component risk analysis

Transcription

1 273: Urban Systems Modeling Lec. 3 component risk analysis instructor: Matteo Pozzi 273: Urban Systems Modeling Lec. 3 component reliability

2 outline risk analysis for components uncertain demand and uncertain capacity; multivariate normal distribution: definition; properties; transformation to standard normal space; design point; multivariate log normal distribution; First Order Reliability Method; sensitivity analysis. 273: Urban Systems Modeling Lec. 3 component reliability 2

3 example of components, general framework A structure, say a bridge. A road segment. An electrical component. A pump in a water system. A component is modeled by a set of random variables, describing loads, demands, capacity, resistance, features affecting the behavior. These variables are modeled by a joint distribution: The functioning of the component is described by a binary variable: the safe, of functioning state, and the failure. Task: computing the probability of failure, Pfailure 9% 273: Urban Systems Modeling Lec. 3 component reliability 3

4 PART I approaches to component risk analysis 273: Urban Systems Modeling Lec. 3 component reliability 4

5 reliability with uncertain load and resistance: method I, the load: ;,, the resistance: independence: s ;, p(s), p(r) r, s find safe condition limit state function: failure p(s) p(r) P var var var always true The difference between two normal rv.s is a normal rv. [to be proved later] ;, Φ ;, standard normal cdf Φ think of as the residual capacity P if resistance R is known Φ 273: Urban Systems Modeling Lec. 3 component reliability

6 reliability with uncertain load and resistance: method II 2 joint prob. in vector notation exp p(g).2 exp 2.3.2, g P with correlated random variables:, x 2 = r 2 4 x = s 6 273: Urban Systems Modeling Lec. 3 component reliability 6

7 example of reliability problem : multivariate normal variable [defined later] vector notation: KN 2.KN.KN 3% KN KN 2 2.KN reliability index [defined later]: probability of damage: Φ.89% 273: Urban Systems Modeling Lec. 3 component reliability 7

8 example of reliability, changing correlation : multivariate normal variable [defined later] varying the correlation coefficient: ; 2 2 = = -.3 sr x 2 = r x 2 = r P f.4.2, 2, = 2 2 = x 2 = r x 2 = r sr, 2 x = s, 2 x = s 273: Urban Systems Modeling Lec. 3 component reliability 8

9 reliability with uncertain load and resistance: method III p(x,x 2 ).3.2. x 2 = r 2 4 x = s 6 x 2 = r ,, safe domain failure domain, x = s p(x,x 2 ), g(x,x2) ,, classical reliability problem: solve an integral., x 2 = r 2 4 x = s 6 273: Urban Systems Modeling Lec. 3 component reliability 9

10 reliability with uncertain load and resistance: method III 7,, 7,, 6 6 u 2 x 2 = r 4 3 x 2 = r 4 3 u 2 x 2 2 x 2 x x = s, x transformation: orthogonal to,, x = s transformation to standard normal space: then it is easy standard normal var.: 273: Urban Systems Modeling Lec. 3 component reliability

11 reliability with uncertain load and resistance: method IV x 2 = r,, x 2 x x = s,,, conditional cumulative distribution s prob. that resistance is lower than load, for load equal to. 273: Urban Systems Modeling Lec. 3 component reliability

12 reliability with uncertain load and resistance: method IV p(s), p(r) F(s), F(r) p(s) p(r) r, s 2 r, s,, prob. that load is higher than resistance, for resistance equal to. alternative formulation: s, conditional cumulative distribution prob. that resistance is lower than load, for load equal to. 273: Urban Systems Modeling Lec. 3 component reliability 2

13 towards multivariate normal distribution 2 exp 2 2 exp 2 2 exp 2 2 2, number of dimension exp 2 determinant parameters ;, multivariate normal distribution 273: Urban Systems Modeling Lec. 3 component reliability 3

14 PART II Gaussian model for component risk analysis 273: Urban Systems Modeling Lec. 3 component reliability 4

15 multivariate normal distribution pdf: ;, 2 exp 2 vector of random variables = 4, 2 = 6, = 2, 2 = 3, 2 =.4 mean vector covariance matrix : E : var, : p(x,x 2 ) x x.3 273: Urban Systems Modeling Lec. 3 component reliability

16 multivariate normal distribution in log. scale pdf: gradient: log ;, Hessian matrix: maximum: uniform curvature, always negative definite. So MVN is log concave. 2 p(x,x 2 ) = 4, 2 = 6, = 2, 2 = 3, 2 =.4 Simplest case: the standard MVN: log ;, 2 2 x x 273: Urban Systems Modeling Lec. 3 component reliability 6

17 multivariate normal distribution: contour plot pdf: log ;, 2 contour line: const. = 4, 2 = 6, = 2, 2 = 3, 2 = lines are ellipses centered in the mean = 4, 2 = 6, = 2, 2 = 3, 2 =.4 x 2 = 4, 2 = 6, = 2, 2 = 3, 2 = -.9 x x examples, changing the correlation only x x - - x 273: Urban Systems Modeling Lec. 3 component reliability 7

18 multivariate normal distribution: eigenvalues pdf: log ;, 2 contour line: const. eigen value decomposition: = 4, 2 = 6, = 2, 2 = 3, 2 =.4 : eigenvector matrix Eigenvectors form an ortho normal base: i. e. : eigenvalue matrix eigen problem: x x 273: Urban Systems Modeling Lec. 3 component reliability 8

19 multivariate normal distribution: eigenvalues pdf: log ;, 2 contour line: const. eigen value decomposition: = 4, 2 = 6, = 2, 2 = 3, 2 =.4 principal components : / / x 2 in terms of variables, the contour line (surfaces) are circles (spheres): is standard MVN - - x 273: Urban Systems Modeling Lec. 3 component reliability 9

20 multivariate normal distribution: eigenvalues principal components: / inverse relation: original rand. var.s as a function of the components: = 4, 2 = 6, = 2, 2 = 3, 2 =.4 / basic idea of eigen values: to change point of view: from canonical base to an ortho normal base centered in the mean. Now variables looks uncorrelated. Re scale using : now variable has also unit variance. x 2 Matlab: [m_v,m_l]=eig(m_sigma), eig - - x 273: Urban Systems Modeling Lec. 3 component reliability 2

21 example of: eigen value decomposition you may check that: covariance matrix: eigenvector matrix: = 4, 2 = 6, = 2, 2 = 3, 2 = eigenvalue matrix: Λ x 2 Length of ellipse s principal axes: x 273: Urban Systems Modeling Lec. 3 component reliability 2

22 covariance matrix properties: symmetry:, positive definitiveness: : :correlation matrix = 4, 2 = 6, = 2, 2 = 3, 2 = = 4, 2 = 6, = 2, 2 = 3, 2 =.6 = 4, 2 = 6, = 2, 2 = 3, 2 = -.9 x 2 x 2 x x x x 273: Urban Systems Modeling Lec. 3 component reliability 22

23 properties of MN: marginalization ;, parameters 2 exp 2 \ ;, rand. vars. ;, marginal is normal = 99% = % same, same marginal = -3% same marginal 273: Urban Systems Modeling Lec. 3 component reliability 23

24 properties of MN: marginalization [cont.] ;, 2 exp 2 partition: marginal probability: ;, Marginalization may be computationally expensive in general. But if a vector of rand. vars. is jointly normal, any subset is jointly normal as well, and parameters can be directly read in those of the joint set. 273: Urban Systems Modeling Lec. 3 component reliability 24

25 properties of MN: conditional ;, 2 exp 2 After observing, the conditional distribution of is still normal: ;, the reduction of variance does not depend on the value observed if uncorrelated, : For jointly normal rand. vars., uncorrelation and independence are equivalent. 273: Urban Systems Modeling Lec. 3 component reliability 2

26 properties of MN: conditional [cont.] ;, 2 p(x 2 x ) p(x,x 2 ) p(x ) x 2. p(x =.3,x 2 ) p(x,x 2 =.4).2.4 x.6 p(x 2 ).8 p(x x 2 ) x 2. x 2 x x.. 273: Urban Systems Modeling Lec. 3 component reliability 26

27 example of marginalization/conditional ;, marginalization vector of random variables mean vector 6 4 ;, covariance matrix conditional suppose to observe reduction of uncertainty ;, : Urban Systems Modeling Lec. 3 component reliability 27

28 recap: transformation of random variables in d p z (z) F z (z) z = f(x) x x = f - (z) = g(z) random variable, transformation new random variable, z z : monotonically increasing inverse p z (z), F z (z) conservation of probability p x (x), F x (x) x p x (x) F x (x) 273: Urban Systems Modeling Lec. 3 component reliability 28

29 transformation of multivariate rand. vars. vector of rv.s joint probability,,, dim: inverse map: invertible map find:,,, : Jacobian Jacobian of the inverse map dim.: when determinant of the Jacobian general formula: 273: Urban Systems Modeling Lec. 3 component reliability 29

30 transformation of multivariate rand. vars. [cont.].8.2 x area( ) area( ) 2.2 y x area( ) 4% area( ).2.. y 2. e.g.: uniform 2... x.. x 2 equal probability (volume). y.. y : Urban Systems Modeling Lec. 3 component reliability 3

31 sign of the determinant of the Jacobian the map preserves orientation x 2 y 2 y we are only interested in the ratio between areas, hence we take the absolute value of the determinant. x de/22/8/26/32/ y 2 the map inverts orientation y 273: Urban Systems Modeling Lec. 3 component reliability 3

32 linear transformation of mult. rand. vars. linear transformation:.4 x x y y inverse transformation: for a linear transformation, the Jacobian (and consequently its determinant) is uniform. 273: Urban Systems Modeling Lec. 3 component reliability 32

33 example of linear transformations: x x pure rotation cos /3 sin /3 sin /3 cos /3 y y diagonal (no rotation)...7 y y general y y 273: Urban Systems Modeling Lec. 3 component reliability 33

34 linear transformation of jointly normal rand. vars. ;, 2 exp 2 linear invertible transformation: ;, exp 2 exp 2 ;, same as for mean vector and covariance matrix of every multivariate rand. var.s this proves that: A linear combination of jointly normal rand. var.s is also jointly normal. This is true in general, also for a transformation to a smaller space, e.g. from vector to scalar (proved by marginalization). 273: Urban Systems Modeling Lec. 3 component reliability 34

35 summary on jointly normal rand. vars. ;, the joint probability is completely defined by mean vector and covariance matrix, which are the parameters of the distribution. the conditional distribution, given any subset of variable, is also jointly normal. each subset of is jointly normally distributed, and marginalization is computationally trivial (just copy part of and ). note: if the marginal probability of each variable is normal, this does not imply that the set of variables is jointly normal. any linear transformation of the variables is jointly normal: ;, ADVANCED the variables can be easily mapped into the «standard normal space». 273: Urban Systems Modeling Lec. 3 component reliability 3

36 linear transformation of jointly normal rand. vars. ;, distribution of : ;, from linear transformation rule sum: 2 difference: 2 example: many loads, many resistances: limit state function resistances loads : Urban Systems Modeling Lec. 3 component reliability 36

37 sum of two random variables in the general case joint probability,,,,,, convolution integral, hopefully it can be solved for specific distributions,. if independency,, second moment representation: always true difference: 273: Urban Systems Modeling Lec. 3 component reliability 37

38 reliability for normal vars., with linear limit state func. linear limit state function: safe condition failure.3 P.2 distribution of : ;, p(x,x 2 ), g(x,x2). -. probability of failure : Φ reliability index : x 2 = r 2 4 x = s 6 273: Urban Systems Modeling Lec. 3 component reliability 38

39 why do we assume a MVN model? consider ~ [not necessarily ] does and exist (can be computed)? consider linearly related to : : safe condition so that failure p norm. appr. can we compute and var? why we need ~? Because we get ~ and we can easily compute P p(x) F(g).2.. =44% appr. =27% - g 273: Urban Systems Modeling Lec. 3 component reliability 39

40 PART III transformation of the Gaussian model 273: Urban Systems Modeling Lec. 3 component reliability 4

41 transformation to standard normal space Given and, find and so that: ;, d: ;, Cholesky decomposition: Given any matrix (positive definite), chol is a lower triangular matrix so that. standard normalization: Eigenvalue analysis: Given any matrix, is orthonormal matrix, is a diagonal matrix so that. not the same map. / Cholesky is simpler. / Matlab: m_l=chol(m_sigma,'lower') 273: Urban Systems Modeling Lec. 3 component reliability 4

42 example of transformation to standard normal space ;, inverse relation: Eigenvalue Cholesky Cholesky chol Eigenvalue / : Urban Systems Modeling Lec. 3 component reliability 42

43 p(u,u 2 ) density in the standard normal space ;, polar coord u 2 p(u,u 2 =) exp 2 p(u =,u 2 ) u 2 u u exp () ;, Maximum density in the origin, fast decay in radial direction. Radial symmetry: density only dependents on. ().3.2. pdf cdf 273: Urban Systems Modeling Lec. 3 component reliability 43

44 rotation in the standard normal space ;, polar coord. 2 new reference system: for an ortho normal system: new coordinates: distribution in new coordinates exp 2 u exp 2 ;, 273: Urban Systems Modeling Lec. 3 component reliability u ;, ;, () () pdf cdf 2 3 the distribution is invariant respect to rotation 44

45 properties of the standard normal space ;, polar coord. 2 exp 2 2 exp 2 ;, 2 for each number of dimensions, there is just one standard normal space; the distribution is invariant respect to rotation; the origin,, is the mean vector and it is the (only) mode (i.e. maximum); each variable is scaled to (zero mean and) unit standard deviation; each variable is independent from the others; all marginal distributions are the same:, (standard normal); the density at one point () depends only by the distance from the origin () and the number of dimensions (). 273: Urban Systems Modeling Lec. 3 component reliability 4

46 reliability in the standard normal space transformation from standard normal to physical space: limit state function in the standard normal space linear limit state functions stay linear: as expected, reliability in standard normal space and in the physical space are equivalent: probability of failure : 273: Urban Systems Modeling Lec. 3 component reliability 46

47 design point in the standard normal space design point design point: the most dangerous condition: arg max in standard normal space: Failure Domain exp 2 log 2 arg min the design point is the point in the failure domain closest to the origin. If : arg min origin in the safe domain, i.e. low probability of failure. design point: it belongs to the failure domain (it is on the edge safe/failure); it has a high probability (the highest in the failure domain); it can be found by solving a constrained optimization problem; for linear limit state functions, the solution is very simple. 273: Urban Systems Modeling Lec. 3 component reliability 47

48 design point in the standard normal space [cont.] limit state function design point conditions to find design point vector of norm coordinates of the design point 273: Urban Systems Modeling Lec. 3 component reliability 48

49 reliability using design point, in stand. normal space design point coordinates of the design point reliability index Φ [check that this is consistent with previous result] once you have found the design point, you can measure how far the failure domain is from the origin, and compute the probability of failure. Summary: Transform your belief in the standard normal space, and define the new limit that function. Find the design point, measure how far is it from the origin to get the reliability index. 273: Urban Systems Modeling Lec. 3 component reliability 49

50 design point in the physical space [normal variables] 4 3 standard normal space 2 =, 2 =, = 2, 2 =., 2 =.6 2 u 2 x 2 = r u arg max physical space 2 x = s arg max The design point is the most dangerous scenario: it is a (incipient) failure condition, and it is the scenario with highest probability in the failure domain. If the map is linear, the design point in the physical coordinates is. If it is not linear, previous equation can be only approximate. 273: Urban Systems Modeling Lec. 3 component reliability

51 reliability index it gives the order of magnitude of the probability of failure: Φ Φ example, problem: Φ Φ [actually, if the set is an hyper plane, and is regular, continuous] IF the limit state function is linear in standard normal space, than the reliability index is the distance between the origin and the design point. IF NOT, it is not necessary. ;, 2 P f : Urban Systems Modeling Lec. 3 component reliability %.6% % 3.3%

52 log normal multivariate distribution log scale ;, exp log linear scale ln;, Jacobian determinant of the Jacobian exp exp exp log normal density log ;, 2 exp 2 log log ln;, 273: Urban Systems Modeling Lec. 3 component reliability 2

53 moments to parameters for log norm. mult. distr. ln;, ;, parameters: log linear scale log scale moments of : relations: ln for small ln as for d ln for small ln for small 273: Urban Systems Modeling Lec. 3 component reliability 3

54 properties of log normal multivariate distribution =, 2 =., =., 2 =.8, xx2 =.6 =, 2 =., =., 2 =.8, xx2 =.6 p(z,z 2 ) z z z completely defined by mean vector and covariance matrix; marginal and conditional distributions of any subset of rand. vars. are lognormal; product functions are jointly lognormal; uncorrelation implies independence. z 273: Urban Systems Modeling Lec. 3 component reliability 4

55 r s problem with log normal rand. var.s load and resistance: ln;, consider the load to resistance ratio: safe condition if log failure if log log log log log limit state function: log log log safe condition failure in the logarithm scale, the formulation is equivalent to that with normal rand. vars. log ;, log log 273: Urban Systems Modeling Lec. 3 component reliability

56 reliability problem with log normal rand. var.s ln;, limit state function log vector of random variables log log we can re shape the problem as that of a linear limit state function on a jointly normal distributed variable: example: failure failure with, > log log ;, log : Urban Systems Modeling Lec. 3 component reliability 6

57 PART IV general approach and FORM 273: Urban Systems Modeling Lec. 3 component reliability 7

58 general reliability problem given 8 6 joint probability limit state function compute When many random variables are involved in the problem (high dimensional space), it is expensive to compute the integral. x 2 4 No analytical solutions are generally available x The integral can be solved numerically, by counting along a grid (but that method is ineffective because of the course of dimensionality). Approximate solutions are provided by reliability methods (FORM: first order reliability method). Or by simulations (Monte Carlo). For reliability methods and for simulations it is convenient to formulate the problem in the standard normal space. 273: Urban Systems Modeling Lec. 3 component reliability 8

59 general reliability problem in stand. norm. space given joint probability limit state function compute x 2 u x u find transformation to the standard normal space: : ;, 273: Urban Systems Modeling Lec. 3 component reliability 9

60 going to the standard normal space It can be easily done for any jointly normal distribution (e.g. using Cholesky). Also for any jointly log normal distribution (taking the log, and using Cholesky). It can also be done for any distribution (Rosenblatt transformation), but it may complicate. u why we prefer this space: once here, distribution is very simple u variables are uniform, in the same scale, uncorrelated. you can easily generate samples from the distribution, you can approximate the solution finding the design point (FORM) 273: Urban Systems Modeling Lec. 3 component reliability 6

61 First Order Reliability Method (FORM) find design point: in standard normal space design point arg max arg min compute approximate reliability index: Φ approximate limit state (linear approximation at the design point) 273: Urban Systems Modeling Lec. 3 component reliability 6

62 FORM in d p(u), g(u) standard normal var.: non linear limit state function: is in the safe domain design point: : approximation: Φ safe failure u find design point find zero Newton Raphson method start at approximate (Taylor) derivative (gradient in higher dim.) repeat until convergence 273: Urban Systems Modeling Lec. 3 component reliability 62

63 FORM in d [cont.] p(u), g(u) standard normal var.: non linear limit state function: is in the safe domain design point: : approximation: Φ safe failure u find design point find zero Newton Raphson method start at approximate (Taylor) derivative (gradient in higher dim.) repeat until convergence 273: Urban Systems Modeling Lec. 3 component reliability 63

64 FORM for more than variable probability limit state function ;, [generally non linear] [standard normal space] linear approximation around [Taylor] gradient design point: distant from the origin: 273: Urban Systems Modeling Lec. 3 component reliability 64

65 FORM iterative method select repeat, from compute until convergence at set Φ 273: Urban Systems Modeling Lec. 3 component reliability 6

66 example of FORM 2 2 g(u,u 2 ) - -2 g(u,u 2 ) u u -3 u u 273: Urban Systems Modeling Lec. 3 component reliability 66

67 example of FORM 2 2 g(u,u 2 ) - -2 g(u,u 2 ) u u -3 u u 273: Urban Systems Modeling Lec. 3 component reliability 67

68 example of FORM u # iter Monte Carlo: 4. % u 273: Urban Systems Modeling Lec. 3 component reliability 68

69 gradient following a transformation invertible map : inverse map: limit state function proof Suppose we have the Jacobian, and the gradient, we can compute the gradient as chain rule, multiplying Jacobian and gradient. 273: Urban Systems Modeling Lec. 3 component reliability 69

70 Jacobian of a composed map invertible maps : : proof Suppose we have the Jacobian, and that for, we can compute the Jacobian as chain rule, multiplying the two matrices. 273: Urban Systems Modeling Lec. 3 component reliability 7

71 example of reliability problem by FORM ln;, failure normal space exp ;,.4.2 7% 7% 2% limit state function 3 exp exp exp exp exp standard normal space ;, 273: Urban Systems Modeling Lec. 3 component reliability 7

72 example of reliability problem by FORM limit state function is not linear in the standard normal space (from integration and Monte Carlo).97% 2 design point u g(u,u2) u - u u2 273: Urban Systems Modeling Lec. 3 component reliability 72

73 reliability problem by FORM: iterative scheme limit state function is not linear in the standard normal space.97% (from integration and Monte Carlo) g(u,u 2 ) 2 - u k = u - u u.64 % 9 273: Urban Systems Modeling Lec. 3 component reliability 73

74 reliability problem by FORM: iterative scheme limit state function is not linear in the standard normal space.97% (from integration and Monte Carlo) g(u,u 2 ) 2 - u k = u - u u : Urban Systems Modeling Lec. 3 component reliability 74

75 reliability problem by FORM: iterative scheme limit state function is not linear in the standard normal space.97% (from integration and Monte Carlo) g(u,u 2 ) 2 - u k = u - u u % 273: Urban Systems Modeling Lec. 3 component reliability 7

76 reliability problem by FORM: iterative scheme limit state function is not linear in the standard normal space.97% (from integration and Monte Carlo) g(u,u 2 ) 2 - u k = u - u u %.4% 273: Urban Systems Modeling Lec. 3 component reliability 76

77 reliability problem by FORM: iterative scheme limit state function is not linear in the standard normal space.97% (from integration and Monte Carlo) g(u,u 2 ) 2 - u k = u - u u %.48% 273: Urban Systems Modeling Lec. 3 component reliability 77

78 reliability problem by FORM: iterative scheme limit state function is not linear in the standard normal space.97% (from integration and Monte Carlo) g(u,u 2 ) 2 - u k = u - u u %.% 273: Urban Systems Modeling Lec. 3 component reliability 78

79 reliability problem by FORM: iterative scheme limit state function is not linear in the standard normal space.97% (from integration and Monte Carlo) g(u,u 2 ) 2 - u k = x u - u u %.2% 273: Urban Systems Modeling Lec. 3 component reliability 79

80 reliability problem by FORM: iterative scheme limit state function is not linear in the standard normal space.97% (from integration and Monte Carlo) g(u,u 2 ) 2 - u k = x u - u u %.2% compare with Monte Carlo result:.97% 273: Urban Systems Modeling Lec. 3 component reliability 8

81 FORM importance measures in the stand. norm. space importance measure: : is a load : is a capacity : irrelevant is a capacity is more important that linearized limit state function: design point: direction of the design point: 273: Urban Systems Modeling Lec. 3 component reliability 8

82 FORM importance measures in the stand. norm. space importance measure: : is a load : is a capacity : irrelevant is a capacity is more important that linearized limit state function: direction of the design point: stand. norm. space: ;, importance measure gives the importance of variable in the problem. 273: Urban Systems Modeling Lec. 3 component reliability 82

83 FORM importance measures in physical space linearized map linearized inverse map linearized limit state function constants stand. dev. of : contribution to the uncertainty (variance) of : we assume correlation matrix, because we are not interested in the correlation, to define the importance singular variables. normalized vector gives the importance measures of variables in. 273: Urban Systems Modeling Lec. 3 component reliability 83

84 example of importance measures in physical space design point: Standard normal space Normal space Physical space % exp exp exp 2 S % % 273: Urban Systems Modeling Lec. 3 component reliability 84

85 PART V further remarks on component reliability 273: Urban Systems Modeling Lec. 3 component reliability 8

86 general transformation to the standard normal space w p w (w) u x = F x (w) w = F x (x).4.2 x a) every rand. var. distributed by:, can be transformed into a uniform rand. var. by transformation. ;, x. p x (x) 273: Urban Systems Modeling Lec. 3 component reliability 86

87 w general transformation to the standard normal space a) every rand. var. distributed by:, can be transformed into a uniform rand. var. by transformation. ;, b) uniform rand. var. can be transformed into rand. var., distributed by,, through transformation u, x hence can be derived by through. In particular, can be mapped into the standard normal distribution by transformation Φ. p w (w) u u = - (w) w = (u) x = F x - (w) w = F x (x) x p u (u), p x (x) multivariate case: independent rand. vars. : Φ dependent rand. vars. Φ Φ Φ Rosenblatt transform. 273: Urban Systems Modeling Lec. 3 component reliability 87

88 properties of limit state function, to use FORM only the sign of the limit state function is relevant. Functions and, so that sign sign, are equivalent. the reliability does not depend on the slope of the gradient, or on the magnitude of : the reliability problem is defined by boundary: [and sign] a linear limit state function is convenient, because local data, at any point define the all boundary. p(u), g(u) u to use FORM (Newton method), we require function to be continuous and differentiable:. 273: Urban Systems Modeling Lec. 3 component reliability 88

89 convergence of Newton Rapshon method The method may not converge in some conditions: To overcome this problem, one may pose a maximum size in the steps taken by the algorithm:. see Wolfe Conditions and Armijo rule: onditions p(u), g(u) u 273: Urban Systems Modeling Lec. 3 component reliability 89

90 SORM Second Order Reliability Method: it approximates the limit state function with a quadratic form around the design point. 2 design point limit state function is approximated taking curvature into account Hessian matrix It is more accurate than FORM, but computationally more expensive because it requires to obtain curvatures ( ) at the design point. It is a further step after FORM: find the design point compute curvature at. 273: Urban Systems Modeling Lec. 3 component reliability 9

91 bounds for FORM and SORM By definition, the design point is the closest to the origin. Hence the all region belongs to the safe domain. Let us define. FORM: upper bound to the probability of failure: P χ Cumulative Chi squared distribution, with degrees of freedom. design point χ [no lower bound] P f FORM upper bound * 273: Urban Systems Modeling Lec. 3 component reliability 9

92 note about design point and reliability index definition of reliability index: Φ Φ distance of the design point from the origin: FORM approximation: design point arg max arg max design point for a non linear map, the design point in the stand. norm. space is not necessary mapped into the max. of the physical space in the failure domain. analogy: the mode of the (e.g. uni variate) normal distr. is not mapped into the mode of the log normal. 273: Urban Systems Modeling Lec. 3 component reliability 92

93 refereces on wikipedia: Cholesky decomposition Eigenvalues and eigenvectors Gradient Jacobian Positive definite matrix Multivariate normal distribution Newton's method Chi squared distribution Wolfe Condition Chain rule Barber, B. (22). Bayesian Reasoning and Machine Learning. Cambridge UP. Downloadable from Section 8.4 on Multivariate Gaussian. Der Kiureghian, A. (2) "First and Second Order Reliability Methods", in book: E. Nikolaidis, D.M. Ghiocel, S. Singhal (Eds), The Engineering design reliability handbook, CRC Press LLC. Ditlevsen, O. and H.O. Madsen. (996). Structural reliability methods. J. Wiley & Sons, New York, NY. Downloadable from HOM StrucRelMeth Ed2.3.7 June September.pdf. Sections 2. 3, 4. 2,. Faber, M. (29) Risk and Safety in Engineering, lecture notes, Lectures 6, available at Sørensen, J.D. (24) "Notes in Structural Reliability Theory And Risk Analysis", notes 3, avail. at eling_waterbouwkunde/sectie_waterbouwkunde/people/personal/gelder/publications/citatio ns/doc/citatie2.pdf 273: Urban Systems Modeling Lec. 3 component reliability 93

94 MW Matlab commands M=zeros(n,m) : it defines matrix M, of size (nm), with all entries zero. length(v) : number of entries in vector v. M=diag(v) : it makes diagonal matrix M, putting vector v on the diagonal. L=chol(M,'lower') : compute lower triangular matrix L, from Cholesky decomp. of M. M.*M2: when matrices (or vectors) M and M2 have the same dimension, it makes matrices M3 as element by element product of M and M2. Similar allowed operations are: M./M2,./M, M.^2. 273: Urban Systems Modeling Lec. 3 component reliability 94