Created by Erik Kostandyan, v4 January 15, 2017

Transcription

1 MATLAB Functions for the First, Second and Inverse First Order Reliability Methods Copyrighted by Erik Kostandyan, Contact: Contents Description... References:... Operational Instructions... Notes regarding ReliabilityByFORM.p, ReliabilityBySORM.p and InverseReliabilityByFORM.p codes... FORM Input... FORM Output... 3 SORM Input... 3 SORM Output... 3 Inverse FORM Input... 3 Inverse FORM Output... 4 Examples... 5 Ex FORM / SORM Solution Ex Inverse FORM Solution Ex Ex FORM / SORM Solution Ex Ex FORM / SORM Solution Ex Inverse FORM Solution Ex Ex FORM / SORM Solution Ex Ex FORM / SORM Solution Ex Inverse FORM Solution Ex Ex FORM / SORM Solution Ex Inverse FORM Solution Ex Description First Order Reliability Method (FORM) is a method of estimating the reliability index ( ), 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 []). 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 []), this method is incorporated by ReliabilityByFORM.p. The probability of failure is estimated approximately using the reliability index by: PFORM ( 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: P ( ). ReliabilityBySORM.p is based on [] suggested formula, where quadratic SORM SORM 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.

2 Inverse FORM is based on the FORM logic with some modification to incorporate search direction (see [3]). Output of the algorithm is a parameter under the interest. If more than parameter under the interest was set up (indicating multiple solutions), then the program estimates the shortest distance point in parameter space and reports those values as well as the line equation based on which multiple solutions might be found. References: [] H.O. Madsen, S. Krenk, N.C. Lind, Methods of structural safety. Prentice-Hall, Englewood Cliffs, London, 986. [] Hohenbichler, M. and Rackwitz, R. (988). Improvement Of Second Order Reliability Estimates by Importance Sampling. J. Eng. Mech.,4(), [3] Der Kiureghian, Armen, Yan Zhang, and Chun-Ching Li. "Inverse reliability problem." Journal of engineering mechanics 0, no. 5 (994): Operational Instructions Save Setup.m, LimitStateFunction.m, ReliabilityByFORM.p, ReliabilityBySORM.p, InverseReliabilityByFORM.p in the same folder. Open Setup.m and LimitStateFunction.m then modify them accordingly then save them and run Setup.m. Please see some examples below. Notes regarding ReliabilityByFORM.p, ReliabilityBySORM.p and InverseReliabilityByFORM.p codes The difference quotient in numerical differentiation is user adjustable, Absolute error_tolerance of limit state value in standardized domain for FORM is user adjustable, as well as for Inverse FORM, Maximum number of iterations in either function is set to 50,000. So if you see that result is reported on the 50,000 iteration, it means your problem has been stopped from looping. Try to change the difference quotient and error_tolerance and try again. If again the result is reported on the 50,000 iteration, it means your problem does not have a solution (at least mathematically). 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(), x(),..., x(n)], with known input vectors {Mean vector, Standard Deviation vector, Correlation Matrix and Distribution vector}, then: o Set up for FORM/SORM: 'Constants' (Constant,Constant,Constant3) are any matrixes or vectors defined in Setup.m that could be declared into the Limit State Function, and FORM/SORM can be called interactively by any loop and results could be stored. o Set up for InverseFORM: 'Constants' are the same as for FORM/SORM set up, 'Parameter' is a Parameter (can be a vector) of the Limit State Function which we wish to find such that target reliability index is Target_HL_ReliabilityIndex in Setup.m. o o Limit State Function for FORM/SORM: 'Parameter' is not used and better to alway set to zero in Setup.m or turned it off. g={any expression with x(i) and Constants}, if Constants exist in Setup.m, otherwise g={any expression with x(i)}, and Constants are not used and better to set them to zero in Setup.m or turned them off. Limit State Function for InverseFORM: 'Parameter' is set to some inital guess in Setup.m. g={any expression with x(i), Parameter and Constants}, if Constants exist in Setup.m, otherwise g={any expression with x(i), Parameter} and Constants are not used and better to set them to zero in Setup.m or turned them off. Note: In any input vector keep the order of variables x(i) as they appear in the vector x=[x(), x(),..., x(n)], see below for the examples. FORM Input If x is a vector of initial variables, x=[x(), x(),..., 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: g={any expression with x(i) and may be Constants} defined in LimitStateFunction.m dz is the difference quotient in numerical differentiation,

3 error_tolerance an absolute error for limit states from adjacent iterations in standardized domain, Mean_X is Mean row vector, SD_X is Standard Deviation row vector, Dist_X is Distribution row vector: o for Normal=, LogNormal=, 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= => Based on EigenDecomposition. Constant, Constant, Constant3 any numerical values that should be considered in LimitStateFunction.m, FORM Output ProbabilityOfFailure_FORM = Probability Of Failure by FORM, HL_ReliabilityIndex = Hasofer-Lind Reliability Index, LimitStateFunctionValue for the final solution NumberOfIterations Alfa_Z = Unit Row Vector Values in Normalized Space, OmissionSensitivityFactor=relative importance on reliability index by assuming stochastic variable i is deterministic, 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. Sign is defined: o if sign= => Alfa_Z and DesignPoint_Z have same direction => B>0 o if sign=0 => Alfa_Z and DesignPoint_Z not same direction => B<0 SORM Input If x is a vector of initial variables, x=[x(), x(),..., 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: g={any expression with x(i) and may be Constants} defined in LimitStateFunction.m, DesignPoint_Z=Coordinates of Design Point in Normalized Space (can be found by FORM), dz is the difference quotient in numerical differentiation, Mean_X is Mean row vector, SD_X is Standard Deviation row vector, Dist_X is Distribution row vector: o for Normal=, LogNormal=, 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= => Based on EigenDecomposition. Constant, Constant, Constant3 any numerical values that should be considered in LimitStateFunction.m, SORM Output ProbabilityOfFailure_SORM = Probability Of Failure by SORM, SORM_ReliabilityIndex = SORM Reliability Index, Alfa_Z = Unit Row Vector Values in Normalized Space, GradientVector = Gradient Row Vector at the DesignPoint_Z, HessianMatrix = Hessian Matrix at the DesignPoint_Z. Inverse FORM Input If x is a vector of initial variables, x=[x(), x(),..., 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: g={any expression with x(i) and Parameter, and may be Constants} defined in LimitStateFunction.m dz is the difference quotient in numerical differentiation,

4 error_tolerance an absolute error for limit states from adjacent iterations in standardized domain, Mean_X is Mean row vector, SD_X is Standard Deviation row vector, Dist_X is Distribution row vector: o for Normal=, LogNormal=, 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= => Based on EigenDecomposition. Constant, Constant, Constant3 any numerical values that should be considered in LimitStateFunction.m, Parameter is what we would like to estimate in limit state function such that reliability index is equal to Target_HL_ReliabilityIndex. Parameter might be a vector, in this case indexing should be used in LimitStateFunction.m and Setup.m. Also, parameter values in Setup.m is the initial guess values. Target_HL_ReliabilityIndex, is a value that we are aiming for the reliability index. Inverse FORM Output DesignConstant estimated parameter or parameters (if more than one parameter was requested to be found then the closest to the origin form the parameter space is reported) DesignConstantEquation a line equation from which parameter(s) might be estimated, reported in equation format such as: size{parameters Space} i= C i Parameter i + Constant = 0, where Ci is constant(s). Iteration= number of iteration until the solution is found

5 Examples Ex. Limit state function 48x x 3600x 3 3 x ~ Normal( mean, sd 0.6) x ~ Normal( mean *0 ^ 7, sd 3*0 ^ 6) x ~ Normal( mean *0 ^ 5, sd *0 ^ 6) All x s are independent. FORM / SORM Solution Ex. LimitStateFunction: g=48*x()*x(3)-constant*x(); dz=0^-3; error_tolerance=0^-; Mean_X=[ *0^7 *0^-5]; SD_X=[0.6 3*0^6 *0^-6]; Dist_X=[ ]; Cor_X=[,0,0; 0,,0; 0,0,]; Constant=[3600]; Constant=0; Constant3=0; ProbabilityOfFailure_FORM = e-04 HL_ReliabilityIndex = LimitStateFunctionValue = NumberOfIterations = 5 Alfa_Z = OmissionSensitivityFactor = DesignPoint_Z = DesignPoint_X =.0e+07 * GradientVector =.0e+03 * sign = ProbabilityOfFailure_SORM = e-04 SORM_ReliabilityIndex = Alfa_Z = GradientVector =.0e+03 * HessianMatrix =.0e+0 * Inverse FORM Solution Ex.

6 LimitStateFunction: g=parameter*x()*x(3)-constant*x(); dz=0^-3; error_tolerance=0^-; Mean_X=[ *0^7 *0^-5]; SD_X=[0.6 3*0^6 *0^-6]; Dist_X=[ ]; Cor_X=[,0,0; 0,,0; 0,0,]; Constant=[3600]; Constant=0; Constant3=0; Parameter=[]; Target_HL_ReliabilityIndex= ; DesignConstant = DesignConstantEquation =.0e+04 * Iteration = 4 Ex. Limit state function x x x ~ Uniform( mean 50, sd 8.87) x ~ Weibull( mean.5, sd.5) or x ~ Exponential ( 0.08) x x Cor _ x x 0.5 x FORM / SORM Solution Ex. LimitStateFunction: g=x()-x(); dz=0^-5; error_tolerance=0^-3; Mean_X=[50.5]; SD_X=[8.87.5]; Dist_X=[5 4]; Cor_X=[ 0.5; 0.5 ]; NatafTransform= see below; Constant=0; Constant=0; Constant3=0; Output for NatafTransform = 0 ProbabilityOfFailure_FORM = HL_ReliabilityIndex = LimitStateFunctionValue = e-05 NumberOfIterations = 6

7 Alfa_Z = OmissionSensitivityFactor = DesignPoint_Z = DesignPoint_X = GradientVector = sign = ProbabilityOfFailure_SORM = SORM_ReliabilityIndex = Alfa_Z = GradientVector = HessianMatrix = Output for NatafTransform = ProbabilityOfFailure_FORM = HL_ReliabilityIndex = LimitStateFunctionValue = e-05 NumberOfIterations = 6 Alfa_Z = OmissionSensitivityFactor = DesignPoint_Z = DesignPoint_X = GradientVector = sign = ProbabilityOfFailure_SORM = SORM_ReliabilityIndex = Alfa_Z = GradientVector = HessianMatrix = Ex. 3 Consider the beam: p The load p is uniformly distributed and the maximum bending moment is m 9 8 max pl. The failure condition is max F m m.

8 m F p, l, and are outcomes of uncorrelated Normally distributed variables P, L, and deviations: E[P] =.0 kn/m SD[P]= 0.4 kn/m E[L] = 4.0 m SD[L]= 0.4 m E[ ] = 5.0 knm SD[ ]= 0.4 knm M F M F M F with expected values and standard Calculate the reliability index, the annual probability of failure and the - values? 9 Given: Limit state function x x x 8 x ~ Normal( mean, sd 0.4) x ~ Normal( mean 4, sd 0.4) x ~ Normal( mean 5, sd 0.4) 3 3 FORM / SORM Solution Ex. 3 LimitStateFunction: g=x(3)-(9/8)*x()*(x()^); dz=0^-5; error_tolerance=0^-3; Mean_X=[ 4 5]; SD_X=[ ] ; Dist_X=[ ]; Cor_X=[,0,0; 0,,0; 0,0,]; Constant=0; Constant=0; Constant3=0; ProbabilityOfFailure_FORM = HL_ReliabilityIndex = LimitStateFunctionValue = e-04 NumberOfIterations = 3 Alfa_Z = OmissionSensitivityFactor = DesignPoint_Z = DesignPoint_X = GradientVector = sign = ProbabilityOfFailure_SORM = SORM_ReliabilityIndex = Alfa_Z = GradientVector = HessianMatrix =

9 Inverse FORM Solution Ex. 3 LimitStateFunction: g=parameter*x(3)-(9/8)*x()*(x()^); dz=0^-5; error_tolerance=0^-3; Mean_X=[ 4 5]; SD_X=[ ] ; Dist_X=[ ]; Cor_X=[,0,0; 0,,0; 0,0,]; Constant=0; Constant=0; Constant3=0; Parameter=[]; Target_HL_ReliabilityIndex=3.99; DesignConstant = DesignConstantEquation = Iteration = 6 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/m, s=0 kn/m ) X is Model uncertainty with Normal distribution (μ=, s=0.) G is Annual maximum Load with Gumbel distribution (μ= 00 kn/m, s=30 kn/m ) The stochastic variables are considered independent. Calculate the reliability index, the annual probability of failure and the - values? Given: Limit state function x xx3 x ~ LogNormal ( mean 400, sd 0) x ~ Normal( mean, sd 0.) x ~ Gumbel( mean 00, sd 30) 3 FORM / SORM Solution Ex. 4 LimitStateFunction g=x()-x()*x(3); dz=0^-5; error_tolerance=0^-3; Mean_X=[400 00]; SD_X=[ ]; Dist_X=[ 3] Cor_X=[,0,0; 0,,0; 0,0,] NatafTransform=0 Constant=0; Constant=0; Constant3=0;

10 ProbabilityOfFailure_FORM = e-06 HL_ReliabilityIndex = LimitStateFunctionValue = e-05 NumberOfIterations = 5 Alfa_Z = OmissionSensitivityFactor = DesignPoint_Z = DesignPoint_X =.0e+0 * GradientVector =.0e+0 * sign = ProbabilityOfFailure_SORM = e-06 SORM_ReliabilityIndex = Alfa_Z = GradientVector =.0e+0 * HessianMatrix = Ex. 5 Given: Limit state function x x x x ~ LogNormal ( mean 400, sd 0) x ~ Normal( mean, sd 0.) x ~ Gumbel( mean 00, sd 30) FORM / SORM Solution Ex. 5 LimitStateFunction: g=(x()^constant)-(x()^constant)*(x(3)^constant3); dz=0^-4; error_tolerance=0^-3; Mean_X=[400 00]; SD_X=[ ]; Dist_X=[ 3]; Cor_X=[,0,0; 0,,0; 0,0,]; Constant=; Constant=; Constant3=4; ProbabilityOfFailure_FORM = HL_ReliabilityIndex = LimitStateFunctionValue = e-04

11 NumberOfIterations = Alfa_Z = OmissionSensitivityFactor = DesignPoint_Z = DesignPoint_X =.0e+0 * GradientVector =.0e+05 * sign = 0 ProbabilityOfFailure_SORM = SORM_ReliabilityIndex = Alfa_Z = GradientVector =.0e+05 * HessianMatrix =.0e+05 * Inverse FORM Solution Ex. 5 LimitStateFunction: g=parameter*(x()^constant)- (x()^constant)*(x(3)^constant3); dz=0^-4; error_tolerance=0^-3; Mean_X=[400 00]; SD_X=[ ]; Dist_X=[ 3]; Cor_X=[,0,0; 0,,0; 0,0,]; Constant=; Constant=; Constant3=4; Parameter=[]; Target_HL_ReliabilityIndex=3; DesignConstant = e+04 DesignConstantEquation =.0e+09 * Iteration = 4 Ex. 6 Given: Limit state function 377x x x x 4 4 3

12 x ~ LogNormal( mean 400, sd 0) x ~ Normal( mean, sd 0.) x ~ Gumbel( mean 00, sd 30) 3 x ~ Normal( mean 0, sd 3) 4 FORM / SORM Solution Ex. 6 LimitStateFunction: g=377*(x()^constant)+x(4)- (x()^constant)*(x(3)^constant3); dz=0^-4; error_tolerance=0^-3; Mean_X=[ ]; SD_X=[ ]; Dist_X=[ 3 ]; Cor_X=eye(length(Mean_X)); Constant=; Constant=; Constant3=4; ProbabilityOfFailure_FORM = HL_ReliabilityIndex = LimitStateFunctionValue = e-04 NumberOfIterations = 8 Alfa_Z = OmissionSensitivityFactor = DesignPoint_Z = DesignPoint_X =.0e+0 * GradientVector =.0e+09 * sign = ProbabilityOfFailure_SORM = SORM_ReliabilityIndex = Alfa_Z = GradientVector =.0e+09 * HessianMatrix =.0e+09 * Inverse FORM Solution Ex. 6 dz=0^-4; error_tolerance=0^-3; Mean_X=[ ]; SD_X=[ ];

13 Dist_X=[ 3 ]; Cor_X=eye(length(Mean_X)); Constant=; Constant=; Constant3=4; Parameter=[ ]; Target_HL_ReliabilityIndex=; DesignConstant = DesignConstantEquation =.0e+09 * Iteration = 6