Open University, From Sept to Dec -2013

OverviewOverview11

How to measure risks?How to measure risks?2

Reliability Index and Probability of FailureReliability Index and Probability of Failure3

Reliability Analysis ProceduresReliability Analysis Procedures14 Reliability Analysis ProceduresReliability Analysis Procedures4

ConclusionsConclusions5

Buildings

Offices Residential structures Hospitals .....

Hydraulic structuresPipelines Bridgesyp g

y Man-made causes+ Design phase: approximation errors,calculations errors lack of knowledge

y Natural causes(wind, hurricanes, floods,

calculations errors, lack of knowledge+ Construction phase: use of inadequatematerials, methods of construction, badconnections changes without analysis

tornados, major storms, snow,earthquakes, wave)

connections, changes without analysis.+ Operation/use phase: overloading,inadequate maintenance, misuse, vehiclecollisions vessel collisions terroristC f U t i ti collisions, vessel collisions, terroristattacks)

Causes of Uncertainties

in the building process

Uncertainties in Load and Uncertainties in Load and i ( di ( d C i C i )C i C i )Resistance (Load Resistance (Load Carrying Capacity)Carrying Capacity)

Load & Resistance parameters must be treated as random variables

y Occurrence probability (return period)y Magnitude (mean values, coefficient of variation)ag tude ( ea va ues, coe c e t o va at o )=> Structures must be designed to serve their function with

a probability of failure

Load and Resistance are Random Variables

y Dead load, live load, dynamic loady Natural loads temperature, water pressure, earth pressure, wind, snow,

th k iearthquake, ice

y Man-made causes collisions (vehicle, vessel), fire, poor maintenance, humanerrors, gas explosion, terrorist acts

y Load effects analytical models, approximationsy Load combinationsy Material properties concrete, steel, wood, plastics, compositesy Dimensions

Consequences of Uncertainties

y Deterministic analysis and design is insufficienty Probability of failure is never zeroy Design codes must include a rational safety reserve (too safe too costly,

otherwise too many failures)

y Reliability is an efficient measure of the structural performance

Reliability and Risk Reliability and Risk ( b bili f il )( b bili f il )(Probability of Failure)(Probability of Failure)

y Reliability = probability that the structure will perform itsfunction during the predetermined lifetimefunction during the predetermined lifetime

y Risk (or probability of failure) = probability that thestructure will fail to perform its function during thepredetermined lifetimepredetermined lifetime=> How to measure risk?=> How to measure risk?

y Course: Reliability of Structures

y 50 hours = 4 hours/1 week * 12 weeksy Evening of Monday, from 6 PM to 9 PM

16/9/2013 6/12/2013y From 16/9/2013 to 6/12/2013

y Exercises: 30% of the final resulty Examination: 70% of the final result

y Book: Andrzej S Nowak Kevin Collins Reliability of Structures 2000y Book: Andrzej S. Nowak, Kevin Collins, Reliability of Structures , 2000

y Book: Andrzej S. Nowak, Kevin Collins, Reliability of Structures, 2000

y Chapter 1: Introductiony Chapter 2: Random variablesy Chapter 3: Functions of Random variablesy Chapter 3: Functions of Random variablesy Chapter 4: Simulation Techniquesy Chapter 5: Structural Safety Analysisy Chapter 6: Structural Load modelsy Chapter 7: Models of Resistancey Chapter 8: Design Codesp gy Chapter 9: System Reliability

y Identify the load and resistance parameters (X1, , Xn)

y Formulate the limit state function, g (X1, , Xn), such that g < 0 for failure, andg 0 for safe performance

y Calculate the risk (probability of failure, PF,

g R Q=

PF = Prob (g < 0)

Fundamental CaseSafety Margin, g = R Q

what is the probability g < 0?

Probability Density Function (PDF)

Figure: PDF of load, resistance and safety margin

Fundamental case

The limit state function, g = R - Q, the probability of failure, PF, can be derivedconsidering the PDFs of R and Q

Figure: PDFs of load (Q), and resistance (R)

Th f il h h l d d h i h h b bili f

Fundamental case

The structure fails when the load exceeds the resistance, then the probability of

failure is equal to the probability of Q>R, the following equations result

( )( ) ( | ) ( )

1 ( ) ( ) 1 ( ) ( )

f i i i i

f Q i R i i Q i R i i

P P R r Q r P Q R R r P R r

P F r f r dr F r f r dr+ +

= = > = > = == = ( )( ) ( ) ( ) ( )

( ) ( | ) ( )

f Q i R i i Q i R i i

f i i i i

f f

P P Q q R q P R Q Q q P Q q

+= = < = < = = ( ) ( )f R i Q i iP F q f q dq

= Too difficult to use, therefore, other procedures are used

Fundamental case

y Space of State Variables

Figure: Safe domain and failure domain in a two-dimensional state space

Fundamental case

y Space of State Variables

Figure: Three-dimensional sketch of a possible joint density function fRQ

Reliability Index,

Mean (RMean (R--Q)Q)

Figure: PDFs of load, resistance and safety margin

Reliability Index, For a linear limit state function, g = R Q = 0, and R and Q are both normalrandom variables

( )2 2

R Q =2 2R Q +

mR = mean resistancemQ = mean loadsR = standard deviation of resistances = standard deviation of loadsQ = standard deviation of load

Reliability Index, PF beta1010--11 1.281.28

1010--22 2.332.33

33

beta beta = = -- FF--11(P(PFF) )

1010--33 3.093.09

1010--44 3.713.71Probability of failure, PF

1010--55 4.264.26

1010--66 4.754.75

PPFF = F (= F (--beta)beta)

1010--77 5.195.19

1010--88 5.625.62

1010--99 5.995.99

Typical values of

y Structural components (beams, slabs, columns), beta = 3-4

y Connections:y Connections: y Welded, beta = 3-4y Bolted, beta = 5-7

y Structural systems (building frames, girder bridges), beta = 6-8

y Closed-form equations accurate results only for special cases

y First Order Reliability Methods (FORM), reliability index is calculated by iterations

y Second Order Reliability Methods (SORM), and other advanced procedures

y M t C l th d l f d i bl i l t d ( t dy Monte Carlo method - values of random variables are simulated (generated by computer), accuracy depends on the number of computer simulations

Reliability Index - Closed-Form Solution

y Lets consider a linear limit state function

g (X X X ) = a + a X + a X + + a Xg (X1, X2, , Xn) = a0 + a1 X1 + a2 X2 + + an Xn

y Xi = uncorrelated random variables, with unknown types of distribution, but i h k l d d d d i iwith known mean values and standard deviations

0 i

n

i Xa a +01

2

1

( )

i

i

i Xi

n

i Xi

a

=

=

=

Reliability Index for a Non-linear Limit State Function

y Lets consider a non-linear limit state function

g (X1, , Xn)

y Xi = uncorrelated random variables, with unknown types of distribution, but with known mean values and standard deviationsknown mean values and standard deviations

y Use a Taylor series expansion

( ) ( ) ( )* * *1 2 11

, ,..., ,...,n

n n i ii i

gg X X X g x x X xX=

+ where the derivatives are calculated at (X1*, , Xn*)

Reliability Index for a Non-linear Limit State Function

01

i

n

i Xi

a a =

+=

2

1

( )i

n

i Xi

a

=

where ii

gaX= calculated at (Xcalculated at (X11

**, , , , XXnn**). ).

But how to determine (XBut how to determine (X11**, , , , XXnn**)? )? It is called the It is called the design point.point.

Monte Carlo simulations

y Given limit state function, g (X1, , Xn) and cumulative distribution functionfor each random variable X1, , Xn

y Generate values for variables (X1, , Xn) using computer random numbergeneratorgenerator

y For each set of generated values of (X1, , Xn) calculate value of g (X1, ,Xn), and save it

Monte Carlo simulations

y Repeat this N number of times (N is usually very large, e.g. 1 million)

y Calculate probability of failure and/or reliability indexy Count the number of negative values of g, NEG, then

PF = NEG/NFy Plot the cumulative distribution function (CDF) of g on the normal

probability paper and either read the resulting value pf PF and b directly from the graph, or extrapolate the lower tail of CDF, and read from the graph.

Load and resistance parameters are random variables, therefore, reliability canserve as an efficient measure of structural performance

Reliability index is usually used in practice Reliability index is usually used in practice

Reliability methods are available for the analysis of components and complexReliability methods are available for the analysis of components and complexsystems: FORM, SORM, Monte Carlo