1. EARTHQUAKE GEOTECHNICAL ENGINEERING Lecture 2 – Strong Motion Parameters

2. Contents

3. Just a reminder…………..  Frequency, f (aka temporal frequency)  number of occurrences of a repeating event per unit time  SI Unit: Hertz  Period, T  time for 1 cycle  Inverse of f  SI unit: second  Angular Frequency, ω  Measure of rotation rate  SI unit: radians/second

4. 1.0 Strong Motion Parameters  Earthquake Description  3 components translation  3 components rotation (typically neglected)

5. 1.0 Strong Motion Parameters  Amplitude  Peak Acceleration  Peak Velocity  Peak Displacement  Frequency Content  Duration

6. 1.1 Amplitude  Peak Horizontal Acceleration  Associated with high frequency  Largest value from accelogram  Vector sum of 2 orthogonal components  Dynamic force in structures  Peak Vertical Acceleration  Associated with intermediate frequency  PVA = 2/3 PHA  Usually less critical  Peak Displacement  Associated with lower frequency components  Difficult to determine, long period, noise  Less common

7. 1.1 Amplitude Effective Acceleration (Newark and Hall)  “That acceleration …most closely related to structural response and damage potential…  less than peak free-field ground acceleration.  Function of  size of loaded area,  frequency content of excitation, which in turn depends on closeness to earthquake source, weight, embedment, damping characteristics and stiffness of the structure and foundation”

8. 1.1 Amplitude a)Peak at high frequency – little effect on buildings with lower frequency b)Same peak as (a), but more cycles

9. 1.1 Amplitude  Sustained Maximum Acceleration and Velocity (Nuttli 1979)  Defined for 3 (or 5) cycles as the 3 rd (or 5 th ) highest value of acceleration in time history  Effective Design Acceleration  Many definitions  Eg 1.25 x 3 rd highest peak acceleration, after filtering (Kennedy, 1980)

10. 1.2 Frequency  Damage response – frequency sensitive  Earthquakes – range of frequencies  Determine frequency content of time history… how?

11. 1.2.1 Frequency - Fourier Analysis Joseph Fourier (1768 – 1830), France Son of a tailor, orphaned at 9 rejected from Science course due parentage, so studied maths French Revolution Made Governor of Egypt by Napolean Translation of Rosetta Stone Propagation of Heat Discovered Green House effect

12. 1.2.1 Frequency – Fourier Analysis Many applications in science and engineering eg Fourier analysis has many scientific applications – in physics, signal processing, imaging (JPEG files), probability, differential equations, acoustics, oceanography, sonar………& earthquake engineering ! http://www.fourier-series.com/ Every period can be approximated by sum of simple harmonic terms

13. 1.2.1 Frequency – Fourier Analysis  General Form  Fourier Coefficients

14. 1.2.1 Frequency – Fourier Analysis  Other forms utilising  trigonometric identities  Euler’s formula

15. 1.2.1 Frequency – Fourier Analysis  Example Using this form Coefficients

16. 1.2.1 Frequency – Fourier Analysis  Fourier Amplitude Spectrum  Fourier Phase Spectrum Amplitude vs Frequency Phase angle vs Frequency

17. 1.2.1 Frequency – Fourier Analysis Earthquake Engineering Applications Loading Response

18. 1.2.1 Frequency – Fourier Analysis  Time Domain to Frequency Domain  Domain  argument  Earthquake Geotechnical Engineering Application Example  1 Dimensional Response Analysis Time domain Frequency domain Time domain Frequency domain

19. 1.2.1 Frequency – Fourier Analysis  Fourier Transform  Continuous variant of Fourier Series  Function that converts signals from “time domain” to “frequency domain”  Finite data points– Discrete Fourier Transform  Fast Fourier Transform  http://www.physik.uni- kl.de/fileadmin/beigang/Vorlesungen/WS_07_08/Fouri er_Transforms_Rick_Trebino.pdf

20. 1.2.2 Frequency – Amplitude Spectra  Fourier Amplitude Spectrum  Clearly distinguish frequency content

21. 1.2.2 Frequency – Amplitude Spectra Earthquake Characteristic Shape Corner Frequency – inv proportional to cube root of seismic moment (larger earthquake, lower freq) Cutoff frequency – constant for geographic region

22. 1.2.2 Frequency - more parameters  Phase Spectra  Gives time of peak harmonic motion  Influences variation of ground motion with time  Predominant Period  Period of vibration corresponding to Fourier amplitude spectrum  Bandwidth  Gives information on dispersion  Level where power of spectrum = ½ max = Fourier Amplitude

23. 1.2.2 Frequency – more parameters  Power Spectral density  Gives intensity of ground motion  Either time domain or frequency domain  Central Frequency  Frequency where power spectral density greatest  Shape Factor  Dispersion of power spectral density

24. 1.3 - Duration  Eq damage related to duration  Damage depends on load reversal  Strong-motion portion of accelogram  Bracket duration

25. 1.4 – Other parameters  Arias Intensity  Strength of ground motion g = accel due gravity Td = duration of signal above threshold

26. 1.5 Predictive Equations  Estimation of Amplitude, Freq, Duration etc parameters  Include Attenuation Equations (PHA, PVA) aka Ground Motion Prediction Equations (GMPE)  Y = f(M,R,Pi)  Ground motion, Y  Magnitude, M  Distance, R  Other parameters (eg source, wave path, site cond), Pi  Regression analysis  Recorded earthquakes

27. 1.5 Predictive Equations  Functional Form 1. Peak values log normally distributed, ln Y 2. Magnitude defined as log of peak motion, therefore ln Y proportional to M 3. Body wave attenuation 1/R, Surface waves 4. Larger M  larger fault rupture area  greater distance R 5. Energy exponentially decreases through material damping 6. Source or site Characteristics

28. 1.5 Predictive Equations  Peak Acceleration  Campbell (1981) PHA Sites < 50km fault rupture M 5.0 to 7.7 World wide data

29. 1.5 Predictive Equations  Campbell and Bozorgnia (1994)  M 4.7 to 8.1  F - 0 for strike slip& normal, 1 for reverse, reverse- oblique, thrust  S SR = 1 soft-rock sites, S HR =1 hard rock sites, S SR = S HR = 0 alluvium sites

30. 1.5 Predictive Equations

31. 2.1 Modes of Vibration  Mode

32. 2.1 Modes of Vibration  Building Modes

33. 2.1 Modes of Vibration Alcoa Building (26 storey), San Francisco Fund. Period of Vibration: N-S = 1.67sec E-W= 2.21 sec Torsional (vert axis) = 1.12 San Francisco Bridge (main spain1.3km) Fund Period of Vibration: Transverse=18.2 sec Vertical=10.9 sec Longitudional=3.81sec Torsional=4.43 sec Transamerica Building (60 storey), San Francisco Fund. Period of Vibration: N-S =2.9sec E-W=2.9sec

34. 2.1 Modes of Vibration  Most structures have multiple modes of vibration  Each mode has a period  Fundamental mode  aka natural period of vibration  longest period (lower freq)  Higher the mode, the shorter the period (higher freq)  Buildings Rule of Thumb  Fundamental period ≈ No of storeys / 10 (moment frame)  http://www.youtube.com/watch?v=cDjcpPgHrbo

35. 2.2 Degrees of Freedom  Motion of a ship (6 degrees)  Buildings  MDOF – multi degree of freedom  SDOF – single degree of freedom

36. 2.3 SDOF  Single Degree of Freedom Systems  Buildings often modelled as SDOF systems  Dynamic Response of SDOF systems  Solve differential equations for different conditions

37. 2.3.1- Condition 1 - Undamped Free Vibration  Undamped free vibration, c = 0, Q(t) =0  Equation of motion  Solve the differential equation  Initial conditions  Solution  where Natural circular freq of vibration Natural Period of vibration Natural cyclic freq of vibration Note: Natural freq depends on: mass stiffness

38. 2.3.2 Condition 2 - Viscously Damped Free Vibration  Damped (underdamped), free vibration, c>0, Q(t)=0  Equation of motion  Solution  where  ζ = damping ratio  Natural period of damped vibration  Natural period

39. 2.3.3 – Condition 3 Undamped Forced Vibration (sinusoidal)  Undamped, forced, c=0, Q(t)>0  Equation of Motion  Response Harmonic force, with forcing freq, ω Response, with ω < ω n Response, with ω = ω n Resonant Frequency 1.Forcing Freq when ratio of Response amplitude/Initial Amplitude = max 2.For undamped system Resonant Freq = ω n Response amp/initial amp ratio is unbounded

40. 2.3.4 Condition 4 - Viscously Damped Forced Vibration (Sinusoidal)  Undamped, forced, c>0, Q(t)>0  Equation of Motion  Response Harmonic force, with forcing freq, ω Response, with ω < ω n Response, with ω = ω n, s Response, with ω = ω n, various damping ratios

41. 2.2.5 – SDOF solution subject to varying force (with time)  Solution for SDOF equation of motion with varying force / ground acceleration eg  Numerical finite difference or “time-stepping” procedures http://en.wikipedia.org/wiki/Finite_difference_method  Solution  Determined at discrete time instances  Depends on prior history  Equation of Motion

42. 3.0 Response Spectrum  Core Concept in Earthquake Engineering  Characterises ground motion effect on structures  Plot of peak response of all possible SDOF (linear) systems to a particular ground motion  Each plot is for SDOF system of fixed damping ratio  Deformation, Velocity, Acceleration Response Spectra  At T = 0 response equal to peak ground response

43. 3.0 Response Spectra  Examples Example 1 Response spectra for various damping ratios Example 2 Design response spectra for various damping ratios

44. 3.0 Response Spectra Examples Example 3 –Design response spectra envelope Example 4 – Design response spectra for various soil categories

45. 3.0 Response Spectra  Examples (cont’d) Example 5 Inelastic Design Response Spectra with Ductility

46. 3.0 Response Spectra  Construct a response spectra  Define ground acceleration ( ∆t = 0.02s)  Select T n, ζ for SDOF system  Solve for u(t) using numerical method  Determine peak response u o  Spectral Ordinates D=u o, V=(2π/T n )D, A=(2π/T n ) 2 D  Design spectra obtained by statistical analysis Tripartite Plot, ζ = 0.02 Design spectra, ζ=0.05

47. 3.0Response Spectra  Deformation, Pseudo Velocity, Pseudo Response  Pseudo Velocity  Pseudo Acceleration  Pseudo vel, pseudo accel approximate system vel, accel

48. 4.0 Summary  Parameters to describe Earthquake Motion  Amplitude  Freq  Duration  Fourier Analysis  Fundamental Mode  SDOF  Response Spectra