Probabilistic design fault tree and reliability analysis of flood defences

Probabilistic design, Fault tree and reliability analysis of Flood defences June 7, 2013 Cong MAI VAN, PhD Water Resources University, Hanoi, Vietnam Emails: CONG.M.V@wru.edu.vn & C.MAIV

Probabilistic design, Fault tree and

reliability analysis of Flood defences

June 7, 2013

Cong MAI VAN, PhD

Water Resources University, Hanoi, Vietnam

Emails: CONG.M.V@wru.edu.vn & C.MAIVAN@yahoo.com

Contents

• Background: overview flood risk

• Reliability analysis: deterministic vs probabilistic

• Probabilistic design: Fault tree & reliability analysis

• Calculation tools: VaP; OpenFTA

• Calculation tools: VaP; OpenFTA

• Exercise: reliability analysis of a simple flood defence system

1 Background

• Recent floods in the world

• Probabilistic design/flood risk analysis, approaches

• Flood defences system: 3 questions

• Focus: reliability analysis of existing flood defences (a part of risk assessment)

Last years:

- New Orleans is still recovering

- River floods in the UK and Eastern Europe

- Serious Floods in Bangladesh, Parkistan; thousands fatalities

compartiment dikes, etc.; debase is going on

=> Much more attention for reducing flood risk

1953 flood, Holland

1953 flood, Holland

Flooding due to huricance Katrina 2005

Flooding in Vietnam 2005 - 2007

Reducing flood risk, how?

Risk = {Probability} x {Consequence}

Scenario 1: Relief centered approach

livestock; accept economic damage

livestock; accept economic damage

=>Evacuation plan; flood proof; spatial planning ect.; Scenario 2: Prevention centered approach

Reducing flood risk, how?

Engineers are challanged to go for Scenario 2 This

requires:

• Good estimate of extreme conditions of nature in

terms of its value and frequency (e.g River discharge,

HW, sea surges and waves 1/100, 1/250, 1/500,

1/1.000, 1/10.000 per year);

1/1.000, 1/10.000 per year);

• Acceptable safety level (required) for a certain

protected region; => target safety

• Reliable design given a target safety

Flood defence system

high ground

city low lying

interested questions

high ground

city low lying

sea dike

Waves

- What is the actual safety? (Q1)

- How safe is safe enough (acceptable risk level)? (Q2)

- What is the best design solution given a target safety? (Q3)

1) Probabilistic assessment of safety level

- for an existing CFD system, what is the actual safety?

dike 1… p1.1 (overtopp.) p1.1 (piping) p1.1(etc.) p1.1(all)

dam … p1.2 (overtopp.) p1.2(piping) p1.2(etc.) p1.2(all)

dune p (overtop.) p (piping) p (etc.) p (all) dune pdune(overtop.) pdune(piping) pdune(etc.) pdune(all) sluice psluice(overtop.) psluice(piping) psluice(etc.) psluice(all) total pall(overtop.) pall(piping) pall(etc.) pall(all)

2) Risk based design: seek for a target safety/

safety standard

this can be found by comparing the cost of protection to

a characteristic value of the consequences of flooding in considered situation e.g present/ future

3) Reliability-based design: reliable design given a target safety

Calculation of failure probability

- a set of acceptable design

(a) (b)

P <P f max(opt.)Estimation of R =f(P )T f Estimation of I

true failse

Steps in a risk analysis

• System description

• inventory of hazards and mechanisms

• models for loads / resistance and deterioration

• reliability calculation (elements, systems)

Top event: most unwanted consequence

• Flood defence -> inundation of polder

• Large dam -> interrupted power supply

• Large dam -> interrupted power supply

-> catastrophic flooding

• System -> loss of function

2 Design approaches: deterministic

vs probabilistic

• Existing design approach & its shortcomings

• Probabilistic design

Conventional design approach

Resistance Factor Design R / γγγγ > S

Allowable stress design σσ < σσa

Load Factor Design γγγγ Sk > Rk

Load & Resistance Factor Design γγγγS Sk > Rk / γγγγm

In general: safety is described by means of partial safety factor: S.F=Strength/Load

Use single characteristic value

Conventional design approach

Shortcomings

- Uncertainties of input variables is not taken into account

- Cannot answer question of How safe the structure is

Probabilistic design approach

- uncertainty of input variables (load and strength) by

considering probability distribution functions

- various failure modes & failure consequences

- safety is assessed in term of probability of failure

- judge its acceptability in a view of the potential

risk/consequences

Probabilistic design

Aims:

to determine the true probability of flooding of a polder and to judge its acceptability in view of the

investment cost and the consequences

3 Reliability &Fault tree analysis

erosion of inner slopes

Overtoping

OR

Failure of dike section # i

dike's slope instability of

instability of instability of

inner slopes outer slopes

instablity of toe structure

scour

instability of protected ele.

too much OR

armour layer damage of

armour layer instability of

AND

Reliability & Fault tree analysis

X2

Limit state function : Z=Strength-Load= R-S

Failure occurs when Z<0

Probability of failure: Pfailure=P(Z<0)

failure

Level I:

semi-probabilistic

Level II: method of FORM

First Order Reliability Method

• Analytical solutions are possible if:

The Limit State Function is linear

• The Limit State Function is linear

• Variables are normally distributed

• A FORM is based on the above property

Level II principle

S R

Z = µ − µ = − =

µ

To determine the reliability index and failure probability: transform to a standard normal variable uZ

thus:

6

3 12 1

3 12 1

From observed data:

S= d(m) = Normal (5.0, 0.5)

Corresponds to 1/20 year load)

R=h(m)= Normal (6.5, 0.1)

0.02 0.03 0.04 0.05 0.06 0.07 0.08

=>P(Z<0)=???

S

R Z=R-S

This means:

- Average failure probability per year (failure rate)

- Return period of the failure event:

Tp=1/p =1/0.0019=526 years

2 R

2

• and:

• α importance factor; is a measure of the contribution

of a variable to the uncertainty in Z (and therefore also

to the failure probability)

Z

S S

2 Z

2 S

2 S Z

R R

2 Z

2 R

2

σ

σ α

σ

σ α

σ

σ α

σ σ

Level III: Monte Carlo simulation

System analysis: questions can be answered:

• What is actual overall failure probability of the system,

Pfsys ?

• How much does each failure mode contribute to Pfsys:

to finding the most dominant failure modes

• Which is the weakest point of the system?

Reliability & Fault tree analysis

• What is the influences of load and strength variables

to the system/component failure?

Basic system models

Parallel systems

R = max (R1 R2)

R = R1 + R2

R1 < S R2 < S P = P(E )P(E | E )P(E | E ,E ) P(E | E E ) f 1 2 1 3 1 2 K n K n 1−

If the events Ei are statistically independent

Combine gates

Summary:

Use of probabilistic design methods:

- Calculation of components/system failure probabilities

- Influences of stochastic variables (load, strength)

- Fault-tree analysis: Monte Carlo simulation to find the contribution of failure modes to the system failure

- Investigation of all possible failure mechanisms,

- Estabilishment of LSE and fault-tree

- Calculation of failure probability of components and system

- Fault-tree analysis to find the contribution of failure modes

to the system failure

Failure matrix [FLOODsite

Reliability & Fault tree analysis

Failure matrix [FLOODsite approach]

Fault tree of flood defence system

structure type 1 fails structure type 2fails other cause

River dike system 2 fails River dike system 1 fails

OR

Inundation of protected area

inner slope erosion of

OR

excess wash away toe sliding/

OR AND

armour layer

instability of

outer slope geo inst.

fails piping

inner slope

geo inst.

Overflowing

Overtoping functional failure of dike section breach

P per year

Fault tree

4 Calculation tools/ softwares

• VaP: Reliability of a element

- Student version freely available

5 Application case: reliability analysis

of a simple flood defence system

• Steps in reliability analysis: a summary

• Case description

• Failure mechanisms

• Fault tree

• Reliability of a element: LSF + solve to find Pf (VaP)

• System reliability: OpenFTA to estimate Pf_system

5 Exercise

A dam system comprise of 3 sections; reliability of

section 2 & 3 is known (i.e Pf-sect2=0.0002; P

f-sect3=0.0005 per year ) while reliability of section 1 is unknown Failure of the dam system happened mainly due to overflowing; instability of revetment amour

units, instability of inner slope and piping Loads and strength parameters of the dam section are given in the below table Safety standard of protected region is 1/100 per year

Failure mechanisms

• Overflow: Z=Zdc –Zwl

Failure mechanisms (contd.)

• Instability of amour layers: Z= (Hs/∆D)R-(Hs/∆D)S

Van der Meer formula

ξ 0 5

2 0 18

0 50

7 8 ]

R D

H

Failure mechanisms (contd.)

• Macro instability of dike’s slopes

Failure mechanisms (contd.)

• Piping: happens when both conditions occur

- Condition 1: Rupturing Z1= ρcgd-ρwg∆H

- Condition 2: Sand Flowing Z2= mLt/c - ∆H

Fault tree

Fault tree Dam system failure

Failure of Section 2

Failure of section 1

Failure of section 3

Overflowing Piping Instability of slope

protection Sliding of inner

slope

Use VaP and openFTA

Failure mode Pf by VaP

Overflowing 0.00233 Piping 1 0.0816 Piping 2 0.0783 Instability of inner slope 2.87e-07 Instability of amour units 0.0527

Instability of amour units 0.0527

Example of a fragility curve

0,5 0,6 0,7 0,8 0,9 1

Actual dike system, constructed after

1992 and upgraded after 1998

1/20 safety require 0

0,1 0,2 0,3 0,4

Reliability based design

Excessive wave overtopping Slope stability

Design of flood defences – integrated model

Design Risk-based design Reliability-based design

Reliability-based design Evaluation of limit states Description of boundary conditions

Input model

PDF & CDF of sea loads (Hs Tp, water levels) alternative geometriesStrength boundary:

Damages/ consequences: inventory/model

Component/system reliability analysis;

Safety assessment (Pfi ; Pfsys )

Failure modes

Limit State Eq.

RSAM

Reliability based design model:

optimal geometry given [Pf]

{from component to system level}

Risk-based design model: optimal level of protection [Pf]

{system level}

Input model

• Allsop, N.W.H., 2006 Failure modes report Technical report FLOODSite Task 4, final draft verion.

• Bakker, W.T and Vrijling, J.K., 1980 Probabilistic design of sea defences

Proceedings International Conference on Coastal Engineering 1980.

• Barlow, R.E., 1998 Engineering reliability Philadelphia, USA: SIAM.

• Ditlevsen, O., 1979 Narrow reliability bounds for structural systems Journal of Structural Mechanics, pp 453-472.

• Oumerarci, H., Allsop, N.W.H., Groot, M.B de, Crouch, R., Vrijling, J.K.,

Kortenhaus, A., Voortman, H.G., 2001 Probabilistic design tools for vertical

breakwaters Balkema, Rotterdam, 2001.

• Vrouwenvelder, A.C.W.M and Vrijling, J.K., 1987 Probabilistic Design (in Dutch: Probabilistisch ontwerpen) Delft University of Technology, Faculty of Civil

Engineering, Delft, September 1987.

• Vrijling, J.K and van Gelder, P.H.A.J.M., 2002 Probabilistic design in Hydraulic Engineering Lecture notes CT5310 Delft University of Technology.

• Mai Van Cong Probabilistic design of coastal flood defences in Vietnam PhD thesis Sieca Repro, the Netherlands (2010) ISBN: 978-90-9025648-1, 249p

• PhD thesis: Google me by title Probabilistic Design

of Coastal Flood Defences in Vietnam; Direct

available for download:

44f4-4453-82d8-c7ec864cf954/

http://repository.tudelft.nl/view/ir/uuid%3Aa7171f84-• Email:

c.maivan9@Gmail.com & Cong.M.V@WRU.Edu.vn

Subject: CHVB ……name