Performance Assessment Through Nonlinear Time History Analysis

Performance Assessment Through Nonlinear Time History Analysis EQ: 11121, Sa: 2.86g EQ: 11122, Sa: 2.32g FEMA 356 Scaling Method: SRSS of Spectra ) 25 N k 2 ( 5 4 e c r o F ] g [S 3 S R as 2 Experimental Results Model Prediction 15 1 5 r a e -5 h S -1 S 1-15 -2.5 1 1.5 2 Period (seconds) 2.5 3 Greg Deierlein Stanford University with contributions by Curt Haselton & Abbie Liel Stanford University -25-15 -1-5 5 Column Top Horizontal Deflection (mm) 1 15

PBEE Current Best Practice: FEMA 273/356 V δ t δ roof Global Pushover Response Nonlinear Pushover Analysis - Modeling Assumptions - Force Distribution - Target Displacement (Sa) Component Modeling Criteria - Backbone Curve Component Acceptance Criteria - Force Controlled Elements - Deformation Controlled Elements 2

Example: Criteria for RC Beams (FEMA 273) 3

Shortcomings of FEMA 273/356 5 ) N 4 k ( Experimental Results 3 e c r o F 2 1 r a e -1 h S -2 Element backbone predicted by FEMA 356 (Table 6-8) -3-4 -5-1 -5 5 1 15 Column Top Horizontal Deflection (mm) Component Backbone Curve: - Overly Idealized - Conservative - Deterministic 4

Shortcomings of FEMA 273/356 Component Force Immediate Occupancy Life Safety Collapse Prevention Component Backbone Curve Component Deformation Over-reliance on idealized (simplified) local component demand indices to predict system response Ambiguous relationships between structural indices and building performance Limited emphasis on static monotonic pushover approach 5

Assessment Using Improved NLTH Analysis Nonlinear Component and System Modeling FEMA 356 Concepts with NLTH Analysis Preview of Comprehensive Collapse Simulation

Structural System & Components GC S J2 C B J1 F Floor Framing Plan Perimeter Frame Elevation Moment Frames Beams, Columns, B-C Joints, and Foundations Gravity Frames Slab/beams, Gravity Columns, S-C Joints, and Foundations Shear Walls (not shown) Issue: Whether or not to consider the lateral resistance of the gravity system in the simulation. There gravity system can provide significant enhancement in a nonlinear assessment.

Deterioration Modes & Collapse Scenarios 1. Deterioration Modes of RC Elements - Simulation vs. Fragility Models 2. Building System Collapse Scenarios - Sidesway Collapse (SC) - Loss in Vertical Load Carrying Capacity (LVCC) 3. Likelihood of Collapse Scenarios - Existing vs. New Construction - Ordinary versus Special seismic design

Simulation Model Gravity Frame(s) Joints with both bond-slip springs and shear springs Lateral Frame Column bondslip springs Gravity Frame(s) Lateral Frame Lumped plasticity beam-columns

More realistic component simulation 1.5 OR OTHER COMPUTER CODES Normalized Moment (M/M y ) 1.5 -.5-1 Non-Deteriorated Backbone -1.5 ) N k ( e c r o F 25 2 15 1 5 Experimental Results Model Prediction -8-6 -4-2 2 4 6 8 Chord Rotation (radians) r a e h S -5-1 -15-2 -25-15 -1-5 5 1 15 Column Top Horizontal Deflection (mm) 1

Illustration: Axial Load & Post-Peak Response P.6 f c Ag M.15 f c Ag Key Parameter: P/P balance

12 Ref. Haselton, Liel, Deierlein (12 story IMF system)

n ti nu um Drift nc e H i n tr a n g te d e Co Physical In e D is la s tr ib tic u te ity d ( fib e r) Co Column Shear Beam-Column Modeling Alternatives Phenomenological

Beam-Column Model Considerations Flexural Deformations - concrete cracking/tension stiffening - reinforcing bar yielding - concrete crushing Shear Deformations - uncracked, cracked Anchorage Bond Slip - pre and post-yield Critical Failure Modes and Deterioration - lateral tie fracture concrete crushing, rebar buckling - longitudinal bar buckling/fracture - PMV interaction

Beam-Column Model Considerations, cont d Definition of Displacements and Deformations - Total Δ = Distortional (or Natural ) Δ + Rigid Body Δ - Total Δ = Clear Story Drift - Damage is typically associated with distortional deformations Δ = Δn Drift from Experiment Δ = Δn+ ΔRBR Drift from Frame Analysis

Standardized Inelastic Component Models 1.5 M, Column Base Moment Θ M 1.5 -.5-1 Concentrated hinge Quasi-elastic (EI, EA) -1.5-8 -6-4 -2 2 4 6 8 _, Chord Rotation Model Input - Physical Design Parameters (material, configuration, geometry, details, ) - Calculated/calibrated backbone parameters (mean and COVs for anchor points and hysteretic response parameters) Model Output: Engineering Demand Parameters (e.g., Θ plastic )

Element Models for Collapse Analyses Montonic Backbone M, Column Base Moment Θ M 1.5 1.5 -.5-1 -1.5-8 -6-4 -2 2 4 6 8 _, Chord Rotation Curve (5 parameters, µ and σ) Cyclic Strength & Stiffness Degradation (µ, σ for significant effects)

RC Beam-Column Simulation Model Calibration "! Cap,pl OVERVIEW OF CALIBRATION EFFORT Basic Hysteretic Model My Ke #Ke 5 parameter backbone curve 2 (x4) hysteretic parameters Previous RC Behavioral Studies! y! -Fardis et al. (Θcap, Θu) ) N k ( e c r o F r a e h S 25 2 15 1 5-5 -1 Experimental Results Model Prediction -Eberhard et al. (EDP criteria for spalling and bar buckling) Current effort: Systematic calibration to 226 flexurally dominated columns -15-2 -25-15 -1-5 5 1 15 Column Top Horizontal Deflection (mm) Goal: Validated model to be vetted through consensus process

RC Beam-Column Parameters ) N k ( e c r o F r a e h S 2 15 1 5-5 -1-15 Experimental Results Model Prediction Θcap Test 19 (kn, mm, rad): K e = 3.1779e+7 K init = 7.424e+7! s =.2! c = -.4 (ND = 1) " y =.91 " cap,pl =.69 (LB = 1) " u,mono,pl =.116 (LB = 1) # = 85, c = 1.2 ispdeltaremoved = 1 Semi-Empirical -- calibrated from tests, fiber analyses, and basic mechanics: Secant Stiffness (EI eff ) Yield Strength (My) Hardening Stiffness -2-1 -5 5 1 15 Column Top Horizontal Deflection (mm) Empirical - calibrated from tests: Capping (peak) point Post-peak unloading (strain softening) stiffness Hysteretic stiffness/strength degradation

Column Models (sample data) "! Cap,pl My Ke #Ke Θ Cap rotation includes bond slip! y! COMPONENT Θ cap,pl (RAD) COV α COV Beam - Conforming.7 6% -.5 6% Beam - Nonconforming.2 to.5 -.15 Column conform, low axial.4 to.5 -.5 Column nonconf, low axial.2 -.15 Column nonconf, med. axial.1 -.15

Phenomenological P-M-V Hinge Element V Shear Spring P P Axial Spring γ M Δ vert Desired Model Features: direct modeling of P-M interaction through limit surface (strength, post-peak softening, hysteretic degradation) DIRECT SIMULATION (as opposed to limit state check) of column shear failure and axial failure (LVCC) More transparent modeling of flexibility introduced by bond slip and shear deformations.

Beam-Column Joint Models Bond slip (pullout?) Concentrated rotations Due to Rebar yielding Shear deformation

Alternative Joint Models Single spring joint Single Spring joint with rigid zones Continuum model Multi spring super element

Idealized Joint Model Panel zone Column Spring Beam Spring Joint Panel Spring Beam-end zone Component Springs Column-end zone Joint Kinematics

Shear Wall Systems Behavior Modes Flexural Behavior - concrete cracking/tension stiffening - reinforcing bar yielding - concrete crushing - tie rupture rebar buckling/fracture Shear Behavior - uncracked, cracked - shear failure Anchorage Bond Slip (base only) - pre and post-yield Coupling Beams Foundations System Compatibility - slab/column & slab/wall - column deformation

Idealization of RC Walls V-Δ s Zer Le M-Θ f V-Δ s X X X X X Zer Le M- φ f Continuum Multi-Spring Concentrated Spring

Shear Wall Modeling and Behavior Squat (short) walls versus tall flexural walls Inelastic Time History versus Equivalent Static Loading - surprising variations in flexural and shear demands - shears can be much higher than predicted by pushover analysis (e.g., Krawinkler & Zareian) Veq 2/3H Veq <1/4H Pushover Time History

Viscous Damping with NLTH Analysis [ M ]{ && x} + [ C]{ x& } + [ K]{ x} =![ M ]{ & x } [ P] g + Raleigh Damping: [ C ] "[ M ] +![ K]; = $ n = +! n 2! n 2 For inelastic analysis, viscous damping should be on the order of ζn = 5%. Need to be careful how [K] is specified, i.e., [K e ] versus [K t ], since the choice will lead to variations in [C] during the analysis (see Bernal). Explicit Damping Elements (preferred?): [ C ] =![ c] i [c] i [c] i configured to represent likely sources of viscous and other incidental damping. # 1 "

Geometric Nonlinear (P-Δ) Effects Negative stiffness effect of P-Δ: V K g = -W/h Δ Increases internal forces associated with overturning: V W h W = ΣP g M ot =Vh + PΔ Key Points W should represent the seismic mass that is being stabilized by the lateral system (not just the tributary gravity load) Linear P-Δ formulations accurate for drift ratios up to about 5-1%; beyond this large rotation (e.g., co-rotational ) formulations should be used to track response.

Dynamic Sidesway Collapse Δ u V w/strain softening Sa = 1.28g w/p-! Δ u Static Pushover Analysis (Critical Story)! u! Sa = 1.23g NLTH Analysis Results For structural systems governed by sidesway collapse, evidence suggests that the dynamic drift capacity is about 2/3 of the static pushover limit (Δ u ) --- provided that the static analysis represents strength degradation due to strain (displacement) softening and P-Δ effects. Collapse point can be very sensitive to ground motion intensity level and other effects.

Assessment Using Improved NLTH Analysis Nonlinear Component and System Modeling FEMA 356 Concepts with NLTH Analysis Preview of Comprehensive Collapse Simulation

FEMA 356 with NL Time History Analysis NL Analysis Model modeling assumptions (role of component backbone curves?) Selection of GM match M, R and fault type/mechanism records from at least 3 events synthetics OK if necessary Scaling of GM to UHS to design EQ 5% damped spectra from SRSS of two orthogonal components Scale such that SRSS spectra > 1.4 UHS for periods between.2t 1 and 1.5T 1 32

Ex. Ground Motion Scaling to 1/5 Hazard 4 3.5.2T 1 FEMA 356 Scaling: Individual Component Spectra 1.5T 1 ] g [ t n e n o p m a c o S 3 2.5 2 1.5 1 UHS.5.5 1 1.5 2 2.5 3 Period (seconds) 2 components for 1 pairs of EQ Records 33

FEMA 356 with NL Time History Analysis Acceptance Critera Demand Parameters < Component Criteria e.g., Θ p < Θ p,limit Evaluation based on either: Maximum demand from results of 3 records Average demand from results of >7 records Concerns: - statistical rationale for acceptance criteria? - implementation (which 3 records)? 34

Enhanced FEMA 356 Realistic Inelastic Model Nonlinear Time History Analysis 2 ground motions (1 pairs) with their geometric mean scaled to hazard at Sa(T1) Statistical evaluation of deformation demands to input ground motions Probabilistic assessment of component acceptance criteria to test data Probability[Θ p >Θ p,limit-state ] = X 35

Illustration 4 Story SMF Building Office occupancy Los Angeles Basin Design Code: 23 IBC / 22 ACI / ASCE7-2 Perimeter Frame System Maximum considered EQ demands: S s = 1.5g; S 1 =.9g S a(2% in 5 yr) =.82g Design V/W of.94g Maximum inelastic design drift of 1.9% (2% limit) 36

RC Beam & Column Component Models NL Joints Lumped plasticity beamcolumn elements OpenSees Model Lumped plasticity beams, columns, and joints with strength/stiffness degradation Geometric NL (P-Δ) 2 ground motions (1 pairs) " My! y Ke! Cap,pl #Ke! Θ pl,cap of conforming members: Columns (low axial) Mean =.5 rad COV = 4% Beams Mean =.65 rad COV = 4%

Peak Interstory Drift Demands ] g [ ) c e s 1 = T (. m a ġ S 1.2 1.8.6.4.2 At 2% in 5 year (MCE) Sa: IDRmax =.16 to.5 Mean IDR max =.28 COV = 37%.2.4.6.8 Peak Story Drift Ratio (story three) 2 time history analyses at each of 5 hazard levels peak inter-story drift ratio from each time history analysis ground motions are scaled to hazard spectra over the region.2t 1 to 1.5T 1. 38

Probabilistic Measures of Drift Demand ] g [ 1.2 ) c e 1 s 1.8 = T. m.6 ( a ġ S y t i s n e D y t i l i b a b o r P 45 4 35 3 25 2.4.2 5.2.4.6.8 Peak Story Drift Ratio (story three) Mean IDR =.28.1.2.3.4.5.6.7 Interstory Drift Ratio, Story Three Fitted Lognormal PDF 15.3 P(D<.2) P(D<.2) =.15 1.2 PDF (probability density function) At 2% in 5 year (MCE) Sa: IDRmax =.16 to.5 Mean IDR max =.28 COV = 37% ] t f i r d < T F I R D [ P 1.9.8.7.6.5.4 P(D<x) =.5 median.1 Empirical CDF Fitted Lognormal CDF.1.2.3.4.5.6.7 Interstory Drift Ratio, Story Three CDF (cumulative density function) 39

Beam and Column Plastic Rotation Demands ] g [ ) c e s 1 = T. m ( a ġ S ] g [ ) c e s 1 = T (. m a ġ S 1.2 1.8.6.4.2.2.4.6.8.1 Peak Beam Plastic Rotation (rad) 1.2 1.8.6.4.2 At 2% in 5 year (MCE) Sa: Beams: Θ p,max =.12 to.45 Mean Θ p,max =.25 COV = 43% (vs. FEMA 356 Θ cp <.25) Columns: Θ p,max = to.3 Mean Θ p,max =.1 COV = 11% (vs. FEMA 356 Θ cp <.2).1.2.3.4.5.6 Peak Column Plastic Rotation (rad) 4

Probabilistic Limit State Assessment t i s n e D y t i l i b a b o r P 9 8 7 6 5 4 3 2 1 Θ pl,max Column Plastic Rotation Demand Column Plastic Rotation Capacity " My Ke Θ pl,cap! Cap,pl #Ke Θ pl,cap! y! Beams Demand (Θ pl,max ) µ σ ln.25.43 Capacity (Θ pl,cap ) µ σ ln.65.4.1.2.3.4.5.6.7.8.9.1 Peak Column Plastic Rotation Demand Col s.1 1.1.5.4 P ln, z [ D ) C] = 1' ( $! % * ln, z " where: & ' µ µ! ln, z = µ ln, D µ ln, C! + # 2 2 2 ln, z =! ln, D! ln, C Φ (standard normal table) Component Limit State Checks: Beams P[D>C] = 5.6% Columns P[D>C] = 5.6% (just a coincidence that they turn out the same) 41

Comments Advantages -More transparent and rigorous assessment of component limit state criteria -Framework to incorporate available test data (outside the scope of FEMA 356) Limitations and Issues - Requires judgment to select appropriate limit states and the probabilistic acceptance criteria, i.e., P[D>C] at some hazard level -Still limited by assumptions between component and system performance. - Does not incorporate variability and uncertainties in structural system behavior. 42

Assessment Using Improved NLTH Analysis Nonlinear Component and System Modeling FEMA 356 Concepts with NLTH Analysis Preview of Comprehensive Collapse Simulation 43

Performance-Based Earthquake Engineering O P E N O P E N O P E N Base Shear Damage Threshold Collapse Onset Deformation PBEE today PEER & ATC 58 vision IO LS CP 25% 5% 1% FEMA 356 Performance Levels $, % replacement..1.1.1.25 1 7 3 18 Casualty rate Downtime, days

PBEE COLLAPSE (SAFETY) Assessment Decision Variable Damage Measure Engineering Demand Parameter Intensity Measure DV: COLLAPSE DM: Loss of Vertical Carrying Capacity (LVCC) EDPs: Deformations & Forces EDP: Interstory Drift Ratio IM: Sa(T 1 ) + Ground Motions 45

Incremental Dynamic Analysis Concept Sa T1 1. Given: Inelastic Analysis Model IM Sa sc Sa LVCC X X 2. Select and scale earthquake ground motion to specified earthquake intensity (IM) 3. Perform nonlinear time history analysis 4. Record and plot engineering demand parameter (EDP) IDR LVCC IDR sc EDP Peak Interstory Drift Ratio 5. Repeat steps 2-5 until system collapse is observed through analysis 6. Perform check for local LVCC conditions that are not simulated in analysis 46

Sidesway Collapse Simulation 4 Story SMF 47

Incremental Dynamic Analysis Collapse ] g [ ) s. 1 = T (. m a ġ S 4 3.5 3 2.5 2 1.5 1 2% in 5 year =.82g.5 Capacity Stats.: Median = 2.2g σ LN =.36 Median col = 2.2g σ LN, col =.36g IDR col = 7-12%.82g is 2% in 5 year motion.5.1.15 48

g Sidesway Collapse Modes EQ: 1121, Sa: 2.52g EQ: 11151, Sa: 2.51g EQ: 1191, Sa: 2.19g EQ: 11131, Sa: 2.19g EQ: 11122, Sa: 2.32g 4% of collapses EQ: 11161, Sa:.66g EQ: 1111, Sa: 1.52g EQ: 11141, Sa: 1.79g 17% of collapses 5% of collapses EQ: 1122, Sa: 2.12g EQ: 11152, Sa: 2.26g EQ: 1192, Sa: 3.6g EQ: 11132, Sa: 2.12g EQ: 11141, Sa: 1.79g EQ: 11142 27% of collapses EQ: 11162, Sa:.72g EQ: 1112, Sa: 1.6g EQ: 11142, Sa: 1.32g 12% of collapses 2% of collapses 49

Incremental Dynamic Analysis Collapse ] g [ ) s. 1 = T (. m a ġ S 4 3.5 3 2.5 2 1.5 1 Capacity Stats.: Median = 2.2g σ LN =.36 Median col = 2.2g σ LN, col =.36g.82g is 2% in 5 year motion.5.5.1.15 5

Collapse Capacity Empirical and Lognormal Fit Median Collapse, Sa (T=1sec) = 2g 2% in 5 yrs P[collapse 2% in 5 yrs] < 1% 51

Uncertainty Plastic Rotation Capacity 1.2 1.2 1. 1. Normalized Moment (M/My).8.6.4.2 Mean (µ) Plastic Rotation Capacity Normalized Moment (M/My).8.6.4.2 Mean minus standard Reduced deviation (lognormal) for both plastic (µ σ) rotation capacity and Plastic post-capping Rot. stiffness Cap....2.4.6.8.1 Total Chord Rotation (radians)...2.4.6.8.1 Total Chord Rotation (radians) ] g [ ) s. 2 = T ( p m o a c S 1.4 1.2 1.8.6.4 ] g [ ) s. 2 = T ( p m o a c S 1.2 1.8.6.4.2.2.5.1.15 Maximum Interstory Drift Ratio.5.1.15 Maximum Interstory Drift Ratio 52

Collapse Capacity with Modeling Uncert. σ LN, modeling =.5g (vs..34g R-R) 2% in 5 yrs P[collapse Sa =.82g] = 1% (increased variability) 53

Combined Sidesway and Vertical (LVCC) Collapse P[ C IM = im] = P[ C IM = im] + Total Collapse Probability SIM ] + P[ C NC, IM = im]! P[ NC IM = im] DM SIM SIM = Sidesway Collapse Probability of LVCC Probability at IM + i (given drift ratio) X Probability of no SS Collapse at IM i S a (g) at (T 1 ) Peak Interstory Drift Ratio P (DM EDP) Probability of LVCC 1..9.8.7.6.5.4.3.2.1. % 1% 2% 3% 4% 5% 6% EDP (IDR) LVCC (e.g., slab failure, column shear-axial failure) Peak Interstory Drift Ratio Plot is shown for illustration purposes; not calibrated to test data. 54

Collapse Capacity Simulation + LVCC P[LVCC MCE] MCE (2/5) Plot is shown for illustration purposes; not calib rated to test or analysis data. 55

Concluding Remarks Benefits of Assessment by NLTH Analysis More explicit simulation of cyclic and dynamic effects Transparent and extendable to innovative systems and materials Challenges with NLTH Analysis Calibration/Validation of Hysteretic Component Models Selection and Scaling of Input Ground Motions Computational hurdles (convergence, runtime, post-processing) Standardization of Structural Component Models & Criteria Simulation & fragility models Statistically neutral models i.e., mean and COV Important role for material standards organizations (e.g., ACI) Future Vision -- Explicit Assessment of Collapse Risk 56

References Deierlein, G.G. and C.B. Haselton (25). Benchmarking the Collapse Safety of Code-Compliant Reinforced Concrete Moment Frame Building Systems. ATC/JSCA US-Japan Workshop on Improvement of Structural Design and Construction Practices, Proceedings of an International Workshop, Kobe, Japan, October 17-19, 25, 12 pp. (in press) Deierlein, G.G. and C.B. Haselton (25), Developing Consensus Provisions to Evaluate Collapse of Reinforced Concrete Buildings, US-Japan DaiDaiToku/NEES Workshop on Seismic Response of Reinforced Concrete Structures, PEER Center, Berkeley, California, July 7-8, 25, 17 pp. Haselton, C.B. AND G.G. Deierlein (26), Toward the Codification of Modeling Provisions for Simulating Structural Collapse, 8NCEE, San Francisco, CA, April 18-22, 26, 1 pp. (in press). Fardis, M.N. and D.E. Biskinis, 23. Deformation Capacity of RC Members, as Controlled by Flexure or Shear, Otani Symposium, 23, pp. 511-53. Krawinkler, H., Miranda E., Bozorgnia, Y. and Bertero, V. V., 24. Chapter 9: Performance Based Earthquake Engineering,, Earthquake Engineering: From Engineering Seismology to Performance-Based Engineering, CRC Press, Florida. Filippou, F., Fenves, G., Bozorgnia, Y. and Bertero, V. V., 24. Methods of Analysis for Earthquake Resistant Structures,, Earthquake Engineering: From Engineering Seismology to Performance-Based Engineering, CRC Press, Florida. Panagiotakos, T.B. and M.N. Fardis, 21. Deformations of Reinforced Concrete at Yielding and Ultimate, ACI Structural Journal, Vol. 98, No. 2, March-April 21, pp. 135-147. PEER, 25. Pacific Earthquake Engineering Research Center: Structural Performance Database, University of California, Berkeley, http://nisee.berkeley.edu/spd/ (March 1, 25). Ibarra, L.F., R.A. Medina, and H. Krawinkler 25. Hysteretic models that incorporate strength and stiffness deterioration, Earthquake Engrg. and Structural Dynamics, Vol. 34, pp. 1489-1511. Open System for Earthquake Engineering Simulation (Opensees). (25). PEER Center, University of California, Berkeley, http://opensees.berkeley.edu/. Bernal, D. (1994). "Viscous damping in inelastic structural response", Journal of Structural Engineering, ASCE, Vol. 12, No.4, pp.124-1254. 57