Lecture 3 sapienza 2017

Academic excellence for business and the professions
Lecture 3:
Response spectrum compatible simulation
techniques of strong ground motions
Lecture series on
Stochastic dynamics and Monte Carlo simulation
in earthquake engineering applications
Sapienza University of Rome, 19 July 2017
Dr Agathoklis Giaralis (agathoklis@city.ac.uk)
Visiting Professor for Research, Sapienza University of Rome
Senior Lecturer (Associate Professor) in Structural Engineering,
City, University of London

The need to consider simulation techniques
For linear structures and for relatively simple (e.g. stationary) stochastic excitation, we do
have input/output (or excitation/response) relationships either exact (e.g. for the variance)
or accurate semi-empirical (e.g. for the peak response)
But for more complex stochastic excitations (e.g., non-Gaussian, non-stationary, etc.) and/or
for nonlinear structures, exact input/output relationships are not easy to obtain or are not
always accurate.
In principle, we would like to have an appreciation of what should is the “exact” solution to
benchmark against by taking the frequentist approach
Monte Carlo simulation is the only way around this problem
(at the expense of being computationally demanding…)

Problem description (Monte Carlo simulation)
Given PSD
Realizations of time-limited
stationary input process
“Structure”
(or Rx(τ))
Realizations of stationary response
process acquired by response history
analysis (deterministically)
“Spectrum
compatible
simulation” Response statistics estimation
or
(ergodicity)
or
(ergodicity)
ALSO
Spectral estimation
(or auto-correlation
estimation)
‘mean’
‘var’
‘pwelch’
‘xcorr’
Stationary case

Problem description (Monte Carlo simulation)
Given EPSD
Realizations of non-stationary
input process
“Structure”
Realizations of non-stationary response
process acquired by response history
analysis (deterministically)
“EPSD compatible
simulation”
Response statistics estimation
Evolutionary
spectral estimation
requires wavelet
analysis or some
other time-
frequency signal
representation
Non-stationary case
Time-dependent mean
Time-dependent rms

Spectral density estimation (for stationary processes):
the “Forward problem”
PSD
Given: Find:
A “naïve” estimator comes from the very definition of the PSD (the “Periodogram”)
It leads to “jagged” estimates. Also, the variance of each spectral
ordinate does not reduce as more samples are included…
‘fft’
‘periodogram’

Welch estimator
Smoothing/weighting through some envelop window and
averaging in time with overlapping windows to reduce variance
where
‘pwelch’
There are several other
algorithms for spectral
estimation: Fourier-
based (non-parametric)
and non Fourier-based
(parametric)
Spectral density estimation (for stationary processes):
the “Forward problem”
Marple 1987
Kay 1987

“Random number generation”
‘rand’
‘randn’
MATLAB command rand which yields
an array of random numbers uniformly
distributed on [0,1] (=U(0,1))
Matlab command randn provides an array
of random numbers following N(0,1). In
order to obtain normal variables following
any N(μ, σ2), we may use the fact that given
Χ~Ν(0, 1) then X1= μ +σ·Χ ~ Ν(μ, σ2)

Power spectrum compatible simulation (for stationary processes):
the “inverse problem”
PSD
Given: Find:
There are a number of different methods to address this problem (Spanos and Zeldin 1998)
We shall briefly discuss the most popular one:
And the most efficient one:
Spectral representation method
ARMA filtering method

The spectral representation method for stationary
spectrum compatible simulation
Define:
Let: Δω->0
and Κ->inf
Vanmarcke 1983

Shinozuka and Deodatis 1991
Zerva 1992
For one-sided spectra: An= 2𝑆(𝜔 𝑛) ⋅ 𝛥𝜔
Ergodic/periodic samples To= 2π/Δω ; asymptonically Gaussian as N->inf

White noise filtering methods for stationary
Intuition: All we need are filters such that |H(ω)|2 is equal to the PSD
BUT: we have to work in discrete-time, so
filters need to be in discrete-time “digital”!
Continuous-time filters Discrete-time filters (T==dt)
Differential equations Difference equations
Z- transform
FRFs: ratios of polynomials in ω
FRFs: ratios of
polynomials in z
0
2
1
( ) ,
1
i T
k
k
k
z
H z z e
c z




 
 
Laplace transform
     
 
2
2 1 2n ny n y n y n
u n
     

     
2
1
k
k
y n c y n k u n

   
 
   
2
1/
1 / 2 /n n
k
H
i

   

 
     
 
2
2 n ny t y t y t
u t
   

or

Non-recursive or
moving average (MA)
filters
or finite impulse
response (FIR) filters
Recursive filters of
infinite impulse
response (IIR)
filters
1 0
p q
n k n k l n l
k l
y c y d w 
 
   
     , 0,1,2,..., 1
q
v q
q
n v n v
v q
y n d v w n v n N
y d w



   


or   0
1
1
q
li T
l
i T l
p
ki T
k
k
d e
H e
c e











 
q
i T i vT
v
v q
H e d e 

 
Discrete-time domain
FRF (z-domain)
Discrete-time domain
FRF (z-domain)
(notation often used)

  0
1
1
q
l
l
l
p
k
k
k
b z
H z
c z







     2 b ijE w i w j  
     
1 0
p q
k l
k l
y r c y r k b w r l
 
     
Find the filter coefficients ck and bl such that:
Input: Clipped
white noise
ARMA (p,q) filter
Output: difference equation in
discrete-time domain
   
2
;i T
b
S H e T 



 w n
Moving Average
(MA)
Problem statement: MA part
polynomial of in the numerator
roots are the “zeros” of the filter
𝑧 = 𝑒 𝑖𝜔𝑇
Auto-regressive
(AR)
Auto-regressive moving average
(ARMA)
AR part
polynomial of in the denominator
roots are the “poles” of the filter
𝑧 = 𝑒 𝑖𝜔𝑇

MA White noise filtering method for stationary

AR White noise filtering method for stationary

ARMA White noise filtering method for stationary
The ACM method defines first a large AR filter. Then, it matches the auto and the cross-
correlations of this AR and the sought ARMA filter. The latter has a significant smaller order
than the auxiliary AR.
Spanos, P. D., Mushung, L. J., Nelson, D. A., and Hamilton, D. A. (1990). "Low-Frequency Spectral
Representation of Space Shuttle Flight Data. Journal of Aerospace Engineering, 3(2), 137-154.

Example 1: Derivation of response spectrum compatible
evolutionary power spectra (non-stationary processes) via
Monte Carlo-based peak factor estimation
     gu t A t y t
     
2
,EPS t A t Y In the frequency domain:
Theory of uniformly modulated evolutionary power
spectra (EPSs) (Priestley, 1965)
Consider a zero-mean non-stationary stochastic process
In the time domain:
Slowly-varying in time modulation function
Evolutionary power spectrum of the
non-stationary process
Power spectrum of the
stationary process gu t  y t
“Fit” an EPS to a given response spectrum
 gu t

Spectral form of the stationary part
(Clough/Penzien, 1975)
Assumed parametric form for the EPS
(Giaralis/Spanos, 2009)
Time-dependent envelop function
(Bogdanoff/Goldberg/Bernard, 1961)
  exp
2
b
A t Ct t
 
  
 
 
2 2
2
2 22 2 2 2
2 2
1 4
;
1 4 1 4
g
f g
b
f g
f f g g
Y
 

 
  
   
 
   
   
      
    
          
                                 
Parameters to be determined:
C, b, ζf, ωf , ζg, ωg

       2
2
0
exp 2 exp 2 ,
t
a n n n
n
t t EPS d

      

  
where σα is given as (Spanos/Lutes, 1980):
Equation relating the EPS to a given
displacement response/design spectrum Sd
      , , , , max , , ,d n n a n
t
S EPS p t EPS       
           2
2 ; 0 0 0n n gx t x t x t u t x x      
     cos nx t a t t t    
It is the variance of the response amplitude of a SDOF quiescent oscillator
with damping ratio ζ and natural frequency ωn subject to the ground
motion process :
Assuming that the input process is relatively broadband compared to the transfer
function of the oscillator (ζ<<1), the relative displacement x(t) of the oscillator is
well approximated by the process:
 gu t

Equation relating the EPS to a given
displacement response/design spectrum Sd
      , , , , max , , ,d n n a n
t
S EPS p t EPS       
η is the so-called “peak factor” :
the critical parameter establishing
the equivalence between the given
design spectrum Sd and the EPS in a
statistical manner.

 
2
2
1
min
M
j j
j
S 

 
 
 

  2
, 1,...,
0 , 1,...,2
d n j
j
S j M
S
j M M
 
 
 
 
  
     
*
*
2 2 *2
3
2 * *2 *
, 1,...,
2
2 2 1 2 4 , 1,...,2
j
j M j M
bt
j
n j
n jj
t
j M j M j M j M j M n j M
C t e
G j M
t bt bt b e j M M




    


     

 
 

      
 2 .k n k
b  
Number of unknowns to be determined: M+6. These are the M {t*k}; k=1,..,M
time instants at which the amplitude of the SDOF oscillator of natural frequency
ωn(k) reaches an absolute maximum, plus the EPS parameters C, b, ζf, ωf , ζg, ωg.
Number of equations: 2M
Over-determined non-linear least-square optimization
problem (Giaralis/Spanos, 2009)
; where:
and

  0
1
1
q
l
l
l
p
k
k
k
b z
H z
c z







     2 b ijE w i w j  
     
1 0
p q
k l
k l
y r c y r k b w r l
 
     
Compute the filter coefficients ck and bl such that:
Input: Clipped
white noise
ARMA (p,q) filter Output: difference equation in
discrete-time domain
   
2
;i T
b
Y H e T 


 
The corresponding non-stationary samples are then obtained by:
     gu r A r y r
 w n
The auto/cross correlation matching procedure (ACM) is employed
(Spanos/Zeldin, 1998)

Numerical experiment
A total number of 15 EC8 compatible EPSs are considered derived by
assuming a constant peak factor η= (3π/4)1/2.
Three different damping ratios ζ (=2%, 5%, 8%); five soil conditions, as
prescribed by the EC8, for each damping ratio; PGA= 0.36g.
For each EPS a suite of 10000 spectrum compatible non-stationary
accelerograms are generated.
Each suite is “fed” to a series of 200 SDOF with natural periods Tn= 2π/ωn
ranging from 0.02sec to 6sec and for each oscillator 15 response ensembles
x(k)(t); k=1,2,…,10000 are obtained.
For each such ensemble peak factor populations η(k); k=1,2,…,10000 are
computed by (Spanos/Giaralis/Jie, 2009)
 
 
 
  
 
   2
max , , ,
, ,
max , , ,
k
nk t
n
k
n
t
x t T EPS
T EPS
E x t T EPS

 


 
 
 

Maximum variances and time
instants at which these are attained
 
 
 
  
 
   2
max , , ,
, ,
max , , ,
k
nk t
n
k
n
t
x t T EPS
T EPS
E x t T EPS

 


 
 
 
ζ= 5%
ζ= 5%

Ratio of t{x=max|x|} over
t{var(x)=max(var(x))}
 
 
 
  
 
   2
max , , ,
, ,
max , , ,
k
nk t
n
k
n
t
x t T EPS
T EPS
E x t T EPS

 


 
 
 ζ= 5%
ζ= 5%
 
 
11
/ , exp , 0
z
f z z z

 
  
  
   
  
    1
0
exp t t dt



  
Gamma distribution provided
the best fit in the standard
maximum likelihood context

Peak factors  
 
 
  
 
   2
max , , ,
, ,
max , , ,
k
nk t
n
k
n
t
x t T EPS
T EPS
E x t T EPS

 


 
 
 
Type III generalized extreme value
distribution (Weibull) provided
the best fit in the standard
maximum likelihood context
(Kotz, Nadarajah,2000)
ζ= 5%
ζ= 5%
 
1/
1 1/
1
/ , , exp 1
1
z
f z
z



   
 




 
  
        
 
 
 
1 0
z 



 where

EC8 compatible peak factors
Polynomial fit to the median peak factor
spectra averaged over the spectra obtained
from the five different soil conditions for
each value of the damping ratio considered.
p0 p1 p2 p3 p4 p5 p6 p7 p8
ζ= 2% 3.3079 -4.9375 8.3621 -8.3368 5.0420 -1.8983 0.4469 -0.0639 0.0051
ζ= 5% 3.1439 -3.9836 5.9247 -5.3470 2.9794 -1.0439 0.2305 -0.0311 0.0023
ζ= 8% 2.9806 -3.2070 4.1190 -3.1733 1.4746 -0.4144 0.0689 -0.0062 0.0002
 
8
0
, 0.02 6secj
j
j
T p T T

  
      , , , , max , , ,d n n a n
t
S EPS p t EPS       
Setting p=0.5: Half of the population of response spectra from
generated EPS compatible accelerograms will lie below the level Sd

EC8 compatible peak factors
      , , , 0.5 max , , ,d n n a n
t
S p t EPS        
Peak ground
acceleration
Soil type
CP power spectrum parameters [Tmin= 0.02,Tmax= 10] (sec)
C (cm/sec2.5)
b
(1/sec)
ζg ωg (rad/sec) ζf
ωf
(rad/sec)
αg= 0.36g
(g= 981 cm/sec2)
A 8.08 0.47 0.54 17.57 0.78 2.22
B 17.76 0.58 0.78 10.73 0.90 2.33
C 19.58 0.50 0.84 7.49 1.15 2.14
D 30.47 0.50 0.88 5.34 1.17 2.12
E 20.33 0.55 0.77 10.76 1.07 2.03

Pulse-like
accelerograms
PL HF LFEPSD EPSD EPSD 
Simulation techniques for
stationary processes
+GLF(ω)
aLF(t)
GHF(ω)
aHF(t)
Fully non-stationary stochastic
process for modelling
pulse-like time-historiesHF
accelerogram
2
( ) ( )HF HF HFEPSD a t G 
LF
accelerogram
2
( ) ( )LF LF LFEPSD a t G 
a(t) – envelope functions
g(t)– stationary zero-mean
processes with the power
spectrum distribution G(ω)
(See also Spanos &Vargas Loli 1985, Conte & Peng 1997)
( ) ( )HF HFa t g t ( ) ( )LF LFa t g t( )PLy t 
-Seismological models
- Phenomenological models
   
2 2
2
2
1
( ) ( ) ( ) (, ( ))HF HF L rF r
r
LFA t G A t Gt A t GS   

  
+
Example 2: A non-stationary stochastic model for
pulse-like ground motions (PLGMs)

2
( ) ( )
HFb t
HF HF HF HFa t C h t C te

 
 
4 2
2
,2 22 2 2 2
2 2
1 4
,
1 4 1 4
g
f g
HF c HF
f g
f f g g
G
 

 
  
   
 
   
   
      
     
          
                                 
High frequency content
 Time varying envelope
 Power spectrum density  Clough Penzien
CHF  peak amplitude
bHF  effective duration
ωg , ζg – soil characteristics
ωf , ζf – filter characteristics
- Phenomenological models  reproduce features of the recorded time-histories,
focusing less on the mechanism causing them and more on obtaining similar
structural effects (i.e. Clough Penzien, Kanai Tajimi spectra)
- Seismological models  source properties, rupture velocity, magnitude, soil
conditions, epicentral distance... (Boore 2003)

 
2
0
1
( )
2
( )
p
t t
LF LF LF LFa t C h t C e


 
   
 
 
   
   
1 2
1 cos
2 2
1 1
LF
COS
p
p
p p
G
for

 



   
  
     
  
   

Low frequency content
 Time varying envelope
 Power spectrum density
CLF  peak amplitude ; t0  peak location;
ωp/γ  effective duration
ωp pulse dominant frequency;
α  bandwidth

Imperial Valley 1979 (array #6) pulse-like ground motion

Lecture 3 sapienza 2017

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