The Fifth Tongji-UBC
Symposium on Earthquake Engineering
PDEM-based Stochastic Seismic Response and Reliability
Evaluation of Nonlinear Structures
Ph. D. Student: Junyi Yang
Supervisor: Prof. Jianbing Chen
Department of Structural Engineering, Tongji University
Facing Earthquake Challenges Together - May 7, 2015 Shanghai China
Contents
The Fifth Tongji-UBC Symposium on Earthquake Engineering
Part 1
Background
random
nonlinear
Reliabil
ity
The Fifth Tongji-UBC Symposium on Earthquake Engineering
Background
5.12 Wenchuan Earthquake
Duration: > 2 minutes
Magnitude: 8.0 Ms
Depth: 19 Kilometers
Dead: 19, 195
Missing: 18, 392
3.11 Tōhoku earthquake and tsunami
Magnitude: 9.0 Mw
Dead: about 15, 884
Missing: 2, 633
Randomness of earthquakes: Magnitude, Duration, Frequency spectrum, etc.
The Fifth Tongji-UBC Symposium on Earthquake Engineering
Background
Uncertainties of structural properties
Nonlinearity
800
线性
非线性
恢复力 ( kN )
600
400
200
 Coupling of randomness and nonlinearity
 Stochastic seismic response of nonlinear
0
-200
structures
-400
-600
Stochastic dynamical system
-0.05
0
0.05
层间位移 ( m )
0.1
 Different failure modes
The Fifth Tongji-UBC Symposium on Earthquake Engineering
Background
Available approaches:
Difficulties in analysis of stochastic seismic response and
evaluation of dynamic reliability of multi-degree-of-freedom
nonlinear structures
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
 Monte Carlo Simulation (MCS)
random nature of convergence
huge costs in computational resource
0
2
0
-2
-3
-2
0
1
2
3
Probabilistic information of
structural properties and earthquakes
PDEM
 Random Perturbation Method
0.028
Inter-story drift ( m )
secular term, small variability
available for static problems
0.022
0.016
0.010
0.004
-0.002
 Orthogonal Polynomials Expansion
available for linear systems
-1
8.8
9
9.2
Time ( sec )
9.4
9.6
Probabilistic information of nonlinear seismic response
 Probability Density Evolution Method (PDEM)
resolve the problems due to coupling of randomness and nonlinearity
The Fifth Tongji-UBC Symposium on Earthquake Engineering
Part 2
Fundamentals of PDEM
Inter-story drift ( m )
0.028
0.022
0.016
0.010
0.004
-0.002
8.8
9
9.2
Time ( sec )
9.4
9.6
The Fifth Tongji-UBC Symposium on Earthquake Engineering
Fundamentals of PDEM
Principle of preservation of probability
X (t1 )
If no new random factors arise nor existing factors
vanish, the probability will be preserved in the
evolution process of the stochastic system.
T he random event
at t ime inst ant t 1
t2
xi
Random event description: Pr X0  t   Pr Xt  t 
0

t0
pX (x 0 , t0 )dx 0   pX (x, t )dx
t
T he same random event
at t ime inst ant t 2
D
pX (x, t )dx  0
Dt t
X (t 2 )
Random event description of principle of
preservation of probability
Generalized Density Evolution Equation (GDEE)
X  f (X, Θ, t ), X(t0 )  x0
Physical equation (2):
Z  H (X,X), Z(t0 )  z 0
GDEE:
m 1
pZΘ (z, θ, t ) m
p (z, θ, t )
  Z j (θ, t ) ZΘ
0
t
z j
j 1
pZΘ ( z , θ, t )
p ( z , θ, t )
 Z (θ, t ) ZΘ
0
t
z
Marginalization:
pZ( z, t ) =
ò
WQ
pZQ ( z, q, t )d q
PDF at 3.39 sec
PDF at 4.57 sec
PDF at 5.56 sec
100
PDF
Physical equation (1):
150
50
0
-0.07
-0.05
-0.03 -0.01
0.01
Inter-story drift ( m )
0.03
0.05
Evolution
PDF
of nonlinear
seismic
response
Probability
density
function(PDF)
of nonlinear
seismic response
Joint
PDFofof
nonlinear
seismic
response
The Fifth Tongji-UBC Symposium on Earthquake Engineering
D
(b
Fundamentals of PDEM
PDEM-based reliability evaluation
Absorbing boundary:
Extreme value distribution (EVD) theory:
Once the criterion is violated, the corresponding
probability could not return to the safety domain.
By constructing equivalent extreme value events,
PDEM could be employed to obtained the EVD
of the seismic response of nonlinear structure .
GDEE:
pZΘ ( z , θ, t )
p ( z , θ, t )
 Z (θ, t ) ZΘ
0
t
z
(
Boundary condition: pZQ ( z, q, t )= 0, for z Î Wf
Reliability: R (t )   pZ (z, t )dz
Equivalent extreme value event:
Z ( )  h  X max (Θ, t ), 
Z ( ) |
 
X
c
max
(t )
EVD: p X max ( x, t )  pZ ( z  x, ) |  c
b
Reliability: R(t )  Pr{X max (t )  b}   p X ( x, t )dx
s
max
absorbing boundary
absorbing boundary
EVD
The Fifth Tongji-UBC Symposium on Earthquake Engineering
cumulative distribution function of EVD
Fundamentals of PDEM
Strategy of selection of representative points
Criterion of selection of representative points：
Partition of probabilityassigned space and point
selection
Theory of discrepancy
1
0.9
 Star discrepancy
0.8
0.7
 F-discrepancy (Fang and Wang, 1994)
0.6
0.5
 EF-discrepancy (Li and Chen, 2009)
0.4
0.3
 GF-discrepancy (Chen and Zhang, 2014)
0.2
Solve the deterministic
dynamic equation
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Partition
of probability-assigned
space
Selection
of representative points
Generalized F-discrepancy (GF-discrepancy):
Solve the GDEE
Maximum of the marginal discrepancies along all the
dimensions

DGF  Pn   max sup Fn ,i  x   Fi  x 
1i  s
Marginalization

Fn ,i  x    pq  I  xq ,i  x
n
q 1
Numerical algorithm of PDEM
pq   pΘ  θ  d θ
Vq
The Fifth Tongji-UBC Symposium on Earthquake Engineering
Marginal discrepancy
Fundamentals of PDEM
Strategy of selection of representative points
Extended Koksma-Hlawka inequality ：

CS
n
f (x)dx   pq f (x q )  s  DGF (Pn )  TV( f )
q 1
0.9
10 dimensions
0.20
0.10
Mean_error
Relative error
0.7
3 dimensions
5 dimensions
10 dimensions
0.5
0.3
5 dimensions
0.05
3 dimensions
0.01
0.1
0
0.05
0.1
s × DGF
0.15
0.2
0.02
0.05
0.10
Mean_DGF × s
The Fifth Tongji-UBC Symposium on Earthquake Engineering
0.15
Part 3
Numerical Examples
600
1-th
2-th
3-th
4-th
5-th
6-th
7-th
8-th
9-th
10-th
20
15
200
PDF
恢复力 ( kN )
400
0
-200
10
5
-400
-0.05
0
0.05
层间位移 ( m )
0.1
0
0
0.2
0.4
0.6
0.8
Dimensionless inter-story drift
1
The Fifth Tongji-UBC Symposium on Earthquake Engineering
Numerical Examples
1. a MDOF structure with different distributions and C.O.Vs
lognormal
stiffness
C.O.V
Weibull
0.1
1500
Mean ( m )
Case1-1
lumped mass
1000
Restoring force ( kN )
Cases
500
0.05
lognormal
Weibull
0.2
Case1-3
lognormal
Weibull
0.3
Case2-1
uniform
uniform
0.1
-1500
Case2-2
uniform
uniform
0.2
-2000
-0.08
Case2-3
uniform
uniform
0.3
0
MCS (100 000 points)
0
-0.05
0
Case1-2
PDEM (512 points)
5
10
Time ( sec )
Std.D ( m )
-0.06 -0.04 -0.02
0
0.02
Inter-story drift of first floor ( m )
A typical hysteretic curve
0.02
MCS (100 000 points)
0.01
0
0
0.04
PDEM (512 pionts)
5
10
Time ( sec )
2
rPDEM  rMCS
rMCS
2
er 

2
rPDEM  rMCS
rMCS
15
20
Mean and standard deviation of seismic response
Relative errors of results of PDEM (512 points) by comparing with MCS (10, 000 points)
er 
20
-500
-1000
(Annotation: C.O.V stands for coefficient of variation)
Relative error:
15


The Fifth Tongji-UBC Symposium on Earthquake Engineering
Numerical Examples
PDFs of responses under different distributions and different coefficients of variation at 4.42 sec
概率密度函数
50
60
δ = 0.1
δ = 0.2
δ = 0.3
50
概率密度函数
60
40
30
20
10
δ = 0.1
δ = 0.2
δ = 0.3
40
30
20
10
0
-0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02
底层层间位移/m
0
-0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02
底层层间位移/m
0
lognormal & Weibull
0
uniform & uniform
PDFs of responses under different distributions and different coefficients of variation at 9.72 sec
90
δ = 0.1
δ = 0.2
δ = 0.3
概率密度函数
概率密度函数
90
60
30
0
-0.02
0
0.02
0.04
底层层间位移/m
0.06
lognormal & Weibull
0.08
δ = 0.1
δ = 0.2
δ = 0.3
60
30
0
-0.02
0
0.02
0.04
底层层间位移/m
0.06
uniform & uniform
The Fifth Tongji-UBC Symposium on Earthquake Engineering
0.08
Numerical Examples
2. Nonlinear structures with different DOF
PDEM (300 points) & MCS (10, 000 points)
PDEM (512 points) & MCS (10, 000 points)
The Fifth Tongji-UBC Symposium on Earthquake Engineering
Numerical Examples
2. Nonlinear structures with different DOF
1
200
PDE at 1.85 sec
PDF at 5.56 sec
PDF at 9.04 sec
20 dimensions (300 points)
150
0.8
CDF
CDF
0.6
100
9.04 s
0.4
50
0
0.2
-0.04
-0.02
0
0.02
Inter-story drift ( m )
0.04
0.06
Cumulative distribution function of seismic response
1
150
PDF at 3.39 sec
PDF at 4.57 sec
PDF at 5.56 sec
50 dimensions (512 points)
5.56 s
0
-0.06 -0.04 -0.02
0
0.02 0.04
Inter-story drift ( m )
0.06
Probability density function of seismic response
3.39 s
0.8
100
PDF
CDF
0.6
0.4
(Annotation: The dots stand for
MCS with 10, 000 points, the
lines stand for PDEM)
1.85 s
50
4.57 s
5.56 s
0.2
0
-0.07
-0.05
-0.03 -0.01
0.01
Inter-story drift ( m )
0.03
0.05
Probability density function of seismic response
0
-0.08 -0.06 -0.04 -0.02
0
0.02
Inter-story drift ( m )
0.04
Cumulative distribution function of seismic response
The Fifth Tongji-UBC Symposium on Earthquake Engineering
Numerical Examples
3. Reliability evaluation of nonlinear structure with high dimension
Failure probability
1
Failure probability
CDF
0.8
0.6
Threshold
levels
PDEM
(512 points)
MCS
(100, 000 points)
0.032
1.6653×10-3
1.9039×10-3
0.030
3.7518×10-3
3.6777×10-3
0.028
6.6499×10-3
6.5994×10-3
0.026
1.1184×10-2
1.1043×10-2
0.024
1.7240×10-2
1.7888×10-2
Reliability
0.4
0.2
PDEM (512 points)
MCS (100 000 points)
0
0
0.01
0.02
0.03
Inter-story drift angle of first floor
0.04
Cumulative distribution function of EVD
of nonlinear seismic response
The Fifth Tongji-UBC Symposium on Earthquake Engineering
Numerical Examples
Conclusions:
 PDEM is capable of resolving the difficulties due to the coupling of randomness and
nonlinearity.
 The probability density function (PDF) of stochastic seismic responses of nonlinear
structures is ever-changing, and accompanied with a significant characteristic of
evolution, which is of great importance in the reliability evaluation.
 PDEM could obtain the instantaneous PDF of stochastic seismic responses of nonlinear
structures and capture the evolution of the PDF. By adopting representative points with
minimum GF-discrepancy, PDEM could provide enough accuracy and demand much less
computational efforts compared to MCS.
The Fifth Tongji-UBC Symposium on Earthquake Engineering
The Fifth Tongji-UBC
Symposium on Earthquake Engineering
Thank you for your kind attentions!
Junyi Yang
Department of Structural Engineering
Tongji University
No.1239, Siping Road, Shanghai 200092, China
E-mail: [email protected]
Facing Earthquake Challenges Together - May 7, 2015 Shanghai China