Sensitivity analysis in probabilistic structural design: A comparison of selected techniques

sustainability Article Sensitivity Analysis in Probabilistic Structural Design: A Comparison of Selected Techniques Zdenˇek Kala Department of Structural Mechanics, Faculty of Civil Engineering, Brno University of Technology, 602 00 Brno, Czech Republic; [email protected].cz Received: 10 May 2020; Accepted: 5 June 2020; Published: 11 June 2020   Abstract: Although more and more reliability-oriented sensitivity analysis (ROSA) techniques are now available, review and comparison articles of ROSA are absent. In civil engineering, many of the latest indices have never been used to analyse structural reliability for very small failure probability. This article aims to analyse and compare different sensitivity analysis (SA) techniques and discusses their strengths and weaknesses. For this purpose, eight selected sensitivity indices are first described and then applied in two different test cases. Four ROSA type indices are directly oriented on the failure probability or reliability index beta, and four other indices (of a different type) are oriented on the output of the limit state function. The case study and results correspond to cases under common engineering assumptions, where only two independent input variables with Gaussian distribution of the load action and the resistance are applied in the ultimate limit state. The last section of the article is dedicated to the analysis of the different results. Large differences between first-order sensitivity indices and very strong interaction effects obtained from ROSA are observed for very low values of failure probability. The obtained numerical results show that ROSA methods lack a common platform that clearly interprets the relationship of indices to their information value. This paper can help orientate in the selection of which sensitivity measure to use. Keywords: sensitivity analysis; uncertainty modelling; load action; resistance; limit states; stochastic simulation; failure probability; structural reliability; correlations 1. Introduction Evaluating the reliability of building structures is a problem whose final goal remains a decision-making process [ 1 ]. In a probabilistic framework, the basic characteristic of engineering reliability is the probability of failure P f , which represents the key quantity of interest in decision-making processes [ 2 ]. It is recommended by the best practices that such a report is supplemented with sensitivity analysis (SA), which describes the effect of changes in model inputs on the measure of reliability [3]. A classical measure of change in P f is the derivative ∂ P f / ∂µxi with respect to the mean value µ of input variable X i [ 4 – 7 ]. A drawback of the derivative-based SA is that it cannot detect interactions between input variables. Since only one µxi is varied at a time while others are fixed, it can be labelled as the One-At-a-Time (OAT) method or local SA (at point µxi ). The aforementioned drawback can partially be overcome by using the factorial experiment, where SA is computed using two-level changes of µxi for all X i in combinations, which permit the computation of interaction effects [ 8 ]. However, only absolute change of the distribution parameter µxi on P f is investigated, not the relative influence of the random variability of Xion Pf. For structural reliability, it is better to prefer such SA types that can compute the effects of the random variabilities of input variables and their interactions on P f and not just changes in distribution parameters. Compared with the local SA, global SA [ 9 ] can measure the effect of input variables on the model output in their entire distribution ranges and provide the interaction effect among different input Sustainability 2020,12, 4788; doi:10.3390/su12114788 www.mdpi.com/journal/sustainability Sustainability 2020,12, 4788 2 of 19 variables. In the literature, many global SA techniques, such as the non-parametric techniques [ 10 ], screening approaches [ 11 ], Sobol’s variance-based (ANOVA) methods [ 12 , 13 ] and moment-independent methods [ 14 , 15 ], can be found, among which the variance-based method has gained the most attention. Variance is an important component of reliability analysis, but is insufficient on its own for the analysis of structural reliability; see, for e.g., [ 16 ]. A more general approach, which generalized Sobol’s sensitivity indices, was introduced by Fort et al. [ 17 ]. These indices are generally applicable (goal-oriented) because they can analyze various key quantities of interest, including Pf. The selection of SA methods that focus on P f or design quantiles is usually based on a stochastic model with binary output failure/nonfailure, 1/0 [ 18 ], but it is not a necessity. In first-order reliability method (FORM), P f can be replaced by reliability index β [ 19 ], which is computed using the first two moments of resistance, and load action and can be applied as an alternative measure of reliability; see Figure 1. However, global SA of βhas not yet been developed. Sustainability 2020, 12, x FOR PEER REVIEW 3 of 18 2. Design Reliability Conditions In limit state design, the resistance of a structure R must be greater than the load action F with a predetermined probability [19]. Structural reliability can also be assessed by comparing the lower quantile of R with the upper quantile of F [19], where the quantiles represent alternative key quantities of interest. The decision-maker who develops or implements stochastic models is expected to provide a forecast of structural reliability, which can be performed by estimating the failure probability, quantiles or other computational statistics related to limit states of structure. Let the reliability of building structures be a one-dimensional random variable Z, which is a function of random variables. () ( ) M XXXgXgZ ...,,, 21 == . (1) The reliability assessment of load-bearing structures is based on a semi-probabilistic approach of standard [19], which falls into the category of FORM methods [47]. Structural reliability is often expressed as a limit state function of random resistance R and random load action F: 0≥−= FRZ , (2) where R and F are statistically independent variables for which Gauss probability density functions (pdfs) are assumed with mean values μ R, μ F and standard deviations σ R, σ F. If R and F have Gauss pdfs, then Z has a Gauss pdf with mean value μ Z and standard deviation σ Z: FRZ μ μ μ −= , (3) 2 F 2 RZ σσσ += . (4) The transformation of Z into a normalized Gaussian pdf of U with mean value μ U = 0, and standard deviation σ U = 1 is written as Z Z σ μ − =Z U. (5) The probability of failure (key quantity of interest) can be expressed as () ()() ββ σ μ −Φ=−<=        −<=<= U UPUPZPP Z Z f0, (6) where ΦU(•) is the cumulative distribution function of normalized Gaussian pdf and μ Z/ σ Z is the so-called reliability index β ; see Equation (7) and Figure 1. Figure 1. Illustration of the reliability index β . It is assumed that β >0. Standard [19] verifies reliability by comparing the obtained reliability index β with the target reliability index β d. Figure 1. Illustration of the reliability index β. From a computational point of view, probabilistic sensitivity measures have been comprehensively studied; however, from a decision-analytic point of view, they remain much less understood [ 20 ]. Their relationship to information value has not yet been particularly established. Linking the information value, SA, and forecasting with scoring rules remains a subject of research [ 20 ]. For a particular reliability task, it is necessary to look for means to select the most appropriate sensitivity measure with common rationale for this selection. Incivilandconstructionengineering, thescientificcommunity usesSAinstructuralmechanics [21,22], geotechnics [ 23 , 24 ], landscape water management [ 25 ], building performance analysis [ 26 ], multi-criteria decision making (MCDM) [ 27 ], sustainable development of the building sector [ 28 ] or sensitivity audits to assess sustainability [ 29 ], but with a lower publication frequency than in basic sciences, such as chemistry, economics or mathematics [ 30 ]. In structural reliability, research deals with limit states [ 31 ] or the verification of partial safety factors of Eurocode standards [ 32 ] using various types of global SA based, for example, on the variance of model outputs [33,34]. The term “sensitivity analysis” can be understood differently in civil engineering than in basic sciences, where local and global SA types with random inputs are well established. For example, very specific (non-stochastic) SA methods based on advanced non-linear models are sometimes used for structures susceptible to buckling when the subject of interest is the stability (or potential energy) of structures [ 35 , 36 ] or imperfection sensitivity [ 37 , 38 ], whereby the main objective of these methods is to increase the stability limits of the structures through the variation of suitable design variables. In stochastic systems, stability often means insensitivity or low sensitivity of the output characteristics to the shapes of some input distributions [ 39 ]. In construction engineering, it is necessary to focus more on cooperation and integration of SA development [ 3 ] with reliability analysis tools [ 31 , 40 ] and decision-making processes [41,42]. Sustainability 2020,12, 4788 3 of 19 This paper compares several existing sensitivity measures in the context of structural reliability in civil engineering. For this purpose, eight selected sensitivity indices are first described and then applied in two different test cases. Four indices are oriented on the probability of failure or reliability index β , another four on the distribution or some moments of the output from the ultimate limit state function Equation (2). The reason for the inclusion of the second group of indices is their common use in the analysis of limit states, despite being only sensitive to reliability; see, for e.g., [ 43 , 44 ]. Correlations present a typical example; however, sensitivity techniques based on fuzzy probability analysis of constructions [ 45 , 46 ] are no exception. These alternative types of SA are not directly focused on the probability of failure, but they provide basic insight into the behaviour of computational models, their structures and their reactions to changes in model inputs. The presented article deals with four ROSA type SA and four other SA, which are empathetic to reliability in civil engineering. Special attention is paid to small failure probabilities, which are relevant for assessing the engineering reliability of structures using design reliability conditions. 2. Design Reliability Conditions In limit state design, the resistance of a structure Rmust be greater than the load action Fwith a predetermined probability [ 19 ]. Structural reliability can also be assessed by comparing the lower quantile of Rwith the upper quantile of F[ 19 ], where the quantiles represent alternative key quantities of interest. The decision-maker who develops or implements stochastic models is expected to provide a forecast of structural reliability, which can be performed by estimating the failure probability, quantiles or other computational statistics related to limit states of structure. Let the reliability of building structures be a one-dimensional random variable Z, which is a function of random variables. Z=g(X)=g(X1,X2,. . . ,XM). (1) The reliability assessment of load-bearing structures is based on a semi-probabilistic approach of standard [ 19 ], which falls into the category of FORM methods [ 47 ]. Structural reliability is often expressed as a limit state function of random resistance Rand random load action F: Z=R−F≥0, (2) where Rand Fare statistically independent variables for which Gauss probability density functions (pdfs) are assumed with mean values µR , µF and standard deviations σR , σF . If Rand Fhave Gauss pdfs, then Zhas a Gauss pdf with mean value µZand standard deviation σZ: µZ=µR−µF, (3) σZ=qσ2 R+σ2 F. (4) The transformation of Zinto a normalized Gaussian pdf of Uwith mean value µU =0, and standard deviation σU=1 is written as U=Z−µZ σZ. (5) The probability of failure (key quantity of interest) can be expressed as Pf=P(Z<0)=P U<−µZ σZ!=P(U<−β)=ΦU(−β), (6) where ΦU ( • ) is the cumulative distribution function of normalized Gaussian pdf and µZ / σZ is the so-called reliability index β; see Equation (7) and Figure 1. Sustainability 2020,12, 4788 4 of 19 It is assumed that β >0. Standard [ 19 ] verifies reliability by comparing the obtained reliability index βwith the target reliability index βd. β=µZ σZ ≥βd, (7) For instance, the reliability index has a target value of βd =3.8 (P fd =7.2 · 10 −5 ), provided that we consider the ultimate limit state for common design situations within the reference period of 50 years; see Table C2 in [19] or [48]. Equation (8) can be written to obtain σZas σZ=qσ2 R+σ2 F=σ2 R+σ2 F qσ2 R+σ2 F =σR qσ2 R+σ2 F σR+σF qσ2 R+σ2 F σF=αRσR+αFσF, (8) where αF , αR are values of sensitivity coefficients (weight factors) according to the FORM method, which [ 19 ] introduces with constant values αF =0.7, αR =0.8. Substituting Equations (3) and (8) into Equation (7), we can write β=µR−µF αRσR+αFσF ≥βd. (9) Equation (9) is the design reliability condition with formally separated random variables that can be expressed as µF+αFβdσF≤µR−αRβdσR. (10) where the left-hand side represents the design load F d and the right-hand side the design resistance R d ; see Figure 2. The basic reliability targets for design values in the ultimate limit state recommended in [ 19 ] are based on the semi-probabilistic approach in Figure 2, with the target value of reliability index βd =3.80 for a 50 years reference period [ 48 , 49 ]. For βd =3.8, R d can be approximately computed as 0.1 percentile [ 40 ]. Standard [ 19 ] enables the determination of design values F d ,R d not only from a Gauss pdf but also from a twoor three-parameter lognormal (for resistance) or Gumbel or Gama (for load) pdfs. The probability of failure for non-Gaussian Rand Fcan be estimated using Monte Carlo (or quasi-Monte Carlo) methods. Sustainability 2020, 12, x FOR PEER REVIEW 4 of 18 d Z β σ μ β ≥= Z , (7) For instance, the reliability index has a target value of βd = 3.8 (Pfd = 7.2·10-5), provided that we consider the ultimate limit state for common design situations within the reference period of 50 years; see Table C2 in [19] or [48]. Equation (8) can be written to obtain σ Z as FFRF FR F R FR R FR FR FRZ σασασ σσ σ σ σσ σ σσ σσ σσσ += + + + = + + =+= R 222222 22 22 , (8) where α F, α R are values of sensitivity coefficients (weight factors) according to the FORM method, which [19] introduces with constant values α F = 0.7, α R = 0.8. Substituting Equation (3) and Equation (8) into Equation (7), we can write d FFRR FR β σασα μ μ β ≥ + − =. (9) Equation (9) is the design reliability condition with formally separated random variables that can be expressed as RdRRFdFF σ β α μ σ β α μ −≤+ . (10) where the left-hand side represents the design load Fd and the right-hand side the design resistance Rd; see Figure 2. The basic reliability targets for design values in the ultimate limit state recommended in [19] are based on the semi-probabilistic approach in Figure 2, with the target value of reliability index β d = 3.80 for a 50 years reference period [48,49]. For βd = 3.8, Rd can be approximately computed as 0.1 percentile [40]. Standard [19] enables the determination of design values Fd, Rd not only from a Gauss pdf but also from a twoor three-parameter lognormal (for resistance) or Gumbel or Gama (for load) pdfs. The probability of failure for non-Gaussian R and F can be estimated using Monte Carlo (or quasi-Monte Carlo) methods. Figure 2. Illustration of the design condition of reliability. 3. Selected Types of Sensitivity Analysis Methods In reliability engineering, SA methods quantify the effects of input variables on the failure probability, reliability index β or design quantiles. However, other statistical model-based inferences sensitive to reliability are often used. In this chapter, we present selected formulae of selected types of sensitivity measures in forms that are adapted to structural reliability analysis. Cramér–von Mises indices [50]. Input random variables in Equation (1) are assumed to be statistically independent. Let ΦZ be the distribution function of Z: () ( ) ( ) tZZ EtZPt ≤ =≤=Φ 1 for Rt ∈, (11) and i Z Φ is the conditional distribution function of Z conditionally on Xi: () ()() itZi i ZXEXtZPt ≤ =≤=Φ 1 for Rt ∈. (12) Figure 2. Illustration of the design condition of reliability. 3. Selected Types of Sensitivity Analysis Methods In reliability engineering, SA methods quantify the effects of input variables on the failure probability, reliability index β or design quantiles. However, other statistical model-based inferences sensitive to reliability are often used. In this chapter, we present selected formulae of selected types of sensitivity measures in forms that are adapted to structural reliability analysis. Cram é r–von Mises indices [ 50 ]. Input random variables in Equation (1) are assumed to be statistically independent. Let ΦZbe the distribution function of Z: ΦZ(t)=P(Z≤t)=E(1Z≤t)for t∈R, (11) Sustainability 2020,12, 4788 5 of 19 and Φi Zis the conditional distribution function of Zconditionally on Xi: Φi Z(t)=P(Z≤t|Xi)=E(1Z≤t|Xi)for t∈R. (12) The first-order Cram é r–von Mises index G i is based on measuring the distance between probability ΦZ(t) and conditional probability Φi Z(t) when an input is fixed [50]. Gi=Z R EΦZ(t)−Φi Z(t)2 ΦZ(t)(1−ΦZ(t)) dΦZ(t). (13) The second-order Cramér–von Mises index Gican be expressed, on the basis of [50], as Gij =Z R EΦZ(t)−Φij Z(t)2 ΦZ(t)(1−ΦZ(t)) dΦZ(t)−Gi−Gj, (14) where Φij Zis the conditional distribution function of Zconditionally on Xi,Xj, for i<j: Φij Z(t)=PZ≤tXi,Xj=E1Z≤tXi,Xjfor t∈R. (15) Integration Equation (13) and Equation (14) respect t. Equation (13) is not oriented to one failure probability value P f , but, depending on t, integrates the averages of squared values from the differences of all probabilities Equations (11) and (12) normalized by F(t)(1 − F(t)). The same applies to other higher-order indices [ 50 ]. Indices G i ,G ij , etc., are based on Hoeffding decomposition; therefore, the sum of all indices is equal to 1 [ 50 ]. It can be noted that Cram é r–von Mises indices can be formulated in copula theory framework [51]. Sensitivity indices subordinated to contrasts associated with probability [ 17 ] (in short, Contrast P f indices). These indices measure the distance between probability P f and the conditional probability P f |X i using the contrast function in Equation (16). The input random variables in Equation (1) are assumed to be statistically independent. ψ(θ)=E(ψ(Z,θ)) =E(1Z<0−θ)2. (16) The first-order probability contrast index C i is defined as Equation (17), where the contrast min θψ(θ)is computed for probability estimator θ*=Argmin ψ(θ)=Pf. Ci= min θψ(θ)−EminE θ(ψ(Z,θ)|Xi) min θψ(θ). (17) The second term in the numerator in Equation (17) is computed as the average value of the conditional contrast functions whose probability estimator is P f |X i . The second-order probability contrast index Cij can be expressed as Cij = min θψ(θ)−EminE θψ(Z,θ)XiXj min θψ(θ)−Ci−Cj, (18) where i<j. Indices of the third and higher orders are computed similarly [ 17 ]. Sensitivity indices subordinated to contrasts are based on decomposition; therefore, the sum of all indices must be equal to one. Examples of the computation of indices using the Latin Hypercube Sampling method (LHS) [52,53] in engineering applications are in [54,55]. Sustainability 2020,12, 4788 6 of 19 Sensitivity indices subordinated to contrasts associated with α -quantile [ 17 ]. The contrast function ψ associated with α-quantile can be written with parameter θas [17]: ψ(θ)=E(ψ(Z,θ)) =E((Y−θ)(α−1Y<θ)), (19) where the input random variables in Equation (1) are assumed to be statistically independent. The first-order quantile contrast index Qiis defined as Qi= min θψ(θ)−EminE θ(ψ(Y,θ)|Xi) min θψ(θ), (20) where min θψ(θ)is the contrast computed for the estimator of α-quantile θ*=Argmin ψ(θ). Qij = min θψ(θ)−EminE θ(ψ(Y,θ)|Xi) min θψ(θ)−Qi−Qj. (21) The second-order quantile contrast index Q ij is defined as Equation (21), where i<j. Indices of the third and higher orders are computed in a similar manner [ 17 ]. Sensitivity indices subordinated to contrasts are based on decomposition; therefore, the sum of all indices must be equal to one. In engineering applications, the random variable Yis, for example, the load action For resistance R[ 56 ]; see Figure 1. Borgonovo moment independent importance measure [14] (in short, Borgonovo indices). The sensitivity indices described in [ 14 ] are defined by introducing a moment-independent uncertainty indicator that looks at the entire input/output distribution and whose definition is well-posed also in the presence of correlations among the input parameters. Bi=1 2EZϕZ(z)−ϕZ|Xi(z)dz, (22) where ϕZ (z) is the pdf of Zand ϕZ|Xi (z) is the conditional pdf of Zgiven that one of the parameters, X i , assumes a fixed value [ 14 ]. Fixing pairs X i ,X j , leads to the second-order index B ij , where i<j. Fixing triplets X i ,X j ,X k leads to the third-order index B ijk , where i<j<k, etc. The sum of all indices is not equal to one. As a general rule, 0 ≤Bi≤Bij≤..≤B1,2, . . . ,M ≤1 [14]. Reliability sensitivity index defined by Xiao et al. [ 57 ] (in short, Xiao indices). All the input variables are independent of each other. The first-order index Simeasures the individual effect of Xion Pf. Ki=1 2Pf EPf−Pf|Xi, (23) where |P f – P f |X i |measures the absolute difference between the unconditional failure probability P f and the conditional failure probability P f |X i . The second-order interaction indices K ij , where i,j , are asymmetrical: Kij =1 2E       Pf|Xi Pf −Pf|Xi,Xj PfXj        . (24) Sustainability 2020,12, 4788 7 of 19 Kij may or may not be equal to Kji. Third-order and higher-order indices are not defined in [57]. Reliability sensitivity index defined by Ling et al. [ 58 ] (in short, Ling indices). The first-order index is the same as in Equation (23) L i =K i . Fixing pairs X i ,X j , leads to the second-order index L ij , where i<j: Lij =1 2Pf EPf−Pf|Xi,Xj. (25) Fixing triplets X i ,X j ,X k leads to the third-order index L ijk , where i<j<k, etc. The sum of all indices defined by Ling et al. [58] is not equal to one. As a general rule [58], 0 ≤Li≤Lij≤...≤L1,2, ... ,M≤1. Sobol’s sensitivity indices [ 12 , 13 ] (in short, Sobol’s indices). Sobol’s first-order sensitivity indices can be written in the form: Si=V(Z)−E(V(Z|Xi)) V(Z)=V(E(Z|Xi)) V(Z)=corr2(Z,E(Z|Xi)), (26) where corr is Pearson correlation coefficient. Fixing pairs X i ,X j , leads to the second-order index S ij , where i<j. Fixing triplets X i ,X j ,X k leads to the third-order index S ijk , where i<j<k, etc.; see, for example [ 9 ]. The sum of all indices is equal to one. It can be noted that Sobol’s indices present a special case of sensitivity indices subordinated to contrasts in which the contrast function is associated with variance ψ(θ)=E(Z−θ)2[17]. Omission sensitivity factor [ 59 ] (in short, Madsen’s factor). The omission sensitivity factor O i is defined as the ratio between the conditional reliability index β|Xi=µxi and the reliability index β(7). Oi= βXi=µXi β. (27) Random variable X i is fixed at its mean value µxi in the numerator in Equation (27), but the possibility of fixing at the characteristic value [60] or median [3] is also indicated. The indices described above can be divided into two groups. The first group (Sobol, Borgonovo and Cram é r–von Mises) focuses on the distribution or some moments of the output function Z, while the second group (Xiao, Ling, Contrast, Madsen’s) considers P f , β or quantiles as the quantity of interest and thus can be referred to as reliability analysis indices. The first group can be classified as global SA, while the second group can be classified as reliability-oriented sensitivity analysis (ROSA) [ 3 ], of which Xiao, Ling and Contrast indices can terminologically [ 54 , 57 , 58 ] be classified as global ROSA. It can be noted that Xiao, Ling and Contrast ROSA indices are typical examples of ambiguous “local–global” indices [ 3 ]. On one hand, they can be considered as global since they are based on changes of P f with regard to the variability of the inputs over their entire distribution ranges and they provide the interaction effect between different input variables. On the other hand, they can be considered as local in the sense of regional SA since they are based on the frequency of failures from the random realization in “region” of pairs of large load actions and small resistances. Correlations. The last SA methods used are the analysis of the correlation between the input X i and output Zaccording to Pearson, Spearman and Kendal Tau. 4. Case Studies Many new sensitivity indices have been developed, but their ability in applications has not yet been reliably demonstrated. In this article, the properties of the selected sensitivity indices mentioned in Chapter 3 are examined in a case study of the probabilistic analysis of the reliability of a steel bar under axial tension; see Figure 3. A static time-independent study is considered. Sustainability 2020,12, 4788 8 of 19 Sustainability 2020, 12, x FOR PEER REVIEW 7 of 18 where corr is Pearson correlation coefficient. Fixing pairs Xi, Xj, leads to the second-order index Sij, where i<j. Fixing triplets Xi, Xj, Xk leads to the third-order index Sijk, where i<j<k, etc.; see, for example [9]. The sum of all indices is equal to one. It can be noted that Sobol’s indices present a special case of sensitivity indices subordinated to contrasts in which the contrast function is associated with variance ψ( θ ) = E(Z− θ )2 [17]. Omission sensitivity factor [59] (in short, Madsen’s factor). The omission sensitivity factor Oi is defined as the ratio between the conditional reliability index β |Xi = μ xi and the reliability index β (7). () β μβ i Xi i X O= =. (27) Random variable Xi is fixed at its mean value μ xi in the numerator in Equation (27), but the possibility of fixing at the characteristic value [60] or median [3] is also indicated. The indices described above can be divided into two groups. The first group (Sobol, Borgonovo and Cramér–von Mises) focuses on the distribution or some moments of the output function Z, while the second group (Xiao, Ling, Contrast, Madsen’s) considers Pf, β or quantiles as the quantity of interest and thus can be referred to as reliability analysis indices. The first group can be classified as global SA, while the second group can be classified as reliability-oriented sensitivity analysis (ROSA) [3], of which Xiao, Ling and Contrast indices can terminologically [54,57,58] be classified as global ROSA. It can be noted that Xiao, Ling and Contrast ROSA indices are typical examples of ambiguous “local–global” indices [3]. On one hand, they can be considered as global since they are based on changes of Pf with regard to the variability of the inputs over their entire distribution ranges and they provide the interaction effect between different input variables. On the other hand, they can be considered as local in the sense of regional SA since they are based on the frequency of failures from the random realization in “region” of pairs of large load actions and small resistances. Correlations. The last SA methods used are the analysis of the correlation between the input Xi and output Z according to Pearson, Spearman and Kendal Tau. 4. Case Studies Many new sensitivity indices have been developed, but their ability in applications has not yet been reliably demonstrated. In this article, the properties of the selected sensitivity indices mentioned in Chapter 3 are examined in a case study of the probabilistic analysis of the reliability of a steel bar under axial tension; see Figure 3. A static time-independent study is considered. (a) (b) Figure 3. Static model: (a) bar under axial tension; (b) probability density functions of R and F. Figure 3. Static model: (a) bar under axial tension; (b) probability density functions of Rand F. In general, the random load action Fand resistance Rare usually described using appropriate types of distribution functions ΦF (y), ΦR (y) and corresponding pdfs ϕF (y), ϕR (y), where ydenotes a general point of the observed variable (force with the unit of Newton), through which both variables F and Rare expressed; see right part of Figure 3. It is assumed that Fand Rare statistically independent of each other with mean values µF,µRand standard deviations σF,σR. The probability of failure Pf=P(Z<0)=P(R<F)can be computed as the integral: Pf= ∞ Z −∞ ΦR(y)ϕF(y)dy. (28) In the case studies, integration in Equation (28) is performed numerically by Simpson’s rule, using more than ten thousand integration steps over the interval [µZ−10σZ,µZ+10σZ]. Reliability can be assessed by comparing the computed P f in Equation (6) with the target value of P f , where target values for design cases are listed in standard EN1990 [ 19 ]. Target values of P f in Table 1are taken from Table B2 in [ 19 ]. Table 1lists the minimum values of P f (the reliability index β ) for ultimate limit state and 50 years reference period. The description of subsequent classes RC1, RC2, and RC3 with examples of building and civil engineering works are in [19,48]. Table 1. Recommended minimum values of βand related Pf. Reliability Class βPf RC3 4.3 8.5·10−6 RC2 3.8 7.2·10−5 RC1 3.3 4.8·10−4 The aim of the presented study is the SA of the influence of input factors R,Fon the output P f using different types of sensitivity indices and the subsequent comparison of obtained results. Resistance R is the input random variable X1,and load action Fis the input random variable X2. Sustainability 2020,12, 4788 9 of 19 4.1. Computation of Sensitivity Indices This section includes a description of numerical methods for computing the size of sensitivity indices based on numerical integration methods in combination with sampling-based methods or analytical computation. Sensitivity indices were computed for eight SA types. Contrast P f indices [ 17 ] (ROSA). The contrast function Equation (16) is minimum if θ *=P f . By substituting P f into Equation (16), we can write first-order index in Equation (17) using min θψ(θ) =P f (1 −Pf), and similarly for Pf|Xi,we can write minE θ(ψ(Z,θ)|Xi)=(Pf|Xi)(1 −(Pf|Xi)). Ci=Pf1−Pf−EPf|Xi1−Pf|Xi Pf1−Pf. (29) By substituting P f (1 − P f ) and (P f |X i )(1 − (P f |X i )) into Equation (17), we can derive Equation (29) for practical use. C i measures, on average, the effect of fixing X i on P f . The estimate of P f is computed as the integral Equation (28). In the first loop, the estimate of P f |X i =P((Z|X i )<0) is computed by numerical integration across z ∈ [ µZ− 10 σZ , µZ +10 σZ ]. In the second loop, E[ • ] is computed by numerical integration of the pdf of X i with a small step ∆ x i taken over [ µXi − 10 σXi , µXi +10 σXi ]. Since the second term in the numerator in Equation (18) is always equal to zero (P f |X 1 ,X 2 is always equal to zero or one), C12 =1−C1−C2. Xiao indices [ 57 ] (ROSA). Indices K 1 ,K 2 are estimated from Equation (23) using double-nested-loop computation. In the outer loop, E[ • ] is computed by numerical integration of the pdf of X i with a small step ∆ x i taken over [ µXi − 10 σXi , µXi +10 σXi ]. Note: the estimate E[ • ] obtained using the LHS method would be inaccurate because it requires an extremely high number of runs for small values of P f . In the nested loop, estimates of P f and P f |X i are computed by integrating according to Equation (28). Indices K12 and K21 defined in Equation (24) are computed in a similar manner. Ling indices [ 58 ] (ROSA). By definition, L 1 =K 1 ,L 2 =K 2 . The computation of L 12 includes an estimate of E[ • ], which is based on double numerical integration. In the outer loop, the pdf of X 2 is numerically integrated with a small step ∆ x 2 taken over [ µX2 − 10 σX2 , µX2 +10 σX2 ]. In the inner loop, the pdf of X 1 is numerically integrated with a small step ∆ x 1 taken over [ µX1 − 10 σX1 , µX1 +10 σX1 ]. During integration, the term Pf|X1,X2can only have a value of 0 or 1. Madsen factor [59] (ROSA). Indices O1,O2are computed using one million LHS runs. Cram é r–von Mises indices [ 50 ]. Indices G 1 and G 2 are computed using Equation (13). Three nested loops are applied. In the first (outer) loop, numerical integration is computed with a small step ∆ t= t l+1− t l , where t=(t l+1 +t l )/2, t ∈ [ µZ− 10 σZ , µZ +10 σZ ], l=1, 2,..., 10000. To each ∆ tbelongs d Φ (t) ≈ P(t l ≤ Z ≤ t l+1 ) and Φ (t) ≈ P(Z ≤ (t l+1 +t l )/2). In the second loop, E[ • ] in the numerator in Equation (13) is computed by numerical integration of the pdf of X i with a small step ∆ x i taken over [ µXi− 10 σXi , µXi+10σXi]. Note: The LHS estimation of E[•] would be numerically very challenging but is possible. In the third (deep) loop, Φi (t) ≈ P(Z ≤ (t l+1 +t l )/2|X i = ξi ) is computed by numerical integration for fixed ξi , where ξi is the middle of interval ∆ x i from the second loop. The index G 12 is computed on the basis of Equation (14) in a similar manner. Borgonovo indices [ 14 ]. Indices B 1 ,B 2 are estimated from Equation (13) using double-nested-loop computation. In the outer loop, 0.5 · E[ • ] is computed using one million runs of the LHS method. In the nested loop, numerical integration | ϕZ (z) – ϕZ|Xi (z)|is taken over [ µZ− 10 σZ , µZ +10 σZ ] using ten thousand runs. B12 =1 in all case studies. Sobol’s indices [ 12 , 13 ]. Sobol’s sensitivity indices are included only for comparison; these indices analyse the influence of the variance of Ror Fon the variance of Z, but not the influence on P f . Sobol’s indices are computed analytically as S 1 = σ2 R /( σ2 R+σ2 F ), S 2 = σ2 F /( σ2 R+σ2 F ), S 12 =0. It can be noted that Sobol’s first-order indices are equal to the squares of the sensitivity coefficients (weight factors) in Equation (8): S1=α2 R,S2=α2 F. Sustainability 2020,12, 4788 16 of 19 contrast P f indices is that the sum of all indices is equal to one. The sum of all Ling or Xiao indices is not equal to one. The Madsen factor values were significantly greater than 1 and therefore cannot be compared in size with Contrast, Xiao and Ling indices. Madsen’s factor does not reflect change in the mean value of input variables, although this change causes a change in Pf. In contrast, Xiao, Ling and Madsen’s indices have correctly identified the order of importance of input random variables to P f ; however, this observation only applies to the presented case studies and cannot be generalized. In the case studies, it is not possible to determine, even approximately, the percentage by which the dominant variable is more influential than the others, so this conclusion is true for each type of SA. Structural reliability lacks a common platform of SA that provides a clear interpretation of the size of sensitivity indices and defines their information value. The other indices (Sobol, Borgonovo and Cram é r–von Mises) and correlation coefficients are not directly addressable to P f and therefore are not generally suitable for the analysis of reliability. As expected, these (out of ROSA type) sensitivity indices do not reflect the change in mean value of input variables, although this change causes a change in P f . This means that the two variables that have a different influence on the reliability may have the same indices. In relation to the reliability of structures, the information value of these indices is not unambiguous. In the case of ROSA, Xiao and Ling indices, no two different P f values exist for which the same sets of sensitivity indices exist, but contrast Pfindices have the same or similar values for unreliability (Pf) and reliability (1 −Pf). There are many engineering reliability assessments in which non-ROSA indices are applied, although the connection with reliability is mentioned. The reason for these applications may be the simplicity of evaluating indices as well as the experience that known indices have at least partial sensitivity for reliability, which, along with other experience, is sufficient for basic decision-making. With the development of ROSA, a gradual transition to new types of reliability-oriented indices can be expected. In connection with sustainable reliability, it is possible to discuss which type of ROSA should be applied and which key quantities of interest ROSA should be oriented to in particular. The Eurocode standards for structural design assess reliability using a so-called semi-probabilistic approach, which is based on design quantiles. The question remains whether P f can be adequately replaced by design quantities, reliability index β or other model-based inferences so that the information value of SA results in relation to reliability is approximately maintained. Design quantiles are an important part of reliability analysis, and SA of the design quantiles may be required to provide results consistent with Pf. In general, ROSA directly addressable to P f may be preferred rather than focusing on the reliability index β or quantiles. Indices with the sum of one and a clear addressability to P f present one SA, an advantage that facilitates the comparison of the results of different probability models. Contrast functions are a more general tool for estimating various parameters associated with probability distributions, and thus the partial consistency of requirements could perhaps be sought on the basis of contrasts. These and other tasks need to be addressed in order to make SA of structural reliability a useful and practical tool. 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