Linear random vibrations analysis for seismically excited structures_Lecture2

1. Academic excellence for business and the professions
Lecture 2:
Linear random vibrations analysis for
seismically excited structures
Lecture series on
Stochastic dynamics and Monte Carlo simulation
in earthquake engineering applications
Sapienza University of Rome, 13 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. 2
     
     
         
       2
0
0
2
I D S
t
eff g
n n g
f t f t f t
mu t cu t ku t
mu t cu t ku t p t mu t
u t u t u t u t 
   
   
     
   
Linear damped SDOF system under ground excitation
(a reminder)
         2 2
2 2t t t
n n n g n gu t u t u t u t u t      Alternatively:

3. Linear damped single degree of freedom oscillator with zero initial conditions
For we get: For we get:
A more concise and efficient approach to exploit the above results requires
complex numbers analysis
Consider
We seek a harmonic/steady state solution of the form:
It can be readily shown (e.g. Chopra 2001):
And (Dynamic amplification factor):
 
    
22 2
1
1 / 2 /st
n n
H
u

   

 
 
   
22
1 1/
1 / 2 /n n
k
H
m i c k i

     
 
    
1/stu k
with:
Steady-state response to harmonic excitation
(a reminder)

4. Assume harmonic ground displacement:
4
  gi t
gu t e


  2 gi t
eff gp t m e

Then: And:
 
 
 
    
2
22 2
/
1 / 2 /
g n
g
g n g n
u t
u t
 
   

 
 
 
    
 22 22
1
1 / 2 /g
n g n g n
u t
H
u t

    
 
 
𝜔 𝑔 𝜔 𝑛𝜔 𝑔 𝜔 𝑛
Steady-state response to harmonic excitation
(a reminder)

5. 5
Any periodic or finite duration function with period
can be written as
with:
OR:
with:
     
   
0
1
i j t i j t i j t
j j j
j j
p t P P e P e P e    
 


 
    
0 0P a
2
j j
j
a ib
P

 *
2
j j
j j
a ib
P P

     
0
oT i j t
jP p t e dt
 
   0 0
1 1
2 2
cos sin cos sinj j j o j o
j jo o
j j
p t a a t b t a a j t b j t
T T
 
 
 
 
 
      
 
 
 p t oT
2 / oT 
         0 0 0 0
1 2 2
; cos ; sin
o o oT T T
j j
o o o
a p t dt a p t j t dt b p t j t dt
T T T
      
1i  where:
Response to arbitrary deterministic excitations
(SDOF systems)

6. where
it can be readily seen that: where
So:
Response to arbitrary deterministic excitations
(SDOF systems)
For aperiodic excitations:

7. Convolution theorem is very general!
Impulse response function
Response by Duhamel’s (convolution) integral
Frequency response function
Response by a simple multiplication
in the frequency domain
   
2
1/
1 / 2 /n n
k
i   

 
Response to arbitrary deterministic excitations
(SDOF systems)

8.  
    
22 2
1
1 / 2 /st
n n
H
u

   

 
Normalised amplification factor
Linear SDOF systems act like “pass-band” filters!

9. Normalised amplification factor
Linear SDOF systems act like “pass-band” filters
and relation to “resonance”

10. Soil acts like a dynamic oscillator and greatly affects the ground motions that a structure
atop the soil column experiences.
Example: Local soil effects (to be revisited later… the Kanai-Tajimi!)
Rock
Equivalent to
Softer and deeper soils will have longer predominant frequency content in terms of
natural period
Certain “softer” soils may also magnify the ground motion
Linear SDOF systems act like “pass-band” filters
and relation to “resonance”

11. Linear SDOF systems under random excitation
Mean value of the response
It holds:
Ensemble averaging:
Change sequence of integration:
Stationarity: or:
Recall

12. Linear SDOF systems under random excitation
Input/output autocorrelation relationship
It holds:
Ensemble averaging:
Change sequence of integration and noting (from stationarity)
we get:

13. Linear SDOF systems under random excitation
Input/output power spectral density relationship
It holds:
and we add 3 exponential terms whose product is “1”:
Therefore:

14. Linear SDOF systems under random excitation
Mean square response (== variance for zero-mean processes)
From the autocorrelation function:
Example: white input noise
From the PSD:

15. Linear SDOF systems under random excitation
A few more important results (e.g. Newland 1993)
2 2
{ ( )} { ( )} { ( )} 0
1 1
{ ( ) ( )} { ( )} { ( )} 0
2 2
d d
E x t E x t E x t
dt dt
d d
E x t x t E x t E x t
dt dt
  
  
Derivatives of stationary stochastic
process (e.g. velocity)
are zero due to time-independence
BUT (for cross-correlations and cross-power spectral densities):
2
( ) { ( ) ( )} [ { ( ) ( )}] ( )
( ) { ( ) ( )} ( )
: ( ) ( ) ( ) ( )
xx xx
xx xx
xx xx xx xx
d d
R t t t t R
d d
d
R t t R
d
and also S i S S S
   
 
  

     
          
      
  
which means (for example)
 2
( ) ( )yy xxS H S   

16. Linear SDOF systems under random excitation
Response to (ideal) white noise (e.g. Crandall and Mark 1963)
Auto-correlation function Power spectrum
== Square magnitude of the FRF (input
base acceleration output displacement)
of a SDOF (times So)
2
3
2
o
y
n
S


Variance of displacement:
Variance of velocity:
2
2
o
y
n
S




17. Linear SDOF systems under random excitation
Response to band-limited white noise White noise is an “approximation”

18.  
2
2
22 2
2
1 4
1 4
g
g
KT
g
g g
S




 

 
 
   
 
    
             
Kanai-Tajimi (KT case)
(Kanai, 1957):
Introduces only two more unknowns (ζg, ωg) in the
optimization problem.
Has strong and clear soil characterization
capabilities: ζg and ωg can be interpreted damping
ratio and natural frequency of the surface soil layers.
Does not suppress the low frequency energy of the
process.
Rock
Equivalent
to
     
   
2
2
2
2
t t t
n n
n g n g
u t u t u t
u t u t
 
 
  

Commonly used power spectra to model seismic processes
Input: white noise acceleration
output: total displacement

19. Commonly used power spectra to model seismic processes
   
2
22 2
2
1 4
f
CP KT
f
f f
S S


 
 

 
 
  
 
    
             
Filters the low frequency energy by a second order
high-pass (H-P) filter
Further increase of the number of parameters to be
defined (ζf, ωf , ζg, ωg).
The values attained by the parameters
corresponding to the bedrock can be unrealistic.
Clough-Penzien (CP case)
(Clough and Penzien, 1975):
Rock
Equivalent
to
Additional filter with little
physical intuition….

20. Commonly used power spectra to model seismic processes
Butterworth filtered Kanai-
Tajimi
   
2
2 2
N
BWKT KTN N
o
S S

 
 


The low frequency content introduced by the Kanai-Tajimi part of
the spectrum can be filtered out of the process more effectively than
in the CP case.
The assets of the Kanai-Tajimi spectral form are maintained; only
two parameters need to be defined which unambiguously reflect on
the soil conditions associated with the form of the design spectrum.
The order (N) and the cut-off frequency (ωo) of the H-
P filter can be judicially selected, so that:
Giaralis and Spanos (2009)

21. Giaralis and Spanos (2009)
Commonly used power spectra to model seismic processes
Manifestation of singularity of the unfiltered Kanai-Tajimi spectrum

22. Commonly used power spectra to model seismic processes

23. Peak response analysis of lightly damped structures to
broadband stationary stochastic processes
           2
2 ; 0 0 0n n gy t y t y t u t y y      
           cos sinn n ny t a t t t and y t a t t t              
Assuming that the input process is relatively broadband compared to the transfer
function of the oscillator (ζ<<1), the relative displacement y(t) of the oscillator is
well approximated by the process:
This is a “Pseudo-harmonic” response and the
envelop represents well local peak responses.
Expected frequency in terms of
moments of the PSD (Rice 1946)

24. Peak response analysis of lightly damped structures to
broadband stationary stochastic processes
For narrow-band processes
It can be proved that it follows a Rayleigh
distribution for Gaussian input processes

25. We are after “positive” up-crossings of a threshold α
Peak response analysis of lightly damped structures to
broadband stationary stochastic processes
We focus on the statistics of να
+ ->

26. Core equation relating a Sd for Gaussian input stochastic process of finite
duration Ts characterized by the power spectrum G(ω) (Vanmarcke 1976)
The peak factor ηj is the constant by which
the standard deviation of the response of a
linear SDOF oscillator must be multiplied
to predict the level Sd below which the
peak response of the considered oscillator
will remain, with probability p, throughout
the duration of the input process Ts.
,0,d j j GS  
Peak response analysis of lightly damped structures to
broadband stationary stochastic processes
where  
   
2
2 22 2
1
,
2
s
j j j
H T
    

   1 exp 2
j
j sT




 
and
     
2
, ,, ,
0
; ,m
s j m G sj m G
T G H T d     

  

27. where
The response spectral moments are given as (Vanmarcke 1976)
,0,d j j GS  
 
   
2
2 22 2
1
,
2
s
j j j
H T
    

   1 exp 2
j
j sT




 
and
     
2
, ,, ,
0
; ,m
s j m G sj m G
T G H T d     

  
Peak response analysis of lightly damped structures to
broadband stationary stochastic processes

28. Core equation relating a Sd for Gaussian stochastic process of finite duration
Ts characterized by the power spectrum G(ω) (Vanmarcke 1976)
and
 
2*
1,2, ,1,
,0, ,0, ,2,
ln ; 1
2
j G j Gs
j j
j G j G j G
T
v p q
 
   

   
 
 
,0,*
,0,
exp 2 1
/ 2
s j G
s s
s j G
T
T T
T


  
    
  
  
where
   1.2
; 2ln 2 1 exp ln 2j j j jv q v    
  
,0,d j j GS  
Peak response analysis of lightly damped structures to
broadband stationary stochastic processes
   
2
G H 

29. Definition of the response spectrum
    , maxd n
t
S x t  Displacement response spectrum:
   , ,v n n d nS S    Pseudo-velocity response spectrum:
Pseudo-acceleration response spectrum:    2
, ,a n n d nS S    
Kramer, 1996

30. Core equation relating a Sd for Gaussian input stochastic process of finite
duration Ts characterized by the power spectrum G(ω) (Vanmarcke 1976)
2
,0,
2
,a j j j G
j
S

   

 
  
 
and
 
2*
1,2, ,1,
,0, ,0, ,2,
ln ; 1
2
j G j Gs
j j
j G j G j G
T
v p q
 
   

   
 
 
,0,*
,0,
exp 2 1
/ 2
s j G
s s
s j G
T
T T
T


  
    
  
  
where
   1.2
; 2ln 2 1 exp ln 2j j j jv q v    
  
The peak factor ηj is the constant by which
the standard deviation of the response of a
linear SDOF oscillator must be multiplied
to predict the level Sd below which the
peak response of the considered oscillator
will remain, with probability p, throughout
the duration of the input process Ts.
Reponse spectrum compatible stationary processes

31. where ω0 is such that
 
 
 
2 1
02
11
0
2 / ,4
;
4
0 ; 0
k
kk
i k N
ik k k k k
k
S
G
G
   
    
     
 


  
     
   

 

Approximate numerical scheme to recursively evaluate G(ω) at a specific set
of equally spaced by Δω (in rad/sec) natural frequencies ωj= ω0+ (j-0.5)Δω; j=
1,2,…,N (Cacciola et al., 2004; Giaralis and Spanos, 2010)
    1.2
0 min ln 2 1 exp ln 2 0
k
k k kv q v

       
 2 ln 0.5
s
k k
T
v 

 
2
1
2 2
1 2
1 1 tan
1 1
kq

  

 
   
   
and the peak factors can be computed by assuming stationary white noise
input (Der Kiureghian 1980)
   1.2
2ln 2 1 exp ln 2k k k kv q v      
andwhere
Reponse spectrum compatible stationary processes

32. Reponse spectrum compatible stationary processes
   
 
2
1
,
,
a jv v
j j v
j
S
G G
A
 
 
 

              
Iterative modification of the obtained discrete power spectrum G[ωj] to
improve the matching of the associated response spectrum A[ωj,ζ] with the
target design spectrum Sα (e.g. Sundararajan, 1980; Gupta and Trifunac, 1998)
2
,0,
2
, j j j G
j
A

   

 
  
 
 
 
 
2*
1,2, ,1, ,0,*
,0, ,0, ,2, ,0,
ln ; 1 ; exp 2 1
2 / 2
sj G j G j Gs
j j s s
j G j G j G s j G
TT
v p q T T
T
 
    

  
        
  
  
   1.2
; 2ln 2 1 exp ln 2j j j jv q v    
  
where
Moments can be numerically computed very efficiently (Di Paola & La Mendola 1992)

33. Reponse spectrum compatible stationary processes
EC8 design spectrum considered: ζ=5%; PGA= 0.36g; soil conditions B
p= 0.5; Ts= 20sec
p= 0.5; Ts= 20sec

34. Giaralis and Spanos (2010)
Reponse spectrum compatible stationary processes
Sensitivity analysis for the assumed input power spectral shape