Matlab codes for First and Second Order Reliability Methods Copyrighted by Erik Kostandyan, contact:
[email protected] Introduction First Order Reliability Method (FORM) is a method of estimating the reliability index ( β ), and failure function is approximated by its linear form. Second Order Reliability Method (SORM) approximates failure function by its quadratic form. FORM allows estimating invariant reliability index irrespective to the limit state function (safety margin) formulation. This index also known as the Hasofer and Lind reliability index (see [1]). Stochastic variables that construct limit state function are transformed to the standard normal stochastic variables. Based on these variables, the limit state function is transformed to the standardized domain. In standardized domain, the reliability index is defined as the shortest distance from the origin to the limit state function, and this solution is termed a design point in standardized domain. To find the shortest distance, the numerical differentiation with iterative algorithm is applied (see [1]), this method is incorporated by ‘ReliabilityByFORM.p’. The probability of failure is estimated approximately using the reliability index by: PFORM ≈ Φ ( − β FORM ) . Once the design point is estimated, SORM might be applied by ‘ReliabilityBySORM.p’ at the design point, and probably of failure and reliability index by SORM might be estimated: PSORM ≈ Φ ( − β SORM ) . ‘ReliabilityBySORM.p’ is based on [2] suggested formula, where quadratic of the approximated failure function is rotated (via Gram–Schmidt) such that alpha vector is on the first axis and the rest main curvatures are estimated based on Hessian matrix. Operational Instructions Save ‘Setup.m’, ‘ReliabilityByFORM.p’ and ‘ReliabilityBySORM.p’ in the same folder. Open ‘Setup.m’ and modify it accordingly and then run it. Please see some examples below. Notes regarding ‘‘ReliabilityByFORM.p’ and ‘ReliabilityBySORM.p’ codeS • The difference quotient in numerical differentiation is user adjustable, • Absolute error for reliability index estimation in FORM is user adjustable, • For the given vector ‘X’ with marginal distributions, means, standard deviations, and linear correlation structures, Nataf transformation (Gaussian Copulas) is used to estimate the ‘fictive correlation matrix’, and then either the Cholesky or Eigen decomposition is used, this selection is user adjustable, • If ‘x’ is a vector of initial variables, x=[x(1), x(2), ... , x(N)], with known input vectors {Mean vector, Standard Deviation vector, Correlation Matrix and Distribution vector}, then the defined Limit State Function has to be in the following format: LimitStateFunction=@(x) {and any expression with x(i)}. Note: In any input vector keep the order of variables x(i) as they appear in the vector x=[x(1), x(2), ... , x(N)], see below for the examples. FORM Input If ‘x’ is a vector of initial variables, x=[x(1), x(2), ... , x(N)], with Mean vector, Standard Deviation vector, Correlation Matrix, Distribution vector, then the defined Limit State Function has to be in the following format: • LimitStateFunction=@(x) {and any expression with x(i)}, • dz is the difference quotient in numerical differentiation, • dB is an absolute error for reliability index estimation, • Mean_X is Mean row vector, • SD_X is Standard Deviation row vector, • Dist_X is Distribution row vector: • for Normal=1, LogNormal=2, GumbelMax=3, WeibullMin=4, Uniform=5, • Cor_X is Correlation Matrix, • NatafTransform is a binary variable for the decomposition method in Nataf transformation: o NatafTransform=0 => Based on CholeskyDecomposition, o NatafTransform=1 => Based on EigenDecomposition.
Created by Erik Kostandyan, v3
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July 25, 2013
FORM Output • ProbabilityOfFailure_FORM = Probability Of Failure by FORM, • HL_ReliabilityIndex = Hasofer-Lind Reliability Index, • Alfa_Z = Unit Row Vector Values in in Normalized Space, • DesignPoint_Z = Coordinates of Design Point in Normalized Space, • DesignPoint_X = Coordinates of Design Point in Initial Space, • GradientVector = Gradient Row Vector at the DesignPoint_Z. SORM Input If ‘x’ is a vector of initial variables, x=[x(1), x(2), ... , x(N)], with Mean vector, Standard Deviation vector, Correlation Matrix, Distribution vector, then the defined Limit State Function has to be in the following format: • LimitStateFunction=@(x) {and any expression with x(i)}, • DesignPoint_Z=Coordinates of Design Point in Normalized Space (can be found by FORM), • dz is the difference quotient in numerical differentiation, • dB is an absolute error for reliability index estimation, • Mean_X is Mean row vector, • SD_X is Standard Deviation row vector, • Dist_X is Distribution row vector: • for Normal=1, LogNormal=2, GumbelMax=3, WeibullMin=4, Uniform=5, • Cor_X is Correlation Matrix, • NatafTransform is a binary variable for the decomposition method in Nataf transformation: o NatafTransform=0 => Based on CholeskyDecomposition, o NatafTransform=1 => Based on EigenDecomposition. SORM Output • ProbabilityOfFailure_SORM = Probability Of Failure by SORM, • SORM_ReliabilityIndex = SORM Reliability Index, • Alfa_Z = Unit Row Vector Values in in Normalized Space, • GradientVector = Gradient Row Vector at the DesignPoint_Z, • HessianMatrix = Hessian Matrix at the DesignPoint_Z. TERMS OF USE The functions (‘Setup.m’ and ‘ReliabilityByFORM.p’) are copyrighted © by Erik Kostandyan. It is provided for educational purposes only, and prior use of these functions for any business-oriented activities in order to generate any type of income, should be contacted to the author Erik Kostandyan at “
[email protected]”. In addition, I have the following request to the users of these functions, if any scientific publication(s) has been made by using these functions, would you please make references to the following papers: Kostandyan E.E., Sørensen J.D., January 2013, “Reliability Assessment of IGBT Modules Modeled as Systems with Correlated Components”, Proceedings on the 2013 Annual Reliability and Maintainability Symposium (RAMS 2013), Orlando, Florida, USA, pp. 1-6. Digital Object Identifier: 10.1109/RAMS.2013.6517663. Kostandyan E.E., Sørensen J.D., June 2012, “Structural Reliability Methods for Wind Power Converter System Component Reliability Assessment “, Proceedings on the 16th IFIP WG 7.5 Conference on Reliability and Optimization of Structural Systems, Yerevan, Armenia, pp. 135. References: [1] H.O. Madsen, S. Krenk, N.C. Lind, Methods of structural safety. Prentice-Hall, Englewood Cliffs, London, 1986. [2] Hohenbichler, M. and Rackwitz, R. (1988). ”Improvement Of Second Order Reliability Estimates by Importance Sampling.” J. Eng. Mech.,114(12), 2195–2199.
Created by Erik Kostandyan, v3
[email protected]
July 25, 2013
Examples Ex. 1 Limit state function = 48 x2 x3 − 3600 x1
x1 ~ Normal ( mean = 2,= sd 0.6) x2 ~ Normal = ( mean 2= *10 ^ 7, sd 3*10 ^ 6) x3 ~ Normal ( mean= 2 *10 ^ −5, sd= 2 *10 ^ −6)
All x’s are independent. Solution Ex. 1 Input: LimitStateFunction = @(x)48*x(2)*x(3)-3600*x(1) Mean_X = 1.0e+007 * [0.000000200000000 2.000000000000000 SD_X = 1.0e+006 * [0.000000600000000 3.000000000000000 Dist_X = [1 1 1] Cor_X = 1 0 0 0 1 0 0 0 1 NatafTransform=0
0.000000000002000] 0.000000000002000]
Output: ProbabilityOfFailure_FORM = 8.211149681132511e-004 HL_ReliabilityIndex =3.148300115056224 Alfa_Z = [0.603188890834175 -0.713954939341735 -0.355572083498939] DesignPoint_Z = [1.899712924431826 -2.245513249059412 -1.122744257392733] DesignPoint_X = 1.0e+007 * [0.000000313982775 1.326346025282177 0.000000000001775] ProbabilityOfFailure_SORM = 8.998000187836356e-004 SORM_ReliabilityIndex = 3.121454583486132 GradientVector = 1.0e+003 * [-2.160000000003493 2.556649653888599 1.273292184305319] HessianMatrix = 1.0e+002 * 0 0 0 0 0 2.879960447899066 0 2.879960447899066 0
Ex. 2 Limit state function= x1 − x2 = 50, = x1 ~ Uniform( mean sd 28.87)
= x2 ~ Weibull= sd 12.5) ( mean 12.5, or x2 ~ Exponential (λ = 0.08) x1 x2 1 0.5 = Cor _ x = x 1 0.5 1 0.5 1 x2 0.5 1
Created by Erik Kostandyan, v3
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July 25, 2013
Solution Ex. 2 Input: Mean_X = [50.000000000000000 12.500000000000000] SD_X = [28.870000000000001 12.500000000000000] Dist_X =[ 5 4] Cor_X = 1.000000000000000 0.500000000000000 0.500000000000000 1.000000000000000
Output for NatafTransform = 0 ProbabilityOfFailure_FORM = 0.049173446692940 HL_ReliabilityIndex = 1.652921241883278 DesignPoint_Z =[ -1.287254491217664 1.036882108394638] DesignPoint_X =[ 9.896825956482255 9.896319175746006] GradientVector = 11.193070388237915 -8.970379309580778 ProbabilityOfFailure_SORM = 0.047307442339634 SORM_ReliabilityIndex = 1.671540912438320 Alfa_Z = [-0.780325271708731 0.625373864446456] GradientVector = [11.192971675022536 -8.970351473713834] HessianMatrix = 19.742643075915115 -3.866618136783019 -3.866618136783019 -5.567173388953962
Output for NatafTransform = 1 ProbabilityOfFailure_FORM = 0.049172214283120 HL_ReliabilityIndex = 1.652933351432254 Alfa_Z = -0.931437085911594 0.363902397613039 DesignPoint_Z = -1.541715663120039 0.596071705726249 DesignPoint_X = 9.795655201308385 9.795251449325061 ProbabilityOfFailure_SORM = 0.047274707057584 SORM_ReliabilityIndex = 1.671872766022835 GradientVector = 13.260760928979211 -5.180846764130110 HessianMatrix = 19.628245695457739 4.030793476772488 4.030793476772488 -5.496350041767074
Ex. 3 Consider the beam:
p
The load p is uniformly distributed and the maximum bending moment is mmax =
9 pl 2 . The 128
failure condition is mmax ≥ mF . Created by Erik Kostandyan, v3
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p , l , and mF are outcomes of uncorrelated Normally distributed variables P , L , and M F with expected values and standard deviations: E[P] = 2.0 kN/m SD[P]= 0.4 kN/m E[L] = 4.0 m SD[L]= 0.4 m E[ M F ] = 5.0 kNm SD[ M F ]= 0.4 kNm Calculate the reliability index β , the annual probability of failure and the α - values ? Given: Limit state function= x3 − 9 x1 x22 128
x1 ~ Normal ( mean = 2,= sd 0.4) x2 ~ Normal ( mean = 4,= sd 0.4) x3 ~ Normal ( mean = 5,= sd 0.4)
Solution Ex. 3 Input: LimitStateFunction = @(x)x(3)-(9/128)*x(1)*(x(2)^2) Mean_X = 2 4 5 SD_X = 0.400000000000000 0.400000000000000 0.400000000000000 Dist_X = 1 1 1 Cor_X = 1 0 0 0 1 0 0 0 1 NatafTransform = 0
Output: ProbabilityOfFailure_FORM = 0.001141766761700 HL_ReliabilityIndex = 3.050641104568201 Alfa_Z = 0.613967817675150 0.696505455633671 -0.371380760314460 DesignPoint_Z = 1.875995477291760 2.122350642563959 -1.132554576199448 DesignPoint_X = 2.750398190916704 4.848940257025584 4.546978169520221 ProbabilityOfFailure_SORM = 0.001053620339267 SORM_ReliabilityIndex = 3.074686299745241 GradientVector = -0.661281232927635 -0.750179053765621 0.400000000011502 HessianMatrix = 0.000008881784197 -0.109101616629914 0 -0.109101616629914 -0.061861626932114 0 0 0 -0.000008881784197
Ex. 4 In reliability, assessment of a structural element the following limit state function (failure function) is used: g = R − X×Q where R is Strength with Log Normal distribution (μ= 400 kN/m2 , s=20 kN/m2) X is Model uncertainty with Normal distribution (μ= 1, s=0.1) G is Annual maximum Load with Gumbel distribution (μ= 100 kN/m2, s=30 kN/m2) The stochastic variables are considered independent. Created by Erik Kostandyan, v3
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July 25, 2013
Calculate the reliability index β , the annual probability of failure and the α - values ? Given: Limit state function= x1 − x2 x3 x1 ~ LogNormal (= mean 400, = sd 20) x2 ~ Normal ( mean = 1,= sd 0.1) x3 ~ Gumbel (= mean 100, = sd 30)
Solution Ex. 4 Input: LimitStateFunction = @(x)x(1)-x(2)*x(3) Mean_X = 400 1 100 SD_X = 20.000000000000000 0.100000000000000 Dist_X = 2 1 3 Cor_X = 1 0 0 0 1 0 0 0 1 NatafTransform = 0
30.000000000000000
Output: ProbabilityOfFailure_FORM = 5.905731498137945e-006 HL_ReliabilityIndex = 4.381039176063910 Alfa_Z = -0.158975201668832 0.283031622933809 0.945843531285068 DesignPoint_Z = -0.695953693834499 1.240774875899398 4.143625275719523 DesignPoint_X = 1.0e+002 * [ 3.858467004427526 0.011240774875899 3.432569415928690] ProbabilityOfFailure_SORM = 5.529149814693902e-006 SORM_ReliabilityIndex = 4.395372422327726 GradientVector = 1.0e+002 * [ 0.192802935998770 -0.343256941562231 -1.147105016372052] HessianMatrix = 0.962927515502088 0 0 0 0 -10.204814770986561 0 -10.204814770986561 -25.671056391729515
Ex. 5 Given: Limit state function= x12 − x2 2 x34 x1 ~ LogNormal (= mean 400, = sd 20) x2 ~ Normal ( mean = 1,= sd 0.1) x3 ~ Gumbel (= mean 100, = sd 30)
Solution Ex. 5 Input: LimitStateFunction = @(x)(x(1)^2)-(x(2)^2)*(x(3)^4) Mean_X = 400 1 100 SD_X = 20.000000000000000 0.100000000000000 30.000000000000000 Dist_X = 2 1 3 Cor_X = 1 0 0 0 1 0 0 0 1 Created by Erik Kostandyan, v3
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July 25, 2013
NatafTransform =
0
Output: ProbabilityOfFailure_FORM = 0.999999950731279 HL_ReliabilityIndex = -5.329400583117265 Alfa_Z = -0.066952269459627 0.145324616109531 0.987116077048096 DesignPoint_Z = 0.349686579147423 -0.779341255114353 -5.260499698676125 DesignPoint_X = 1.0e+002 * [ 4.065429262459056 0.009220658744886 0.209981433395055] ProbabilityOfFailure_SORM = 0.999999947899903 SORM_ReliabilityIndex = -5.319242464068313 Alfa_Z = -0.066952647094378 0.145325429486438 0.987115931687681 GradientVector = 1.0e+005 * [0.165173992354539 -0.358521767557249 -2.435241718281759] HessianMatrix = 1.0e+005 * 0.016481499187648 0 0 0 -0.038885627873242 -0.528214150108397 0 -0.528214150108397 -3.061845200136304
Ex. 6 Given: Limit state function = ( x2 x3 ) 4 − x12 / x4 4 = x1 ~ LogNormal (= mean 400, sd 20) x2 ~ Normal (= mean 200, = sd 10) = x3 ~ Gumbel (= mean 100, sd 30) x4 ~ Uniform( mean = 3,= sd 1.5)
All x’s are independent , besides Cor ( x3 ; x4 ) = 0.3 Solution Ex. 6 Input: LimitStateFunction = @(x)((x(2)*x(3))^4)-((x(1)^2)/(x(4)^4)) Mean_X = 400 200 100 3 SD_X = 20 10 30 1.5 Dist_X = 2 1 3 5 Cor_X = 1 0 0 0 1 0 0 0 1 0 0 0.3 NatafTransform =
0 0 0.3 1 0
Output: ProbabilityOfFailure_FORM = 4.449754828986080e-018 HL_ReliabilityIndex = 8.587354009541601 Alfa_Z = 0.001170982129692 -0.002356167467331 -0.999984551681932 0.004896291780764 DesignPoint_Z = 0.009256808292931 -0.020769617905101 -8.587231609437875 -0.039813383849037 DesignPoint_X = 1.0e+002 * 3.996857680642813 1.997923038209490 0.002402691706526 0.004169254302136 ProbabilityOfFailure_SORM = 4.243166227317571e-018 SORM_ReliabilityIndex = 8.592816070921446 Created by Erik Kostandyan, v3
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July 25, 2013
GradientVector = 1.0e+008 * -0.005283633372746 0.010631350906100 0.022092525444925 HessianMatrix = 1.0e+010 * [-0.000005271285772 0 0.000007380731404 0.000022078631446
Created by Erik Kostandyan, v3
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0 0.000015963800251 0.009015864692628 0
4.510636410610751
0.000007380731404 0.009015864692628 2.868967331945897 0.000154769048095
0.000022078631446 0 0.000154769048095 0.000462969765067]
July 25, 2013
