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System Reliability-Based Design Optimization of Structures Constrained by First Passage Probability
Junho Chun*University of Illinois at Urbana-Champaign, USA
June 17th, 2015
Junho SongSeoul National University, Korea
Glaucio H. PaulinoGeorgia Institute of Technology, USA
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Structural Engineering under Natural Hazards and Risks
Random Excitations
Random processNon-deterministic excitationsMany possibilities of the process
San Francisco Earthquake, 1907
0 5 10 15 20 25 30 35 40 45 50-600
-400
-200
0
200
400
600
800HYOGOKEN NANBU EQ - KOBE-JMA3.EW 1/17/1995 DT=0.02 Amax=617.14gal
x - time / DT = 0.02
Accele
rati
on
(g
al)
1. http://www.documentingreality.com
2. Photograph: Kimimasa Mayama/Reuters
1 2
Kobe Earthquake, 1995
One of the most fundamental requirements on building structures is to withstand variousuncertain loads such as earthquake ground motions, wind loads and ocean waves.
The structural design needs to ensure safe and reliable operations over a prolonged period oftime despite random excitations caused by hazardous events.
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Motivation – Reliable Structural Design under Stochastic Excitations
Structural systemCourtesy of Skidmore, Owing and Merrill, LLP
Structural elements optimization Structural performance optimizationA
ccel
era
tio
n
Time, sEl
evat
ion
, mStory Displacement, mm
Structural Design
Research aims to find the optimal structure and system under stochastic excitations
4
Reliability-Based Design Optimization Formulation / Sensitivity Analysis
Numerical Applications / Discussion
Outline
Discrete Representation Method First Passage Probability / Structural Engineering Constrains
5Der Kiureghian, A. (2000). The geometry of random vibrations and solutions by FORM and SORM. Probabilistic Engineering Mechanics, 15(1),: 81-90.
1
( ) ( ) ( ) ( ) ( )n
T
i i
i
f t t v s t t t
s v
Modeling Ground Excitations - Filtered Gaussian Process
0
1 1
T
0
1
( ) ( ) ( )
( ) ( )
2π / ( ) ( )
t
n n
i i i f i
i i
n
i f i
i
f t v s t d
v s t W h t t t
t v h t t t t
s v
Discrete Representation of Stochastic Excitation
The stochastic excitation is represented by a linear combination of basis functions, s(t), withstandard normal independent random variables, v:
Stochastic ground excitations can be modeled by using a filter representing the characteristic of soil mediumand Gaussian process.
Gaussian process Soil Medium (Filter)Filter parameter: ωg, ζg
Ground acceleration (Filtered Gaussian Process)
6
Discrete Representation of Responses of Linear Structures
The convolution integral for determining the responses of linear systems subjected to thestationary process can be developed with the impulse response function.
0
( ) (τ) ( τ) τ
t
su t f h t d Dynamic Responses
T
1 10
( ) ( ) ( τ) τ ( ) ( )
t n n
i i s i i
i i
u t v s h t d v a t t
a v
Deterministic, time-dependent - filter + structure
Random, time-independent
Instantaneous Failure Probability
Failure event of a linear system at a certain time ti
T
0 0 0: ( , ) 0 : : ( )f i f i f iE g t u E u t u E t u a v
Failure Probability
0 0: ( , ) 0 β ,f f i iP E g t u t u
00β ,i
i
ut u
t
a
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First Passage Probability
In the reliability analysis of dynamic system subjected to stochastic excitations, a significantproblem is to determine the first passage probability that any one of output states of interestexceeds a certain threshold value within a given time duration T.
0 0 0
1
( ) ( max | ( ) |) ( )n
n
fp sys t t i
i
P E P u u t P u t u
First passage probability is defining the problem as a series system problem such as:
Ssiger International Plaza Courtesy of Skidmore, Owing and Merrill, LLP
Stress Displacement
Song, J., and A. Der Kiureghian (2006). Joint first-passage probability and reliability of systems under stochastic excitation. J. Engineering Mechanics,
ASCE, 132(1):65-77.
Fujimura, K. and A. Der Kiureghian (2007). Tail-Equivalent Linearization Method for Nonlinear Random Vibration. Probabilistic Engineering Mechanics,
22: 63-76
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Reliability-Based Design Optimization under Constraints on First Passage Probability
Optimization Formulation
target
,
1
min ( )
. ( , ) : ( , ) 0 , 1,...,
with ( ) ( , ) ( ) ( , ) ( ) ( , ) ( , )
t
i isys
obj
n
fp i f k i k f c
k
lower upper
i
f
s t P E t g t P i n
t t t t
E
dd
d d
d d d
M d u d C d u d K d u d f d ( , )= ( ) ( )= ( ) ( )gt u t f t f d M d l M d l
Probabilistic Constraints in Structural Engineering
Stress Maximum Displacement Inter-Story Drift Ratio
Hearst Tower (New York City)http://www.sefindia.org/
Chun, J., Song, J., Paulino, G.H. System reliability-based design/topology optimization of structures constrained by first passage probability. In preparation.
Objective function
Probabilistic constraints
9
1
2
ue1,y
ue1,x
ue2,y
ue2,x
ul
e1
Ae, L
e, D
ene
θe
x
y ul
e2
cosθ
sinθ
e
e
e
n1, 2,1
1 2
1, 2,2
, , g
e x e xg ge
e e ege y e ye
u u
u u
uu u u
u
2 1
2 1
( , ) ( ( , ) ( , ))
( ( , ) ( , ))
g ge ee e e e e e
e e
l lee e
e
D Dt t t
L L
Du t u t
L
d n u d u d B u
d d
T T
2 1( , ) ( ( , ) ( , ) )ee e e
e
Dt t t
L d a d v a d v
Stress Maximum Displacement
Probabilistic Constraints in Structural Engineering - Detail
Inter-Story Drift Ratio
Engineering constraints can be expressed in terms of the discrete representation form as:
( , ) : ( , ) 0 ( , ) : ( , ) 0fe k e k fe k oe e kE t g t E t t d d d d
tip
( , ) : ( , ) 0
( , )( , ) : 0
f k k
k
f k o
E t g t
tE t u
H
d d
dd
1
( , ) : ( , ) 0
( , ) ( , )( , ) : 0
i
i
f k i k
i k i kf k o i
i
E t g t
t tE t u
H
d d
d dd
1
T T
1 2
β ,( , )( , ) ( , )
e oe e oee k fe
e e ke k e k e
L Lt P
D tD t t
d
b da d a d
( , ) :Stress( , ) β ,fe fe k k e kP E t t t d d d
_ ,fp e nt eP β R
10
Finite difference method1
1
( )
f sysP E
A
3
2
( )
f sysP E
A
4
4
( )
f sysP E
A
Sensitivity Analysis of Probabilistic Constraint (Stress, Time duration = 8 secs, σoe=35MPa)
Adjoint Method Verification
Adjoint method
( ) ,fp sys n
i i
P E
d d
β R
11
Numerical Application 1 – 2D Bracing System Optimization
target
,
1
2 2
min ( )
. ( , ) : ( , ) 0 , 1,...,
0.02m 1m
with ( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( )
t
i i
obj
n
fp i f k isysi k f c
k
f
s t P E t g t P i nE
t t t f t
dd
d d
d
M d u d C d u d K d u d M d l
2
2 2
2
(2 1)( ) exp( ) sin( 1 ) exp( )2 cos( 1 )
1
f f
f f f f f f f f f f f
f
h t t t t t
Kanai-Tajimi Filter
Image courtesy of SOM
Volume
Stress / Maximum Drift Ratio/ Inter-Story Drift Ratio
12
Numerical Application 1 – 2D Bracing System Opt. (Stress Constraint)
Optimized Structures
Pftarget=0.0668 Pf
target=0.0062 Pftarget=0.00023
Φo ωf ζf t (sec) ∆t (sec)Init. Bars
(m2)Threshold E
0.2 5π 0.4 6.0 0.06 0.5 σoe = 35 MPa 20,000 MPa
Convergence HistoryInitial area = 0.5m2
ω fζ foeou
0.462
0.276
0.123
13
Numerical Application 1 – 2D Bracing System Opt. (Stress Constraint)
Dynamic Response Comparison
Initial System Optimized System
Dynamic Behavior
14
Numerical Application 1 – 2D Bracing System Opt. (Inter-Story Drift constraints)
Φo ωf ζf t (sec) ∆t (sec)Init. Bars
(m2)Threshold E
1.0 5π 0.4 6.0 0.06 0.3 uoΔ = 1/50 20,000 MPa
Pftarget=0.0668 Pf
target=0.0062 Pftarget=0.00023
Optimized Structures Convergence HistoryInitial area = 0.3m2
15
Numerical Application 1 – 2D Bracing System Opt. (Inter-Story Drift constraints)
Pftarget=0.0668 Pf
target=0.0062 Pftarget=0.00023
Optimized Structures Dynamic Response Comparison (Pftarget=0.00023)
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Concluding Remarks
New framework integrating random vibration theories into structural optimization was developed.
First passage probability was incorporated into structural optimization.
SCM enables for an efficient and accurate computation of the failure probability of a large-size system reliability problem.
Efficient method of sensitivity calculation was derived.
Developed framework identified optimal bracing systems that can resist future realization of stochastic processes with a desired level of reliability.
Junho [email protected]
Thank you for your attention
Junho [email protected]
Acknowledgement
• National Science Foundation (NSF) - CMMI 1234243
BACK-UP SLIDES
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Numerical Application 1 – 2D Bracing System Opt. (Drift constraints)