research into stochastic dynamic testing and reliability ... · research into stochastic dynamic...

Research into stochastic dynamic testing and reliability model updating

C S ManoharDepartment of Civil Engineering

IISc, Bangalore

AcknowledgementCollaborator: V S Sundar, PhD studentFunding: BRNS

Symposium: Greek tradition

…where men gather to drink, eat, contemplate, and ponder over life…

3

Multi-physics problems

Organization

• Overview of treatment of uncertainties in structural engineering

• Testing of highly reliable engineering systems functioning under random dynamic environment

• Existing instrumented structures and model updating

5

Hazards

• Earthquake• Wind• Waves• Vehicles• Blast• Impact• Fire

Undesirable consequence

http://betterplan.squarespace.com/todays-special/2011/5/3/5211-blade-failure-and-wind-project-residents-worries-and-tu.html

http://www.ribapix.comhttp://www.ecy.wa.gov

http://www.thehindu.com/news/national/article2900321.ece

Total no. of slides: 75Extremes

6

Irreducible

Limited knowledge Modeling approximat

Aleatoric

Epistemic

Black swan

ions Reducible

"unknown unHuman errors

Bona fide Mal

kno

a

wns"

fide

•

•

•

•

Uncertainties

http://www.geneticsandsociety.org

Alia: rolling of dice

7

Aleatoric or Epistemic?

http://www.eoearth.org/article/Earthquake

A K Chopra, Dynamics of structures.

1 12 23 34 45 56 6

×

When does the nature roll its dice?And, when it rolls, what happens?

Mavroeidis & Papageorgiu, BSSA, 2003

Near fault

Far field

SiteResponseBased model

8

Peak Ground Accelerationcontours with 10% probability of exceedance in 50 years(Type A sites)

Development of probabilsitic seismic hazard map of India, Technical report of WCE, NDMA, 2010

Long rangeuncertainties

Average shear wave velocity in top 30 m greater than 1.5 km/s Total no. of slides: 75

9

A Kareem (1987)

Short rangeuncertainties

Aleatoric uncertainties in art

11

JCSS (2002)

Description COVYield strength 0.07Ultimate tensile strength

0.04

Young’s modulus 0.03Poisson’s ratio 0.03Ultimate strain 0.06

−−

=

10100160.000145.00075.01

ρ

Steel as a 5-dimensional random variable

Distribution: Multivariate lognormal random variable

12

Load effectsMoments in framesAxial forces in framesShear force in framesMoments in platesForces in platesStresses in 2D solidsStresses in 3D solids

DistributionLNLNLNLNLNNN

Mean1.01.01.01.01.000

COV0.10.050.10.20.10.050.05

JCSS recommendations on model uncertainties

For “more or less standard FE models”

13

Pushover Results by Participants

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 100 200 300 400 500 600 700

Displacement (mm)

Lo

ad (

kN)

NPCIL1NPCIL2IITGIITBIITRTyagarajar1Tyagarajar2BARCSERC1CPRI+NITKSERC2SERC3AERBIITDThapar1Thapar2IITM

P

2P

3P

4P

ROUND ROBIN EXERCISE ON PUSHOVER EXPERIMENTS AND ANALYSIS OF PROTOTYPE STRUCTURE (BARC-CPRI Joint work)

1ST AND 2ND MAY, 2008

Data provided by Dr G R Reddy, BARC

14

D J Ewins, 1982 F Fahy, 1995

Total no. of slides: 75

15

Mail online, 12, Sept 2012

The “unknown unknowns” Relative to the observerNo data for prognosis••

Frameworks for modeling uncertainties

• Probability theory• Interval analysis• Convex sets• Fuzzy set• Hybrid models

16

ChallengeHow to combine these tools with structural analysis methods?

Structuralsystems

Uncertain actions

UncertainSystem parameters

Uncertain outputs

Propagation of uncertainties must be consistent with the laws ofmechanics

18

Intensity Measure (IM)Engineering Demand Parameter (EDP)

Treated as a set of random variables Damage Measure (DM)Decision Variable (DV)

Grouping of basic variables

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

, , , ( , , , )

| , , | , |

| | |

DV DM EDP IM

DV DM EDP

DV DM EDP

p dv dm edp im

p dv dm edp im p dm edp im p edp im p im

p dv dm p dm edp p edp im p im

=

≈

FUNDAMENTAL ASSUMPTION

Performance Based Structural Engineering (PBSE)

19

Stress analysisDynamics Fracture FatigueStabilityCorrosion…

Plate tectonicsState of stressFaultingWave propagationSite amplificationLiquefaction[Instrumental data]

RepairRetrofitInjuriesLoss of lifeDelayed damages

Finite element methodMonte Carlo SimulationsStochastic differential equationsLaboratory and field testingNonlinear behavior

Imperfection sensitivityGeometric complexityControls: passive/active

( ) ( ) ( ) ( ) ( )1 || |Vim dm

DMe

Edp

DPp edp id im

DV dv P dv p dm edpdm m dimdimλ

λ > = − ∫ ∫ ∫Client specification Fragility analysis Response analysisLoss analysis

Hazard

20

Mathematical models

Experimental models

Can we combine them?

What are the issues?

Studies on existing structures

21

1962 1966IS 1893

1970 2002

1967 Koyna1988 Bihar-Nepal1991 Uttarakashi1993 Killari1997 Jabalpur1999 Chamoli2001 Bhuj

What happens to existing structures?

Railway bridgesGauge conversionLocomotion Axle loads

22

Sensors• Strain gauges• LVDT-s• Accelerometers (uni-axial / tri-

axial, translation / rotation)• Tilt• TemperatureLoading• Static / Dynamic• Measured / Unmeasured• Diagnosis and Performance

assessment• NDT & acoustic emission• Cores and samples

http://www.goabest.com/WondersOfGoa/Dudhsagar-Falls-WondersofGoa.asp

Condition monitoring of existing railway Bridges

•Heavier axle loads •Longer trains •Higher speeds

Funding: Indian railways. Collaborators: J M Chandra KishenAnanth Ramaswamy

Total no. of slides: 75

Numerical models

Measurements

Data assimilation•Bayes’ theorem•Markov property

Predictive tools

Physical laws

•Synthetics•Laboratory •Field

Impe

rfec

t

•FEM•Spatio-temporal discretization•Limits on scales

Choice?

You cannot doubt everything and function; you cannot believe everything and survive. (Taleb)

24

( ) ( ) ( ) ( ) ( )0 0 0

1 | | |DV DM EDP

d imDV dv P dv dm p dm edp p edp im dim

dimλ

λ∞ ∞ ∞

> = − ∫ ∫ ∫

PBSE format extension to instrumented structures

( ) ( ) ( ) ( ) ( )

Damage model updating Fragility model updating Reliability model updatingLoss model updating Hazard model updat

|| 1 | , | , | ,DV DM EDP

d im DDV dv D P dv dm D p dm edp D p edp im D

dimλ

λ > = −

( ) ( ) ( )Reliability model upd

0 0

ating

0

System identification

ing

| , | , , |EDP EDP Xp edp im D p edp x im D p x D

di

dx

m∞ ∞ ∞

=

∫ ∫ ∫

∫ ∫ ∫

Line of action: Apply Bayesian tools

The PBSE format

The extension: D=measurement data

2525

Estimation of hidden states

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

, , , ; 0 , 0

, , ; 1,2, ,k k k k k

M U t C U t K U t F U t U t t G t t U U

y t H U t U t t k N

θ θ θ θ θ ξ

θ ε

+ + + = Γ + = + =

Given

( ) ( ) 1:, | ; 1,2,k k kp U t U t y k N = To find

Problem I

[ ]1: 1 2k ky y y y=

STRUCTURAL SYSTEM IDENTIFICATION

System

Input Output

Given GivenTo be determined

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

, , , ; 0 , 0

, , ; 1,2, ,k k k k k

M U t C U t K U t F U t U t t G t t U U

y t H U t U t t k N

θ θ θ θ θ ξ

θ ε

+ + + = Γ + = + =

Given

( )| ; 1,2,kp y t k Nθ = To find

Problem II

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( )

0 0Process equation , ; (0) ; 0

Response Measurement , , ; 1, 2, ,

Input measurement ( ) ; 1, 2,... ; 1, 2, ,k k k k k k k k

r k r k r k

MU t F U t U t p t t U U U U

y t f U t U t q U t U t k N

t p t t r NDOF k N

ς

ν

ρ ε

+ = + = = = + = = + = ≤ =

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )0

, , 0 0, | , ; 1,2, ,

max , , 0 | , ; 1,2, ,

0 | , ; 1,2, ,

p k k

k kt T

m k k

P h U t U t t t T y t t k N

h U t U t t y t t k N

h y t t k N

ρ

ρ

ρ< <

= < ∀ ∈ = = < = = < =

Given

To find

Reliability model updatingProblem III

Performance function could involve statesthat are not measured.

Remark

Two themes

• Random vibration testing with controlled samples

• Updating reliability models of instrumented dynamical systems

V S Sundar and C S Manohar• 2013, Int. Jl. of Non-linear Mechanics, 520, 32-40• 2013, Str. Safety, 40,21-30.• 2013, Structural Control & Health Monitoring, to appear

FE Model

•ParticleFiltering•MCMC

samplingMeasurements

FE modeling &Bayesian tools

Matlab

Commercial codes•Nisa•Ansys

•Laboratory•Field•Synthetics

Batch files in MS DOS

30

( )( ) ( )

1Process equation: , ; 0,1, 2,: , , ; 1,Measurement equat 2ion ,

k k k k k

k k k k k k

x h x w ky f x G x k

θ γθ θ ν

+ = + == + =

Nonlinear dynamic state estimation

( ) ( ) ( )

( ) ( ) ( )( ) ( )

1: 1 1 1 1: 1 1

1: 11:

1: 1

Prediction | | |

| |Updating |

| |

k k k k k k k

k k k kk k

k k k k k

p x y p x x p x y dx

p y x p x yp x y

p y x p x y dx

− − − − −

−

−

=

=

∫

∫

Kalman filter. Exact solution to the state estimation problem.

Use Monte Carlo simula

⇒

⇒

Linear systems with additive Gaussian noises

Nonlinear systems, non - Gaussian noises, multiplicative noises, ...tions

Particle filters.

Time• • •

• • •

Mathematicalmodel

Measurements

Chapman-Kolmogorov equationFPK equation

Markov process

Bayes' theorem

1kt − kt 1kt +

1ky − ky 1ky +

3232

( )

( )

Consider the problem of evaluation of the definite

integral ( ) .

This can be re-written as

1( ) ( ) ( ) ( )

1where ; is now interpreted

as the pdf of

b

a

b b

Xa a

X

I f x dx

I b a f x dx b a f x p x dxb a

p x a x bb a

=

= − = − −

= < < −

∫

∫ ∫

a random variable that is uniformly distributed in to . a b

33

( )

( )

1

Following this, the integral is now interpreted as an expectation

( - )

where the expectation is evaluated with respect to . Furthermore, is now approximated by

1ˆ ( )

where -s are u

X

N

ii

i

I

I b a f X

p x I

I f XN

X=

=

= ∑

( )niformly distributed random numbers

samples from .Xp x

34

100

101

102

103

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

sample size *100

Est

imat

e of

Iestimateexact

12

0

Evaluation of 1 / 3I x dx= =∫

3535

0 50 100 150 200 250 300 350 400 450 5000.2

0.25

0.3

0.35

0.4

0.45

0.5

run

Est

imat

e of

I

estimateexact

500 runs with 500 samples

12

0

Evaluation of 1 / 3I x dx= =∫

3636

0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.370

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normal PDFEmpirical PDF

Estimate of PDF

12

0

Evaluation of

1 / 3I x dx= =∫

( )

( ) ( ) ( )2

2

Remarksˆ lim with probability 1 Law of large numbers,

ˆlim 0, Central limit theorem

ˆ is a consistent estimator with minimum variance

In the evaluation of multi-fold inte

N

N

I I

N I I N

IN

σ

σ

→∞

→∞

• =

• − →

• =

• grals, the sampling variance is independent of dimension of the integral.

3838

( ) ( ) ( )( )

( ) ( )

12

01 1 2 2

2

0 0

2

11

.

Here a valid pdf defined over 0 to 1.

1ˆ where are samples drawn from .

.

NNi

i iii

I x dx

x XI x dx x dxx X

x

XI X xN X

π

ππ π

π

ππ =

=

=

= = =

=

=

∫

∫ ∫

∑

revisitedEvaluation of

39

( )

( )

( )

2

1 2

20

2

21

2

Let 3 ;0 1.

3

1 1ˆ = for any value of and hence for =1.33

3 ;0 1 is the ideal ispdf.Catch: the definition of this ispdf requires the knowledge of being evaluate

Ni

i i

x x x

xI x dxx

XI N NN X

x x x

I

π

π

π=

= < ≤

=

=

= < ≤

∫

∑

d.

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

x

pdf

Given pdfideal ispdf

41

( )

( ) .3111

10;

1

0

2

1

0

21

0

2

==⇒=⇒=

<<=

∫∫∫

dxxdxxdxx

xxx

ααπ

απ

( ) ( ) ( ) ( )

( )

( ) ( )

1

0 0

*

0

0

, ,

, 1, 2, ,

Let it be required to find P max , 0

max , highest response

q

i i ij jj

i i

F t T

t T

dX t A t t dt t t dB t

X t X i d

P h h t t

h t t

σ=

≤ ≤

≤ ≤

= +

= =

= − ≤

=

∑

The idea of "importance sampling" for dynamical systems

X X

X

X

( )

( ) ( )

( )

1

*

*

011

1

over a time period

permissible value of the response.1ˆ max , 0

ˆ

1ˆVar

ˆHow to reduce Var ?

Increase .

Ni

F t Ti

F F

F FF

F

h

P I h h t tN

P P

P PP

N

P

N

≤ ≤=

=

= − ≤

• =

−• =

∑ X

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( )

1

1

0 0 0 0

*

0

1

*

Alternative: Girsanov's transformation

, ,

, , 1, 2, ,It can be shown that

max

,

, 0 m

q

i i ij jj

q

ij jj

q

j jj

i i

t T

t t u tdX t A t t dt t t dB t

d t t u t dB t

X t X t

d

i d

I h h t t T h

t

I

σ σ==

=

≤ ≤

= + +

Γ = −Γ

= Γ = Γ =

− ≤ = Γ −

∑ ∑

∑

X

X

XX

( ) ( ) ( )

( )

2

00

*

012 0

ax , 0

1 max , 0

: Radon-Nikodym derivative

t T

Ni

F t Ti

h t t

P T I h h t tN

t

≤ ≤

≤ ≤=

≤ Γ

= Γ − ≤ Γ

Γ

∑

X

X

( )

( )

( ) ( )

2

*

1

Variance of the estimator depends upon sample size and the controls

Idea: Use to control the sampling variance.Importantly, there exists an ideal control.

1 1 , , 2

d

j kj

j

k

j

k

N

u t t

u t

t

t jX

u

ψψσ

=

∂ = − ∂∑

X

( ) ( ) ( ) ( )

*

0

*

1, 2, ,

max , 0

Sampling variance goes to zero.t T

j j

q

T I h h t t

u t u t

ψ≤ ≤

=

= Γ − ≤

⇒ = ⇒

X

( )

( ) ( )

( ) ( ) ( ) ( )

*

1

*

0

*

Idea: Use to control the sampling variance.Importantly, there exists an ideal control.

1 1 , , 1, 2, ,2

max , 0

Sampling variance goes to ze

d

j kjk k

t T

j j

j

u t t t j qX

T I h h t t

u t

u

u

t

t

ψσ

ψ

ψ=

≤ ≤

∂ = − = ∂

= Γ − ≤

⇒ = ⇒

∑

X

X

ro.: The construction of the ideal control requires the

knowledge of the very quantity being sought.: construct suboptimal contr

ols.

,ψCatch

Strategy

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )

( ) ( ) ( )

0

Idea: find the controls by solving a deterministic problem

Take , & ,

and consider the deterministic system

0

m mi ij j

t t t t t t

d tt t t

dt

V h t t uτ τ α

= =

= +

=

⇒ = −

Distance minimizing suboptimal controlA

A

A X X X

VV u

V X

σ

σ

σ

( )

( )

1 0

1

, 1, 2, , ; 1, 2, ,

where &

, 1, 2, , ; 1, 2, , impulse response functions

mq

jj

r

lj jl

mij

t dt i r j q

h t i r j q

τ

α

τ

=

=

= =

=

− = = =

∑ ∫

∑

σ

( )( )

( ) ( )

( )

2 2

1 0

2 *

1

2 *

1 0 1 1 0

Define

Find by solving the optimization problem:

minimize subject to the constraint 0

0 and

q mm

jkj k

rm m

i ii

q qm r mm

jk i ij k jk jkj k i j k

u

u t

h V

L u h h t u

L L

β τ

β τ ψ τ

λ ψ τ α

λ

= =

=

= = = = =

= ∆

− =

⇒

= ∆ + − − ∆

∂ ∂

=∂ ∂

∑∑

∑

∑∑ ∑∑∑

( )

( ) ( )

*

1

2

1 0 1 1

0, 1, 2, , ; 0,1, ,

,

1, 2, , ; 0,1, ,

jk

rm

jk i ij ki

jk q m r rm m

jk k i i j ki ijj k i i

j q k mu

h h tu

h t h t

j q k m

α ψ τ

α ψ τ ψ τ

=

= = = =

= = =

⇒

− − =

− − − −

= =

∑

∑∑∑∑

Nonlinear systems• In absence of mathematical model: use impulse response function

• If mathematical model is available, use the model to derive the distance minimizing controls.

• Use alternative variance reduction schemes (currently under development)

( )( ) ( ) ( ) ( )( ) ( )

( ) ( )

( )

1 2

2 3

0

0 1

0 0

2

*

1

2

2.31, 0.7 0.2,0 0

0.06, 4 , 1, 10

P ma

, 0 0

x 0F

t

t

tz z z z F t

e t A e eF t F t e t w t

Az

P h z

z

t

α αηω ω

η

α

βα α ω π β σ

≤ ≤

− −

= = =

+ + + = = −+ =

= − = == ==

= −

≤

0 1 2 3 4 5 6 7 8 9 10-2

0

2

4

6

t s

u(t)

m/s2

0 1 2 3 4 5 6 7 8 9 10-2

0

2

4

t s

u(t)

m/s2

0 1 2 3 4 5 6 7 8 9 10-2

0

2

4

t s

u(t)

m/s2

OptimizationApproximation



α = 105

α = 106

α = 104

To implement the Girsanov transformation we need to establishSuboptimal controls

Distance minimizing control in terms of impulse response function of the systemThe Radon-

•

•

Nikodym derivative

Important observation

We do not need a mathematical model for the structureto establish these quantities

We could think of a experimental reliability testing method

( )gx t

( )gy t

5252

Kanai Tajimi & Clough and PenzienPower spectral density function modelsfor free field earthquake ground acceleration

−

Bed rock

Ground

Soil layer

( )bx t

( )u t

Local site condtionsare accounted for( ) : White noisebx t

53

( ) ( )( )

( ) ( )( )

( )

( )( )

( )( ) ( )

4 2 2 2

22 2 2 2 2

4 2 2 22

22 2 2 2 2High pass filter

44 2 2 2

2 22 22 2 2 2 2 2

High pass filter

4

4

4| |

4

4 /

4 1 / 4 /

g g g

g g g

g g gf

g g g

g g g f

g g g f f f

S I

S I H

I

ω η ω ωω

ω ω η ω ω

ω η ω ωω ω

ω ω η ω ω

ω η ω ω ω ω

ω ω η ω ω ω ω ς ω ω

+=

− +

+=

− +

+=

− + − +

Clough and Penzien model

545454

0 5 10 15 20 25 30-1.5

-1

-0.5

0

0.5

1

1.5

time t

acce

lera

tion

555555

( )Strategy: Use a deterministic modulaitng function.

( ) ( )

( ) determigX t e t S t

e t

=

=

How to allow for nonstationary nature of ground accelerations?

Nonstationarity : in amplitude modulation & frequency content.

( ) ( )0

0

nistic envelope function( )=zero mean stationary Gaussian random process

(with PSD given by Kanai-Tajimi or Clough and Penzien models)

( ) exp exp ; 0

( )

S t

e t A t t

e t A

α β α β = − − − > > =

Examples

( ) ( )1 expA t tα+ −

56

0 5 10 15 20 25 30-1.5

-1

-0.5

0

0.5

1

1.5

time t

acce

lera

tion

57

( ) ( )

( )( )( )

21 1 1 1 1 1

2 22 2 2 2 2 2 1 1 1 1 1

2

2

2

1 1

2 1

3 2

4 2

2

2 2

Ground displacementGround velocity

Ground acceleration

Introduce

y y y e t s t

y y y y y

y ty ty t

x yx yx yx y

η ω ω

η ω ω η ω ω

+ + =

+ + = +

=

=

( ) ( )

1 12

2 1 1 1 2

3 32 2

4 1 1 1 2 2 2 4

0 1 0 0 02 0 0 1

0 0 0 1 02 2 0

x xx x

e t s tx xx x

ω ηω

ω η ω ω η ω

− − ⇒ = + − −

58

( )

( )

( ) ( )

( ) ( )

2

2

0 1

for 0 4s4

=1 for 4 24s1=exp 24 for 24 s2

( ) exp exp ; 0

( ) exp

te t t

t

t t

e t a t t

e t A At t

α β α β

α

= < < < <

− − >

= − − − > >

= + −

Examples for envelope function

59

0 5 10 15 20 25 30-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

time s

disp

lace

men

t m

60

0 5 10 15 20 25 30-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

time s

velo

city

m/s

61

0 5 10 15 20 25 30-5

-4

-3

-2

-1

0

1

2

3

4

5

time s

acce

lera

tion

m/s

/s

0 2 4 6 8 10-0.5

0

0.5

1

1.5

2

2.5

3

t s

Bas

e di

spla

cem

ent m

m

Determination of impulse response function(Impulse at the bed rock level)

Motion to be applied at the shake table top

0 1 2 3 4 5 6 7 8 9 10-15

-10

-5

0

5

10

15

t s

h(-t)

µ-s

train

Measured impulse response for strain(shown reversed in time)

0 1 2 3 4 5 6 7 8 9 10-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

t s

h(-t)

mm

Measured impulse response for inter storey drift(shown reversed in time)

0 1 2 3 4 5 6 7 8 9 10-6

-4

-2

0

2

4

6

t s

u(t)

m/s2

Control force; performance function defined with respect to strain

* 325 -strain; 2.5 s.mh µ τ= =

0 1 2 3 4 5 6 7 8 9 10-8

-6

-4

-2

0

2

4

6

8

10

t s

u(t)

m/s2

Control force; performance function defined with respect to inter-storey drift

* 1.3 mm; 2.5 s.mh τ= =

0 1 2 3 4 5 6 7 8 9 10-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

t s

Bas

e ac

cele

ratio

n m

/s2

UnbiasedBiased, h* = 325 µ-strain

Sample excitation

0 1 2 3 4 5 6 7 8 9 10-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

t s

Bas

e ac

cele

ratio

n m

/s2

UnbiasedBiased, h* = 1.3 mm

Sample excitation

0 0.5 1 1.5 2 2.5 310-5

10-4

10-3

10-2

10-1

100

101

t s

Γ(t)

Realization 1Realization 2Realization 3

Radon-Nikodym derivative

50 100 150 200 250 300 350 40010-3

10-2

10-1

100

h* µ-strain

P F

Method 1Method 2

Linear frame, uniaxial ground motions

Method 1Brute force : 3000 samples

Method 2 250 samples

0.2 0.4 0.6 0.8 1 1.2 1.4 1.610-3

10-2

10-1

100

h* mm

P F

Method 1, case (i)Method 1, case (ii)Method 2, case (i)Method 2, case (ii)

Nonlinear frame, uniaxial ground motions



0 50 100 150 200 250 30010-6

10-5

10-4

10-3

10-2

10-1

100

h* µ-strain

P F

Method 1Method 2

Nonlinear frame, biaxial ground motions



Extension to automotive applications

BiSS (Private) Limited

Current developments

• How to derive controls/importance sampling strategies that explicitly takes into account nonlinearity?

• How to deal with material nonlinearity?

Way out: particle splitting methods?

Reliability model updating in instrumented dynamical systems

Two steps procedure

(a) Structural system identification(b) Reliability model updating

StrategyStructural system identification• Maximum likelihood estimation

Reliability model updating• Modify the scope of the Bayesian filtering tools

( ) ( ) ( ) ( )( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

0

Process equation

0Measurement equation

, 0Quantitites to be determinedˆ ,0

ˆ ˆ ,0t

d t t dt t dt d t

t t t t T

t t T

t t t t t T

τ τ

τ τ

= + +

=

= + ≤ ≤

= ≤ ≤

= − − ≤ ≤

Linear - Gaussian state space model

A

H

P

X X f B

X X

Z X

X X Z

X X X X Z

µ

( ) ( ) ( ) ( ) ( ) ( )

( )( )

( )

( ) ( ) ( )

( ) ( )

0

0

0

1

1

1

Kalman's filterˆ

ˆ ˆ

ˆ ˆ0

0

0 , 0

t t

t

t t

d tt t t t t

dt

d tdt

t t t

−

−

−

= + − +

=

= + − +

=

=

=

−

= =

g

g

A K H

PAP PA PH R HP Q

P P

K PH R

P

= A + Q

= H RH AP I

XX Z X f

X X

Ω Ψ

Ω Ω Ψ

Ψ Ω Ψ

Ω Ψ

( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( )

( )

0

0

1 1

: future excitationsWhat is the reliability against future loading actions?Idea: Interpret Kalman filter equations as SDE-s.

ˆ ˆ ˆ

ˆ ˆ0

0 ,

t

t t

f t

d t t dt t t t t dt d t− − = + − +

=

−

=

A H R H

= A + Q

= H RH AP

X X Z X B

X X

Ω Ψ

Ω Ω Ψ

Ψ Ω Ψ

Ω Ψ( )

( ) [ ] ( ) ( ) [ ]

( )

*|

*

*

0

0

P 0, ,0

ˆP 0,

ˆP max

S

t T

t

t

t

P t h t T Z T

t h t T

t h

τ τ

≤ ≤

=

= ≤ ∀ ∈ ≤ ≤

= ≤ ∀ ∈

= ≤

I

Z X

X

X

φ

φ

φ

( )

( ) ( )

( ) ( )

*| | 0

*| 0

1

| ||

ˆ1 max 0

1ˆ ˆmax 0

1ˆVar

Back to the same dilemma!How to reduce the sampling variance?

s

F S t T

Nl

F t Tls

F FF

s

t

t

P P I h t

P I h tN

P PP

N

φ

φ

≤ ≤

≤ ≤=

= − = − ≤

= − ≤

−=

∑

Z Z

Z

Z ZZ

X

X

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )

( ) ( )

( ) ( ) ( )

( )

0 0

0

* *00 0

*| 0

1 1

Girsanavo's transformation

0 , 0

0 , 0

ˆmax 0 max 0

1 max

t T t T

Fg

t

t

t t

t t

d t t dt t t t t dt t t dt t d t

d t t t d t

I h t T I h t

P T I hN

φ φ≤ ≤ ≤ ≤

− − = + − + +

Γ = −Γ

= Γ = Γ

−

= =

− ≤ = Γ − ≤ Γ

= Γ −

A H R H

= A + Q

= H RH AP I

Z

X X Z X u B

u B

X X

X X

Ω Ψ σ σ

Ω Ω Ψ

Ψ Ω Ψ

Ω Ψ

( ) ( ) 01

0gN

l

t Tl

t tφ≤ ≤

=

≤ Γ ∑ X

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )( )

2

1 0

0

Controls: Derive from the Kalman's filter equationTakes into account the measurements

; 0

0

mm

mj

j

u t dt

d t t dt t t t dt t t dt t T

τ

β τ=

⇒

=

= + − + ≤ ≤ =

∑ ∫

gA K HV V Z V u

V X

σ

( ) ( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( )

( ) [ ] ( )

0

*|

Process equation, ,

0

Measurement equation, 0

Quantity to be determined

P 0, ,0S

d t t dt t d t t d t

t t t t T

P h t h t T Tτ τ

= + +

=

= + ≤ ≤

= ≤ ∀ ∈ ≤ ≤

Nonlinear state space model

H

Z

X A X X B B

X X

Z X

X Z

σ σ

µ

• Girsanaov’s transformation• Distance minimizing controls• Modified state space model• Bootstrap based Monte Carlo filter

( ) ( ) ( )

( ) ( )

21 2

2 2

2

2 2s s s s s s

f f f f f f s s s s s

f

y y y e t w t w t

y y y y y

y t t

η ω ω

η ω ω η ω ω

+ + = +

+ + = +

+ + = − +M C K M

Y Y Y Ω ξ

( )1 1 1 5 5 5 1 2 15

Step-1System identification: 32 parameters

, , , , , , , , , , , ,ts a s a

x x x x s ak k k k k kθ θ ρ ρ η η η ψ =

0.2 0.4 0.6 0.8 1 1.2 1.4 1.610-3

10-2

10-1

100

h* mm

P F

ExperiementalAnalytical

0 1 2 3 4 5 6 7 8 9 10-0.015

-0.01

-0.005

0

0.005

0.01

0.015

t s

u(t)

m/s2

*

Control force1.6 mm; 4.0 s.mh τ= =

0.8 1 1.2 1.4 1.6 1.8 210-6

10-5

10-4

10-3

10-2

10-1

100

h* mm

P F

Method 1Method 3 Method 1

Proposed method500 samples

Method-3Brute force10E+05 samples

Closure

• Combining computing and experimental hardware in structural dynamic testing.

• Instrumented structures and ageing infrastructure.

Strong-Floor Reaction-wall

Net height : 5m Net Length: 4m Net Width: 3mFloor Thickness: 1m Wall Thickness : 1m

research into stochastic dynamic testing and reliability ... · research into stochastic dynamic...

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