Inelastic Design Response Spectra

Transcription

1 Inelastic Design Response Spectra Developed by statistical processing of actual nonlinear spectra for ground motions, hysteretic characteristics, ξ and damage indices of interest. η plastic /g µ=3 Constant Ductility Spectra for various records Period Developed by modification of an elastic design spectrum. /g R µ=6 Elastic Response Spectrum µ=3 µ=6 µ=3 Period Period U.C. Berkeley Spring 2009 UC Regents 9-1

2 Scope of Discussions Prefer: Actual spectra when site specific motions used. Empirical modification factors R and γ when elastic response spectrum specified. Look at typical empirical modification methods for ideal EPP systems Newmark and Hall (et al) Multi-linear relations (Hildalgo, Riddell, AC-32) Continuous functions (Miranda, Krawinkler, et al) Next Section: Examine briefly effect of and modifications for: P-Δ effects Shape of hysteretic loops Special ground motion characteristics soft soils near fault motions Viscous damping Duration of shaking other damage indices Compare to code provisions hen: Extend to multiple degree of freedom systems U.C. Berkeley Spring 2009 UC Regents 9-2

3 Log S V Newmark and Hall IDRS Method Acceleration/Yield Force Spectrum Log D Log S Log D D LogS D C B C Log S B v R E A Short Periods Acceleration Preserved R = 1 γ = µ Amplified Acceleration Energy Preserved R = 2µ! 1 " = µ 2µ! 1 A Log Log Max. Displacement Spectrum Long Periods Log Displacement D LogS D Preserved C R=µ B Log S v γ=1 γ E A E Log U.C. Berkeley Spring 2009 UC Regents 9-3

4 Newmark and Hall IDRS (Continued) his can easily be plotted using conventional axes. B C R = 2µ! 1 R = µ A E o get maximum displacement d max.nl = µ d y = γ d elastic γ 1 γ=µ µ! = 2 µ "1 γ=1 Note: Q y =S a nl m So: d y = S a nl / ω 2 S D Inelastic Elastic S D =constant S D S D =d g Q y d y d max U.C. Berkeley Spring 2009 UC Regents 9-4

5 A Note on Displacement Estimates d max = µd y = µq y /K So: = µ(q elastic /R)/K = (µ/r) ( elastic M)/K = (µ/r) S D elastic = γ S D elastic γ = µ / R hus, if we have a relation that defines R as a function of µ [R=f(µ,Τ)], we can estimate the maximum displacement using γ = µ/ f(µ,τ), However, the R function does not exactly fit the data (µ actual µ target ), so if we use γ = µ target /R our results will be slightly in error. R 1 R = f(µ,) Actual Ave. Data Period We only have a few analytical expressions for γ = f(µ, Τ) U.C. Berkeley Spring 2009 UC Regents 9-5

6 Empirical Modification Factors for IDRS Empirically derived equations have been developed to estimate R for a given displacement ductility demand on a structure. Comprehensive review by: E. Miranda and V. Bertero in Evaluation of Strength Reduction Factors for Earthquake-Resistant Design, Earthquake Spectra, EERI, Vol. 10, No. 2, µ 1 R = /ω 2 = µ /R = 1 / ω 2 X µ = S D γ S D X = U.C. Berkeley Spring 2009 UC Regents 9-6

7 Some R factors for µ = 5 6 R 6 R 6 R Newmark/Hall Riddell/Newmark Period, Sec Riddell, Hildalgo, Cruz Vidic, Fajfar, Fishinger Mohraz Period, Sec Miranda Krawinkler Hildalgo/Arias Period, Sec. Simplified form used in several codes and guidelines Analytical format good for use with statistical relationships for elastic spectrum U.C. Berkeley Spring 2009 UC Regents 9-7

8 Miranda s Relationships for R Acceleration Modification Factors See: E. Miranda, Site Dependent Strength Reduction Factors, Journal of Structural Engineering, ASCE, Vol. 119, No. 12, Dec R = µ! 1 " + 1# 1 where" is a function of µ, and soil conditions, such that 1 " = 1+ 10! µ! 1 2 exp (! 3 $ 2 ln! 3 & 2+ * ) % 5' -, " = ! µ! 2 5 exp (!2 $ ln! 1 & * ) % 5' 2+ -, for rock sites for alluvial sites " = 1+ S 3! 3 ( S 4 exp!3 $ ln! 1 & 2+ *. / - for soft soil sites % s 4' ), U.C. Berkeley Spring 2009 UC Regents 9-8

9 Miranda Data for Rock / Alluvial Sites U.C. Berkeley Spring 2009 UC Regents 9-9

10 CoV for Rock/Alluvial Sites U.C. Berkeley Spring 2009 UC Regents 9-10

11 Effect of Distance & Magnitude U.C. Berkeley Spring 2009 UC Regents 9-11

12 Near-Fault and Soft Soil Sites U.C. Berkeley Spring 2009 UC Regents 9-12

13 Some Applications Simplified Linear Relations B R = 2µ! 1 1/ B C R R = µ A E E Newmark & Hall Continuous Relations Elastic Spectrum ( =f(m,r,soil,etc.)) R E U.C. Berkeley Spring 2009 UC Regents 9-13

14 Displacement Estimates γ = µ /R Displacement Amplification Factors See: Miranda, E., Estimation of Maximum Interstory Drift Demands in Displacement-Based Design, Seismic Design Methodologies for the Next Generation of Codes, Balkema, Rotterdam, # #! = 1 + % 1 $ µ "1 & % ( exp "12µ "0.8 $ ' γ γ = d plastic /d elastic 1 ( ) U.C. Berkeley Spring 2009 UC Regents 9-14 & ( ' "1 Weakest (µ.>>1) Weak Elastic (µ>>1) Note: his is simply an equation that satisfies the general boundary conditions. It does not satisfy the requirement that γ = µ /R

15 Newer Displacement Amplification Factors For a comparison of various methods of estimating nonlinear displacements from elastic response spectra, see: Statistical Evaluation of Approximate Methods for Estimating Maximum Deformation Demands on Existing Structures, S. Akkar and E. Miranda, Journal of Structural Engineering, ASCE, Jan Compares several classical and newer methods based on γ - factor approach and use of effective stiffness and periods Iwan and Guyander (2002) Ruiz-Garcia and Miranda (2003) (C R =γ) U.C. Berkeley Spring 2009 UC Regents 9-15

16 Comparison of Predictions Newmark & Hall Ruiz-Garcia & Miranda Iwan & Guyander Note: S d, R and γ spectra from BiSpec are exact for selected motions CoV s are around 30-40% U.C. Berkeley Spring 2009 UC Regents 9-16

17 Comments on Formats Acceleration/Yield Force Spectrum Log D LogS D Log S B C v R A E Log Max. Displacement Spectrum Log D LogS D C B Log S v A γ E Log - and S D - Formats Elastic Spectrum S D S D -S D Format Yield Point Spectrum NL Disp. Spectrum U.C. Berkeley Spring 2009 UC Regents 9-17

18 Some Example Applications Design - Use N & H methods = 1.2 sec (const. velocity range) elastic = 0.6g µ= 4 nl = 0.6g / 4 = 0.15g Q y = S a M = 0.15gM = 0.15W nl d y = nl /ω2 = 0.15 (386.4 in/sec 2 )/ [2π/(1.2sec)] 2 = 2.11 in. R=µ γ 1.4g 0.6g 1.2 sec d max = elastic x 1 = 4 x 2.11 = 8.44 inches => d plastic = 6.33 µ = 2 => Q y = 0.3W & d max = 8.44 inches => d plastic = 4.22 U.C. Berkeley Spring 2009 UC Regents 9-18

19 Some Example Applications (2) Design - displacement reduced to 7.5 guess = 1.2 sec & elastic = 0.6g d max 7.5 inches max = 7.5 (1.2sec) / 8.44 = 1.07 sec. Still on velovity segment of curve? o s /2π=S v1.2 = 1.2 (1.2sec)/2π o = 0.6g (1.2sec) / 1.4g = sec. OK So, if M is constant, K new = (1.2/1.07) 2 K old = 1.27 K old = 0.6g (1.2 s)/1.07 s. = 0.67g S a elastic Q y = S a M = 0.67W/4 = 0.17W nl d max = S d x 1 = 7.5 inches nl 1.4g? 0.6g S D 1.2 sec? 1.2 sec d plastic = = 5.6 U.C. Berkeley Spring 2009 UC Regents 9-19

20 Evaluation Some More Applications Structural Analysis 20kips Q 6.0 = 1.2 sec > sec (displ. preserved) R = elastic /nl = Q elastic /Q y =0.6g*m /20kips = 0.6(100k)/20k = 3 µ = R = 3 (for this period range) d d max = γd elastic = 1 S a elastic /ω2 W=100kips k = 7.11 k/in. 0.6g 0.2g 8.44 S aplastic /g = 20k/gM = sec = 1(0.6)(386.4 in/sec 2 )/ [2π/(1.2sec)] 2 = 8.44 in.> 6 in. NG S D 1.2 sec U.C. Berkeley Spring 2009 UC Regents 9-20

21 Comment Note: for other period ranges, other relationships need to be used for the Newmark and Hall method. See previous equations relating µ and γ as a function of R It is straight forward to use empirical modification factors other than developed by Newmark and Hall In these other relationships, it may not be as simple to assess design changes necessary to change displacement or ductility to desired values. U.C. Berkeley Spring 2009 UC Regents 9-21

22 Ductility Versus Displacement Consider two systems having the same stiffness designed to have the same ductility demand. Q 1 2 If we are in the displacement preserved range Q 1 2 d max If µ 1 = µ 2, then d max.1 /dy 1 = d max.2 /d max.2 => d max1 d max2 d if Q y1 = 2Q y2, => µ 1 = µ 2 /2 but... d plastic = d max - d y so µ = 1 + d plastic /d y if µ 1 = 3 and µ 2 = 6, then µ => +100% d pl => +25% d plastic.1 = 4 d y2 and d plastic.2 = 5 d y2 U.C. Berkeley Spring 2009 UC Regents 9-22

23 Displacement vs. Ductility If we double the stiffness For the energy preserved range, similar calculations Q 1 can be done! Q Elastic 2 d 1 max increases slower than µ d y1 / 2 d y1 0.7d max1 d max1 Assuming Displacement Preserved hus, µ 2 = 0.7d max1 /0.5d y1 = 1.4µ 1 but d max2 = 0.7 d max 1 and d plastic2 = 0.7d max1 -d y1 /2 so, for µ = 3, d plastic2 = 0.53d plastic1 Q 2 d Elastic 1 d max may decrease 2 d U.C. Berkeley Spring 2009 UC Regents 9-23

24 Summary While there are many different equations used to represent the variations of R and γ with, µ and soil effects, they exhibit similar trends and all give similar results. Many codes are beginning to incorporate these factors directly or indirectly Increased focus on displacements Next Other factors: Hysteretic and viscous damping P-Δ effects Near-fault motions Pending factors for PBEE: Cummulative low-cycle fatigue damage Residual displacements U.C. Berkeley Spring 2009 UC Regents 9-24