# Sensitivity Analysis in Probabilistic Structural Design: A Comparison of Selected Techniques

## Abstract

**:**

## 1. Introduction

_{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].

_{f}is the derivative ∂P

_{f}/∂μ

_{xi}with respect to the mean value μ of input variable X

_{i}[4,5,6,7]. A drawback of the derivative-based SA is that it cannot detect interactions between input variables. Since only one μ

_{xi}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 μ

_{xi}). The aforementioned drawback can partially be overcome by using the factorial experiment, where SA is computed using two-level changes of μ

_{xi}for all X

_{i}in combinations, which permit the computation of interaction effects [8]. However, only absolute change of the distribution parameter μ

_{xi}on P

_{f}is investigated, not the relative influence of the random variability of X

_{i}on P

_{f}. 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.

_{f}.

_{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 β has not yet been developed.

## 2. Design Reliability Conditions

_{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}:

_{U}= 0, and standard deviation σ

_{U}= 1 is written as

_{U}(•) 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.

_{d}.

_{d}= 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 σ

_{Z}as

_{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 Equations (3) and (8) into Equation (7), we can write

_{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 β

_{d}= 3.80 for a 50 years reference period [48,49]. For β

_{d}= 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 two- or 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.

## 3. Selected Types of Sensitivity Analysis Methods

_{Z}be the distribution function of Z:

_{i}:

_{i}is based on measuring the distance between probability Φ

_{Z}(t) and conditional probability ${\mathsf{\Phi}}_{Z}^{i}$(t) when an input is fixed [50].

_{i}can be expressed, on the basis of [50], as

_{i}, X

_{j}, for i < j:

_{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 Cramér–von Mises indices can be formulated in copula theory framework [51].

_{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.

_{i}is defined as Equation (17), where the contrast $\underset{\theta}{\mathrm{min}}\mathsf{\psi}\left(\theta \right)$ is computed for probability estimator θ* = Argmin ψ(θ) = P

_{f}.

_{f}|X

_{i}. The second-order probability contrast index C

_{ij}can be expressed as

_{i}is defined as

_{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 Y is, for example, the load action F or resistance R [56]; see Figure 1.

_{Z}(z) is the pdf of Z and ϕ

_{Z│}

_{Xi}(z) is the conditional pdf of Z given 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 $\le $ B

_{i}$\le $ B

_{ij}$\le $..$\le $ B

_{1,2,…,M}$\le $ 1 [14].

_{i}measures the individual effect of X

_{i}on P

_{f}.

_{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 $i\ne j$, are asymmetrical:

_{ij}may or may not be equal to K

_{ji}. Third-order and higher-order indices are not defined in [57].

_{i}= K

_{i}. Fixing pairs X

_{i}, X

_{j}, leads to the second-order index L

_{ij}, where i < j:

_{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 $\le $ L

_{i}$\le $ L

_{ij}$\le $...$\le $ L

_{1,2,…,M}$\le $ 1.

_{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].

_{i}is defined as the ratio between the conditional reliability index β|X

_{i}= μ

_{xi}and the reliability index β (7).

_{i}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.

_{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.

_{i}and output Z according to Pearson, Spearman and Kendal Tau.

## 4. Case Studies

_{F}(y), Φ

_{R}(y) and corresponding pdfs ϕ

_{F}(y), ϕ

_{R}(y), where y denotes a general point of the observed variable (force with the unit of Newton), through which both variables F and R are expressed; see right part of Figure 3. It is assumed that F and R are statistically independent of each other with mean values μ

_{F}, μ

_{R}and standard deviations σ

_{F}, σ

_{R}.

_{Z}− 10σ

_{Z}, μ

_{Z}+ 10σ

_{Z}].

_{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 1 are taken from Table B2 in [19]. Table 1 lists 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].

_{f}using different types of sensitivity indices and the subsequent comparison of obtained results. Resistance R is the input random variable X

_{1}, and load action F is the input random variable X

_{2}.

#### 4.1. Computation of Sensitivity Indices

_{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 $\underset{\theta}{\mathrm{min}}\mathsf{\psi}\left(\theta \right)$ = P

_{f}(1 − P

_{f}), and similarly for P

_{f}|X

_{i}, we can write $\underset{\theta}{\mathrm{min}E}\left(\psi \left(Z,\theta \right)|{X}_{i}\right)$ = (P

_{f}|X

_{i})(1 − (P

_{f}|X

_{i})).

_{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∈[μ

_{Z}− 10σ

_{Z}, μ

_{Z}+ 10σ

_{Z}]. In the second loop, E[•] is computed by numerical integration of the pdf of X

_{i}with a small step Δx

_{i}taken over [μ

_{Xi}− 10σ

_{Xi}, μ

_{Xi}+ 10σ

_{Xi}]. 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), C

_{12}= 1 − C

_{1}− C

_{2}.

_{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 Δx

_{i}taken over [μ

_{Xi}− 10σ

_{Xi}, μ

_{Xi}+ 10σ

_{Xi}]. 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 K

_{12}and K

_{21}defined in Equation (24) are computed in a similar manner.

_{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 Δx

_{2}taken over [μ

_{X2}− 10σ

_{X2}, μ

_{X2}+ 10σ

_{X2}]. In the inner loop, the pdf of X

_{1}is numerically integrated with a small step Δx

_{1}taken over [μ

_{X1}− 10σ

_{X1}, μ

_{X1}+ 10σ

_{X1}]. During integration, the term P

_{f}|X

_{1},X

_{2}can only have a value of 0 or 1.

_{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 Δt = t

_{l}

_{+1}− t

_{l}, where t = (t

_{l}

_{+1}+ t

_{l})/2, t∈[μ

_{Z}− 10σ

_{Z}, μ

_{Z}+ 10σ

_{Z}], l = 1, 2,..., 10000. To each Δt belongs dΦ(t)≈P(t

_{l}$\le $ Z $\le $ t

_{l}

_{+1}) and Φ(t)≈P(Z $\le $ (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 Δx

_{i}taken over [μ

_{Xi}−10σ

_{Xi}, μ

_{Xi}+10σ

_{Xi}]. Note: The LHS estimation of E[•] would be numerically very challenging but is possible. In the third (deep) loop, Φ

^{i}(t)≈P(Z $\le $ (t

_{l+1}+ t

_{l})/2|X

_{i}= ξ

_{i}) is computed by numerical integration for fixed ξ

_{i}, where ξ

_{i}is the middle of interval Δx

_{i}from the second loop. The index G

_{12}is computed on the basis of Equation (14) in a similar manner.

_{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 |ϕ

_{Z}(z) – ϕ

_{Z}

_{|}

_{Xi}(z)| is taken over [μ

_{Z}− 10σ

_{Z}, μ

_{Z}+ 10σ

_{Z}] using ten thousand runs. B

_{12}= 1 in all case studies.

_{f}. Sobol’s indices are computed analytically as S

_{1}= ${\sigma}_{R}^{2}$/(${\sigma}_{R}^{2}+{\sigma}_{F}^{2}$), S

_{2}= ${\sigma}_{F}^{2}$/(${\sigma}_{R}^{2}+{\sigma}_{F}^{2}$), 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): S

_{1}= ${\alpha}_{R}^{2}$, S

_{2}= ${\alpha}_{F}^{2}$.

_{1}, Z) and corr(X

_{2}, Z) are evaluated even though direct sensitivity to P

_{f}is not measured by correlation. Pearson, Spearman and Kendal Tau correlation coefficients between input X

_{i}and output Z are computed using one hundred thousand runs of the LHS method.

_{f}requires extremely high numbers of simulation runs and is numerically challenging.

#### 4.2. Case Study 1

_{f}. Let R (resistance) and F (load action) be statistically independent variables X

_{1}, X

_{2}with Gauss pdfs, where μ

_{R}= 412.54 kN, σ

_{R}= 34.132 kN, σ

_{F}= 34.132 kN, and mean value μ

_{F}is the parameter; see Figure 4. Let parameter μ

_{F}change with the step Δμ

_{F}= 10 kN and gradually attain the values 92.54 kN, 102.54 kN,..., 722.54 kN. The sensitivity indices are plotted in dependence on P

_{f}, where P

_{f}= Φ

_{U}(−(412.54 − μ

_{F})/(2

^{0.5}·34.132)) is a function only of parameter μ

_{F}. If μ

_{F}decreases, then P

_{f}decreases.

_{1}, Z) and corr(X

_{2}, Z) are added to Figure 5. Only indices within the interval [0, 1] are depicted. Madsen’s factors are not plotted because they have a constant value O

_{1}= O

_{2}= 1.415 for all P

_{f}(μ

_{F}).

_{f}(μ

_{F}) influences only indices C

_{1}, C

_{2}, C

_{12}, indices K

_{1}, K

_{2}, K

_{12}, K

_{21}and indices L

_{1}, L

_{2}, L

_{12}; see Figure 5 and the left part of Figure 6. For the other five types of SA, it was observed that two variables that have a different influence on the output have the same indices. This demonstrates properties of sensitivity indices that will prove useful in the interpretation of the result.

_{f}. Xiao asymmetrical interaction indices are identical: K

_{12}= K

_{21}. Contrast-based sensitivity indices have values of C

_{1}= C

_{2}= C

_{12}= 0.33 for P

_{f}= 0.5, but, otherwise, decrease with absolute distance from P

_{f}= 0.5. Approaching P

_{f}→ 0 or P

_{f}→ 1 leads to C

_{1}= C

_{2}→ 0 and C

_{12}→ 1. Change in mean value μ

_{F}has no influence on the values of Sobol’s indices, which are functions of only the variance and therefore remain constant S

_{1}= S

_{2}= 0.5, S

_{12}= 0. Kendall’s tau coefficient is approximately equal to 0.5 for all P

_{f}(μ

_{F}). Spearman’s and Pearson coefficients confirm the dependence between the inputs R, F and the output Z. Borgonovo and Cramér–von Mises first-order indices have approximately the same value B

_{1}= 0.306, G

_{1}= 0.286, while the second-order indices are B

_{12}= 1.0 and G

_{12}= 1.0 − G

_{1}− G

_{2}= 0.428.

#### 4.3. Case Study 2

_{1}, X

_{2}with Gauss pdfs, where μ

_{R}= 412.54 kN, σ

_{R}= 34.132 kN and mean value μ

_{F}is the parameter, while variation coefficient of F is constant v

_{F}= v

_{R}= 34.132/412.54 = 0.0827 and thus σ

_{F}= v

_{F}·μ

_{F}; see Figure 7. Let parameter μ

_{F}change with the step Δμ

_{F}= 12.89 kN and gradually attain values of 0.06 kN, 12.95 kN,..., 902.36 kN.

_{f}, where P

_{f}= Φ

_{U}(−(412.54 − μ

_{F})/(34.132

^{2}+ (μ

_{F}·34.132/412.54)

^{2})

^{0.5}) in Equation (6) is a function of only parameter μ

_{F}, where P

_{f}decreases if μ

_{F}decreases.

_{1}, Z) and corr(X

_{2}, Z) are added to Figure 8. Only indices within the interval [0, 1] are depicted. Omission sensitivity indices are not plotted because they have a value greater than one. The curves meet the expectation that small P

_{f}(due to small μ

_{F}and small σ

_{F}) are less sensitive to F and more sensitive to R.

_{f}(μ

_{F}) influences the values of all indices. All first-order indices of variable X

_{1}(R) are axially symmetrical to indices X

_{2}(F) along the vertical axis P

_{f}= 0.5 with the exception of Ling and Xiao indices. Xiao asymmetrical interaction indices are K

_{12}≠ K

_{21}with the exception of P

_{f}= 0.5 where K

_{12}= K

_{21}. The plots of Borgonovo and Cramér–von Mises first-order indices are similar; the second-order indices are B

_{12}= 1.0 and G

_{12}= 1.0 − G

_{1}− G

_{12}. The plots of Kendall’s tau coefficient and plots of Sobol’s indices are similar. The plots of Spearman’s and Pearson coefficients are also similar. On the left side of the graphs, contrast P

_{f}indices reach their extreme at point P

_{f}= 3.216·10

^{−10}, C

_{1}= 0.06, C

_{12}= 0.94, but no extreme on C

_{2}. On the right side of the graphs, the extreme is at point P

_{f}= 1 − 3.216·10

^{−10}, C

_{2}= 0.06, C

_{12}= 0.94, but no extreme on C

_{1}. Ling and Xiao indices have an extreme at P

_{f}= 3.216·10

^{−10}, K

_{2}= L

_{2}= 0.82, K

_{12}= 0.94; other extremes of Ling and Xiao indices are difficult to identify numerically (compared to other indices) because they quickly attain relatively small or large values for large or small P

_{f}.

_{f}= 7.2·10

^{−5}we obtain K

_{1}= L

_{1}= 0.97, K

_{2}= L

_{2}= 0.76, K

_{12}= 0.91, K

_{12}= 0.994, L

_{12}= 0.99993. A detail of the plot of sensitivity indices for P

_{f}< 1·10

^{−4}is depicted on the right part of Figure 9. For instance, for P

_{f}= 7.2·10

^{−5}, we obtain O

_{1}= 1.88, O

_{2}= 1.18, or for P

_{f}= 8.5·10

^{−6}, we obtain O

_{1}= 1.99, O

_{2}= 1.16.

## 5. Observation, Discussion and Questions

_{f}indices have relatively small values of first-order indices and high values of second-order indices for small P

_{f}. The numerical results of reliability engineering tasks [54,55] with five input random variables have shown that the smaller P

_{f}is, the smaller the values of first-order indices and the higher the values of higher-order indices. Change in the mean value or standard deviation of the dominant variables had a clear effect on P

_{f}, confirming the rationality of the contrast indices applied in [54]. The sum of all indices is equal to one, which makes it easier to compare SA results for different P

_{f}associated with different reliabilities, for e.g., different design conditions, different stages of the structural life or different loading conditions. For very small values of P

_{f}, Equation (29) can be written approximately as:

_{f}is evident from Equation (30) and Equation (31). If the binary random variable 1

_{Z<0}is considered, Equation (30) can then be written as:

_{f}) and reliability (1 − P

_{f}) because C

_{i}in Equation (29) is computed from the values of the contrast functions P

_{f}(1 − P

_{f}) and (P

_{f}|X

_{i})(1 − (P

_{f}|X

_{i})). If 1

_{Z}

_{<0}is rare, the evaluation of Equation (32) using Monte Carlo type methods requires an extreme number of samples.

_{f}, the values of the first-order indices are relatively high; moreover, the second-order index is always greater than the first-order index, which complicates the comparison of the influence on P

_{f}. The advantage of computing Ling indices is that the computation of the higher-order indices does not depend on the computation (accuracy) of lower-order indices; therefore, their computation may be performed parallelly on multiple processor cores.

_{f}level because Equations (13) and (14) are integrated over all dΦ

_{Z}(t), i.e., over all t (which means over all P

_{f}). This is also the reason that, at intervals relevant to reliability, index values are not extremely high or low. The advantage is that a zero value of the Cramér-von Mises index clearly means that the input is not important. Triple-nested-loop computation makes these indices very numerically challenging. Nevertheless, numerous effective approaches to reduce this computational complexity already exist; see, for e.g., [61].

_{f}.

_{Z}

_{<0}as the quantity of interest, Sobol’s indices can be an interesting reliability-oriented sensitivity technique [62].

_{f}< 0.5. The disadvantage of Madsen’s factor is that factor O

_{i}can have values significantly greater than 1. For example, in Case study 2, for P

_{f}= 6.14·10

^{−32}we obtain O

_{1}= 32. A model with one random variable would theoretically lead to O

_{1}= ∞. A significant computational problem can occur in non-linear problems when fixing to the mean value X

_{i}leads to the limit case of a given physical phenomenon. For example, the amplitude of the axial curvature of a slender bar subjected to buckling has a mean value equal to zero, which means a perfectly straight bar [63]. The resistance of such a perfectly straight (unrealistic) bar is always higher than the resistance of a bar with any non-zero imperfection [64], and thus the mean value of zero is not suitable for fixing in reliability analysis or SA. The modification of Equation (27) to the form E(β|X

_{i})/β can be discussed, but with the proviso that fixing X

_{i}must not lead to negative values of β. So far there is no global SA based on β, and it is questionable whether the first two statistical moments are sufficient to describe the influence on reliability.

_{f}but can be used as sensitivity indicators if the output (Z) is monotonically dependent on the input variables R and F. Correlation points to dependence, but the opposite is not true. The advantage of correlation coefficients is their availability in computer software and they are relatively computationally inexpensive in simulation approaches.

_{f}but not on the distribution of Z and conversely. Nevertheless, there is relatively good agreement between contrast Cramér–von Mises indices, Borgonovo indices and P

_{f}indices in the interval of approximately P

_{f}$\u03f5$ (0.1, 0.9). However, common building structures are designed with a reliability of P

_{f}< 4.8·10

^{−4}; see Table 1. For such small P

_{f}, only Cramér–von Mises indices and Borgonovo indices have similar values. The values of the other sensitivity indices are considerably different.

_{F}< σ

_{R}(σ

_{F}> σ

_{R}), the sensitivity of P

_{f}to R is higher (lower) than the sensitivity of P

_{f}to F. It was confirmed that only ROSA-type indices are suitable for probability-based reliability assessment. The results of Case study 1 showed that change in μ

_{F}together with σ

_{F}= const. changes only the values of contrast P

_{f}indices and Ling and Xiao indices, the other indices remain unchanged. Furthermore, change in σ

_{F}together with μ

_{F}= const. changes the values of all indices, except of course B

_{12}= 1 and S

_{12}= 0. The results of Case study 2 showed that changes in σ

_{F}and μ

_{F}with the condition v

_{F}= σ

_{F}/μ

_{F}= const. causes changes in the values of all indices; therefore, none of these indices is a pure indicator of the influence of v

_{F}on P

_{f}.

_{f}be influenced by the skewness and kurtosis values of the input variables or by correlations between them, and what is its importance for the analysis of reliability? For instance, the values of Sobol’s indices change when the kurtosis changes, but not when the skewness changes [34]. Of the SA types presented here, only Borgonovo indices [14] have the ability to have correlations between input variables. This ability must also be sought in other indices suitable for structural reliability analysis.

_{f}; therefore, P

_{f}should be the overall objective of SA. However, the concept of Eurocodes [19] assesses reliability according to the limit states using the so-called semi-probabilistic method, which compares the design values (quantiles) of resistance and load. Because probabilistic reliability analysis would be too expensive in common engineering practice, design values are usually computed deterministically according to design standards. These design values can be verified using the lower quantity of resistance (for e.g., 0.1 percentile) and upper quantity of load, where resistance and load are functions of other random variables; see Figure 2. Another useful property of SA could be that sensitivity indices oriented to P

_{f}and design quantiles form pairs based on the same theoretical basis. For example, global SA subordinated to contrasts can be associated with both P

_{f}and quantities R

_{d}and F

_{d}; see Figure 2. However, the question is whether there is a link between indices Equations (17), (18) and (20), (21) when the contrast functions Equations (16) and (19) are different. Preliminary studies show that partial similarity can be expected between the total indices.

## 6. Conclusions

_{f}, which is lower than 4.8·10

^{−4}.

_{f}indices have relatively small values of first-order indices and high values of second-order indices for small P

_{f}. Ling or Xiao indices have relatively high values of first-order indices, but also high values of second-order indices for small P

_{f}. For instance, in the first case study, if P

_{f}→ 0 then C

_{1}= 0, C

_{2}= 0, C

_{12}= 1 and L

_{1}= 1, L

_{2}= 1, L

_{12}= 1. In civil engineering, P

_{f}is generally very small, and extreme values of sensitivity indices estimated by ROSA can be expected. The advantage of 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 P

_{f}.

_{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.

_{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 P

_{f}indices have the same or similar values for unreliability (P

_{f}) and reliability (1 − P

_{f}).

_{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 P

_{f}.

_{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.

## Funding

## Conflicts of Interest

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**Figure 6.**Case study 1: (

**a**) second-order sensitivity indices of R, F; (

**b**) sensitivity analysis (SA) results related to structural reliability.

**Figure 9.**Case study 2: (

**a**) Second-order sensitivity indices of R, F; (

**b**) SA results related to structural reliability.

Reliability Class | β | P_{f} |
---|---|---|

RC3 | 4.3 | 8.5·10^{−6} |

RC2 | 3.8 | 7.2·10^{−5} |

RC1 | 3.3 | 4.8·10^{−4} |

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Kala, Z.
Sensitivity Analysis in Probabilistic Structural Design: A Comparison of Selected Techniques. *Sustainability* **2020**, *12*, 4788.
https://doi.org/10.3390/su12114788

**AMA Style**

Kala Z.
Sensitivity Analysis in Probabilistic Structural Design: A Comparison of Selected Techniques. *Sustainability*. 2020; 12(11):4788.
https://doi.org/10.3390/su12114788

**Chicago/Turabian Style**

Kala, Zdeněk.
2020. "Sensitivity Analysis in Probabilistic Structural Design: A Comparison of Selected Techniques" *Sustainability* 12, no. 11: 4788.
https://doi.org/10.3390/su12114788