Reliability of structures chapter 1

Open University, From Sept to Dec -2012 OverviewOverview 1 1 How to measure risks?How to measure risks? 2 Reliability Index and Probability of FailureReliability Index and Probability of Failure 3 Reliability Analysis ProceduresReliability Analysis Procedures 1 4 Reliability Analysis ProceduresReliability Analysis Procedures 4 ConclusionsConclusions 5 Buildings Offices Residential structures Hospitals H y draulic structures Pi p elines Brid g es y p g 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 inadequate materials, methods of construction, bad connections changes without analysis ma j o r s t orms, snow, earthquakes, …) connections , changes without analysis . + Operation/use phase: overloading, inadequate maintenance, misuse, vehicle collisions vessel collisions terrorist CfUtiti collisions , vessel collisions , terrorist attacks) C auses o f U ncer t a i n ti es in the building process Uncertainties in Load and ResistanceUncertainties in Load and Resistance (dCiC i)(dCiC i) (L oa d C arry i ng C apac i ty )(L oa d C arry i ng C apac i ty ) Load & resistance parameters have to be treated as random variables 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, th k i ear th qua k e, i ce y Man-made causes – collisions (vehicle, vessel), fire, poor maintenance, human errors, gas explosion, terrorist acts y Load effects – analytical models, approximations y Load combinations y Material properties – concrete, steel, wood, plastics, composites y Dimensions 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 Reliability and Risk ( b bili f il )( b bili f il ) (P ro b a bili ty o f F a il ure )(P ro b a bili ty o f F a il ure ) y Reliability = probability that the structure will perform its function during the predetermined lifetime function during the predetermined lifetime y Risk (or probability of failure) = probability that the structure will fail to perform its function during the predetermined lifetime predetermined lifetime => How to measure 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 (X 1 ,…,X n ) y Formulate the limit state function, g (X 1 ,…,X n ), such that g < 0 for failure, and g ≥ 0 for safe performance y Calculate the risk (probability of failure, P F , gRQ = − P F =Prob(g<0) [...]... + σQ mR = mean resistance mQ = mean load sR = standard deviation of resistance sQ = standard deviation of load Reliability Index, β PF = F (-b) (- 1. 28 10 -2 2.33 10 -3 3.09 10 -4 Probability of failure, PF b 10 -1 b = - F -1( PF) PF 3. 71 10-5 4.26 10 -6 4.75 10 -7 5 .19 10 -8 5.62 10 -9 5.99 Typical values of β Structural components (beams, slabs, columns), b = 3-4 Connections: welded b = 3-4 bolted b = 5-7... with unknown types of distribution, but with known mean values and standard deviations Use a Taylor series expansion ( g (X1, X 2 , , Xn ) ≈ g x , , x * 1 * n n ) + ∑ (X i =1 where the derivatives are calculated at (X1*, …, Xn*) i −x * i ) ∂g ∂X i Reliability Index for a Non-linear Limit State Function n β= a 0 + ∑ ai μX i =1 n ∑ (a σ i =1 where ∂g ai = ∂X i i Xi i )2 calculated at (X1*, …, Xn*) But how... (X1*, …, Xn*) But how to determine (X1*, …, Xn*)? It is called the design point Monte Carlo simulations Given limit state function, g (X1, …, Xn) and cumulative distribution function for each random variable X1, …, Xn Generate values for variables (X1, …, Xn) using computer random number generator For each set of generated values of (X1, …, Xn) calculate value of g (X1, …, Xn), and save it Monte Carlo... linear limit state function g (X1, X2, …, Xn) = a0 + a1 X1 + a2 X2 + … + an Xn Xi = uncorrelated random variables, with unknown types of distribution, but with known mean values and standard deviations ihk l d d dd i i n β= a 0 + ∑ ai μX i =1 n (ai σX )2 ∑ i =1 i i Reliability Index for a Non-linear Limit State Function Let’s consider a non-linear limit state function g (X1, …, Xn) Xi = uncorrelated random... Repeat this N number of times (N is usually very large, e.g 1 million) Calculate probability of failure and/or reliability index Count the number of negative values of g, NEG, then PF = NEG/N 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... Figure: PDF of load, resistance and safety margin Fundamental case The limit state function, g = R - Q, the probability of failure, PF, can be derived considering the PDF’s of R and Q Figure: PDFs of load (Q), and resistance (R) Fundamental case The structure fails when the load exceeds the resistance, then the probability of Th f il h h l d d h i h h b bili f failure is equal to the probability of Q>R,... results only for special cases First Order Reliability Methods (FORM), reliability index is calculated by iterations Second Order Reliability Methods (SORM), and other advanced procedures Monte Carlo th d M t C l method - values of random variables are simulated (generated l f d i bl i l t d( t d by computer), accuracy depends on the number of computer simulations Reliability Index - Closed-Form Solution... case Space of State Variables Figure: Three-dimensional sketch of a possible joint density function fRQ Reliability Index, β β Mean (R-Q) (RFigure: PDFs of load, resistance and safety margin Reliability Index, β For a linear limit state function, g = R – Q = 0, and R and Q both being normal random variables (μ β= R − μQ ) 2 2 σR + σQ mR = mean resistance mQ = mean load sR = standard deviation of resistance... CDF, and read from the graph • Load and resistance parameters are random variables, therefore, reliability can serve as an efficient measure of structural performance • Reliability methods are available for the analysis of components and complex systems • Target reliability indices depend on consequences of failure or costs ... equations result Pf = ∑ P (R = r i ∩ Q > ri ) = +∞ i i +∞ ∫ (1 − F (r )) f (r )dr = 1 − ∫ F (r ) f (r )dr = ∑ P (Q = q ∩ R < q ) = ∑ P (R < Q |Q = q )P (Q = q ) Pf = Pf ∑ P (Q > R |R = r )P (R = r ) Q i R i i −∞ Q i R i −∞ i i +∞ Pf = i ∫ F (q ) f R i Q (qi )dqi −∞ Too difficult to use, therefore, other procedures are used i i Fundamental case Space of State Variables Figure: Safe domain and failure domain . 33 3.093.09 10 10 44 3. 713 . 71 Probability of failure, P F 10 10 55 4.264.26 10 10 66 4.754.75 PP FF = F (= F ( b)b) 10 10 77 5 .19 5 .19 10 10 88 5.625.62 10 10 99 5.995.99 Typical values of β y Structural. deviation of resistance s = standard deviation of load s Q = standard deviation of load Reliability Index, β P F b 10 10 11 1. 2 81. 28 10 10 22 2.332.33 33 b b = = FF 11 (P(P FF ) ) 10 10 33 3.093.09 10 10 . hours /1 week * 12 weeks y Evening of Thursday, from 6 PM to 9 PM y From 13 /9/2 012 to 6 /12 /2 012 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