Reliability of structures chapter 1

Uncertainties in Load and ResistanceLoad Carrying Capacity Load & resistance parameters have to be treated as randomvariables y Occurrence probability return period y Magnitude mean valu

Open University, From Sept to Dec -2012

Reliability Analysis Procedures

14 4 Reliability Analysis Procedures

Conclusions

5

Offices Residential structures Hospitals

Hydraulic structures

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

y Natural causes (wind,

hurricanes, floods, tornados,

j t calculations errors, lack of knowledge

+ Construction phase: use of inadequatematerials, methods of construction, badconnections changes without analysis

major storms, snow,

earthquakes, …)

connections, changes without analysis.+ Operation/use phase: overloading,inadequate maintenance, misuse, vehiclecollisions vessel collisions terrorist

attacks)

Causes of Uncertainties

in the building process

Uncertainties in Load and Resistance

(Load Carrying Capacity)

Load & resistance parameters have to be treated as randomvariables

y Occurrence probability (return period)

y Magnitude (mean values, coefficient of variation)

=> 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 load

y Natural loads – temperature, water pressure, earth pressure, wind, snow,

Consequences of Uncertainties

y Deterministic analysis and design is insufficient

y Probability of failure is never zero

y 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

=> How to measure risk?

y 50 hours = 4 hours/1 week * 12 weeks

y Evening of Thursday, from 6 PM to 9 PM

y From 13/9/2012 to 6/12/2012

y Exercises: 30% of the final result

y Presentation: 20% of the final result

y Examination: 50% of the final result

y Examination: 50% of the final result

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

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

y Formulate the limit state function, g (X1, …, Xn), such that g < 0 for failure, and

g ≥ 0 for safe performance

y Calculate the risk (probability of failure, PF,

g R Q = −

PF = Prob (g < 0)

Fundamental CaseSafety Margin, g = R – Qwhat is the probability g < 0?

Probability Density Function (PDF)

Figure: PDF of load, resistance and safety margin

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

sR = standard deviation of resistance

s = standard deviation of load

sQ = standard deviation of load

Reliability Index, β P

10 11 1.28

10 22 2.33 33

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 d

y 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 Let’s consider a linear limit state function

( )

i

i

i X i

n

i X i

Reliability Index for a Non-linear Limit State Function

y Let’s 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 deviations

known mean values and standard deviations

y Use a Taylor series expansion

Reliability Index for a Non-linear Limit State Function

0

n

i X i

i

n

i X i

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 numbergenerator

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 index

y Count the number of negative values of g, NEG,

then PFF = NEG/N

y 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 methods are available for the analysis of components and complexReliability methods are available for the analysis of components and complexsystems

• Target reliability indices depend on consequences of failure or costs