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Introduction to stochastic processes and modelling of seismic strong ground motions_Lecture1

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- 1. Academic excellence for business and the professions Lecture 1: Introduction to stochastic processes and modelling of seismic strong ground motions Lecture series on Stochastic dynamics and Monte Carlo simulation in earthquake engineering applications Sapienza University of Rome, 12 July 2017 Dr Agathoklis Giaralis Visiting Professor for Research, Sapienza University of Rome Senior Lecturer (Associate Professor) in Structural Engineering, City, University of London
- 2. INPUT: Time-varying excitation loads SYSTEM: Civil Engineering structure OUTPUT: Structural response -Non-linear Σκυρόδεμα καθαρότητος +0,00 m -3.85 m αντισεισμικός αρμός Ισόγειο (Pilotis) Υπόγειο ελαστομεταλλικό εφέδρανο 2000150800 Overview of challenges in Earthquake Engineering: A structural dynamics perspective
- 3. INPUT: Time-varying excitation loads SYSTEM: Civil Infrastructure OUTPUT: Structural response -Non-stationary -Uncertain Typical earthquake accelerograms exhibit a time-evolving frequency composition due to the dispersion of the propagating seismic waves, and a time-decaying intensity after a short initial period of development. 1st~2nd second: 15 zero crossings 6th~7th second: 7 zero crossings -Outcome of a Non-linear “process” -Non-stationary - Non-linear -Uncertain NPEEE, 2005 Overview of challenges in Earthquake Engineering: A structural dynamics perspective
- 4. INPUT: Time-varying excitation loads SYSTEM: Civil Infrastructure OUTPUT: Structural response -Non-stationary -Uncertain -Outcome of an Inelastic process -Non-stationary -Inelastic -Uncertain Joint time-frequency representation of ground motions Random process theory or (probabilistic methods) Evolutionary power spectrum concept and (probabilistic seismic hazard analyses) Incorporation of hysteretic models -Monte Carlo simulation -Statistical linearization -(Performance-based earthquake engineering) Overview of challenges in Earthquake Engineering: A structural dynamics perspective
- 5. Uniform hazard linear response spectrum Recorded ground motions Equivalent linear single mode or multi-mode analyses Modal analysis Elastic or inelastic response-history analysis (Numerical integration of governing equations of motion)Modal combination rules Absolute maximum response quantities of interest to the structural design process Time-histories of response quantities of interest to the structural design process InputAnalysisOutput (Evolutionary) power spectrum Seismological parameters: Source-Path-local Site effects or artificially generated accelerograms e.g. The “Specific Barrier Model” (Papageorgiou /Aki, 1983) e.g. The “Stochastic Method” (Boore, 2003) Viable methods to represent the seismic input action Proper record selection and scaling seismic hazard analysis
- 6. “Response/ Design” spectrum Recorded ground motions Equivalent linear static or dynamic analysis Modal analysis Elastic or inelastic response-history analysis (Numerical integration of governing equations of motion)Modal combination rules Absolute maximum response quantities of interest to the structural design process Time-histories of response quantities of interest to the structural design process InputAnalysisOutput (Evolutionary) power spectrum (Non-) stationary linear or non-linear (e.g. statistical linearization) random vibration analyses Statistical moments of response quantities of interest to the structural design process or artificially generated accelerograms Response spectrum compatible (evolutionary) power spectrum representation Viable methods for aseismic design of new structures
- 7. The classical (not so random) experiment! Experiment: We throw a six-sided die up in the air which bumps on a table We decide to model the outcome “event” of this experiment using a “random variable (rv)”; our aim is to describe the outcome in probabilistic terms… We do statistical characterization of outcomes (the “frequentist interpretation of probability theory”) and use a probability density function (pdf) to represent this Uniform distribution…. Constant pdf for a typical discrete rv Outcome of interest: The colour of the side that looks to the sky after die stops moving Uncertainty of the outcome (?): Conditions and “chaos” We assign a scalar number to each outcome as part of the modelling Prob(green)= Prob(X=1)≈ 1 1 lim t x x n t t n n n n
- 8. Random (or stochastic) process: The engineering “temporal” interpretation Experiment: We monitor the sea elevation in a small Mediterranean port under perfect weather conditions for 10mins using a buoy Outcome of interest: A time-history of sea elevation 10mins long Uncertainty of the outcome (?): Conditions and “chaos” We decide to model the outcome “event” of this experiment using a “stochastic process”; our aim is to describe the outcome in probabilistic terms… We register a time-history to each outcome as part of the modelling: realization of the stochastic process. And we need consider infinite many realizations for a “frequentist approach”….
- 9. Random (or stochastic) process visulization Engineering (temporal) view: (infinite) many time-histories running in parallel along time which may be analog (continuous set) or digital (discrete set) Mathematical (ensemble) view: (inifinite) many rvs “living” at different times up to infinity and taking different values across the ensemble of the (infinite) many time-histories. These times are “indexed” continuously (continuous time) or discretly (discrete time) Stationary and non-stationary processes… Ergodic and non-ergodic stationary processes…
- 10. (Empirical) temporal calculation of the probability density function p(x) for an ergodic stationary random process
- 11. We can “scan” all possible infinitesimally small intervals of dx length to evaluate (continuously) the full pdf as a function of x values attained. The probability density function (pdf) First axiom of probability Second axiom of probability Note:
- 12. The probability density function (pdf) computed from discrete-time signals
- 13. The Normal (Gaussian) distribution
- 14. The “moments” of the pdf: central tendency and dispersion Mean value for discretely distributed rvs Mean value for continuously distributed rvs Variance Standard deviation Coefficient of variation (COV): σx/μx
- 15. Properties of Normal (Gaussian) distribution m=μ (mean) and σ= standard deviation Central Limit Theorem: when independent random variables are added, their sum tends toward a normal distribution even if the original variables themselves are not normally distributed.
- 16. Properties of Normal (Gaussian) distribution Central Limit Theorem: when independent random variables are added, their sum tends toward a normal distribution even if the original variables themselves are not normally distributed.
- 17. The log-normal distribution
- 18. The Rayleigh distribution
- 19. Temporal Averaging (along a realization) Very heuristic math here: recall that Ensemble average == temporal average (ergodic process in mean) (expected value) (statistics)
- 20. Temporal Averaging (along a realization) Ensemble mean square == temporal mean square (ergodic process in mean square) (expected value) (statistics) For zero mean processes: mean square== variance
- 21. “Ensemble moments” vs “temporal moments” BUT: there must be some correlation (statistical relationship) between the x(t1) and x(t2) random variables…..
- 22. Second order pdfs! Interpretation: x -> x(t1) y-> x(t2) As before: And: With:
- 23. Marginal probabilities: Interpretation: x -> x(t1) y-> x(t2) Averages: Second order pdfs!
- 24. Second order pdfs! Interpretation: x -> x(t1) y-> x(t2)
- 25. Second order pdfs! Interpretation: x -> x(t1) y-> x(t2) Conditional pdf: x and y are independent <=> If x and y are independent
- 26. Second-order Gaussian distribution
- 27. Higher-order Gaussian distribution (multi-variate)
- 28. Auto-correlation function of stationary stochastic process Auto-correlation function: Stationarity further implies: So: or:
- 29. Auto-correlation function of stationary stochastic process All important properties A zero-mean stationary Gaussian random process can be fully represented by the auto-correlation function!
- 30. An aside on Fourier Analysis 4 types of signals 4 types of Fourier Transform Continuous/Finite length (duration T) OR Periodic Continuous/Infinite length (duration) Discrete/ Infinite length (duration) vector Discrete/Finite N-length (duration) vector Continuous-time Fourier Series (CTFS) OR Fourier Series (FS) Fourier Transform (FT) Discrete-Time Fourier Transform (DTFT) Discrete Finite Transform (DFT)
- 31. Continuous/ Finite length (duration) OR Periodic Continuous-time Fourier Series (CTFS) Orthogonality of basis functions Synthesis/Reconstruction/Decomposition Analysis/Projection An aside on Fourier Analysis
- 32. Continuous/ Finite length (duration) OR Periodic Continuous-time Fourier Series (CTFS) Convergence issues- Gibbs phenomenon Convergence in the “mean sense” or in “energy”- ENERGY OF A SIGNAL! Energy conservation- Parseval’s theorem An aside on Fourier Analysis
- 33. Continuous/ Finite length (duration) OR Periodic Continuous-time Fourier Series (CTFS) Synthesis/Reconstruction/Decomposition Analysis/Projection Euler’s equation: Orthogonality of basis functions An aside on Fourier Analysis
- 34. Some important definitions Time domain Frequency domain Energy 1 ˆ 2 i t g t G e dt “Duration” as the moment of energy in the time-domain 2 E g t dt 2 ˆE G dt “Bandwidth” as the moment of energy in the frequency-domain Uncertainty principle For E=1 e.g. Chui 1992 “Centralized Values”
- 35. Power spectral density (PSD) function: definition The Fourier integral of a stationary stochastic process cannot be defined (infinite energy…) BUT Define
- 36. Power spectral density (PSD) function: definition BUT
- 37. PSD is a Fourier pair with the auto-correlation function (Wiener-Khinchin theorem) Power spectral density (PSD) function: properties Most important property
- 38. Gaussian White noise process
- 39. Gaussian wideband process 2So(ω2-ω1)
- 40. Gaussian narrowband process
- 41. Frequency content of recorded ground motions
- 42. Typical earthquake accelerograms exhibit a time-evolving frequency composition due to the dispersion of the propagating seismic waves, and a time- decaying intensity after a short initial period of development. 1st~2nd second: 15 zero crossings A paradigm of a non-stationary signal 6th~7th second: 7 zero crossings
- 43. The ordinary Fourier Transform (FT) provides only the average spectral decomposition of a signal. Limitations of the Fourier transform
- 44. STFT- analysis The Gaussian envelop Optimizes the product: Short-Time Fourier Transform (Gabor, 1946)
- 45. Short-Time Fourier Transform (Gabor, 1946)Frequency Time “Tiling” the time-frequency domain using STFT STFT- reconstruction
- 46. Short-Time Fourier Transform
- 47. El Centro, 1940(N-S); Ms=7.1 (Far field)
- 48. Hachinohe, 1968(N-S); Ms=7.9 (Far field)
- 49. Uniformly modulated non- stationary stochastic process y(t) (Priestley, 1965) Quasi-stationary stochastic process g(t) Frequency content: Power spectral density function G(ω) Time variation: A rectangular window of finite duration Ts (Boore, 2003) (Kanai, 1957) Evolutionary Power Spectrum (EPSD) of the separable kind Time variation: envelop A(t) y t A t g t zero-mean stationary stochastic process Slowly-varying in time modulation function 2 ,EPSD t A t G Phenomenological stochastic characterization of seismic ground motions
- 50. 2 , ,EPS t A t G Separable Evolutionary Power Spectrum (EPS) (Priestley 1965) “Fully” non-stationary stochastic processes: Time-frequency domain definitions 2 2 0.15 2 5 , 5 t t EPS t e t e e.g. Lin, 1970 Sum of uniformly modulated stochastic processes (e.g. Spanos/Vargas Loli 1985, Conte/Peng 1997) 2 1 , ( ) . N r r rGEPS t A t e.g. Conte/Peng, 1997 ElCentro (1940) compatible EPSD- N=21 Phenomenological stochastic characterization of seismic ground motions

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