Guideline for Offshore Structural Reliability Analysis - General 3. RELIABILITY ANALYSIS 38

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1 FEBRUARY 20, RELIABILITY ANALYSIS General Variables Events Event Probability The Reliability Index The Design Point Transformation of Variables Analytical Methods The First Order Reliability Method, FORM The Crude First Order Reliability Method, CRUDE-FORM The Second Order Reliability Method, SORM Simulation Methods Monte-Carlo Simulation Directional Simulation Axis-Orthogonal Simulation Choice of Method FORM CRUDE-FORM SORM Monte-Carlo Simulation Directional Simulation Axis-Orthogonal Simulation Evaluation of Analysis Results General Interpretation of Analysis Analysis Models Analysis Methods Distribution Interpretation of Calculated Reliability Measures Fulfillment of Assumptions for Analysis Conditional Probability Parametric Sensitivity Factor The Uncertainty Importance and Omission Sensitivity Factors Time Dependent Reliability Analysis Introduction First Passage Failure Probability Random Process Simulation Outcrossing Rate Application Response Surface Methods Introduction Classical Response Surface Adaptive Response Surface 75 REFERENCES 76

2 3. Reliability Analysis 3.1 General The basic problem in Structural Reliability Analysis (SRA) may be formulated as the problem of calculating the small probability that 2 g( X ) < 0 (3. 1) where X is a vector of basic variables (cfr. Section 3.1.2). g( X ) is referred to as the limit state function. In most cases in SRA, g( X ) may be split into a load term l and a strength term s such that g( X) = s( X) l( X) (3. 2) and failure is defined by the event of loads exceeding the strength, implying g( X ) < 0, which is the convention for definition of failure. The failure event may be a combination of several events Variables A stochastic model is defined through variables. The variables describe functional relationships in the physical model and the randomness of parameters in the model. A parameter of a variable may be a function of coordinates of other variables so that a network structure for dependencies between variables can be defined. Additional statistical dependence between the variables can be modeled through correlations. A variable X may be a vector of dimension n and coordinates X, i i = 1,..., n. A variable in a probabilistic model can be assigned: a numeric constant a probability distribution a function Numeric Constant The variable is a constant value, for example 0.5. A variable with this type attribute may be assigned to parameters of every other variable. It permits sensitivity factor calculations and parameter studies for parameters which enters the physical model more than one place. Probability Distribution The variable is assigned a probability distribution (e.g., the exponential distribution, the normal distribution etc.). The distribution describes the randomness of a parameter in the physical model. For further details see Chapter 4. A variable which has this type attribute may be assigned to parameters of any other variable.

3 3 Function The variable can be a user defined function of other variables (numeric constants, distributions, etc.). More advanced calculations may need variables to be defined as event probabilities (in nested reliability analysis) or generated distributions. See Tvedt (1993) for more details Events An event, E( X ), is a subset of the sample space for the stochastic process involved, i.e., a subset of all the possible outcomes of the stochastic process. An event may be defined through a functional relationship { g } E( X) = x; ( x) 0 (3. 3) The event identifies the outcomes of interest while the random variables X defines the nature of the stochastic process. The following types of events are described single event union of events (series system) intersection of events (parallel system) conditional event Single Event The purpose of the single event is to define a functional relationship which identifies a subdomain of the state space. The single event E SE ( X ) is defined by { } ESE( X) = x; g( x) θ (3. 4) where g( x ) is a function in the functions library and θ is the threshold value. The event is an inequality event which identifies a volume in the n-dimensional x-space. Alternatively the event is defined as { g } ESE( X) = x; ( x) =θ (3. 5) The event is an equality event which identifies an (n 1)-dimensional surface in the n- dimensional x-space. To be meaningful the definition of an equality event requires the definition of a limit process. If a measurement variable is assigned to the event, the limit process is on the measured value. If not, the limit process is on θ, i.e., on the threshold value. The event function is gse ( x, θ) = g( x) θ (3. 6)

4 4 In structural reliability the function g SE ( x, θ ) is denoted as a limit state function. The domain of Eq. (3. 4), excluding the boundary, is denoted the failure domain. The boundary itself is denoted the failure boundary. The domain of E C SE ( X ) is the complement of E SE ( X ) and is denoted the safe domain. A simple single event function is the load-resistance function g(,)= r l r l shown in Figure l Failure Surface FAILURE DOMAIN r SAFE DOMAIN Figure 3. 1 Load-Resistance Limit State Function Union of Events The union of m events E i ( X ) is the collection of outcomes which are in at least one subevent E i ( X ). E U ( X) = E ( X) (3. 7) =1 i m i The subevent E i ( X ) may be a single event, a union of events or an intersection of events. In structural reliability the union of single events is used to model series systems. Intersection of Events The intersection of m events E i ( X ) is the collection of outcomes in the sample space which are common to all the subevents. A special case is the intersection of single events E ( X) = E ( X) (3. 8) =1 I i m i

5 E ISE ( X) = E ( X) (3. 9) =1 i m The significance of this event is that it is the only intersection system for which the probability can be calculated directly using FORM or SORM. In structural reliability the intersection of single events is used to model parallel systems. SE, i 5 Conditional Event The conditional event E C ( X ) is the collection of outcomes in E 1 ( X ) conditioned on the occurrence of event E 2 ( X ). EC ( X) = ( E1 E2 )( X) (3. 10) The subevent E 1 ( X ) or E 2 ( X ) is either a single event, a union of single events or an intersection of single events. The complexity of the subevent depends on the calculation method used. If the FORM is used, the intersection of E 1 ( X ) and E 2 ( X ) must be an intersection of single events. In structural reliability the conditional event is used to model inspection no-find situations (inequality events) and together with the measured value variable (involves equality events) to model inspection find situations (Section 3.6) Event Probability The event probability, P E, is the probability that an outcome of the stochastic process X yields the event E, PE = P( E( X )) (3. 11) The Reliability Index The reliability index β R is defined to be the argument of the standard normal distribution which yields one minus the event probability, i.e., = Φ ( 1 P ) = Φ ( P ) (3. 12) 1 1 β R E E see Madsen et al. (1986). It is customary to distinguish between the FORM reliability index, β FORM, where P E is obtained through a FORM approximation method, the SORM reliability index, β SORM, where P E is obtained through a SORM approximation method and the Generalized reliability index, β R, where P E is the exact event probability.

6 While the event probability is most often a rapidly changing nonlinear function of the distribution parameters, θ, the reliability index β R is most often a more "linear" function of θ. This implies that the parametric sensitivity factors of β R (Section 3.5.2) most often are predictive in a wider range of θ than those of P E. 6 The other reason for using the reliability index is historically motivated. In early structural engineering applications the safety was measured in terms of an index, Cornell (1969), Hasofer and Lind (1974), Ditlevsen (1979a), corresponding to β R for special cases The Design Point In the transformed space of standard normally distributed variables, see Section 3.1.6, the design point is defined as the point on the failure surface with the smallest distance from the origin. Reference is made to Figure The design point corresponds to the most probable combination of the stochastic variables causing a failure. u 2 SAFE SET FAILURE SET u * Design Point u 1 Figure 3. 2 The Design Point Transformation of Variables This basic problem may be transformed into an equivalent problem where the basic stochastic variables are transformed into a standard-normal-space, i.e. the space of decorrelated normally distributed variables with zero mean and unit standard deviation. This transformation (Rosenblatt,1952) is

7 7 1 u1 = Φ ( F( x1))... 1 ui = Φ ( F( xi x1, x2,..., xi 1))... 1 un = Φ ( F( xn x1, x2,..., xn 1)) (3. 13) where x = ( x1, x2,..., x j,..., xn) is the basic vector and u = ( u1, u2,..., ui,... un) is the transformed vector, Φ is the standard-normal distribution. It is customary to use the term x-space for the space of random basic variables and the term u- space for the space of independent standard normal random variables. The usefulness of this mapping is due to Rotational symmetry: The probability content of domains in the u-space is invariant with the rotation of the domains about the origin u = 0. Exact results: The probability content in u-space can be obtained exactly for the linear domain, the elliptic and the hyperbolic domains, Rice (1980) and Hellstrom (1983), the parabolic domain, Tvedt (1988) and the domain obtained through a second-order Taylor expansion of the event function, Tvedt (1990). Approximations: The probability content in u-space can be obtained approximately for a convex domain bounded by linear surfaces, i.e. a domain defined by a parallel system of single events with linear event functions. The method used is the approximation method for the cumulative multinormal distribution, Hohenbichler (1984), and Gollwitzer and Rackwitz (1988). If the intersection of the single event yields a parabolic surface in directions orthogonal to the subspace spanned by the gradients of the active constraints (see below), the probability content of the corresponding domain is approximated by a split of the linear parallel system and the parabolic domain, Hohenbichler (1984). Asymptotic justification: Breitung's theorem, Breitung (1984a), states that the domain limited by a parabolic approximation to the single event boundary obtained by fitting the curvatures at the design point (the approximation point closest to the origin), asymptotically, i.e., as β = u *, yields the true probability of the event. A similar result is valid for a small intersection parallel system, Hohenbichler (1984). The rapid decay of the standard normal distribution away from the origin implies that the asymptotic approximations are useful also for moderate β, say β 2.. It follows from this that, if an event can be approximated by one of the above mentioned domains, or a composition of such domains, then an approximation of the probability content of the domain can be derived. Simulation methods: The unbiased Directional simulation method, Bjerager (1988) and the Axis-Orthogonal importance sampling method, Schall et al. (1988), require a formulation in terms of u-space variables.

8 8 3.2 Analytical Methods The First Order Reliability Method, FORM The basic idea of FORM is to approximate the system event boundary by linear surfaces at a selected set of points. The computational methods available for linear and piecewise linear event are then applied to obtain an estimate (possibly in terms of bounds) of the event probability. The selection of linearisation points depends on the system configuration. Theoretical support for the FORM is offered by Breitung (1984a) and Hohenbichler (1984). Their theorems state that the system reliability index obtained using the FORM, asymptotically (i.e., as the norm of each design point u * tends to infinity) yields the true reliability index β FORM * β, u (3. 14) R under certain restrictions on the curvatures. Single Event The principles are illustrated in Figure 3. 3 for two stochastic variables. The boundary of the single event is linearised at the point of maximum likelihood in u-space, the design point u *. u 2 g( u ) = 0 g( u ) > 0 g( u ) < 0 FORM α = * g( u ) * g( u ) u * = βα β = u * u 1 Figure 3. 3 Linearisation of single event boundary in u-space The FORM reliability index is

9 β FORM The corresponding probability of failure is = u * sign( g( 0 )) (3. 15) 9 P FORM = Φ( β ) (3. 16) FORM Intersection (Parallel System) of Single Events The term small intersection denotes an intersection of single events where the event function is nonnegative at u = 0. The linearisation of small intersection is illustrated in Figure Three limit state functions gh( u1, u2), g i( u1, u2), g j( u1, u2) define three boundary curves for the failure domain (g( u ) = 0 ). u 2 g( u ) < 0 u h * u * α j g j ( u ) = 0 α i β j g i ( u ) = 0 β i u 1 β h α h g h ( u ) = 0 Figure 3. 4 Linearisation of small intersection in u-space The steps in the linearisation procedure are: Step 1, Active constraints: Step 2, Inactive single events: The first linearisation point is the point of maximum likelihood, the design point u *, on the event boundary of the event function. The single events which have zero valued functions at this point are denoted active. The single events which are not active, are denoted inactive. The inactive single events are linearised in succession, if possible. The linearisation point of an inactive single event is the point of maximum likelihood on the part of its event

10 boundary that intersects the event defined by the previous linearisations. If a single events boundary does not intersect this event, it is deleted from the further calculations. 10 The term large intersection denotes an intersection of single events where the event function is strictly negative at u = 0. The linearisation of small intersection is illustrated in Figure Three limit state functions gh( u1, u2), g i( u1, u2), g j( u1, u2) defines three boundary curves for the failure domain (g( u ) = 0 ). u 2 g h ( u ) = 0 g j ( u ) = 0 α i g i ( u ) = 0 u i * u h * α h β i β h β j u j * α j u 1 Figure 3. 5 Linearisation of large intersection in u-space The steps in the linearisation procedure are: Step 1, Active single events: Step 2, Inactive single events: The single events are first linearised at their separate design points. The single events that have design points on the event boundary of the linearised large intersection are denoted active. The single events which are not active, are denoted inactive. The inactive single events are linearised in succession, if possible. The linearisation point of an inactive single event is the point of maximum likelihood on the part of its event boundary that intersects the event defined by the previous linearisations. If a single events boundary does not intersect this event, it is deleted from the further calculations.

11 11 Denoting the linearisation of single event no. i u = β + a u (3. 17) T g i ( ) i i the linear surfaces defines a point, β, on the cumulative multivariate normal distribution, with T mean 0, standard deviation 1 and correlations R = ( ρ ) = ( α α ). Thus ij i j P FORM = Φ( β; R ) (3. 18) where the correlation matrix is denoted R. The corresponding reliability index is β FORM = Φ 1 ( P ) (3. 19) FORM Union (Series System) of Single Events Rather than computing the union of single events, one may choose to solve the complementary problem which is an intersection of single events. The relation is m m { 0} 1 { 0} P( u; g( u) ) = P( u; g( u ) ) (3. 20) i i= 1 i= 1 i The intersection of single events is computed using the methods described above. The second alternative is to use bounds for the failure probability. Each single event is linearised separately. An upper bound (P U ) and a lower bound (P L ) to the event probability of the linearised system is computed making use of simple bounds or Ditlevsen bounds. Simple bounds: First orders bounds using the individual event probabilities (P s i' ) P L n = = max Pi (3. 21) i 1 P U n = min Pi, 1 (3. 22) i= 1 Ditlevsen bounds: Second order bounds (Ditlevsen (1979b), Kounias (1968)) that also uses the intersection probabilities P = P( E E ) ij i j n i 1 PL = P1 + max 0, Pi Pij (3. 23) i= 2 j= 1

12 i 1 { i ij j= 1 } P = P + P max P (3. 24) U 1 n i= 2 12 The bounds on the reliability index are β FORM, L = ( P U ) β FORM, U = ( P L ) Φ 1 (3. 25) Φ 1 (3. 26) The Crude First Order Reliability Method, CRUDE-FORM The CRUDE-FORM method replaces the event functions by their separate FORM linearisations prior to the computation of the event probability. The original event functions are again used with the calculation of sensitivity factors. The motivation for using the CRUDE-FORM method is that a single event which is a member of more than one subsystem is linearised only once. The computational effort using the CRUDE- FORM is thus generally less than the computational effort using the FORM. If the system is a union of single events for which the bounds option is used or if the system is one single event, the FORM and the CRUDE-FORM gives identical results. If the system is an intersection of single events, the difference in the reliability estimates using the FORM and the CRUDE-FORM could be considerable The Second Order Reliability Method, SORM The basic idea of the Second Order Reliability Method, SORM, is to approximate a single event boundary in u-space by a quadratic surface and an intersection of single events by a linear/parabolic surface. In each case the quadratic approximation is made at the design point. The computational methods available for quadratic forms and for linear and piecewise linear are then applied to obtain an estimate of the event probability (possibly in terms of bounds). The SORM for unions makes use of the SORM for single events and intersections of single events. The SORM extends significantly the class of problems that can be treated by approximation methods because even highly curved event boundaries can be well represented by such methods. Theoretical supports for the SORM are offered, in the case of a single event and a union of single events, by Breitung (1984a), and in the case of intersections of single events and unions of intersections of single events, by Hohenbichler (1984). Their theorems state that the event probability obtained using the SORM, asymptotically, i.e. as the norm of each design point u * tends to infinity, under certain restrictions on the curvatures, yields the true event probability. Below, only the essential differences from the FORM are included.

13 13 Single Event Three methods for approximating the boundary of the single event are: Parabolic approximation of the main curvatures of the boundary at u *. The exact result for the probability of the parabolic domain is available, Tvedt (1988) and (1990), together with an asymptotic result by Breitung (1984a). Approximation derived from the second order Taylor expansion of the event function at u *. The event is either a parabolic quadratic form, an elliptic quadratic form or a hyperbolic quadratic form, Fiessler et al. (1979). The exact result for the quadratic form is available, Rice (1980), Hellstrom (1983), Tvedt (1990). Parabolic approximation derived from the diagonal of G( u * * ). G( u) is the matrix of the second derivatives of the event function evaluated at the design point u *. The SORM event probability is computed using the exact results for the parabolic quadratic form, Tvedt (1989). The approximation is a modified version of a suggestion by Der Kiureghian et al. (1987). Intersection (Parallel System) of Single Events Small Intersection: The SORM approximation is here used to derive a correction factor to the FORM probability of failure, Hohenbichler (1984). The intersection of the active constraints describes a nonlinear surface. This surface is approximated by a parabolic surface. The SORM probability of failure is in this case an asymptotic result for the parabolic/linear failure domain P SORM = P FORM QSORM Φ( β) (3. 27) where Q SORM is the separate probability of the parabolic quadratic form and β is the design point reliability index. Large Intersection: The single events are linearised separately. The linearised version of a single event is as in FORM, however, in SORM the distance parameter is scaled to account for the second order probability. 3.3 Simulation Methods Monte-Carlo Simulation The Monte-Carlo simulation method samples from the joint distribution of the n random basic variables X. The indicator function I( x)

14 14 Ix ( ) 1for g( x) 0 = 0 for g( x) > 0 (3. 28) is evaluated at each sampled point and the probability is estimated as N 1 PE = Ix ( i) N i= 1 (3. 29) An advantage of the method is that it makes use of point values of the event function only. Thus, the event function is not required to be a smooth function of its variables. Also, the estimate on the event probability is unbiased. The disadvantage is the long computational time for small probabilities (see Section 3.4.4). An illustration of the Monte-Carlo method is given in Figure x j g( x ) > 0 g( x ) < x i Figure 3. 6 The Monte-Carlo Method Directional Simulation The idea of directional simulation was put forward by Deak (1980) who used it to sample the multinormal cumulative probability function. The method is a conditional expectation simulation method that samples directions uniformly distributed on the surface of the n-dimensional unit sphere centered at the origin of the u-space, Figure 3. 7.

15 15 u j g( u ) < 0 g( u ) > 0 r χ q ( r ) f A ( a) a u i Ω q Figure 3. 7 Directional Simulation in u-space The method assembles the contributions to the probability integral conditioned on the sampled directions. A number of variance reducing methods, i.e., sampling of random orthogonal sets of directions were proposed. Application of the method to general systems with nonlinear event boundaries was suggested by Bjerager (1988). The motivation for the method is that in many cases it allows unbiased and efficient sampling of small probabilities, provided that n is not too large. The probability of the event can be formulated as P = f () v dvdω (3. 30) E χ n 2 gva ( ) 0 where v is a χ 2 n -distributed random variable, a is a unit vector and dω is the surface element of the n-dimensional unit sphere. The unit directions a are sampled and the function χ 2 n ( v) conditioned on a is integrated. This may be done by identifying the upper bounds v( u i ) and the lower bounds v( l i ) of the intervals where g( va ) < 0 and to sum up the contributions to the integral

16 m { 2 n i χ 2 n i } P() a = χ (( v u)) (()) v l i= 1 16 (3. 31) assuming there are m such intervals. The estimator for the probability is: N 1 PE = P( a i) (3. 32) N i= 1 where a i, i = 1,.., N are the simulated directions. The simplest method that reduces the variance of the sampled set is to replace P( a ) by 1 P1 ( a) = ( P( a) P( a) ) (3. 33) 2 A further reduction of the variance of the sampled set is obtained through sampling of a randomly oriented orthonormal system of vectors b1,.., bn. A new system of vectors is formed as n! follows: To each set { bi 1, bi2,.., bik} of k out of n vectors, there are m = such sets, the k!( n k)! following 2 k vectors, ai ( b), are formed with s =±, j = i, i,..., i. One point is sampled as j k a ( b ) = 1 i ( bi bi kbik) k s + s + + s (3. 34) O = 1 m m P( a( b )) (3. 35) { ss ; j =±, j=,,..., k} k k i 2 i= The estimator of the probability based on sampling of N orthonormal systems of vectors, is now N Ek, = Ok, j N j= 1 P 1 (3. 36) Taking advantage of symmetry, the number of actual calculations is half the amount indicated by the above methods. Thus r = n! k k!( n k)! 2 1 (3. 37)

17 The number of evaluations of the event function is thus approximately rn a, where n a is the average number of event function calls needed to perform the summation of Eq. (3. 31). The following estimators are recommended dependent on the number of stochastic variables n: 17 n 2 or 16 n 50 Use P E 1 ( a), n= 3 or 11 n 15 Use P E 2 ( a ) (3. 38) 4 n 10 Use P E, 3 ( a) n > 50 Use P 1 ( a), Axis-Orthogonal Simulation Importance sampling around the design point has been suggested by Shinozuka (1983) and also by Harbitz (1983). The recommended method is, however, based on the ideas of Hohenbichler and Rackwitz (1988) and Schall et al. (1988). The idea is to define an axis for a small intersection domain, i.e., an intersection of single events where the u-space origin is in the complementary event, and to define the sampling density in a plane orthogonal to this axis. The method assumes that the true boundary of the event is suitably approximated by a set of linear surfaces obtained using the FORM linearisation procedure for a small intersection domain. The idea is to calculate the probability of the linearised domain by methods available for the cumulative multinormal integral and to obtain an estimate on the probability of the difference of the true event and the linearised event using the Axis-Orthogonal sampling method. The principles are illustrated in Figure Φ( u n ) g( u ) < 0 g( u ) > 0 u * u n v i v

18 Figure 3. 8 Axis Orthogonal Simulation in u-space 18 Selection of Axis Let a i, i = 12,,..., n be the set of normalised gradients of the single events active at the design point of the parallel system. The axis is the direction of the averaged gradient, pointing into the interior of the linearised event at the design point. a a i = = i 1 i i= 1 a a i i (3. 39) The coordinate system is rotated so that the new coordinate u n is in the direction of the axis while u ~ = ( u 1, u 2,..., u n 1 ) is the coordinates normal to u n. The axis is now defined as a= u0 + u n e n (3. 40) where e n is a unit vector in the direction of u n. Sampling of a Multiplicative Correction Factor The sampling density, H M, is in this case as follows: a) In the space spanned by the gradients of the m single events active at the design point, the m-dimensional standard normal density, conditioned on the linearized event, is used, Bjerager (1988). b) In the directions normal to this space, and normal to the axis, n m 1 independent standard normal random variables centered on the axis are sampled from. Thus P E = P L g( u) P ( u ~ ) H du~ = PC A M L ( un( ~ (3. 41) 0 Φ u) ) where P L is the probability of the linearised domain, u n ( u ~ ) is the intersection of the linearised domain and the line l = u ~ + un e n (3. 42)

19 19 P A ( u ~ ) is the true probability of the event conditioned on the line defined by l. The estimator of C is N 1 P ( ~ A ui) C = N Φ( u ( u ~ )) i= 1 n i (3. 43) Sampling of an Additive Correction The density function sampled from is in this case the (n 1)-dimensional standard normal density function φ( ~ ν ) with u ~ = u ~ + ~ 0 ν, i.e., with center on the axis a. The linearised domain is bounded by the linear approximations of the single events active at the design point and the linear T approximations of the constraints having a positive a i a. The difference of the probability of the true event and the probability of the linear approximation of the event is estimated. Thus PE = PL + { PA( ~ ) ( un( ~ φ( u } ~ ) u Φ u)) d PL + A ( ~ u~ = (3. 44) φν) g( u) 0 with u ~ = u ~ + ~ 0 ν, and the other symbols explained above. The estimator of A is: N 1 ( ~ ) ( ( ~ φ( ~ ) A = PA un )) N u u ( ~ ui i Φ i φ ν ) i= 1 i (3. 45) Computation of P A ( u ~ ) The line integral P ( u ~ ) = φ ( u ) du ( u ~, ) 0 A n n g u n (3. 46) is evaluated once for each simulation of ~ u. Because this is the time-consuming part of the calculations, a number of options are made available to reduce the effort in the simple cases. The general integration method is to search for the points un( il ) and un( iu) denoting the lower and upper bounds of the intervals where g( u ~, u n ) is less than zero, and then sum up the contributions using the formula m { Φ } PA( u ~ ) = Φ( un( iu)) ( un( il )) i= 1 (3. 47) assuming there are m such intervals.

20 Choice of Method The applicability of a particular method depends, as in all mathematical modelling, on the problem at hand and on the objective of the analysis. In the following some important characteristics of the methods are tabulated to assist the user to choose the method appropriate to the problem at hand. The following issues are important to the selection of a method: The objective of the analysis. The number of random variables involved. The computational cost of evaluating the event (e.g. limit state) function. The properties of the event function, e.g. existence, continuity and differentiability. The reliability level of interest FORM The motivations for using the FORM are: Almost linear event boundary. It is often found that the distributions used to model random basic variables are narrow, i.e., the coefficients of variation have small values. In such cases, the event boundary in u-space is often significantly less curved than the event boundary in x-space. Because of this, the desired results can be obtained by the FORM with a quality sufficient for the intended application. Comparison of probabilistic models. An interesting application of probabilistic methods is to compare probabilistic models under variation of the model assumptions. In such cases the accuracy required is one that permits comparison rather than exact results. Computational efficiency. In many applications in engineering, the calculation of a variable may require computationally costly subroutines, e.g., the value of the variable may require use of the finite element method. In such cases, the use of other methods, e.g., simulation methods, may be computationally prohibitive for small failure probabilities. Systems: Event function: Results: The method is applicable to single events, union and intersections of single events and union of intersections of single events. In principle the event function must be continuously differentiable. However, in practical applications it may be sufficient that this is the case near the design point. Probability, parametric sensitivity factors and uncertainty importance factors.

21 Applicability: The method is particularly useful for the high reliability problems often encountered in engineering. The method assumes that each single event boundary is linear or nearly linear in the neighborhood of the approximation points. The method is also quite useful if qualitative statements about the model are sufficient, e.g., if the results are used to compare models. 21 Check: Computation cost: If the applicability of the FORM to a specific model is uncertain, the FORM results should be compared for selected cases with the results obtained using simulation or SORM. In many cases a comparison with SORM results suffices. The cost of a FORM calculation is essentially the cost of solving a number of optimisation problems, one for each subsystem in the model. Assume that the system is composed of m subsystems, that subsystem no. i is composed of k i single events and that single event no. j of subsystem no. i to have n ij random variables. If solving the optimisation problem of subsystem no. i requires r i iterations, then the computational effort is of k i m i ij + i= 1 j= 1 the order r ( n 1). This implies that the solution time is not a function of the probability. It is linear in number of stochastic variables and events. n ij +1 refers to the calculation of gradients g( x ). In some cases gradients may be derived analytically or by efficient numerical algorithms (e.g. Adjoint method in FEM). In such cases FORM will be particularly efficient CRUDE-FORM The CRUDE-FORM linearises each single event boundary separately prior to the probability calculation. Then the FORM is applied to the linearised system. The applicability of the method is as follows: Systems: Event function: Results: Applicability: The method is applicable to single events, union and intersections of single events and union of intersections of single events. In principle the event function must be continuously differentiable. However, in practical applications it may be sufficient that this is the case near the design point only. Probability, parametric sensitivity factors and uncertainty importance factors. The method is identical to the FORM for single event and union of single events systems (if bounds are used). For intersection of single events and union of intersections systems, the method assumes that the boundaries of the event are well approximated by the independent linearisation procedure for the single events. Thus, care should be exercised when using this method. Note that the computational cost is in the order of a FORM computation.

22 22 Check: Computation cost: If the applicability of the CRUDE-FORM to a specific model is uncertain, the CRUDE-FORM results should be compared with the results obtained using FORM. The cost of a CRUDE-FORM calculation is essentially the cost of computing the approximation point of each single event separately. Assume there are m single events, that single event no. i has n i random variables and that the computation of the approximation point needs r i iterations. Then the computational effort is of the order ri( ni + 1) m i= SORM In the case of a single event, the SORM method fits a quadratic surface to the boundary of the event. In case the system comprises more than one single event, the parabolic SORM is used. The SORM is used to compute a correction factor to the FORM. The applicability of the method is as follows: Systems: Event function: Results: Applicability: Check: Computation cost: The method is applicable to single events, union and intersections of single events and union of intersections of single events. In principle the event function must be two times continuously differentiable. However, in practical applications it may be sufficient that this is the case near the design point. Probability, parametric sensitivity factors and uncertainty importance factors. The method assumes that the boundary of the event is closely approximated by a quadratic surface in the neighbourhood of the approximation point. The invariance property of the parabolic quadratic form makes it generally applicable. The diagonal fit is generally applicable if the event function can be written as g( x ) = h( x ) and the random variables X i are mutually independent. It is also applicable if the curvatures of a full parabolic approximation have small values. The lack of invariance implies that care has to be exercised whenever the second order Taylor expansion is used. If the applicability of the SORM to a specific model is uncertain, the SORM results should for selected cases be compared with the results obtained using simulation. The cost of the SORM is the cost of the FORM plus the cost of deriving the quadratic approximation. If the event function has n random variables, the number of extra computations is: n i= 1 i i

23 23 Parabolic: Second order Taylor: Diagonal: n ( n 1) 2 n ( n + 1) (3. 48) 2 2n If n is large, say 100, the simulation methods become competitive also for small probabilities, say For small intersection: If there are k active single events in the intersection system, the number of computations of the event functions to establish the parabolic approximation is k ( n k )( n k 1). For large intersection: number of single events 2 single event SORM Monte-Carlo Simulation The Monte Carlo simulation method samples from the joint distribution of the random basic variables. Systems: Event function: Results: Applicability: Computation cost: All systems except systems involving equality events. The calculation of the event function should be possible at all valid realizations of the basic variables, i.e. all the points where the joint density distribution has a positive value. If the event function cannot be calculated at some point x, then the point contributes zero to the probability. Probability. (Sensitivities may be calculated at extra costs.) The Monte Carlo simulation method gives an unbiased estimate of the probability. The method is efficient for the simulation of the central part of the distribution, i.e., probabilities in the range from 0.1 to 0.9, say. The number of points (evaluations of the g-function) required to obtain a reasonable reliable estimate of the probability P E is approximately 100/P E Directional Simulation The directional simulation method samples directions uniformly distributed on the surface of the n-dimensional unit sphere in u-space and accumulates a weighted sum of the event probabilities conditional on the simulated directions. Systems: Event function: All systems. The event function should be calculable at each point where the density distribution of the random basic variables X is positive.

24 24 Results: Applicability: Computation cost: Probability and parametric sensitivity factors. The directional simulation method gives an unbiased estimate of the probability. The method is recommended for the simulation of probabilities below 0.1 and above 0.9. The computational effort is proportional to the proportion of the area of the surface of the n-dimensional unit sphere that generates directions intersecting the event. As this proportion tends to be reduced with increasing number of random variables, the Monte Carlo method becomes competitive if the number of random variables is large. The Monte Carlo simulation method should be used for large n, n 100, say Axis-Orthogonal Simulation The Axis-Orthogonal simulation is an importance sampling method that samples a correction to a FORM approximation of a single event or an intersection of single events. The number of samples to obtain the correction is small if the intersection system is closely approximated by the FORM. Systems: Event function: Results: Applicability: Computation cost: Single events and intersection of single events (small intersection only). The event functions should be calculable at each point u. If the event function cannot be calculated at some point u, it is assigned the value zero at u. Probability, reliability index. The Axis-Orthogonal simulation method gives an unbiased estimate of the probability in the case there is one local design point only. The method is recommended for the simulation of probabilities smaller than 0.1 and greater than 0.9. The computational effort depends on how well the event boundary is approximated by a (set) of linear surfaces. If well approximated, the number of samples may be down to Evaluation of Analysis Results General The basic results of a probabilistic analysis are a set of design point values of the random values the probability of an event the conditional probability of an event

25 parametric sensitivity factors uncertainty importance factors. Results which are specific for time dependent reliability analysis, i.e., crossing rate and crossing probability, are described in Section 3.6. The event probability and the reliability index are described in Sections and Interpretation of Analysis Reliability analysis should be performed and interpreted with a critical mind. The analysis results should be interpreted critically as the results and corresponding conclusions will significantly depend on analysis model and associated bias distribution types target reliability level An essential result of a structural reliability analysis is the reliability measure. The measure may depend on a number of assumptions adopted by the analyst. The reliability measures shall be interpreted as an engineering factor expressing the current knowledge/information about the structure and its environment under the types assumptions set forth in the analysis (for example assumptions regarding probability distribution types and the structural models adopted). The reliability measure introduces thereby an ordering of the considered structures with respect to reliability that can be used in comparative studies. The reliability measure is often taken as the reliability index defined as a monotone function of the failure probability. There are several reasons for using the reliability index instead of probability directly; the index has a historical significance, it has a clear geometrical interpretation in a number of basic cases, it is an essential component in first order reliability methods and it has certain simple relationships to common partial safety factors. Additional results which may be obtained from the reliability analysis are: sensitivity factors (parameter sensitivities, importance factors) estimation point (most likely failure point). As a guideline for a critical interpretation of analysis the following items are listed: It is important that the selected analysis model reflects the actual failure mode in an appropriate way. The selected distributions should be representative for the considered analysis. It should be assured that assumptions made for reliability analysis of problems in design and in-service life are fulfilled through performance of, e.g., fabrication or in-service inspection. The analysis method should give correct results based on the limit state function and the distributions given as input to the analysis. Calculated probabilities of failure are nominal values. The use of relative values from reliability analysis are preferred above that of absolute values Analysis Models

26 26 Analysis models that reflect the actual failure mode should be selected. Wherever possible, bias and model uncertainty should be estimated and included in the analysis models. Bias and model uncertainty may for example be obtained from comparison of experimental and theoretical values. Bias and model uncertainty can be represented by a model uncertainty factor defined as the ratio between experimental (true) values and theoretical (predicted) values. Bias is defined as the expected estimation error and will therefore be reflected in terms of a non-unity mean of the model uncertainty factor. Variability in the model will be reflected in terms of the standard deviation of the model uncertainty factor Analysis Methods It should be checked that a solution of the limit state function is achieved with sufficient accuracy. This may be checked using the data for the estimation point calculated. For special problems (e.g., if the limit state function is not monotonic in all basic variables, or if one or more distributions are bimodal) it should be checked that a right estimation point is obtained. This may be checked using alternative starting points for the optimisation algorithm for finding the most likely failure point. For problems with limit state surfaces possessing several local minima (of similar magnitude) special care is needed in estimating P F, and FORM/SORM results should be verified by simulation, e.g., by directional simulation. FORM/SORM results are generally highly accurate for practical purposes: i.e., for the small range probabilities, say P F < 10 3, and smooth limit state surface typical of most problems. However they do not directly yield results indicating the accuracy of the reliability estimates. Confidence intervals for the reliability measures may be established by simulation, in particular importance sampling (using the FORM estimation point as the center of the sampling directly) may be efficient for this purpose. If it is difficult to achieve a solution for a limit state function, a reformulation should be considered. Use of logarithms may have a "linearising effect" on the limit state function which may improve FORM/SORM solutions. This is achieved by rewriting the limit state function from to g( x) = R( x) S( x) (3. 49) g( x ) = ln R(x) ln S( x) (3. 50) Distribution It should be checked that the distributions selected for the analysis are representative. Whenever the type of distribution is uncertain, another analysis with a more conservative distribution (a sensitivity study) may be performed for comparison. It is important to have a realistic distribution for the parameters contributing most to the failure probability. The importance of a distribution may be dependent on the estimation point which again is dependent on the reliability level. For example the tails of the distributions becomes more important as the reliability level is increased. An evaluation of appropriate distributions should therefore be performed at a relevant reliability level. Both the distribution type and the parameters describing the distribution should be considered.

27 Interpretation of Calculated Reliability Measures It should be kept in mind that is is difficult to work with absolute values of probability of failure as indicated in Chapter 2 of this Guideline. The calculated reliability measures depend on subjective assessments by the analysts and, as such, should be considered as nominal values. Whenever possible, it is recommended to base conclusions on relative values or comparisons with results from alternative analyses of "well known cases" as recommended in Chapter Fulfillment of Assumptions for Analysis The calculated reliabilities will be dependent on the assumptions made for the analysis. It is important that these assumptions are fulfilled when going from theoretical considerations on safety to that of design, fabrication, and in-service life. I.e. it is important that assumptions on material properties, fabrication tolerances, probability of detecting defects etc. are fulfilled in order to obtain the target safety level for the built structure. This can be achieved through a proper development of specifications for design and fabrication and through a proper development of procedure for in-service inspection Conditional Probability In many cases it is of interest to compute the probability that an event occurs conditioned on the occurrence of another event. The event that a structure fails within a specified number of years may, for example, be conditioned on the detection or nondetection of a number of cracks of given size. This is useful for updating of failure probabilities. If the problem at hand involves no single equality events, then the law of conditional probabilities is used. The probability of event A conditioned on event B is P EA E = B P EA EB P EB (3. 51) If the event E B involves the single equality event, SE, as E = E~ { ; G ( ) = } B B SE x x 0 then P EA EB = PE ( A E~ G B SE + θ 0) θ PE ( ~ G B SE + θ 0) θ = 0 θ θ = 0 (3. 52) where θ is an artificial zero-valued variable added to the single equality event function. If the event E B involves the single equality event SE as a measured value of some quantity, e.g., a measured crack size, then

28 P EA EB PE ( A E~ G B SE( θ) 0) = θ PE ( ~ G B SE( θ) 0) θ = 0 θ θ = 0 28 (3. 53) where θ is an artificial zero-valued variable added to the measured value. Note that most structural reliability textbooks, including those referenced in this document, contain an error (relative to Eq. (3. 53)) in the formulation of P E A E for the case that the single equality event SE B is defined as a measured value of some quantity such as a crack size. One of the most useful applications of SRA is for inspection, maintenance planning, remaining life assessment and life extension. For this purpose it is necessary to calculate conditional probabilities, e.g., (1) probability of fatigue failure conditioned on a crack of a given uncertain size found, or (2) probability of fatigue failure given that no cracks were found in a number of inspections with a certain inspection method. In the first case it is required to calculate Pg ( ( x) < 0 h( x, ad + θ) = 0) Pg ( ( x) < 0 h( x, ad ) = 0) = θ Ph ( ( x, ad + θ) = 0) θ = 0 θ θ = 0 (3. 54) in which h( x, a D + θ ) is the event margin corresponding to detection of a crack a D, where a D may be stochastic due to measurement uncertainties, and g( x ) is the limit state function corresponding to failure. This means that the probabilities can be calculated as the partial sensitivity factors of a parallel system with respect to the observed quantity a D, the detected crack size, which can be a uncertain quantity. In the second case it is required to calculate Pg ( ( x) < 0 h( x) > 0) = P( g( x) < 0 h( x) > 0) Ph ( ( x) > 0) (3. 55) This means that the conditional probability can be calculated by any software that can handle parallel systems, if due considerations are given to the correlation between the variables in the failure event formulation g( x ) < 0 and the inspection event formulation h( x ) > 0. Example 3.1: Example Based on Linear Elastic Fracture Mechanics The motivation for describing the fatigue process as stochastic may be illustrated by Figure 3. 9, where the crack depth in identical specimens as a function of the number of load cycles are plotted. The tests are from the same laboratory, with identical initial crack geometries. In addition the loads, response analysis, local stress analysis, initial crack size are stochastic in a realistic application.

29 29 Figure 3. 9 Crack depth as a function of time for identical specimens In linear elastic fracture mechanics, the increase in a crack depth a in a load cycle is described by da = dn C ( K ) m (3. 56) where ( C, m ) are material parameters estimated from laboratory tests. K is the stress intensity factor usually expressed as K = σ π ay( a) (3. 57) where σ is the far-field stress and Y( a) is the geometry function accounting for the stress concentration due to the presence of the crack, and local geometry effects. This may be reformulated as a limit state function as g( x ) = a c a0 da ( πay( a)) m C i σ i (3. 58) if cycle ordering effects can be neglected. The stochastic variables in a typical case would be represented by x = ( a0, ac, Y, C, m, σ ), the initial crack distribution, critical crack distribution, local geometrical effects, material parameters and far field stress. The calculation of the far field stress may be broken down to a model of random variables representing environmental and other loads, response analysis and local stress analysis. For this illustration we will not focus on these further details, reference is given to Skjong and Torhaug (1991). The inspection event that a crack is found with an uncertain crack size a D, due to the sizing inaccuracy may be formulated as the event