties of inelastic springs used to model ductile and non-ductile RC columns in the first story of a typical 4-story space frame:(a)monotonic behavior;(b)cyclic behaviorJOURNAL OF STRUCTURAL ENGINEERING ©ASCE /APRIL 2011/495D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y S u l t a n Q a b o o s U n i v e r s i t y o n 06/21/14. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .selected subassembly data of joints with minimal amounts of trans-verse reinforcement and other nonductile characteristics.Unfortu-nately,available data on nonconforming joints are limited.Joint shear strength is computed using a modified version of the ACI 318equation (ACI 2005),and depends on joint size (b j is joint width,h is height),concrete compressive strength (f 0c ,units:psi),and confinement (γ,which is 12to 20depending on the configu-ration of confining beams)such that V ¼0:7γffiffiffiffif 0c p b j h .The 0.7modification factor is on the basis of empirical data from Mitra and Lowes (2007)and reflects differences in shear strength between seismically detailed joints (as assumed in ACI 318Chap.21)and joints without transverse reinforcement,of the type consid-ered in this study.Unlike conforming RC joints,which are assumed to behave linear elastically,nonductile RC joints have limited duc-tility,and shear plastic deformation capacity is assumed to be 0.015and 0.010rad for interior and exterior joints,respectively (Moehle et al.2006).For joints with axial load levels below 0.095,data from Pantelides et al.(2002)are used as the basis for a linear increase in deformation capacity (to a maximum of 0.025at zero axial load).Limited available data suggest a negative postcapping slope of approximately 10%of the effective initial stiffness is appropriate.Because of insubstantial data,cyclic deterioration properties are assumed to be the same as that for RC beams and columns.The calculated elastic fundamental periods of the RC frame models,reported in Table 2,reflect the effective “cracked ”stiffness of the beams and columns (35%of EI g for RC beams;35%to 80%of EI g for columns),finite joint sizes,and panel zone flexibility.The effective member stiffness properties are determined on the basis of deformations at 40%of the yield strength and include bond-slip at the member ends.The computed periods are signifi-cantly larger than values calculated from simplified formulas in ASCE (2002)and other standards,owing to the structural modeling assumptions (specifically,the assumed effective stiffness and the exclusion of the gravity-resisting system from the analysis model)and intentional conservatism in code-based formulas for building period.Nonlinear static (pushover)analysis of archetype analysis mod-els shows that the modern RC frames are stronger and have greater deformation capacities than their nonductile counterparts,as illus-trated in Fig.3.The ASCE 7-05equivalent seismic load distribu-tion is applied in the teral strength is compared on the basis of overstrength ratio,Ω,defined as the ratio between the ultimate strength and the design base shear.The ductility is com-pared on the basis of ultimate roof drift ratio (RDR ult ),defined as the roof drift ratio at which 20%of the lateral strength of the structure has been lost.As summarized in Table 3,for the archetype designs in this study,the ductile RC frames have approximately 40%more overstrength and ultimate roof drift ratios three times larger than the nonductile RC frames.The larger structural deformation capacity and overstrength in the ductile frames results from (1)greater deformation capacity in ductile versus nonductile RC components (e.g.,compare column θcap ;pl and θpc in Table 2),(2)the SCWB requirements that promote more distributed yielding over multiple stories in the ductile frames,(3)the larger column strengths in ductile frames that result from the SCWB and joint shear strength requirements,and (4)the required ratios of positive and negative bending strength of the beams in the ductile frames.Fig.3(b)illustrates the damage concentration in lower stories,especially in the nonductile archetype structures.Whereas nonlin-ear static methods are not integral to the dynamic collapse analyses,the pushover results help to relate the dynamic collapse analysis results,described subsequently,and codified nonlinear static assessment procedures.Collapse Performance Assessment ProcedureSeismic collapse performance assessment for archetype nonductile RC frame structures follows the same procedure as in the companion study of ductile RC frames (Haselton et al.2011b ).The collapse assessment is organized using incremental dynamic analysis (IDA)of nonlinear simulation models,where each RC frame model is subjected to analysis under multiple ground motions that are scaled to increasing amplitudes.For each ground motion,collapse is defined on the basis of the intensity (spectral acceleration at the first-mode period of the analysis model)of the input ground motion that results in structural collapse,as iden-tified in the analysis by excessive interstory drifts.The IDA is repeated for each record in a suite of 80ground motions,whose properties along with selection and scaling procedures are de-scribed by Haselton et al.(2011b ).The outcome of this assessment is a lognormal distribution (median,standard deviation)relating that structure ’s probability of collapse to the ground-motion inten-sity,representing a structural collapse fragility function.Uncer-tainty in prediction of the intensity at which collapse occurs,termed “record-to-record ”uncertainty (σln ;RTR ),is associated with variation in frequency content and other characteristics of ground-motion records.Although the nonlinear analysis model for RC frames can simulate sidesway collapse associated with strength and stiffness degradation in the flexural hinges of the beams andcolumnsFig.3.Pushover analysis of ductile and nonductile archetype 12-story RC perimeter frames:(a)force-displacement response;and (b)distri-bution of interstory drifts at the end of the analysis496/JOURNAL OF STRUCTURAL ENGINEERING ©ASCE /APRIL 2011D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y S u l t a n Q a b o o s U n i v e r s i t y o n 06/21/14. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .and beam-column joint shear deformations,the analysis model does not directly capture column shear failure.The columns in the archetype buildings in this study are expected to yield first in flexure,followed by shear failure (Elwood and Moehle 2005)rather than direct shear failure,as may be experienced by short,squat nonductile RC columns.However,observed earthquake damage and laboratory studies have shown that shear failure and subsequent loss of gravity-load-bearing capacity in one column could lead to progressive collapse in nonductile RC frames.Column shear failure is not incorporated directly because of the difficulties in accurately simulating shear or flexure-shear failure and subsequent loss of axial load-carrying capacity (Elwood 2004).Collapse modes related to column shear failure are therefore detected by postprocessing dynamic analysis results using compo-nent limit state ponent limit state functions are devel-oped from experimental data on nonductile beam-columns and predict the median column drift ratio (CDR)at which shear failure,and the subsequent loss of vertical-load-carrying capacity,will occur.Here,CDR is defined similarly to interstory drift ratio,but excludes the contribution of beam rotation and joint deforma-tion to the total drift because the functions are established on data from column component tests.Component fragility relationships for columns failing in flexure-shear developed by Aslani and Miranda (2005),building on work by Elwood (2004),are employed in this study.For columns with nonductile shear design and detailing in this study and axial load ratios of P =A g f 0c between 0.03and 0.35,Aslani and Miranda (2005)predict that shear failure occurs at a median CDR between 0.017and 0.032rad,depending on the properties of the column,and the deformation capacity decreases with increasing axial load.Sub-sequent loss of vertical-carrying capacity in a column is predicted to occur at a median CDR between 0.032and 0.10rad,again depending on the properties of the column.Since the loss of vertical-load-carrying capacity of a column may precipitate progressive structure collapse,this damage state is defined as collapse in this assessment.In postprocessing dynamic analysis results,the vertical collapse limit state is reached if,during the analysis,the drift in any column exceeds the median value of that column ’s component fragility function.If the vertical collapse mode is predicted to occur at a smaller ground-motion intensity than the sidesway collapse mode (for a particular record),then the collapse statistics are updated.This simplified approach can be shown to give comparable median results to convolving the probability distribution of column drifts experienced as a function of ground-motion intensity (engineering demands)with the com-ponent fragility curve (capacity).The total uncertainty in the col-lapse fragility is assumed to be similar in the sidesway-only case and the sidesway/axial collapse case,as it is driven by modeling and record-to-record uncertainties rather than uncertainty in the component fragilities.Incorporating this vertical collapse limit state has the effect of reducing the predicted collapse capacity of the structure.Fig.4illustrates the collapse fragility curves for the 8-story RC space frame,with and without consideration of shear failure and axial failure following shear.As shown,if one considers collapse to occur with column shear failure,then the collapse fragility can reduce considerably compared to the sidesway collapse mode.However,if one assumes that shear failure of one column does not constitute collapse and that collapse is instead associated with the loss in column axial capacity,then the resulting collapse capac-ity is only slightly less than calculations for sidesway alone.For the nonductile RC frame structures considered in this study,the limit state check for loss of vertical-carrying capacity reduces the median collapse capacity by 2%to 30%as compared to the sidesway collapse statistics that are computed without this check (Liel and Deierlein 2008).Table 3.Results of Collapse Performance Assessment for Archetype Nonductile and Ductile RC Frame Structures Structure ΩRDR ult Median Sa ðT 1Þ(g)Sa 2=50ðT 1Þ(g)Collapse marginλcollapse ×10À4IDR collapse RDR collapseNonductile 2S 1.90.0190.470.800.591090.0310.0172P 1.60.0350.680.790.85470.0400.0284S 1.40.0160.270.490.541070.0540.0284P 1.10.0130.310.470.661000.0370.0178S 1.60.0110.290.420.68640.0420.0118P 1.10.0070.230.310.751350.0340.00912S 1.90.0100.290.350.83500.0340.00612P 1.10.0050.240.420.561190.0310.0064S a 1.40.0160.350.490.72380.0560.0244S b 1.60.0180.290.490.60890.0610.02612S a 1.90.0120.330.350.93350.0390.00912S b 2.20.0120.460.351.32160.0560.012Ductile 2S 3.50.085 3.55 1.16 3.07 1.00.0970.0752P 1.80.0672.48 1.13 2.193.40.0750.0614S 2.70.047 2.220.87 2.56 1.70.0780.0504P 1.60.038 1.560.77 2.04 3.60.0850.0478S 2.30.028 1.230.54 2.29 2.40.0770.0338P 1.60.023 1.000.57 1.77 6.30.0680.02712S 2.10.0220.830.44 1.914.70.0550.01812P1.70.0260.850.471.845.20.0530.016a These design variants have better-than-average beam and column detailing.bThese design variants have better-than-average joint detailing.JOURNAL OF STRUCTURAL ENGINEERING ©ASCE /APRIL 2011/497D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y S u l t a n Q a b o o s U n i v e r s i t y o n 06/21/14. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .。
