Prof. A. Giaralis, STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS

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

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

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

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)

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
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

“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
design process
Time-histories of
design process
InputAnalysisOutput
(Evolutionary) power
spectrum
(Non-) stationary linear or
non-linear (e.g. statistical
linearization) random
vibration analyses
Statistical moments of
design process
or artificially generated
accelerograms
Response spectrum compatible (evolutionary) power spectrum representation
Viable methods for aseismic design of new structures

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
 



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”….

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…

(Empirical) temporal calculation of the probability density
function p(x) for an ergodic stationary random process

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:

The probability density function (pdf)
computed from discrete-time signals

The Normal (Gaussian) distribution

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

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.

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.

Temporal Averaging (along a realization)
Very heuristic math here: recall that
Ensemble average == temporal average (ergodic process in mean)
(expected value) (statistics)

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

“Ensemble moments” vs “temporal moments”
BUT: there must be some correlation (statistical relationship)
between the x(t1) and x(t2) random variables…..

Second order pdfs!
Interpretation:
x -> x(t1)
y-> x(t2)
As before:
And:
With:

Marginal probabilities:
Interpretation:
x -> x(t1)
y-> x(t2)
Averages:
Second order pdfs!

Second order pdfs!
Interpretation:
x -> x(t1)
y-> x(t2)

Second order pdfs!
Interpretation:
x -> x(t1)
y-> x(t2)
Conditional pdf:
x and y are independent <=> If x and y are independent

Second-order Gaussian distribution

Higher-order Gaussian distribution
(multi-variate)

Auto-correlation function of
stationary stochastic process
Auto-correlation function:
Stationarity further implies:
So: or:

Auto-correlation function of
All important properties
A zero-mean stationary
Gaussian random process can
be fully represented by the
auto-correlation function!

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)

Continuous/
Finite length (duration) OR Periodic
Orthogonality of basis functions
Synthesis/Reconstruction/Decomposition Analysis/Projection

Continuous/
Convergence issues- Gibbs phenomenon
Convergence in the “mean sense” or in
“energy”- ENERGY OF A SIGNAL!
Energy conservation-
Parseval’s theorem

Continuous/
Synthesis/Reconstruction/Decomposition Analysis/Projection
Euler’s equation:
Orthogonality of basis functions

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”

Power spectral density (PSD) function: definition
The Fourier integral of a stationary stochastic process
cannot be defined (infinite energy…)
BUT Define

Power spectral density (PSD) function: definition
BUT

PSD is a Fourier pair with the auto-correlation function
(Wiener-Khinchin theorem)
Power spectral density (PSD) function: properties
Most important property

Gaussian wideband process
2So(ω2-ω1)

Frequency content of recorded ground motions

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

The ordinary Fourier
Transform (FT) provides
only the average
spectral decomposition
of a signal.
Limitations of the Fourier transform

STFT- analysis
The Gaussian envelop
Optimizes the product:
Short-Time Fourier Transform (Gabor, 1946)

Short-Time Fourier Transform (Gabor, 1946)Frequency
Time
“Tiling” the time-frequency
domain using STFT
STFT- reconstruction

El Centro, 1940(N-S); Ms=7.1 (Far field)

Hachinohe, 1968(N-S); Ms=7.9 (Far field)

Uniformly modulated non-
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

     
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

Prof. A. Giaralis, STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS

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