Using Monte Carlo and Directional Sampling combined with an Adaptive Response Surface

for system reliability evaluation

L. Schueremans, D. Van Gemertluc.schueremans@bwk.kuleuven.ac.be,

dionys.VanGemert@bwk.kuleuven.ac.be

Department of Civil Engineering

KULeuven, Belgium

Praha

Euro-Sibram, June 24 tot 26, 2002, Czech Republic

IntroductionFramework:

– Ph. D. “Probabilistic evaluation of structural unreinforced masonry”,

– Ongoing Research: “Use of Splines and Neural Networks in structural reliability - new issues in the applicability of probabilistic techniques for

construction technology”.

Target: – obtain an accurate value for the global pf, accounting for the exact PDF of

the random variables;

– minimize the number of LSFE, which is of increased importance for complex structures;

– remain workable for a large number of random variables (n). In practice, the number of LSFE should remain proportional with the number of random variables (n).

IntroductionLevel II and Level III methods:

Leve l D efin i tion

III L e v e l I I I m e t h o d s s u c h a s M o n t e C a r l o ( M C ) s a m p l i n g a n d N u m e r i c a lI n t e g r a ti o n ( N I) a r e co n s i d e r e d m o s t a c c u r a t e . T h e y c o m p u t e t h e e x a c tp r o b a b i l i t y o f f a i l u r e o f t h e w h o l e s t r u c t u r a l s y s t e m , o r o f s t r u c t u r a le l e m e n ts , u s i n g t h e e x a c t p r o b a b i li t y d e n s i t y f u n c ti o n o f a ll r a n d o m v a r i a b l e s .

II Leve l II me thods such a s F O R M and S O R M compu te the p robab il i ty o f fa i l ure bym eans o f a n idea l iza t ion o f the l im it s ta te func tion w here the p robab il i ty de ns i tyfunc tions o f a l l ra ndo m var iab les a re app roxi m ated by equiva lent no rmald is tr ibution func tions .

I Leve l I m e thods ve r ify w he the r o r no t the re l iab il i ty o f the s truc ture is suffic ien tins tead o f co m puting the p robab il i ty o f fa i l ure e xp lic i tly. In p rac tice this is o ftenca rr ied out by means o f pa r tia l sa fe ty fac to rs .

T a b l e 1 : Leve ls fo r the ca lc ula tion o f s truc tura l sa fe ty va lues (E C 1 , 1994 ; JC S S , 1982 )

p P g f df

g

X x xX

X

00

. . .

Introduction - reliability methodsR e l i a b i l i t y m e t h o d s - L e v e l ( I , I I , I I I ) - D i r e c t / I n d i re c t ( D , I D )

In tegra tionm e thods

A na lytica l o r N um erica l Integra tion (A I/N I, III, D)

D irec tiona l Integra tion (D I, III, D )

S a m plingm e thods

( Im por tance S a m pling) M onte C ar lo ( IS M C , III, D )

( Im por tance ) D irec tiona l S ampli ng ( ID S , III, D )

F O R M /S O R Mm e thods

F irs t O rde r S econd M om en t re l iab i l i ty m e thod (F O S M , II, D )

F irs t O rde r and S econd O rder R e liab i l i ty M ethod ( F O R M /S O R M , II, D )i n co m bi na tion w i th a sys tem ana lys is ( F O R M /S O R M -S A , III, D )

C o m binedm e thods us i ngA dap tiveR esponseS urfacetechn iques

D irec tiona l A dap tive R esponse surface S ampling ( D A R S , III, D-ID )

M onte C ar lo A dap tive R esponse surface S ampli ng (M C A R S , III, D -ID )

F O R M w ith an A dap tive Response S urface (F O R M A R S , II, D-ID ) i ncombina tion w ith a sys te m ana lys is (F O R M A R S -S A , III, D -ID)

T a b l e 2 : O verv iew o f re l iab i l i ty me thods fo r a leve l III re l iab i l i ty ana lys is

#LSFE~9n

#LSFE~3/pf VI

#LSFE~cte.n

Methods for System Reliability using an Adaptive Response Surface

Real structure: high degree of mechanical complexity, numerical algorithms, non-linear FEM

Response Surface: low order polynomial, Splines, Neural Network,...

Reliability analysis

Optimal scheme: DARS or MCARS+VI

DARS: Matlab 6.1 [Schueremans, 2001], Diana 7.1 [Waarts,2000]

MCARS+VI: Matlab 6.1 [Schueremans, 2001]

u1

u2

g1(u1,u2)<0

g2(u1,u2)<0

g3(u1,u2)<0

g4(u1,u2)<0

unsafe

unsafe

unsafe

unsafe

safeg3>0 g1>0

g4>0g2>0

Component reliability:

pf,g1= 0.0161, g1=2.14pf,g2= 0.0161, g2=2.14pf,g3= 0.0062, g3=2.50pf,g4= 0.0062, g4=2.50

System reliability:

pf= 0.0446=1.70

DARS-Directional Adaptive Response surface Sampling

u1u2 g1(u1,u2)<0

g2(u1,u2)<0

g3(u1,u2)<0

g4(u1,u2)<0

unsafe

unsafeunsafe

unsafe

u2

u1

fU1,U2(u1,u2)

fU1,U2(u1,u2)

g1(u1,u2)<0

unsafesafe

g u u

g u u u uu u

g u u u uu u

g u u u u

g u u u u

1 2

1 1 2 1 22 1 2

2 1 2 1 22 1 2

3 1 2 1 2

4 1 2 2 1

2 0 0 12

2 0 0 12

2 5 22 5 2

, m in

, . .

, . .

, ., .

DARS -Directional Adaptive Response surface Sampling

Step 1: - Evaluate the LSF forthe origin in the u-space;- Search the roots ofthe limit state functionfor the principaldirections in the u-space (n=2):- [1,0];[0,1];[-1,0];[0,-1]With the root-findingalgorithm, this requiresapproximately 3 to 4LSFE

u1

u2

[0,0] =

min

[0,0]

[-0,0]

[0,-0]

N=5=2n+1, #LSFE=21

min = 3.5, =2.85

DARS -Directional Adaptive Response surface Sampling

Step 2: Fit a response surfacethrough these data inthe x-space and theresulting outcome Y,using a least squaresalgorithm.

u1

u2

gRS,1= 1.65-0.13u1

2

-0.13u22

gRS,1 = 0

min = 3.5

add = 3.0

DARS -Directional Adaptive Response surface Sampling

Step 3: iter. procedure required accuracyPerform DS on theResponse Surface:

If i,RS < min+

add

Calculatepi(LSF)=2(i,LSF,n)Update the responsesurface with new data

ElseCalculatepi(RS)= 2(i,RS,n)

u1

u2

gRS,2= 0.92+0.046u1

-0.023u2-0.074u1u2

-0.097u12-0.084u2

2

gRS,2 = 0

min = 2.05

add = 3.0

DARS -Directional Adaptive Response surface Sampling

u1

u2

gRS,3 = 0

pf N = 14

add=3

min=2

p

L S F E

N

f

0 0 2 8

1 9 1

5 1

1 4

2 0 5

.

.

#

.m in

Step 3:

Monte Carlo Variance Increase on the Response Surface (vi).

Sampling function: h=n-0.4

IF |gRS(v i)|<|g,add|

calculate gLSF(vi)

update RS

update g,add

Else

.

u

gLSFRS

RS

add

g,add

g,add

i

gRS,i

gLSF,i

g,i

MCARS+VI Monte Carlo Adaptive Response surface Sampling+Variance Increase

p I g

f

hi LSF i vv

vu

v

0

p I g

f

hi RS i vv

vu

v

0

DARS and MCARS+VI

The number of direct LSFE remains proportional to the number of random variables (n),

There is no preference for a certain failure mode. All contributing failure modes are accounted for, resulting in a safety value that includes the system behavior, thus on level III.

Safety of masonry Arch

R ando mvariab les

P robab il i tydens i tyfunc tion

Mean va lue µ

S tandarddev ia tion σ

C oeffic ient o fva r ia tion V [% ]

x 1 = r 0 [m ] N or m al 2 .5 0 .02 0 .8

x 2 = t [m ] N or m al 0 .16 0 .02 12

x 3= dr [m ] N or m al 0 0 .02 /

x 4 = F [N] Lognormal 750 150 20

T a b le : R andom var iab les and the ir pa ra m ete rs

Safety of masonry Arch

To evaluate the stability of the arch, the thrust line method is used (Heyman, 1982), which is a Limit Analysis. Following assumptions are made:– blocs are infinitely resistant,

– joints resist infinitely to compression

– joints do not resist to traction

– joints resist infinitely to shear

An external program Calipous is used for the Limit State Function Evaluations [Smars, 2000]

Safety of masonry ArchFailure modes - limit states - limit analysis based on thrust lines

g g

S

m in

X

X

1

1

S X 1 S X 1

g X 1 g X 1

Safety of masonry Archprocedure pf #LSFE (time) Accuracyreliability analysis – initial random values: d~N(0.16,(0.02)²)DARSMCARS+VI

1.261.25

0.110.11

43 (17 min)23 (12 min)

()=0.15()=0.15

reliability analysis – increased accuracy on thickness: d~N(0.16,(0.005)²)DARSMCARS+VI

3.553.46

1.9 10-4

2.7 10-434 (13 min)56 (21 min)

V()=0.05V()=0.05

reliability analysis – increased mean value for thickness: d~N(0.21,(0.02)²)DARSMCARS+VI

3.693.67

10-4

1.2 10-445 (17 min)23 (12 min)

V()=0.05V()=0.05

Table 2: Outcome of reliability analysis for masonry arch – initial parameters and update

Safety of masonry Arch

Figure: DARS-outcome - reliability vernus number of samples N

Increased thickness: t: =0.21 m; =0.02 mN=192: =3.72, pf=1.0 10-4

Increased accuracy: t: =0.16 m; =0.005 mN=273: =3.44, pf=2.9 10-4

Initial survey: t: =0.16 m; =0.02 mN=371: =1.26, pf=1.0 10-1

Number of Samples N

95% Confidence interval

T=3.7

Conclusions• Focus was on the use of combined reliability methods to

obtain an accurate estimate of the global failre probability of a complete structure, within an minimum number of LSFE.

• A level III method is presented and illustrated: (DARS/MCARS+VI

• Ongoing research: Splines and Neural Network instead of low order polynomial for Adaptive Response Surface (ARS).

• Acknowlegment: IWT-VL (Institute for the encouragement of Innovation by Science and Technology in Flanders).