Lecture 4 sapienza 2017

Academic excellence for business and the professions
Lecture 4:
Statistical linearization methodologies for inelastic
seismically excited structures
Lecture series on
Stochastic dynamics and Monte Carlo simulation
in earthquake engineering applications
Sapienza University of Rome, 20 July 2017
Dr Agathoklis Giaralis
Visiting Professor for Research, Sapienza University of Rome
Senior Lecturer (Associate Professor) in Structural Engineering,
City, University of London

Overview of Statistical linearization
Stochastic input
process
(Gaussian)
Non-linear system
(elastic or inelastic)
Response statistical
properties
(Non-Gaussian)
Underlying linear system
Response statistical
properties (Gaussian)
of a linear system
Assume Gaussianity
and adopt a
Statistical criterion
For stationary input process (e.g., PSD), the equivalent linear system is time-
invariant
For non-stationary input process (e.g., EPSD), the equivalent linear system is time-
varying
In many statistical linearization techniques, the equivalent linear system does not
have any physical meaning, or it is never explicitly defined
The properties of the equivalent (underlying) linear system depends on the
nonlinear system and on the excitation process

Example of nonlinear elastic behaviour in earthquake
engineering: “seismic pounding”
ωn= (k/m)1/2 ; ζn= c/(2mωn)
Base isolated structures
Monolithic bridges
Linear springs:
Nonlinear springs
(Hertz impact model):
 
0 ;
;
;
x
x x x
x x

  
 
 

  
   
   
 
3/2
3/2
0 ;
;
;
x
x x x
x x

  
 
 

  

    

Examples of nonlinear hysteretic behaviour in
earthquake engineering: “material yielding”
The bilinear hysteretic model is commonly used by
seismic codes of practice to represent inelastic
behaviour in deriving inelastic response spectra.
with
Linear
part
hysteretic
part
Extra state variable z and non-
linear first-order governing
differential equation

where γ, β, n, A are
constant parameters
which control the shape
of the hysteretic loops
defined by the above
differential equation.
The versatile Bouc-Wen model used with a viscously damped nonlinear SDOF oscillator
related to its relative non-dimensional displacement (normalized by a nominal yielding
displacement xy), through a differential equation (Wen, 1976):
               
             
2 2
1
2 1 / ; 0 0 0n n n y
n n
x t x t a x t a z t g t x x x
z t x t z t z t x t z t Ax t
  
 

        

   
Σκυρόδεμα καθαρότητος
+0,00 m
-3.85 m
αντισεισμικός
αρμός
Ισόγειο (Pilotis)
Υπόγειο
ελαστομεταλλικό
εφέδρανο
2000150800

The Bouc-Wen model considers an additional state ( ) in the equation of motion
               
             
2 2
1
2 1 / ; 0 0 0n n n y
n n
x t x t a x t a z t g t x x x
  
 

        

   
z
The bilinear model is a limiting case of the “smooth” Bouc-Wen!

The standard second-order statistical linearization for
SDOF systems with no hysteresis
Nonlinear system
subject to stationary
Gaussian process:
Assumed
equivalent linear
system (ELS)
Assumed error
function to be
minimized
   2 2
, 2 2n n n eq eq eqx x x x           Crandall 2001
Assumed
minimization
criterion
   2 2
2
0 0
eq eq
E and E 
 
 
 
 
Equivalent linear
properties and
And more assumptions: (I) Approximate the (unknown) distribution of x by a Gaussian
distribution in evaluating the expectations; (II) Take the variances of x and y as equal

Equivalent linear
properties become
and
For many different φ functions the above integrals can be computed in
closed-form as functions of the (unknown) variances:
and
4x4 system of
nonlinear
equations that
needs to be
satisfied
simultaneously
(numerical
solution is
required…)
The input PSD appears in the variances….
Most applications focus on these estimated nonlinear response variances

2 3
2 ( ) ( )n n nx x x x w t      
Classical example: white noise excited Duffing oscillator
Roberts and
Spanos 2003

The standard stochastic averaging for weakly nonlinear
SDOF systems with hysteresis
Caughey 1960
Nonlinear system
subject to stationary
Gaussian process:
Assumed
equivalent linear
system (ELS)
           cos sineq eq eqy t A t t t and y t A t t t              Assume a “lightly damped” ELS
so response is approximated as
pseudo-harmonic
Assume constant over one
“cycle of response”
envelop A and phase φ
 
 
cos
sin
eq
eq eq
y t A t
y t A t
 
  
   
    
  
 
  
 
2
2
2
n
eq n
eq eq
eq
E AJ A
E A
E AC A
E A

 
 

 

Equivalent linear
properties
where
(temporal averaging
over one cycle)
   
   
2
0
2
0
1
sin ,
1
cos ,
J A A t d
S A A t d


 

 

 




Caughey 1960
the envelop can be written as:    
 2
2
2
eq
y t
A t y t

 
Using:
And, apparently:  
 2
2
2
eq
E y
E y


Recall that the envelop has a Rayleigh distribution:  
2
2 2
, exp
2y y
A A
f A t
 
 
   
 
which can be used to compute the expected values in the ELPs…
For the bilinear hysteretic oscillator:
           cos sineq eq eqy t A t t t and y t A t t t              
where
3x3 system of
nonlinear
equations that
needs to be
satisfied
simultaneously

α=0.02
Roberts and
Spanos 2003

A stochastic dynamics response spectrum-based analysis framework
(Giaralis and Spanos 2010)
Sα(Τ,ζ): Elastic response spectrum for various
damping ratios ζ
G(ω): Response spectrum compatible
“quasi”-stationary power spectrum
for damping ratio ζn
Vibro-impact SDOF system with
viscous damping ratio ζn and pre-
yield natural period Tn
Second-order
Statistical
Linearization
Solution of an
inverse stochastic
dynamics problem
Sα(Τeq,ζeq):
Peak response of the vibro-
impact system obtained
from the elastic response
spectrum
Step 1
Step 2
Effective linear properties (ζeq, Teq) characterizing an equivalent linear
system (ELS)
A two-step approach
(Spanos and Giaralis 2008;
Giaralis and Spanos 2010)

EC8 design spectrum considered: ζ=5%; PGA= 0.36g; soil conditions B
The thus derived
power spectrum is
used as a surrogate for
determining effective
natural frequency and
damping parameters
associated with various
vibro-impact systems.
p= 0.5; Ts= 20sec
p= 0.5; Ts= 20sec
Step 1: Design spectrum compatible power spectra

Step 2: Statistical linearization of vibro-impact systems
           2
2 ; 0 0 0eq eq eqy t y t y t g t y y       
2 2 2
1
2
eq n na erf

  

  
    
  
 
2
2 2 2
2
3
exp
22
eq n n
u
u du

   


 
    
 

 
   
2
2 22 2
0 2eq eq eq
G
d

 
   


 

n
eq n
eq

 


Substitute the nonlinear equation of the vibro-impact system:
by an equivalent linear:
For the linear springs: For the Hertzian springs:
 
   
2
2 22 2
0 2eq eq eq
G
d

 
   


 

n
eq n
eq

 


 
0 ;
;
;
x
x x x
x x

  
 
 

  
   
   
 
3/2
3/2
0 ;
;
;
x
x x x
x x

  
 
 

  

    
              2
2 ; 0 0 0n n nx t x t x t x g t x x         
Example #1: vibro-impact systems

Effective linear properties
Various vibro-impact systems are considered excited by the EC8 compatible power spectrum
*
1
1
2
1 exp 2
eq
eq
n s
n
T
T

 

 
 
  
 
Select Ts such that:

Use of the effective linear properties in conjunction with the EC8 elastic
design spectrum for various values of damping.
Peak response estimation of the vibro-impact systems

Validation via Monte Carlo analyses
Numerical validation of the proposed approach by considering an ensemble of 250
artificial accelerograms whose average response spectra practically coincides with the
considered EC8 design spectrum (Giaralis and Spanos 2009).

where
3x3 system of
nonlinear
equations that
needs to be
satisfied
simultaneously
Example #2: bilinear hysteretic systems- Caughey’s approach (stochastic averaging)
Various bilinear hysteretic systems are considered excited by the EC8 compatible power spectrum

40 EC8 compatible
accelerograms used
(Giaralis and Spanos 2009)

Derivation of constant strength and constant ductility spectra
without resorting to NRHA
Giaralis and Spanos 2010

Can we do better than this???
Higher-order statistical
linearization!

System of governing differential equations for the bilinear oscillator (Asano
and Iwan, 1984; Lutes and Sarkani, 2004)
              2 2
12 1 , ,n n n yx t x t a x t a f x t z t x g t       
        2 , , yz t x t f x t z t x
           
            
1 2, , , ,y y
y y y
f x t z t x z t f x t z t x
x U z t x U x t U z t x U x t

     
where
                 2 , , 1y y yf x t z t x U z t x U x t U z t x U x t      
State z is considered in addition to x and dx/dt
Example #3: bilinear hysteretic systems- 3rd order statistical linearization
Higher-order statistical linearization for enhanced accuracy

Third-order equivalent linear system
(Asano/Iwan, 1984; Lutes/Sarkani, 2004)
where
   
22
1 2 2 2 2
1
exp
2 1 2 1 2 1
y y yz
x z x z z
x x x
C erfc
 
       
       
        
 2
2 32
2
1 1
1 exp
2 2 1y
z
y
xz
x v
C erf v erf dv C


  
    
                

   
2 22
4 2 2 22 2
1
exp 1 exp
2 2 12 2 1
y x y y yx
z z zz z
x x x x
C erf
    
      
                         
          2 22 2
; ;x z
z x
E x t z t
E x t E z t  
 
  
              2 2
1 22 1n n nx t x t a x t a C x t C z t g t        
     3 4 0z t C x t C z t   Four Linearization coefficients…
… functions of three moments:

              2 2
     3 4 0z t C x t C z t  
Frequency domain statistical linearization formulation
(Spanos and Giaralis 2013)
 
 
 
 
 
 
 
0
x t x t x t g t
z t z t z t
              
         
             
M C K
   2 2 2
1 2
3 4
1 0 2 1 0 1
; ;
0 0 1 0
n n n na C a a C
C C
         
      
    
M C K
 
   
   
 
 
 
0
0 0
xx xz
zx zz
B B G
B B
  
  
 
   
    
  
*
B H H
 
   
   
 
12xx xz
zx zz
H H
i
H H
 
  
 
 
     
 
H M C K  2 2
0
x xxB d   

 
Written in matrix form
Input-output relationship for linear systems in the frequency domain
 2
0
z zzB d  

 

 
 
 
 
2 2
2 2 2 44
3 3
00
0 0
N
k
x k k
j jk
j k j
j j
i Ci C
G d G
i A i A

      
 


 

   
 
    
 
2 2 2 2
0 4 1 1 4 2 3
2
2 4 1 3
; 2 1 1 ;
2 1 ; 1
n n n n n
n n
A a C A a a C C a C C
A C a C A
    
 
      
    
  24
3
z
C
E xz
C
 
 
 
 
 
2 2
2 3 3
3 3
00
0 0
N
z k
j jk
j k j
j j
i C i C
G d G
i A i A
 
    
 


 
 
   
 
where:
Frequency domain statistical linearization formulation
(Spanos and Giaralis 2013)

 
 
 
 
2 2
2 2 2 44
3 3
00
0 0
N
k
x k k
j jk
j k j
j j
i Ci C
G d G
i A i A

      
 


 

   
 
  24
3
z
C
E xz
C
 
 
 
 
 
2 2
2 3 3
3 3
00
0 0
N
z k
j jk
j k j
j j
i C i C
G d G
i A i A
 
    
 


 
 
   
 
   
22
1 2 2 2 2
1
exp
2 1 2 1 2 1
y y yz
x z x z z
x x x
C erfc
 
       
       
        
 2
2 32
2
1 1
1 exp
2 2 1y
z
y
xz
x v
C erf v erf dv C


  
    
                

   
2 22
4 2 2 22 2
1
exp 1 exp
2 2 12 2 1
y x y y yx
z z zz z
x x x x
C erf
    
      
                         
A 7-by-7 system of non-linear equations: Iterative algorithm or optimization routine

       2
2 /eff eff eff yy t y t y t g t x     
Enforce equality of the variances of x and dx/dt with y and dy/dt:
 
   
2
2
2 22 2
0
/
2
y
x
eff eff eff
G x
d

 
    


 

 
   
2 2
2
2 22 2
0
/
2
y
x
eff eff eff
G x
d
 
 
    


 

 
 
2
4
3
0
0
N
k
k
jk
k j
j
i C
G
i A

 



 

              2 2
     3 4 0z t C x t C z t  
by a second-order linear system with effective properties: ζeff and ωeff
Replace the third-order linear system:
Step 3: “Order reduction” to linear SDOF system
(Giaralis/Spanos/Kougioumtzoglou 2011; Giaralis/Spanos 2013)

Sα(Τ,ζ): Elastic response spectrum for various
damping ratios ζ
G(ω): Response spectrum compatible
“quasi”-stationary power spectrum
for damping ratio ζn
Hysteretic SDOF system with
viscous damping ratio ζn and pre-
yield natural period Tn
Higher-order
Statistical
Linearization
Teff and ζeff :effective linear
properties characterizing a
linear SDOF oscillator
Solution of an
inverse stochastic
dynamics problem
Sα(Τeq,ζeq):
Peak response of the
hysteretic system obtained
from the elastic response
spectrum
Order reduction
via a statistical
criterion
Step 1
Step 2
Step 3
Spanos, P.D., Giaralis A., (2013),
“Third-order statistical linearization-
based approach to derive equivalent
linear properties of bilinear
hysteretic systems for seismic
response spectrum analysis,”
Structural Safety, accepted.
Giaralis/Spanos/
Kougioumtzoglou 2011,
Kougioumtzoglou/
Spanos 2013
A statistical linearization based framework for response
spectrum compatible analysis using higher-order linearization

Various bilinear hysteretic oscillators are considered excited by an EC8
compatible power spectrum
Response spectrum compatible effective linear properties

Peak inelastic response estimation

Assessment via Monte Carlo analyses

Example #4: Bouc-Wen hysteretic systems
Statistical linearization for the Bouc-Wen model
The extended 3-step framework can also accommodate the Bouc-Wen model
(Wen 1980)
where
1 2 3 4eq eqc F F A and k F F       
 
/2 /2
1 2
1 11
/2 2 /22
3 4
2 1
2 ; 2
2 2
2 1
2 2 1 ; 2
2 2
n n
n nz z
n nn
n nx z x z
n n
F P F
n nn n
F P F
 
 
   
 
 
 
    
      
   
     
         
    
/2 2
1 1
2 sin tann
L
P d and L


 


 
  
 
 

          2 22 2
; ;x z
z x
E x t z t
E x t E z t  
 
  
           2 2
2 1 /n n n yx t x t a x t a z t g t x       
      0eq eqz t c x t k z t   Two Linearization coefficients…
… functions of three moments:
           
             
2 2
1
2 1 /n n n y
n n
x t x t a x t a z t g t x
  
 

      

   
in which

Statistical linearization for the Bouc-Wen model
 
 
 
 
2 2
2 2 2
3 32 2
00
0 0
N
eq eq k
x k
j jky y
j k j
j j
i k i k GG
d
x x
i A i A
  
    
 


 
 
   
 
  2eq
z
eq
k
E xz
c
 
 
 
 
 
2 2
2
3 32 2
00
0 0
N
eq eq k
z
j jky y
j k j
j j
i c i c GG
d
x x
i A i A
  
  
 


 
 
   
 
1 2 3 4eq eqc F F A and k F F       
 
/2 /2
1 2
1 11
/2 2 /22
3 4
2 1
2 ; 2
2 2
2 1
2 2 1 ; 2
2 2
n n
n nz z
n nn
n nx z x z
n n
F P F
n nn n
F P F
 
 
   
 
 
 
    
      
   
     
         
    
/2 2
1 1
2 sin tann
L
P d and L


 


 
  
 in which

       2
2 /eff eff eff yy t y t y t g t x     
Enforce equality of the variances of x and dx/dt with y and dy/dt:
 
   
2
2
2 22 2
0
/
2
y
x
eff eff eff
G x
d

 
    


 

 
   
2 2
2
2 22 2
0
/
2
y
x
eff eff eff
G x
d
 
 
    


 

 
 
2
3
0
0
N
k eq
k
jk
k j
j
i k
G
i A

 



 

by a second-order linear system with effective properties: ζeff and ωeff
Replace the third-order linear system:
           2 2
2 1 /n n n yx t x t a x t a z t g t x       
      0eq eqz t c x t k z t  
Step 3: “Order reduction” to linear SDOF system
(Giaralis/Spanos/Kougioumtzoglou 2011; Giaralis/Spanos 2013)

Response spectrum compatible effective linear properties
for Bouc-Wen oscillators

Peak inelastic response estimation for Bouc-
Wen oscillators

Assessment via Monte Carlo analyses

- Energy distribution over time and frequency  harmonic wavelet transform
(Giaralis & Lungu 2012)
Pulse-freePulse-like
- Pulse(s)  low-frequency energy enrichment of pulse-free earthquakes
Example #5: bilinear hysteretic systems- non-stationary statistical linearization
Pulse-like ground motions (PLGMs)

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
A non-stationary stochastic model for
pulse-like ground motions (PLGMs)
   
2 2
2
2
1
( ) ( ) ( ) (, ( ))HF HF L rF r
r
LFA t G A t Gt A t GS   

  
+

Imperial Valley 1979 (array #6) pulse-like ground motion
Non-stationary input EPSD

4s 7s
About 0.35s period
About 3.8s period
Non-stationary input EPSD

System of governing differential equations for the bilinear oscillator (e.g.
Suzuki and Minai, 1987)
Bilinear hysteretic term
   
    
 
,
2
h
o o PL
f u t u t
u t u t y t
m
    
          , 1 ,hf u t u t aku t a kz t  
               1 1 1yz t u u t H u t H z t H u t H z t        
Additional “state”
       
         
cos
sin
u t A t A t t
u t A A t A t t
 
  
   
    
For “lightly” damped systems (Caughey 1960)
Response envelop:
A non-stationary stochastic linearization approach for
bilinear hysteretic systems

Define equivalent quiescent seismically excited linear SDOF oscillator
with equivalent/effective properties as functions of the envelop A(t):
           2
eq eq PLy t A y t A y t y t    
 
    2 2
0.5sin 2 ;1
;
y
eq o
y
A
A ua k
A a
mA
A A u
  

    
  
 
 
 
 
4
1 ;1
2
0 ;
y y
y
eq o o
eq
y
u u
A ua k
AA
mA A
A u
  

  
    
    
 
where cos(Λ)=1-2uy/A
The above ELPs are non-stationary stochastic
processes themselves since the response
envelop A(t) is a stochastic process…
weq
t( )= EA
weq
A( )é
ë
ù
û
beq
t( )= EA
beq
A( )é
ë
ù
û

Assume that A(t) follows a time-dependent Rayleigh pdf
and one can form and solve a Fokker-Planck equation using stochastic averaging
to retrieve a first-order differential equation for the response variance
weq
t( )= EA
weq
A( )é
ë
ù
û beq
t( )= EA
beq
A( )é
ë
ù
û
 
 
 
 
 
2
2 2
, exp
2u u
A t A t
f A t
t t 
 
  
 
 
      
   
  
2
2 2 2
2 2
,eq u
u eq u u
eq u
G t t
t t t
t
  
   
 
  
             2 2 2
eq u eq u gy t t y t t y t a t      
Kougioumtzoglou and Spanos (2009)
with a time-evolving response variance. The underlying SDOF can be written as:
Use Runge-Kutta for numerical integration
to solve for the response variance… 1
2
3

The equivalent natural frequency ωeq can be interpreted as an instantaneous stiffness index of
the inelastic oscillators since it decreases due to yielding at times where the oscillators are
exposed to the strong ground motion part of the input stochastic process.
4s 7s
It captures the impact of the salient non-stationary features of the input PLGM process
on the inelastic response in both the time and in the frequency domain.
Non-stationary ELPs for pulse-like stochastic
seismic excitation

The stiffer oscillator is
significantly excited
(and yields) relatively
early in time due to
the HF burst of energy
The flexible oscillator
yields approximately 3s
later in time from the
stiff oscillator (as
manifested by a
reduction to the ωeq) as
its response is almost
exclusively governed by
the LF burst of energy.
seismic excitation

It is seen that ξeq has a
reciprocal relationship with
ωeq and is associated with
an instantaneous
hysteretic energy
dissipation
It manifests of the level of
non-linear behavior in
terms of energy dissipation
seismic excitation

Extreme ELP values as functions of
the strength reduction factor
Peak non-linear response
estimation with no NRHA
seismic excitation

Asano, K. and Iwan, W.D. (1984), “An alternative approach to the random response of bilinear hysteretic
systems”, Earthquake Eng. Struct. Dyn., 12, 229-236.
Cacciola, P., Colajanni, P. and Muscolino G. (2004). “Combination of modal responses consistent with
seismic input representation”, J. Struct. Eng., ASCE, 130: 47-55.
Caughey TK. (1960). Random excitation of a system with bilinear hysteresis. Journal of Applied Mechanics,
ASME, 27: 649-652.
Crandall SH. (2001). Is stochastic equivalent linearization a subtly flawed procedure? Probabilistic
Engineering Mechanics, 16: 169-176.
Giaralis, A. and Spanos, P.D. (2009), “Wavelets based response spectrum compatible synthesis of
accelerograms- Eurocode application (EC8)”, Soil Dyn. Earthquake Eng., 29, 219-235.
Giaralis, A. and Spanos P.D. (2010), “Effective linear damping and stiffness coefficients of nonlinear systems
for design spectrum based analysis”, Soil Dyn. Earthquake Eng., 30, 798-810.
Giaralis A and Spanos PD. (2013). Derivation of equivalent linear properties of Bouc-Wen hysteretic
systems for seismic response spectrum analysis via statistical linearization. In: Proceedings of the 10th HSTAM
International Congress on Mechanics (May 25-27, 2013, Chania, Greece) (eds: Beskos D and Stavroulakis GE),
Technical University of Crete Press.
Giaralis, A., Kougioumtzoglou, I.A. and Dos Santos, K. (2017). Non-stationary stochastic dynamics response
analysis of bilinear oscillators to pulse-like ground motions. In: 16th World Conference on Earthquake
Engineering- 16WCEE (January 9-13, 2017, Santiago, Chile), paper #3401, pp.12.
Giaralis A, Spanos PD and Kougioumtzoglou IA (2011), “A stochastic approach for deriving effective linear
properties of bilinear hysteretic systems subject to design spectrum compatible strong ground motions.
Proceedings of the 8th International Conference on Structural Dynamics (EURODYN 2011), Leuven, Belgium,
4-6 July, pp. 2819-2826.
Bibliography

Iwan WD and Lutes LD. (1968). Response of the bilinear hysteretic system to stationary random excitation.
The Journal of the Acoustical Society of America, 43: 545-552.
Kougioumtzoglou, I.A., Spanos, P.D., (2013), “Nonlinear MDOF system stochastic response determination
via a dimension reduction approach,” Computers and Structures, 126, 135-148.
Lungu A and Giaralis A (2013). A non-separable stochastic model for pulse-like ground motions. In:
Proceedings of the 11th ICOSSAR International Conference on Structural Safety and Reliability for Integrating
Structural Analysis, Risk and Reliability (June 16-20, 2013, New York, US) (eds: Deodatis G, Ellingwood BR
and Frangopol DM), paper #136, pp. 1017-1024.
Lutes, L.D. and Sarkani, S. (2004), Random Vibrations: Analysis of Structural and Mechanical Systems,
Butterworth-Heinemann, Oxford.
Roberts, J.B. and Spanos, P.D. (2003), Random Vibration and Statistical Linearization, Dover Publications,
New York.
Spanos, P.D. and Giaralis, A. (2008), “Statistical linearization based analysis of the peak response of inelastic
systems subject to the EC8 design spectrum”, Proceedings of the 2008 Seismic Engineering International
Conference commemorating the 1908 Messina and Reggio Calabria Earthquake, A. Santini and N. Moraci,
Eds., Vol 2: 1236-1244, American Institute of Physics, New York.
Spanos, P.D., Giaralis A., (2013), “Third-order statistical linearization-based approach to derive equivalent
linear properties of bilinear hysteretic systems for seismic response spectrum analysis,” Structural Safety, 44,
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Wen, Y.K., (1976), “Method for random vibration of hysteretic systems,” Journal of the Engineering
Mechanics Division ASCE 102, pp. 249-263.
Wen, Y.K., (1980), “Equivalent linearization for hysteretic systems under random excitation,” J. Applied
Mech., ASME 47, pp. 150-154.
Bibliography

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