risk assessment of systems - Civil and Environmental ......from: Kottegoda and Rosso(2008) "Applied...

Lec. 0512735: Urban Systems Modeling systems

12735: Urban Systems Modeling

instructor: Matteo Pozzi

1

risk assessment of systems

Lec. 05

C1 C2 C3

C1

C2

C3C3

C1 C2


outline

2

‐ definition of system;

‐ classification and representation;

‐ two‐state systems;

‐ cut and link sets;

‐ system reliability;

‐ bounds for system probability of failure;

‐ multivariate cumulative distribution.


what is a system

3

http://bccprepoutdoors.files.wordpress.com/2011/11/9694359‐carbine‐and‐hook‐with‐rope‐in‐stone.jpg?w=400

F

M1

M2M3

components in a single structure

Der Kiureghian, A. (2005) "First and Second‐Order Reliability Methods", in book: The Engineering design reliability handbook, CRC Press LLC.

mechanisms components in a network

12

3

4

5

6 78 9

Google map


system classification and representation

4

components: single, two, multi‐states.

systems: single, two, multi‐states.

e.g. working/not working e.g. working at 0%, 25%, 50%...

representation:

http://dkv.columbia.edu/demo/medical_errors_reporting/site/module2/fault‐vs‐event‐trees.html

fault trees, event trees

C1

C2 C7

C3

C4

C5

C6

C8

C9

C10

source sink

block diagram


two‐state system with two‐state components: example

5

C1

1 component works0 component fails

C2 C7

C3

C4

C5

C6

C8

C9

C10

source sink

Ci

1 systemworks 0 systemfails

component state

system state

convention is opposite respect to component analysis.


two‐state system with two‐state components

6

vector of component

state⋮

system‐components

relation

∶ →Boolean variable

number of components

∶ 2 → 2

a1 1 1 1 1 0 0 0 0a2 1 1 0 0 1 1 0 0a3 1 0 1 0 1 0 1 0

series as 1 0 0 0 0 0 0 0parallel as 1 1 1 1 1 1 1 0

coherent as 1 1 1 0 1 0 1 0non coherent as 0 0 0 1 0 0 0 1

states of components

for components, there are 2 possible relations.

C1 C2 C3

C1

C2

C3

C3

C1 C2

2 2 256 systems

coherent system: ∀ ∶ is non‐decreasing in . That is, if goes safe, it cannot degrade system operation.


famous systems

7

C1 C2 … Cn

min , , … , or

series system:

max , , … ,

1 1

or

parallel system: C1

C2

…

Cn“k‐out‐of‐n” system:

indicator: true 1false 0

1 : parallel system: series system


general system: cut and link sets

8

cut set: a set of components whose joint failure implies failure of the system. E.g. C1,2,3,4 .

minimum cut set: a cut set with no extra component : C1,4 and C2,3,4 .

link set: a set of components whose joint survival implies survival of the system. E.g. C1,4 .

minimum link set: a link set with no extra component : C1,2 , C1,3 and C4 .

example:

C1

C4

C2

C3

C2,3, C2,4 and C3,4are NOT cut sets

C1 and C2 are NOT link sets

Algorithms are available for identifying cut sets and link sets.


min cut set representation

9

example:

C1

C4

C2

C3

a system can be seen a series of its cut sets.

C1

C4C3

C4

C2

1 1∈

function relation for cut set

1 1 11 1 1 1

1 1 1 1 1 1 1

2 cut sets


min link set representation

10

example:

C1

C4

C2

C3

a system can be seen a parallel system of its link sets.

C3

C4

C2C1

C1

1 1

∈

function relation for link set

1 1 1 1

3 link sets


component and system reliability

11

probability of failure: P 0reliability: 1 P 1

1

probability of failure: P 0reliability: 1 P 1

1

component:

system:

1P 1 0P 0 P 1


reliability of series systems of indep. comp.

12

1

1 1 1 ≅

1

≅

independent failures

1 1

C1 C2 … Cn

0 5 10 15 200

0.2

0.4

0.6

0.8

1

n

Ps

P=20%P=10%P=5%P=2%P=1%

≜equally reliable components:

≜ 1


1 2 3 4 5 6 7 8 9 10

10-8

10-6

10-4

10-2

100

n

Ps

P=20%P=10%P=5%P=2%P=1%

reliability of parallel systems of indep. comp.

13

1 1

≜independent failures

equally reliable components:≜ 1

C1

C2

…

Cn


general system: example of network of indep. comp.

14

source

sink

1

2

3

4

1

3

2

4

A B

11

, 1 1 1

reliability probability of failure C4

C3C1

C2

link set A

link set B

1 , 2 for equally reliable components

independence



15

source

sink

1

2

3

4

1

3

2

4

A B

, ⋁ 1 1 1 …


reliability probability of failure

5

15

2

4

C D

45

31 1 11

∗ 2 2 5 2 for equally reliable components

,

⋁ : fail. prob. of C or D.



16

∗ 2 2 5 2source

sink

1

2

3

45

2

source

sink

1

2

3

4

increment of reliability after adding segment 5:

Δ 2 4 2 2 1

example:1%, 99%, 99.96%, ∗ 99.98%, Δ 0.02%

from: Kottegoda and Rosso (2008)"Applied statistics for civil and environmental engineers", Blackwell Pub.


two‐component series system

17

≜ 30%

equally reliable components:

≜ 1 70%C1 C2

| | ⟹ | 1 |37 1 |

independent failures∶ | ⇒ |

| 0 ⇒ |37 43%

| 1 ⇒ | 0 ;

opposite failures negative correlation

identical failures positive correlation


limit cases for a two‐component series system

18


C1 C2

∶ | ⇒ | ∶

30% 70%37 60%

| 1 ⇒ | 0 ; ∶



For series system, positive correlation decrease the probability of failure.

≜ 30%


≜ 1 70%

| 1 |37 1 | |

system failure

1 51%

30%

| 0 ⇒ |37 43% ∶

2


limit cases for a two‐component parallel system

19

C1

C2

≜ 30%


≜ 1 70%


0%



For parallel system, positive correlation increase the probability of failure.

| 1 |37 1 | |

system failure

9%

| 1 ⇒ | 0 ; ∶

| 0 ⇒ |37 43% ∶

∶ | ⇒ | ∶

30%


bounds for series systems

20

→→ ∨ ,

, 0,

,

, 0, min ,

max , min , 1

∀ ∈ 1,2, … , ∶ → max min , 1

two failure events (mechanisms):

many failure events (mechanisms):

when the joint probability for each pair of events, , is known, you can use Ditlevsen’sbounds (see Sørensen, J.D. (2004) "Notes in Structural Reliability Theory And Risk Analysis").

three failure events (mechanisms):

∨ ∨ , , , , ,

Ditlevsen’s bounds require second order probability ( ) and define max and min values for higher order probability (e.g. third order: ).


bounds for parallel systems

21

→→ ,

, 0,

,

, 0, min ,

0 min ,

two failure events (mechanisms):

∀ ∈ 1,2, … , ∶ → 0 minmany failure events (mechanisms):

second order bounds are also available (see Sørensen, J.D. (2004) "Notes in Structural Reliability Theory And Risk Analysis").


system reliability problem

22

0

0

0

00

00 limit state functions

, … ,

standard normal space

0 ↔ componentfailsmechanismactive

the system‐components relation defines the failure of the system, depending on the failures of the components.


system reliability problem: series system

23

0

000

failure domain:Ω 0 ∪ 0 ∪ 0

Ω


000

in a series system, the system fails if at least one component fails.


system reliability problem: parallel system

24

0

000

failure domain:Ω 0 , 0, 0

Ω


00

in a series system, the system fails if at least one component fails.

0


system reliability problem: general system

25

0

000

e.g. failure domain:Ω 0, 0 ∪ 0

Ω


000 C1

C3C2


approaches for system reliability problem

26

Monte Carlo Uniform Sampling:it can be easily implemented, it provides an unbiased estimate.

Monte Carlo Important Sampling:it is not easy to find an appropriate proposal distribution.

FORM:the design point (defined as the failure closest to the origin) may be not differentiable ⟶ no linear approximation. Newton’s method may not work. In any case, linear approximation may be not accurate.

0

000

Ω


000


FORM for series systems

27

0


000

P 0∗ , ∗ : design pointsfor and

∗

∗

series system

≅ 1

limit state functions approximated around the corresponding design point:

≜

≜ ⋯ ⋯≜ ⋯ ⋯

new coordinates:


FORM for series systems

28

0


000

∗ , ∗ : design pointsfor and

∗

∗

≅

≜ ⋯ ⋯

≜ ⋯ ⋯

; , → ; ,

≜∈ 1,1

1 →

; ,

limit values for each coordinate:

limit state function:

matrix of “directions”:

1 Pprobability of failure:


multivariate cumulative normal distribution

29

-50

510

15

-50

510

150

0.5

1

x1

1=4 , 2=6 , 1=2 , 2=3 , 12=0.4

x2

F(x 1,x

2)

-50

510

15

-50

510

150

0.01

0.02

0.03

x1

1 = 4 , 2 = 6 , 1 = 2 , 2 = 3 , 12 = 0.4

x2

p(x 1,x

2)

Φ ; , ≜ P ∞ ,… , ∞

Φ ; , … ; , …

Φ ; , ∈ 0,1

∀ :Φ

; , 0

PDF

CDF

x1

x 2

-5 0 5 10 15-5

0

5

10

15

,

F=mvncdf(v_x',v_mu',m_Var)Matlab


s1

s 2

-2 0 2-3

-2

-1

0

1

2

3

mult. cum. norm. for series systems

30

Φ ; , P ∞ ,… , ∞

,

P safe Φ ; ,

P failure 1 Φ ; ,

recipe:‐ solve “component” FORM for each limit state function

∗ → , ;‐ compute matrix and build ;‐ compute .

; ,

; ,1 Φ ; ,

Formulation in normal space, with linear limit state functions:

Φ ; ,


FORM for parallel systems

31

0


000

∗

∗

linearization around this point

Design points may be too far from failure domain. To improve the quality of the approximation:


FORM for parallel systems

32

s1

s 2

-2 0 2-3

-2

-1

0

1

2

3

,

s1

s 2

-2 0 2-3

-2

-1

0

1

2

3

,

failure domain

equal probability Φ ; ,

; ,

; , Φ ; ,

Formulation in normal space, with linear limit state functions:


example

33

5 10 15 205

10

15

20

x1

x 2

1 = 10 , 2 = 15 , 1 = 2 , 2 = 1.5 , 12 = 0.3 ; ,1015

4 0.90.9 2.25

1 1

0.1 0.9

3

13

; ,

Φ ; , 1.4%

20.5

4.45 1.521.52 1.70

MC : ∈ 0.67%, 1.40%


references

34

Ditlevsen, O. and H.O. Madsen. (1996). Structural reliability methods. J. Wiley & Sons, New York, NY. Downloadable from http://www.web.mek.dtu.dk/staff/Od/books/OD‐HOM‐StrucRelMeth‐Ed2.3.7‐June‐September.pdf. Chapter 14.

Faber, M. (2009) “Risk and Safety in Engineering, lecture notes”, Lecture 8, available at http://www.ibk.ethz.ch/emeritus/fa/education/ws_safety/Non_printable_script.pdf

Sørensen, J.D. (2004) "Notes in Structural Reliability Theory And Risk Analysis", notes 6 and 7, avail. at http://www.waterbouw.tudelft.nl/fileadmin/Faculteit/CiTG/Over_de_faculteit/Afdelingen/Afdeling_Waterbouwkunde/sectie_waterbouwkunde/people/personal/gelder/publications/citations/doc/citatie215.pdf

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risk assessment of systems - Civil and Environmental ......from: Kottegoda and Rosso(2008) "Applied...

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