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8. Numerical methods for reliability computations Objectives • Learn how to approximate failure probability using Level I, Level II and Level III methods 1 Level I, II, III methods Crude, economical • Level I methods: use one parameter to account for the uncertainty in each random variable • Level II methods: use two parameters to account for the uncertainty in each random variable (usually the mean and the standard deviation) • Level III methods: take into account for the probability distributions of the random variables. More accurate, But also more expensive 2 First order, second moment methods • Reduced random variables. Independent, standard normal variables obtained from a transformation of random variables. • Safety index, =distance from straight line representing the boundary between survival and failure and origin in space of reduced variables. Failure probability=(- ). • Of all combinations of the r.v. that entail failure the one corresponding to the most probable failure point is the most likely. 3 z2 Constant joint probability density O OA=safety index OB=safety index if X1 were deterministic OC=safety index if X2 were deterministic X2 more important than X1 C z1 B A, most probable failure point 4 Lessons learnt (refer to the previous slide) • If performance function is linear, it is easy to compute the failure probability exactly. • If we plot the limit state function in the space of the random variables, then this function is represented by a straight line. • The safety index, , is equal to the distance of the straight line representing the limit state function to the origin. The probability of failure is (-). • Of all combinations of values of the values of the random variables, the one corresponding to the most probable failure point, A, is the most likely 5 Finding the most probable point when the performance function is linear and the random variables are independent, standard normal g(z1,z2 )=a0+a1z1+a2z2 z2 z* 1 g ( 0 ) g * 2 z2 g z*1, z*2 g<0 gradient of g g>0 O z1 6 Estimating failure probability using linear approximation of the performance function about MPP: First order methods (FORM) or First order, second moment methods (FOSM) • It is important to approximate g-function accurately in the vicinity of MPP to estimate failure probability accurately. Therefore, we should use linear Taylor expansion about the MPP. • Case A: standard normal, independent random variables • Case B: correlated standard normal random variables • Case C: independent variables with arbitrary probability distributions • Case D: dependent variables with arbitrary probability distributions (beyond the scope of this class) 7 Case A: normal independent random variables Transform variables into standard normal Find MPP: Find z1*,…,zn* To minimize (z1*2+…+zn*2)1/2 So that g(z1*,…,zn*)=0 Find safety index, and P(F) 8 Performance function in the space of reduced random variables Limit state function,g(z)=0 Linear approximation of limit state function, g(z)=0 z2 MPP: z1*, z2* Failure region Gradient of g z1*2 z2*2 O Safe region z1 P( F ) ... f Z1,..., Z n ( z1,..., zn )dz1...dzn ( ) g 0 9 Case B Correlated, standard normal random variables • Need to transform correlated variables into uncorrelated • Transformation: rotation of coordinate system • Y=TTX • T: matrix whose columns are the eigenvectors of the covariance matrix 10 Independent standard normal random variables 11 M Positively correlated standard normal random variables 12 M Case C:Independent random variables with arbitrary probability distribution • Transform random variables to standard normal using Rosemblatt transformation (the same transformation is used for generating random variables with arbitrary probability distributions) ( z ) FX ( x) (z ) z FX (x) x x, z 13 Reliability computations: practical considerations • Available methods: FORM, Monte Carlo, direct integration of PDF • FORM: can be tricky but it can be the only feasible approach for many complex problems – Validate the results using Monte-Carlo for in a few cases – Conduct parametric studies to understand what the important variables are – Understand the behavior of the performance function – Remove the unimportant random variables and repeat the analysis to see if the results change – Repeat the procedure for finding MPP starting from different initial points to see if there are multiple MPPs 14 Suggested reading • Der Kiureghian, “First- and Second-Order Reliability Methods,” Engineering Design Reliability Handbook, CRC press, 2004, p. 14-1. • Thoft-Christensen, “System Reliability,” Engineering Design Reliability Handbook, CRC press, 2004, p. 15-1. 15