Sanaz Rezaeian Postdoctoral Research Fellow, PEER, UC Berkeley Group Members: I.M. Idriss

1. NGA-West 2 Project:Scaling of NGA Models for Damping (December 7, 2010) SanazRezaeian Postdoctoral Research Fellow, PEER, UC Berkeley Group Members: I.M. Idriss YousefBozorgnia Ken Campbell Walter Silva Norm Abrahamson

2. Outline • Review (Literature, Trends in data) • Database • Regression for given T and β • Residual plots  Functional form • Regression coefficients (and sigma) in terms of β

3. Review http://peer.berkeley.edu/ngawest2_wg/workshops_damping_scaling.html

4. Review • Definition: • Existing models have looked at dependence of DSF on • β: damping • T: period (frequency) • D: duration • M: magnitude • R: distance • Site class • Tectonic setting (WUS, CEUS) • PGA  Common factors in modeling  Very few models

5. Review • Two common formulations: Newmark & Hall, 1982 (1978) American building codes Idriss, 1993 EuroCode 8, 2004 Priestley, 2003 2001 Caltrans Tolis & Faccioli, 1999 1990 French, 1994 Spanish code

6. Review • Random Vibrations Theory: • Assuming White-Noise

7. Review (formulas with M or Dur) • Abrahamson and Silva 1996: • Stafford et al. 2008: • x is a measure of duration • Recommended R.V. method by McGuire et al., 2001 (NUREG/CR-6728) 0.2< T <1 : Rosenblueth (1980) T <0.2 : Vanmarke (1976)

8. Review (nonlinear regression, thousands of mathematical eqns) • Lin and Chang, 2003: • Lin and Chang, 2004: a is a function of site and β. b and c depend on the site class. Site class can be neglected if 5% error is acceptable in structural design and if β < 20%. • Hatzigeorgiou, 2010: DSF(T,β) = 1 + (β −5) [1+c1 ·ln(β)+c2 ·(ln(β))2] · [c3+c4 ·ln(T)+c5 ·(ln(T ))2]

9. Review (DSF is Tabulated / Plotted) • Cameron & Green, 2007: • frequency content and durationM, distance, and tectonic setting. • For β>=2%: DSF depends on T, β, M, WUS/EUS, Rock/Soil • For β=1%: Distance (significantly influences duration) • Bommer & Mendis, 2005 • Atkinson & Pierre, 2004

10. Review • Assumed DSF is lognormally distributed: ln(DSF) = µ (β,T,M,…) + ε ; ε~N( 0 , σln(DSF) ) • Note: • Linear combination of independent normally distributed random variables is normal. Normal Normal in GMPE Normal in GMPE

11. Distribution of DSF: (at a given T) T=0.2 s T=1 s T=7.5 s Fitted pdf to ln(DSF) is normal  testing the lognormality of DSF

12. Distribution of DSF: (at a given T and 2% damping) T=0.2 s T=1 s T=7.5 s Fitted pdf to ln(DSF) is normal  testing the lognormality of DSF

13. Distribution of DSF: (at a given T and 20% damping) T=0.2 s T=1 s T=7.5 s Fitted pdf to ln(DSF) is normal  testing the lognormality of DSF

14. Exception At: T = 0.1 s & Very low damping or Very high damping

15. ln(DSF) = µ (β,T,M,…) + ε ; ε~N(0,σ) Functional Form ?

16. Review Influence of Period, T:

17. Review Influence of damping, β: 35

18. Review • Looked at data pattern with Duration, Magnitude, Distance, and Vs30: • There is pattern with ln(Duration). • Increases at long periods. • There is significant dependence on M (linear or quadratic). • Increases at long periods. • Insignificant dependence on Rrup. • Insignificant dependence on Vs30.

19. Database • 6,056 Records (5,819 have PSA at T=0) • 4.265 ≤ M ≤ 7.9 (reported for 5,856) • 0.07 ≤ Rrup ≤ 1529 km (reported for 4,878) • 93.7 ≤ VS30 ≤ 2016.1 m/s (reported for 5,845) • GMRotI50 is used to calculate DSF • Data for Duration: • Average (arithmetic) duration of H1 and H2 components.

20. Database:

21. Database:

22. Database:

23. Database:

24. Regression (Given T and β) Step 0: ln(DSF) = c0+ε ; ε~ N(0,σ) Step 01: ln(DSF) = c0+ c1M+ ε Step 02: ln(DSF) = c0+ c1M+ c2M2+ ε Step 03: ln(DSF) = c0+ c1M+ c2ln(R)+ ε ln(DSF) = c0+ c1M+ c2ln{(R2+c32)1/2}+ ε Step 04: ln(DSF) = c0+ c1ln(D5-95)+ c2ln(D5-95)2+ ε

25. Step 0 (All data are used.) At a given period and damping: ln(DSF) = c0+ ε ; ε~N(0,σ) β = 0.5, 1, 2, 3, 5, 7, 10, 15, 20, 25, 30 % T = 0.1 , 0.4 , 1 , 7 sec

26. ln(DSF) = c0 + ε

27. ln(DSF) = c0 + ε

28. ln(DSF) = c0 + ε

29. ln(DSF) = c0 + ε

30. ln(DSF) = c0 + ε

31. ln(DSF) = c0 + ε

32. ln(DSF) = c0 + ε

33. ln(DSF) = c0 + ε

34. ln(DSF) = c0 + ε

35. ln(DSF) = c0 + ε Conclusion: Residuals have a linear pattern with magnitude. Maybe a break in pattern around M=6.75 Strong pattern with Duration. Mild pattern with Distance.

36. Step 01 (Data with R < 50km are used.) At a given period and damping: ln(DSF) = c0 + c1 M + ε β = 0.5, 1, 2, 3, 5, 7, 10, 15, 20, 25, 30 % T = 0.1 , 0.4 , 1 , 7 sec

37. ln(DSF) = c0 + c1 M + ε

38. ln(DSF) = c0 + c1 M + ε

39. ln(DSF) = c0 + c1 M + ε

40. ln(DSF) = c0 + c1 M + ε

41. ln(DSF) = c0 + c1 M + ε

42. ln(DSF) = c0 + c1 M + ε

43. ln(DSF) = c0 + c1 M + ε

44. ln(DSF) = c0 + c1 M + ε

45. ln(DSF) = c0 + c1 M + ε

46. ln(DSF) = c0 + c1 M + ε

47. ln(DSF) = c0 + c1 M + ε

48. ln(DSF) = c0 + c1 M + ε

49. ln(DSF) = c0 + c1 M + ε

50. ln(DSF) = c0 + c1 M + ε Conclusion: No pattern with Magnitude. There is linear/quadratic pattern with Duration in log space. Mild pattern with Rrup, looks logarithmic (specially at long T). Problem at small T and very high or very low damping (suspect nonlognormality).