2021-Earthquake Engineering - 17.pptx

Earthquake
Engineering
Lecture 17
Jianbing Chen(chenjb@tongji.edu.cn)
School of Civil Engineering, Tongji University
June 21, 2021

1
Chapter 7 Stochastic response
of structures under earthquakes
7.1 Fundamentals of stochastic processes
7.2 Time domain method for linear structures
7.3 Frequency domain method
7.4 Nonlinear stochastic response of structures
7.5 Dynamic reliability of structures

2
7.1.1 Time domain description
7.1 Fundamentals of stochastic process
x
t
o
x
t
o
x
t
o
1
t 2
t j
t

3
 Finite dimensional distributions
 Second order statistics
• mean
• Auto-
Correlation
function

4
 Stationary process
 Definition: strict - distribution
 Definition: weak – second moments
 Properties
• Symmetric
• Bounded
• Asymptotic decaying

5
7.1.2 Frequency domain description
 The Wiener-Khinchin formula
• The power spectral
density function (PSD)
• The auto-correlation
function
Norbert Wiener (1894-1964)
Aleksandr Yakovlevich
Khinchin (1894-1959)

6
 The physical sense of power spectral density function
• From the perspective of average energy
• From the perspective of a sample process

7
Finite Fourier Transform:
Power spectral density function:

8
7.2.1 Time domain method for SDOF systems
 The equation of motion of a SDOF system
 The standardized equation of motion
 Closed-form solution – The Duhamel integral

9
 The mean response
 Special case
If then

10
 The auto-correlation function

11
 Example 1: white noise excited system (Li & Chen 2009)

12
7.2.2 Time domain method for MDOF systems
 Equation of motion
 Closed-form solution – Duhamel integral
 Auto-correlation function matrix
 Direct Matrix Method

13
 Modal Superposition Method
 Modal expansion
 Generalized SDOF systems
 Standardized generalized SDOF systems

14
 Duhamel integral
 The structural response

15
 The structural response
 The auto-correlation function matrix

16

17

18
 Closed-form unit pulse response function matrix

19
7.2.3 Modal decomposition response spectrum method
 Modal Superposition Method
 Assume the responses are stationary

20
 Assume the responses are stationary
 Extreme value and the standard deviation – peak factor
 Then

21
 The CQC (complete quadratic combination)
 The SRSS (Square root of summation of squares)
 What is the correlation coefficients ? [Homework 1
of the Chapter 7]

22
Clough & Penzien (1995)

23
Newmark NN, Rosenblueth E. Fundamentals of Earthquake Engineering, Prentice-
Hall, 1971.

24
Li YG, Fan F, Hong HP. Engineering Structures, 151 (2017) 381-390

25
7.3.1 Frequency domain method for SDOF systems
7.3 Frequency domain method for linear struc
 Taking Fourier transform on both sides
 Frequency transfer function

26
 Frequency transfer function
 Taking the complex conjugate
 Multiplying both sides

27
 Example 2: excited by white noise
Li & Chen (2009)
Half-power method
for damping ratio
estimate:

28
 Example 3: the Kanai-Tajimi spectrum (Homework 2
of Chapter 7)
• Consider a filtered SDOF system
• The Fourier transform yields:
• The Fourier transform of the acceleration response

29

30
7.3.2 Frequency domain method for MDOF systems
 Modal superposition/decomposition method
 Generalized SDOF systems:
 Frequency response functions:
 Modal superposition:

31
 Equation of motion of a MDOF system:
 Pseudo-Excitation method (Lin 1985):

32
 Generalized SDOF systems:
 Power spectral density matrix:

33
 Example 4: Response of MDOF system to white noise
excitation
 The modal matrix and modal mass matrix:
 The frequency response functions:

34
 Example 4: Response of MDOF system to white noise
excitation
 The power spectral density of X1:
 Contribution of
different modes

35
7.4.1 The so-called unclosure problems
7.4 Stochastic response of nonlinear structu
-3 -2 -1 0 1 2 3
-6
-4
-2
0
2
4
6
Displacement (m)
Restoring
force
(kN)
Nonlinear
Linear
-0.02 -0.015 -0.01 -0.005 0 0.005 0.01
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
x 10
4
Inter-story Drift (m)
Restoring
force
(kN)
Linear
Nonlinear

36
 The principle of superposition does not hold！
• Duhamel integral is invalid
• Spectral analysis is invalid

37
 Second order statistics：
Second order statistics of input 
Second order statistics of output
 Linear systems：

38
 Nonlinear systems：
 Second order statistics：
Second order statistics of input ?
Second order statistics of output

39
 Nonlinear systems：
 Statistically equivalent linear systems：
 Discrepancy： The error is random！
7.4.2 Statistical linearization (equivalent linearization)

40
Criterion 1：Minimizing the mean-square error (least
square method)
Criterion 2：Error is orthogonal to the displacement
and velocity

41
Error orthogonal to displacement:
Error orthogonal to velocity：

42
Error orthogonal to displacement:
Error orthogonal to velocity：

43
Stationary processes (zero-mean case)：
Error orthogonal to displacement and velocity:

44
Case 1 – Duffing oscillator：

45
Case 2 – Viscous damping

46
Case 3 – TLCD (Homework
3 of Chapter 7)

47
• Conservation of mass → Continuity equation
• Conservation of momemtum → Equation of motion
• Conservation of energy → Equation of energy
• Principle of Preservation of Probability→ Probability
density evolution equation
Deterministic
systems
Stochastic systems
Li J, Chen JB. Computational Mechanics, 2004, 34: 400-409.
Li, Chen JB. Structural Safety, 2008, 30: 65-77. 47
7.4.3 Probability density evolution method (PDEM)

• Randomness in the initial condition → Liouville equation
• Randomness in external excitation → FPK equation
• Randomness in structural parameters → Dostupov-Pugachev equation
Chen JB, Li J. A note on the principle of preservation of probability and probability density
evolution equation. Probabilistic Engineering Mechanics, 2009, 24(1): 51-59
Change of
probability
density
Change of physical state

 State space description – Liouville equation
Chen JB, Li J. A note on the principle of preservation of probability and probability density
evolution equation. Probabilistic Engineering Mechanics, 2009, 24(1): 51-59

( )
t
X
D
dS
2
( )
t
X
1
( )
t
X
xi
The random event
at time instant t1
dV
(b) A domain in the state space
The same random event
at time instant t2
A certain random event
1
t
2
t
Li J, Chen J, Sun W, Peng Y. Probabilistic Engineering Mechanics, 2012, 28: 132-142. 50
( )
( ) ( )
( )
1 2
1 2
Pr{ ,
Pr{ , } }
t t
X
t t
X q q
=
Î W Î W
´ W ´ W
Q
Q
( ) ( )
1 2
1 2
, ,
, ,
t t
X X
p x dxd
t
d
t
p x xd
q
q
´ W
W
W ´ W
=
ò ò
Q Q
q q
q q
( ) 0
, ,
t
X
d
p x d
t
dt
xd
q
W´ W
=
ò Q
q q
 Preservation of probability – Random event description

( )
( )
( )
( )
( )
( )
0
0
0
, ,
, ,
, ,
, ,
, ,
, ,
t
X
X
X
X
X X
X
X X
X
d
p x dxd
d
p x J dxd
dp x d J
J p x dxd
p p h
h J p x J dxd
x x
p
t
dt
t
dt
t
t
dt d
p h
h p x
t
t
t
t
t
dx
x x
q
q
q
q
W´ W
W´ W
W´ W
W´ W
=
æ ö
÷
ç
= + ÷
ç ÷
ç
è ø
æ
æ ö ö
¶ ¶ ¶
÷ ÷
çç
= + +
÷ ÷
çç ÷ ÷
çç
èè ø ø
¶ ¶
æ
æ öö
¶ ¶ ¶ ÷
÷
çç
= + + ÷
÷
çç ÷
÷
çç
èè ø
¶
ø
¶ ¶ ¶
ò
ò
ò
ò
Q
Q
Q
Q
Q Q
Q
Q Q
Q
q q
q q
q
q q
q q
q
( ) 0
,
t
t
t
X X
X X
t
d
p hp
t
t
dxd
x
p p
h dxd
x
q
q
q
W´ W
W´ W
W´ W
æ ö
¶
=
¶ ÷
ç
= + ÷
ç ÷
ç
è ø
¶
æ ö
¶ ¶ ÷
ç
= + ÷
ç ÷
ç
è ø
¶
¶
¶
ò
ò
ò
Q Q
Q Q
q
q
q q
Li J, Chen JB. Stochastic Dynamics of Structures, John Wiley & Sons, 2009.

52
Physical equation
Generalized density
evolution equation
• Li J, Chen JB. Stochastic Dynamics of Structures. John Wiley & Sons, 2009.
• Li J, Wu JY, Chen JB. Stochastic Damage Mechanics of Concrete Structures. Science Press, 2014.
The Quantity of Interest (QoI) Z can be:
• Macro-scale responses: displacement, shear force,…
• Local quantities: stress, strain, …

Definition of first-passage reliability:
Single-boundary:
Double-boundary:
Envolop:
s
( ) Pr{ ( ) , (0, ]}
R T X t t T
= Î W Î
( ) Pr{ ( ) , (0, ]}
R T X t a t T
= £ Î
1 2
( ) Pr{ ( ) , (0, ]}
R T a X t a t T
= - £ £ Î
( ) Pr{ ( ) , (0, ]}
R T X t a t T
= £ Î
Á
Ã
Â
2
2
2
( )
( ) ( )
X t
X t X t
w
= +
&
Á
Ã
Â
( ) cos( )
( ) sin( )
X t A t
X t A t
w q
w w q
= +
= +
&
7.5.1 Definition

s
( ) Pr{ ( ) , (0, ]}
R T X t t T
= Î W Î
s
( ) Pr{ ( ) }
R T X T
= Î W
%
Definition of first-passage reliability:
7.5.1 Definition

7.5.2 Level-crossing process theory
Basic ideas：
The number of crossing the threshold over [0,T] is a
random variable N.
Pr{N=0} is the reliability of the first-passage problem.
a
( )
x t
t
up-crossing down-crossing
o

 Reliability problem of Poisson processes
The probability of crossing when there is no crossing before t：
( ) ( )
1
( ) { | }
( ) ( )
1 ( )
( )
( )
1 ( )
t
F t f t dt
t
t T t t T t
t
f t f t
F t
f t dt
F t
F t
a
- ¥
+
=
¥
= < + D >
D
ò
= ¾ ¾ ¾ ¾ ¾ ¾
®
-
¢
=
-
ò
Pr
( )
( ) ln[1 ( )] ( )
1 ( )
F t d
t F t t
F t dt
l l
+ +
¢
= ¾ ¾® - = -
-
0
0 0
( )
0
ln[1 ( )] ( )
1 ( )
t
t
t dt
F t C t dt
F t L e
l
l
+
+
-
- = - ¾ ¾®
ò
- =
ò

 Reliability function of Poisson processes
The crossing probibility:
1
( ) { | }
{ , }
1
{ }
[ ( )] [ ( )]
1
( )
( )
( )
t T t t T t
t
T t t T t
t T t
R t t R t
t R t
R t
R t
a = < + D >
D
< + D >
=
D >
- + D - -
=
D
¢
= -
Pr
Pr
Pr
( )
( )
( )
R t
t
R t
a
¢
= - ( )
( )
1 ( )
F t
t
F t
l +
¢
=
-
( ) 1 ( )
R t F t
= -

 Reliability of first-passage problem
0
( )
0
( ) 1 ( )
t
t dt
R t F t L e
a
- ò
= - =
0
( ) ( ) ( , , )
a XX
t t xp a x t dx
a a
+ ¥
+
= = ò &
& &
( )
( )
( )
R t
t
R t
a
¢
= -
Differential equation：
Reliability:
Single boundary：
( )
0
0
( ) ( ) ( )
( , , ) ( , , )
a b
XX XX
t t t
xp a x t dx x p b x t dx
a a a
+ -
-
+ ¥
- ¥
= +
= + - -
ò ò
& &
& & & &
Double boundary：

Figure 8.1 Excursion of a sample process
 Mean crossing rate
“Crossing” event constructs a counting process。
The average times of crossing in unit time is:
a
( )
x t
t
o
1
{ ( ) , ( ) }
X t t a X t a
t
a +
= + D > <
D
Pr

The probability of crossing:
{ }
,
,
0
0
1
{ ( ) , ( ) }
1
{ ( ) ( ) , ( ) }
1
( , , )
1
( , , )
1
( , , )
1
( , , )
( , , )
XX
x x t a x a
XX
x a x a x t
a
XX
a x t
XX
XX
X t t a X t a
t
X t X t t a X t a
t
p x x t dxdx
t
p x x t dxdx
t
dx p x x t dx
t
dx x tp a x t
t
xp a x t dx
a +
+ D > <
< > - D
¥
- D
¥
= + D > <
D
= + D > <
D
=
D
=
D
=
D
= D
D
=
ò
ò
ò ò
ò
&
&
&
&
&
&
&
&
&
& &
& &
& &
& & &
& & &
Pr
Pr
0
¥
ò
( )
x t
( )
x t
&
( )
( )
a x t
x t
dt
-
=
&
0
Equation (8.2.2)

 The meaning of mean crossing rate：
The probability of crossing in unit time:
1
{ ( ) , ( ) }
X t t a X t a
t
a +
= + D > <
D
Pr
{ ( ) , ( ) } ( )
X t t a X t a t t
a +
+ D > < = D
Pr
The probability of crossing once during [t, t+△t] is then：
The probability of happening twice or more during △t is a quantity
of higher infinitesimal (i.e., = 0).
{ ( ) , ( ) }
{ 1}
X t t a X t a
N
+ D > <
= =
Pr
Pr

Therefore,
{ 1}
{ 2} ( )
{ 0} 1
[ ] { 1} 1 { 0} 0
[ ]
N t
N o t
N t
E N N N t
E N
t
a
a
a
a
+
+
+
+
ü
ï
= = D ï
ï
ï
³ = D ï
ï
ý
ï
Þ ï
ï
ï
= = - D ï
ï
þ
Þ
= = ´ + = ´ = D
Þ
=
D
Pr
Pr
Pr
Pr Pr
Thus, mean rate of crossing
= probability of crossing in unit time

 Problem 1: Computation of the mean crossing rate
(1) Probability of crossing in the next unit time：
0
1
{ ( ) , ( ) }
( , , )
XX
X t t a X t a
t
xp x x t dx
a +
+ ¥
= + D > <
D
= ò &
& & &
Pr
(2) The probability of
conditioning on without
crossing: 1
( ) { | }
( )
1 ( )
t T t t T t
t
F t
F t
l +
= < + D >
D
¢
=
-
Pr
( , ) ( , )
1
{ ( ) | ( ) }
{ ( ) , ( ) }
1
{ ( ) }
( , )
a
X
F a t p x t dx
X t t a X t a
t
X t t a X t a
t X t a
F a t
a
a
- ¥
+
+
=
= + D > <
D
+ D > <
=
D <
ò
¾ ¾ ¾ ¾ ¾ ¾ ¾ ®
% Pr
Pr
Pr

 Problem 2: Vanmarcke modification (wide-band
process)
a
( )
x t
t
o
a
T a
T ¢
1
[ ]
a a
E T T
a +
¢
+ =
[ ]
( )
[ ]
a
X
a
a a
E T
p x dx
E T T
+ ¥
=
¢
+ ò
( )
( ) 1
[ ]
ˆ
a
X
X
a
p x dx
F a
E T
a a a
- ¥
+ + +
¢ = = =
ò
ˆ
( )
X
F a
a
a
+
+
=

,
[ ] a
a a
a R
E N r
a
a
+
+
= =
1
1
[ ]
1 exp{ }
a
a
E N
r-
- -
;
1
1 exp{ }
( )
a
a a
X
r
F a
a a
-
+ + - -
=
% 0
( )
0
( ) 1 ( )
t
t dt
R t F t L e
a
- ò
= - =
t
( )
X t
( )
A t
a
o
 Problem 3: Vanmarcke modification (narrow-band
process)

0
a =
2
2
1
exp
2 2
X
a a
X X
a
s
a a
p s s
-
æ ö
÷
ç ÷
ç
= = - ÷
ç ÷
ç ÷
ç
è ø
&
0
1
2
X
X
s
l
p s
=
&
 Mean crossing rate
Stationary Gaussian process：
Non-stationary Gaussian processes:
*2
2
2 2
* *2 *
2 2
1
( ) 1 exp
2 1 2
2 exp
2
2 1
X
a
X X
X X
X
a
t
a a a
s
a r
ps r s
r r
p
s s
s r
ì æ ö
ï ÷
ï ç
ï ÷
ç
= - -
í ÷
ç ÷
ï ç ÷
ç -
è ø
ï
ï
î
ü
æ ö æ öï
÷ ÷
ï
ç ç ï
÷ ÷
ç ç
+ - F ý
÷ ÷
ç ç
÷ ÷
ï
ç ÷
ç
÷
ç -
è ø
è ø ï
ï
þ
&
*
[ ( )], ( )
a a X t t
r r
= - =
E

 Shortcomings
(1) Difficult to compute mean crossing rate；
(2) Only the correlation between two time instances are
used.
s
(0, ]
1, ,
( ) { ( ) , (0, ]}
{ [ ( ) ]}
{ [ ( ) 0]}
t T s
j n j
R T X t t T
X t
R g
Î
=
= Î W Î
= Î W
= >
X
L
I
I
Pr
Pr
Pr

7.5.3 Absorbing boundary method based on PDEM

s
0
( ) Pr{ ( ) , [ , ]}
R t X t
t t
= Î W Î
First-passage problem:
71
射人先射马
Shoot at his horse before the horseman
擒贼先擒王
To catch brigands, first catch their king
Du Fu (712-770)

Dynamic reliability:
Generalized density evolution equation:
Absorbing boundary condition:
s
( ) { ( ) , (0, ]}
R T X t t T
= Î W Î
Pr
( , , ) ( , , )
( , ) 0
X X
p x t p x t
X t
t x
¶ ¶
+ =
¶ ¶
&
Q Q
q q
q
f
( , , ) 0, for
X
p x t x
= Î W
Q q

 
, ,
X
p x t
Θ
θ
Reliability
( , ) ( , , )
X X
p x t p x t d
W
= ò
( (
Q
Q q q
s
( ) ( , ) ( , )
X X
R T p x t dx p x t dx
¥
W - ¥
= =
ò ò
( (
“remaining” probability”

Absorbing boundary condition:
Reliability is:
b
( ) Pr{ ( ) , (0, ]}
R T X t x t T
= £ Î
b
( , , ) 0, for
X
p x t x x
= >
Q q
b
b
s ( , ) ( , )
x
X X
x
F p x t dx p x t dx
¥
- - ¥
= =
ò ò
( (
For the double boundary problem:

(a) Without absorbing boundary condition (b) With absorbing boundary condition
Figure 8.3 Contour of the PDF surface

Figure 8.4 First-passage reliability

Homework 4 of Chapter 7
Compare and discuss the two methods (PDEM and
level-crossing process based) for dynamic
reliability.

2021-Earthquake Engineering - 17.pptx

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