euromem : FROM UNCERTAINTIES TO PARTIAL … · Transformation T From physical space X ......

euromem : FROM UNCERTAINTIES TO PARTIAL SAFETY FACTORS CALIBRATION: SAFETY FACTORS CALIBRATION:

APPLICATION TO TENSILE MEMBRANE STRUCTURES - Discover the birth of a Eurocode

A TRAINING SCHOOL OF COST ACTION TU130329 Sep-1 Oct 2015 NANTES (France)

Franck SCHOEFS, Professor, Universit de Nantes

Deputy director of Scientific Interest Network MRGENCI

Basis of reliability analysis: Reliability Computation

- Concept of load combination factors and partial Safety

factors.

- Illustration on simple examples where G writes R S with R

and S known (pdf).

Yesterday

and S known (pdf).

- Safety assessment for more complex cases (no analytical

solution or non-linearity of the solution: R=A2 Y).

- Practical calibration of partial safety factors

Today

1. Definition of Pf.

Basis of reliability analysis: Reliability Computation

2. How to compute a safety estimate?

3. Another estimates: safety index.

4. Orders of magnitude in the reference case.

5. Examples.

Aim: Evaluate the probability of failure (tension yield limit)

Resistances R Limit State : G(R,S)=R - S

Sollicitations S

> 0 : Safety

0 : Failure

Problem statement

example: beam in tension

1. Definition of Pf

Difficulties and stakes

Numerical Estmation of a probability of 10-5 (ULS)

Example for R deterministic (1st dimension in probabilistic space)

Prob. S

[ Mpa ]

R

Probability of failureS

4

If 2 random variables

S

R

P*: design point (highest probability of occurrence

on the limit state)

2. How to compute a safety

estimate?

5

Df,i

Df,i

fRi,Si Pf,t

fR,S (r,s) = fR(r)*fS(s)

r

s Df

r

r

s Df

r

III.2

If pdf of R and S known and if they are independant


estimate?

Pf = Df fR,S(r,s)dr ds = 1- -+ FS(r) fR(r)dr = -+ FR(s) fS(s)ds

r r

fS(s)fR(r)

E(R)E(S)

FR(s)

r,s

fS(s)ds

r

s Df

s

here, S = (Nx , My , Mz)R = (Np , Mp)

Mz

Mp

Np

R

S

III.2

a xNpN

b y2

M + z2MpM

=0

(< 0: failure)

Real study cases


estimate?

Geometry of beams:

Length: 5 to 20 mDiameter : 0.5 to 1.5 m

Environnement marin

Nx

Np

s

LOADING Np

Displacement

N0

N0Np1

1

s x

y

z

N simulations (computations) : looking for failure cases Xi s.a. = {XiRn|G(Xi)0}

Computation of G(Xi) : analytical/physical response surface / FEM/ SFEM

Pf = card( )/N: thats an approximation of

For these more complex cases (general case)

Monte carlo simulation (non explicit form of G)


estimate?

f

8

Pf = card( )/NError: Shooman

formula

%erreur = fP~

1200

Monte carlo simulation

(non explicit form)


estimate?

% e

rror

%erreur = f

f

P~

N

P1200

( ) ( )f

f

fPN

PPCOV

.

1~ =

9

Number of simulations

For Pf = 10-3

Huge computational cots

Safety index (Rjanytsine 1949)

3. Another estimate: safety index

Failure domain

1. Transformation TFrom physical space X

to standard space U

2. Distance btw O and limitstate (IN STANDARD

Most probable point = design pointStandard space U:

Random variables are standard: Normally

distributed, centered, normalized Failure

domain

10

Hasofer-Lind sefety index: = OP* BUT Pf cannot be deduced direclty (Eurocodes are thus writen by considering )

Algorthms: rackwith-Fliesser / GRaCE

Hyperplane:

approximation

Physical space:

state (IN STANDARDSPACE)

3. APPROXIMATION of Pf

P*: design point (larger probability of occurrence on the

limit state)

: normal cdf

Safety index (Rjanytsine 1949)


Standard space U:

Random variables are standard: Normally

distributed, centered, normalized

Failure domain

Failure domain

1. Transformation TFrom physical space X

to standard space U

2. Distance btw O and limitstate (IN STANDARD

Most probable point = design point

11

P*: design point (larger probability of occurrence on the

limit state)

: normal cdf

Hasofer-Lind sefety index: = OP* BUT Pf cannot be deduced direclty (Eurocodes are thus writen by considering )

Algorthms: rackwith-Fliesser / GRaCE

Hyperplane:

approximation

Physical space:

state (IN STANDARDSPACE)

3. APPROXIMATION of Pf

What is the design point?

S

R

P*: design point (highest probability of occurrence

on the limit state)


12

Df,i

Df,i

fRi,Si Pf,t

Ri,Si t

Safety index, methods

Pf(-))))


13

: multinormal cumulative dentity function

Pf

See Eurocode 0

Simplified relationship btw Pf-

4. Orders of magnitude in the

reference case: (linear limit states)

Source : Corus society

report

Pf(-))))

5. Examples

Academic example: cantilever beam

Limit state on displacement : G(Z)= L/30-(5/48)*(P*L^3/E*I)

First study : sensibility / l

Quantity Mean. m STD

P 4 [kN] 1 [kN]Mfr * 20 [kN.m] 2 [kN.m]L 5 [m] 0

Reference study case: l = 5 m

PL

P

Results obtained with COMREL 7.10Prgm POUTRECONSOLV0.ITI

L 5 [m] 0

I 10-4 [m4] Cov : 20 %

E 2.107 [kPa] Cov : 10 %

FORM-beta= 4.123

Ll. value, -FORM, Param. Sens., Param. Elas.2 4.872 -0.1304 -0.5351E-013 4.705 -0.2053 -0.13094. 4.459 -0.2898 -0.26005 4.123 -0.3821 -0.46346 3.695 -0.4735 -0.76897 3.182 -0.5482 -1.2068 2.609 -0.5918 -1.8159 2.010 -0.6014 -2.69310 1.416 -0.5840 -4.126

P -0.19826E 0.09958I 0.97508PAR1 0.00000Sum of a^2 1.00000

Representative Alphas of Variables FLIM(1) [POUTRECONSOLV0.PTI]

Z

5. Examples


Pf

l. value, -FORM, Param. Sens., Param. Elas.2 4.872 -0.1304 -0.5351E-013 4.705 -0.2053 -0.13094. 4.459 -0.2898 -0.26005 4.123 -0.3821 -0.46346 3.695 -0.4735 -0.76897 3.182 -0.5482 -1.2068 2.609 -0.5918 -1.8159 2.010 -0.6014 -2.69310 1.416 -0.5840 -4.126

5. Examples


Limit state on displacement : FLIM(1){poutre console appui}= l/30-(5/48)*(P*l^3/E*I)

Second study : sensibility / std(P)

(P) P -0.37746E 0.13849


1 [kN] 4,12

2 [kN] 3,9

3 [kN] 3,59

l. value, -FORM, Param. Sens., Param. Elas.2 4.853 -0.1682 -0.6930E-013 4.613 -0.3232 -0.21024 4.193 -0.5182 -0.49445 3.591 -0.6689 -0.93146 2.896 -0.7007 -1.4527 2.220 -0.6412 -2.0228 1.625 -0.5459 -2.6879 1.129 -0.4477 -3.56810 0.7264 -0.3609 -4.968

E 0.13849I 0.91561PAR1 0.00000Sum of a^2 1.00000

P -0.52156E 0.17872I 0.83429PAR1 0.00000Sum of a^2 1.00000


STD(P)= 3 kN

III.2

G= L/30-(5/48)*(P*L^3/E*I)

5. Examples


Quantity Mean. m

P 4 [kN] 1 [kN]

Mfr * 20 [kN.m] 2 [kN.m]

l 5 [m] 0

I 10-4 [m4] Cov : 20 %

E 2.107 [kPa] Cov : 10 %

Industrial Example: corrosion of a pipe

5. Examples

Source : Corus society

report

5. Examples

Industrial Example: corrosion of a pipe

Analyze correlations too

Source : rapport socit

Corus

Probabilistic Design (conservatism of codes or absence of code)

Reliability assessment f structures or structural components designed with a given code ( -> )

Calibration of partial safety coefficients of a code ( -> )

New structures

Formulation and Interest of partial safety factor code format

Summary

Calibration of partial safety coefficients of a code ( -> )

Existing damaged structural assessment

Requalification of existing structures based on decisional appraoches

Inspection Maintenance Repair -> IMR optimisation on a given lifetime

Existing structures

Structural reliability Methods

Mthodes danalyse de risqueMthodes quantitatives

Tutorial (in French)-20 24 heures en format long-12 16 heures en format court-12 16 heures en format court

-.

Objectif : analyse de risque / RBI

Structures Tipode utilises comme torchres

Crte de houle

Direction de la houle

P1 P2

Torchre

M2 TPM - Universit de Nantes

Environnement interaction fluide-structure

Structure fissure dbouchante

Donnes de houle

Efforts rpartis

Fonction de

rponse f1

Fissure de longueur a

matrice de

rigidit

Fonction de rponse

f2

Fonction de rponse f3

Dplacement ou nergie

potentielle totale

Approche physique-mcaniste (modles / instrumentation)

Matriau (pathologie / Inspection) Fiabilit

Risque

Performance/Consquence

IMR

RBI

IllustrationsA : Structures ptrolires

Variables de base Distribution Conditionnement c.o.v

Troncature

H, Hauteur de vague extrme Gumbel sachant (Hs,Tstat, ) 8 % [m-2;m+5]

T, Priode de vague extrme Log-Normale sachant H 10% [m-k ;m+k]k = 3

Coefficients hydrodynamiques CD, CM, CX, CX,

Normale - 35% [m-p;m+p]p = 2Longueur de fissure a Exponentielle - 35% [- ; 2 R]

Probabilits

IMR

Processus, analyse stat.

Fiabilit dpendant

du temps

Lois de dommages

(Paris)


Modlisation des donnes dentre

-Jeu de donnes de houle 2 niveaux :

*Non Expert CS : identifier une distribution H et une distribution T : puis analyse

hyp corrlation. Soit excell et tests aux moindres carrs sur CdF soit Matlab et tests

cdf ou pdf : diffrences.

*Expert CS : identifier une distribution jointe (proj sur chaos)

-Jeu de donnes de fissures

identifier une distribution H et une distribution T : puis analyse hyp corrlation. Soit


identifier une distribution H et une distribution T : puis analyse hyp corrlation. Soit

excell et tests aux moindres carrs sur CdF soit Matlab et tests cdf ou pdf :

diffrences.

05

1015

2025

0

5

10

15

20

250

0.005

0.01

0.015

0.02

0.025

0.03


Modlisation des structures

-Structure tripode:

* Non Expert CS : poutre console. Fissure = ressort + solution RdM.

*Expert CS : modle EF + modle EF fissure



Fiabilit :

* Non Expert CS : monte carlo + FORM + donnes tripodes (rsultats)

* Expert CS: monte carlo + FORM + EFS

-Analyse de sensibilit ltat limite


Limiter le dplacement au sommet :

G = Uc U6

avec Uc = P5P6 / 600 = 0.041 m

= 0

= 90 Structure non-fissure

d

Limiter lnergie de dformation (ou son volution) :

G = c d

Pour comparer la fiabilit : c Uc


RBI

* Niveau I et II (courts): Fourniture des PoD PFA (II) ou Analyse des

graphes (I)

*Niveau III (long) : fourniture des ROC

-Sensibilit aux modles de cot (inspection + cout relatif), la

connaissance experte

-Fiabilit dpendant du temps / optimisation des inspections (fatigue ou hyp fi = f(t), ..)

Illustrations

A : Structures ptrolires

=0.1 20 =0.1 40

0 0.2 0.4 0.6 0.8 1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

=0.1 20

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.05

0.06

0.07

0.08

0.09

0.1

0.11

r

i1

ccc =

f

r2

ccc =

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

=0.1 40

0 0.2 0.4 0.6 0.8 1

euromem : FROM UNCERTAINTIES TO PARTIAL … · Transformation T From physical space X ......

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