**5. Computation of Sensitivity Indices**

The aim of SA in the presented case study is to assess the influence of input quantities *F*1, *F*2, *fy*, *t*, and *b* on the failure probability *P<sup>f</sup>* or design quantiles *F<sup>d</sup>* and *Rd*.

The numerical parameter of the case study is µ*P*, which changes with the step ∆µ*<sup>P</sup>* = 10 kN. Although µ*<sup>P</sup>* is the computation parameter, sensitivity indices are preferably plotted, depending on *Pf* , because *P<sup>f</sup>* has a clear relevance to reliability. The transformation of µ*<sup>P</sup>* to *P<sup>f</sup>* is expressed using Equation (30) (see Figure 6a). *Symmetry* **2020**, *12*, x FOR PEER REVIEW 10 of 20

**Figure 6.** Probability of design *α*-quantiles vs. failure probability *Pf*: (**a**) *μp* vs *Pf*; (**b**) *P<sup>f</sup>* vs *α*-quantiles. **Figure 6.** Probability of design α-quantiles vs. failure probability *P<sup>f</sup>* : (**a**) µ*<sup>p</sup>* vs. *P<sup>f</sup>* ; (**b**) *P<sup>f</sup>* vs. α-quantiles.

steeper slope. In the case study, for *β* = 3.8 (*P<sup>f</sup>* = 7.2 × 10−5), *P*(*F* < *Fd*) = 0.9963, and *P*(*R* < *Rd*) = 0.0036,

where *Fd* = *Rd* = 321.01 kN (*μF* = 229.97 kN, *μR* = 412.68 kN, and *σ<sup>F</sup>* = 33.94 kN ≈ *σR* = 34.057 kN).

*5.1. Local ROSA—Sensitivity Indices Based on Derivatives* 

sensitivity ranking of input variables as *fy*, *F1*, *F2*, *t*, and *b*.

(symmetric pdfs of both *F* and *R*).

the solid curves.

In Figure 6b, the probability of design quantiles *Fd* and *R<sup>d</sup>* is considered under the condition *F<sup>d</sup>* =

Figure 7a shows the partial derivatives of *Pf* with respect to the mean values *μxi*. Although the partial derivative of *P<sup>f</sup>* with respect to *μt* has the greatest value, *t* is not the most influential input variable in terms of the absolute change of *Pf* due to the uncertainty (variance) of the input variable *t*. A better measure of sensitivity is obtained by multiplying the partial derivatives by the standard deviations of the respective input variables (see Figure 7b). Ranking according to *Di* gives the

(**a**) (**b**) **Figure 7.** Derivative-based local SA of failure probability *Pf*: (**a**) Derivatives; (**b**) Sensitivity index *Di*.

The plots in Figure 7 are approximately symmetrical about the vertical axis, but not perfectly symmetrical. The small amount of asymmetry is due to the small skewness of resistance *R* in Equation (1) (see Equation (29)). Perfect symmetry of the curves would occur if *F* and *R* had zero skewness

A small amount of asymmetry is graphically visible upon mirroring the solid curves to the dashed curves (see Figure 8). The dashed curves are artificial, showing the left-right asymmetry of

In practice, the procedure is as follows: The value of µ*<sup>P</sup>* is selected, the sensitivity indices and *P<sup>f</sup>* are computed, and the indices vs. *P<sup>f</sup>* are then plotted. If the design quantiles are the key quantities of interest, then the dependency between *P<sup>f</sup>* and the probabilities of the design quantiles can be considered, according to Figure 6b. (**a**) (**b**)

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 10 of 20

In Figure 6b, the probability of design quantiles *F<sup>d</sup>* and *R<sup>d</sup>* is considered under the condition *F<sup>d</sup>* = *R<sup>d</sup>* in Equation (7) and σ*<sup>F</sup>* = σ*R*. Perfect biaxial symmetry of the curves in Figure 6b is only observed for perfect σ*<sup>F</sup>* = σ*R*; otherwise, the curve of the variable with the smaller standard deviation has a steeper slope. In the case study, for <sup>β</sup> <sup>=</sup> 3.8 (*P<sup>f</sup>* <sup>=</sup> 7.2 <sup>×</sup> <sup>10</sup>−<sup>5</sup> ), *P*(*F* < *Fd*) = 0.9963, and *P*(*R* < *Rd*) = 0.0036, where *F<sup>d</sup>* = *R<sup>d</sup>* = 321.01 kN (µ*<sup>F</sup>* = 229.97 kN, µ*<sup>R</sup>* = 412.68 kN, and σ*<sup>F</sup>* = 33.94 kN ≈ σ*<sup>R</sup>* = 34.057 kN). **Figure 6.** Probability of design *α*-quantiles vs. failure probability *Pf*: (**a**) *μp* vs *Pf*; (**b**) *P<sup>f</sup>* vs *α*-quantiles. In Figure 6b, the probability of design quantiles *Fd* and *R<sup>d</sup>* is considered under the condition *F<sup>d</sup>* = *R<sup>d</sup>* in Equation (7) and *σF* = *σR*. Perfect biaxial symmetry of the curves in Figure 6b is only observed for perfect *σF* = *σR*; otherwise, the curve of the variable with the smaller standard deviation has a steeper slope. In the case study, for *β* = 3.8 (*P<sup>f</sup>* = 7.2 × 10−5), *P*(*F* < *Fd*) = 0.9963, and *P*(*R* < *Rd*) = 0.0036,

## *5.1. Local ROSA—Sensitivity Indices Based on Derivatives* where *Fd* = *Rd* = 321.01 kN (*μF* = 229.97 kN, *μR* = 412.68 kN, and *σ<sup>F</sup>* = 33.94 kN ≈ *σR* = 34.057 kN).

Figure 7a shows the partial derivatives of *P<sup>f</sup>* with respect to the mean values µ*xi*. Although the partial derivative of *P<sup>f</sup>* with respect to µ*<sup>t</sup>* has the greatest value, *t* is not the most influential input variable in terms of the absolute change of *P<sup>f</sup>* due to the uncertainty (variance) of the input variable *t*. A better measure of sensitivity is obtained by multiplying the partial derivatives by the standard deviations of the respective input variables (see Figure 7b). Ranking according to *D<sup>i</sup>* gives the sensitivity ranking of input variables as *fy*, *F*1, *F*2, *t*, and *b*. *5.1. Local ROSA—Sensitivity Indices Based on Derivatives*  Figure 7a shows the partial derivatives of *Pf* with respect to the mean values *μxi*. Although the partial derivative of *P<sup>f</sup>* with respect to *μt* has the greatest value, *t* is not the most influential input variable in terms of the absolute change of *Pf* due to the uncertainty (variance) of the input variable *t*. A better measure of sensitivity is obtained by multiplying the partial derivatives by the standard deviations of the respective input variables (see Figure 7b). Ranking according to *Di* gives the sensitivity ranking of input variables as *fy*, *F1*, *F2*, *t*, and *b*.

**Figure 7.** Derivative-based local SA of failure probability *Pf*: (**a**) Derivatives; (**b**) Sensitivity index *Di*. **Figure 7.** Derivative-based local SA of failure probability *P<sup>f</sup>* : (**a**) Derivatives; (**b**) Sensitivity index *D<sup>i</sup>* .

The plots in Figure 7 are approximately symmetrical about the vertical axis, but not perfectly symmetrical. The small amount of asymmetry is due to the small skewness of resistance *R* in Equation (1) (see Equation (29)). Perfect symmetry of the curves would occur if *F* and *R* had zero skewness (symmetric pdfs of both *F* and *R*). A small amount of asymmetry is graphically visible upon mirroring the solid curves to the The plots in Figure 7 are approximately symmetrical about the vertical axis, but not perfectly symmetrical. The small amount of asymmetry is due to the small skewness of resistance *R* in Equation (1) (see Equation (29)). Perfect symmetry of the curves would occur if *F* and *R* had zero skewness (symmetric pdfs of both *F* and *R*).

dashed curves (see Figure 8). The dashed curves are artificial, showing the left-right asymmetry of the solid curves. A small amount of asymmetry is graphically visible upon mirroring the solid curves to the dashed curves (see Figure 8). The dashed curves are artificial, showing the left-right asymmetry of the solid curves. *Symmetry* **2020**, *12*, x FOR PEER REVIEW 11 of 20

**Figure 8.** Derivative-based local SA of failure probability *Pf*: (**a**) Derivatives; (**b**) Sensitivity index *Di*. **Figure 8.** Derivative-based local SA of failure probability *P<sup>f</sup>* : (**a**) Derivatives; (**b**) Sensitivity index *D<sup>i</sup>* .

In Figure 8a, the dashed curves are lower than the solid curves on the left side of the graph. On

For the case study, contrast *Pf* indices are depicted in Figures 9–13. All contrast *Pf* indices were

In the interval *P<sup>f</sup>* ∈[0.1, 0.9], the plot of *C<sup>i</sup>* is a concave function with approximately left-right symmetry. The sum of indices *C*<sup>1</sup> + *C*<sup>2</sup> is the same as what would have been obtained had we introduced only one random variable for *F* with a Gaussian pdf with a mean value of *μF* = 309.56 kN +*μP* and standard deviation of *σF* = 33.94 kN: *C*<sup>2</sup> + *C*1 = CF. The sum of indices *C<sup>3</sup>* + *C<sup>4</sup>* + *C<sup>5</sup>* is the same as what would have been obtained had we introduced only one random variable for *R* with a threeparameter lognormal pdf with parameters *μR* = 412.68 kN, *σR* = 34.057 kN, and *aR* = 0.111: *C*<sup>3</sup> + *C*<sup>4</sup> + *C*<sup>5</sup>

The slight asymmetry of the indices is of the same type as was described in the previous chapter for indices *Di*. For example, for *Pf* = 0.3, indices *C*1, *C*2, and *C*12 (load action) have slightly smaller values and indices *C*3, *C*4, *C*5, *C*34, *C*35, *C*45, and *C*345 (resistance) have slightly higher values, compared to the

> (**a**) (**b**) **Figure 9.** (**a**) First-order contrast *P<sup>f</sup>* indices and (**b**) second-order contrast *Pf* indices.

In the interval *Pf* ∈ [0.1, 0.9], the first-, fourth-, and fifth-order indices generally have higher

computed numerically using Equation (30) for the interval *P<sup>f</sup>* ∈ [9.35 × 10−8, 1–1.51 × 10−8].

graph, and the opposite would be true on the right sides of the graphs.

perfect symmetry. For the other indices, there is a mix of both influences.

values than the second- and third-order indices.

*5.2. Global ROSA—Contrast Pf Indices* 

= *CR*.

*Symmetry* **2020**

In Figure 8a, the dashed curves are lower than the solid curves on the left side of the graph. On the right side of the graph, the opposite is true. The same is observed in Figure 8b. A small amount of asymmetry occurs due to the small positive skewness of *R*. If *R* had a (theoretically) negative skewness, then the dashed curves would be higher than the solid curves on the left sides of each graph, and the opposite would be true on the right sides of the graphs. symmetry. The sum of indices *C*<sup>1</sup> + *C*<sup>2</sup> is the same as what would have been obtained had we introduced only one random variable for *F* with a Gaussian pdf with a mean value of *μF* = 309.56 kN +*μP* and standard deviation of *σF* = 33.94 kN: *C*<sup>2</sup> + *C*1 = CF. The sum of indices *C<sup>3</sup>* + *C<sup>4</sup>* + *C<sup>5</sup>* is the same as what would have been obtained had we introduced only one random variable for *R* with a threeparameter lognormal pdf with parameters *μR* = 412.68 kN, *σR* = 34.057 kN, and *aR* = 0.111: *C*<sup>3</sup> + *C*<sup>4</sup> + *C*<sup>5</sup> = *CR*.

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 11 of 20

(**a**) (**b**) **Figure 8.** Derivative-based local SA of failure probability *Pf*: (**a**) Derivatives; (**b**) Sensitivity index *Di*.

graph, and the opposite would be true on the right sides of the graphs.

In Figure 8a, the dashed curves are lower than the solid curves on the left side of the graph. On the right side of the graph, the opposite is true. The same is observed in Figure 8b. A small amount of asymmetry occurs due to the small positive skewness of *R*. If *R* had a (theoretically) negative skewness, then the dashed curves would be higher than the solid curves on the left sides of each

For the case study, contrast *Pf* indices are depicted in Figures 9–13. All contrast *Pf* indices were

In the interval *P<sup>f</sup>* ∈[0.1, 0.9], the plot of *C<sup>i</sup>* is a concave function with approximately left-right

#### *5.2. Global ROSA—Contrast P<sup>f</sup> Indices* The slight asymmetry of the indices is of the same type as was described in the previous chapter for indices *Di*. For example, for *Pf* = 0.3, indices *C*1, *C*2, and *C*12 (load action) have slightly smaller values

*5.2. Global ROSA—Contrast Pf Indices* 

For the case study, contrast *P<sup>f</sup>* indices are depicted in Figures 9–13. All contrast *P<sup>f</sup>* indices were computed numerically using Equation (30) for the interval *<sup>P</sup><sup>f</sup>* <sup>∈</sup> [9.35 <sup>×</sup> <sup>10</sup>−<sup>8</sup> , 1–1.51 <sup>×</sup> <sup>10</sup>−<sup>8</sup> ]. and indices *C*3, *C*4, *C*5, *C*34, *C*35, *C*45, and *C*345 (resistance) have slightly higher values, compared to the perfect symmetry. For the other indices, there is a mix of both influences.

**Figure 9.** (**a**) First-order contrast *P<sup>f</sup>* indices and (**b**) second-order contrast *Pf* indices. **Figure 9.** (**a**) First-order contrast *P<sup>f</sup>* indices and (**b**) second-order contrast *P<sup>f</sup>* indices. , *12*, x FOR PEER REVIEW 12 of 20 *Symmetry* **2020**, *12*, x FOR PEER REVIEW 12 of 20

**Figure 10.** (**a**) Third-order contrast *P<sup>f</sup>* indices and (**b**) fourth-order contrast *Pf* indices. **Figure 10.** (**a**) Third-order contrast *P<sup>f</sup>* indices and (**b**) fourth-order contrast *P<sup>f</sup>* indices. **Figure 10.** (**a**) Third-order contrast *P<sup>f</sup>* indices and (**b**) fourth-order contrast *Pf* indices.

**Figure 11.** (**a**) Fifth-order contrast *P<sup>f</sup>* indices and (**b**) all-order contrast *Pf* indices for *P<sup>f</sup>* = 7.2 × 10−5 . **Figure 11.** (**a**) Fifth-order contrast *P<sup>f</sup>* indices and (**b**) all-order contrast *Pf* indices for *P<sup>f</sup>* = 7.2 × 10−5 . **Figure 11.** (**a**) Fifth-order contrast *P<sup>f</sup>* indices and (**b**) all-order contrast *P<sup>f</sup>* indices for *<sup>P</sup><sup>f</sup>* <sup>=</sup> 7.2 <sup>×</sup> <sup>10</sup>−<sup>5</sup> .

(**a**) (**b**) **Figure 12.** Total contrast *Pf* indices: (**a**) All *P<sup>f</sup>* and (**b**) low *Pf*.

(**a**) (**b**) **Figure 12.** Total contrast *Pf* indices: (**a**) All *P<sup>f</sup>* and (**b**) low *Pf*.

**Figure 13.** Details of contrast *Pf* indices for low *Pf*.

**Figure 13.** Details of contrast *Pf* indices for low *Pf*.

In civil engineering, the target values of *Pf* for reliability classes RC1, RC2, and RC3 taken from [4] are 8.5 × 10−6, 7.2 × 10−5, and 4.8 × 10−4 (also see [19]). Figure 11b shows the contribution of all 31 indices for target value *Pf* = 7.2 × 10−5. First-order indices are represented minimally, where ∑*Si* = 0.017.

In civil engineering, the target values of *Pf* for reliability classes RC1, RC2, and RC3 taken from [4] are 8.5 × 10−6, 7.2 × 10−5, and 4.8 × 10−4 (also see [19]). Figure 11b shows the contribution of all 31 indices for target value *Pf* = 7.2 × 10−5. First-order indices are represented minimally, where ∑*Si* = 0.017. *Symmetry* 

(**a**) (**b**)

*P<sup>f</sup> <sup>f</sup>*indices for *<sup>f</sup>* = × 10−5.

(**a**)**b**)

.

**Figure 11.** (**a**) Fifth-order contrast *P<sup>f</sup>* indices and (**b**) all-order contrast *Pf* indices for *P<sup>f</sup>* = 7.2 × 10−5

(**a**) (**b**) **Figure 10.** (**a**) Third-order contrast *P<sup>f</sup>* indices and (**b**) fourth-order contrast *Pf* indices.

(**a**) (**b**)**Figure 10.** (**a**) Third-order contrast *P*indices and (**b**) fourth-order contrast *P* indices.

**Figure 12.** Total contrast *Pf* indices: (**a**) All *P<sup>f</sup>* and (**b**) low *Pf*. **Figure 12.** Total contrast *P<sup>f</sup>* indices: (**a**) All *P<sup>f</sup>* and (**b**) low *P<sup>f</sup>* . *<sup>f</sup>***a**) and **b**) *Pf*.

**Figure 13.** Details of contrast *Pf* indices for low *Pf*. *Pf*indices low *P<sup>f</sup>* indices for low *P<sup>f</sup>* .

In civil engineering, the target values of *Pf* for reliability classes RC1, RC2, and RC3 taken from [4] are 8.5 × 10−6, 7.2 × 10−5, and 4.8 × 10−4 (also see [19]). Figure 11b shows the contribution of all 31 indices for target value *Pf* = 7.2 × 10−5. First-order indices are represented minimally, where ∑*Si* = 0.017. In civil engineering, the target values of*P* for reliability classes RC1, RC2, and RC3 taken from [4] × −6, × 10−5, and × 10−4 (also see contribution indices for target value *<sup>P</sup>f* = 7.2 × 10. First-order indices are represented minimally, where ∑*S* = 0.017. **Figure 13.** Details of contrast *<sup>P</sup><sup>f</sup>* In the interval *P<sup>f</sup>* ∈[0.1, 0.9], the plot of *C<sup>i</sup>* is a concave function with approximately left-right symmetry. The sum of indices *C*<sup>1</sup> + *C*<sup>2</sup> is the same as what would have been obtained had we introduced only one random variable for *F* with a Gaussian pdf with a mean value of µ*<sup>F</sup>* = 309.56 kN +µ*<sup>P</sup>* and standard deviation of σ*<sup>F</sup>* = 33.94 kN: *C*<sup>2</sup> + *C*<sup>1</sup> = CF. The sum of indices *C*<sup>3</sup> + *C*<sup>4</sup> + *C*<sup>5</sup> is the same as what would have been obtained had we introduced only one random variable for *R* with a three-parameter lognormal pdf with parameters µ*<sup>R</sup>* = 412.68 kN, σ*<sup>R</sup>* = 34.057 kN, and *a<sup>R</sup>* = 0.111: *C*<sup>3</sup> + *C*<sup>4</sup> + *C*<sup>5</sup> = *CR*.

The slight asymmetry of the indices is of the same type as was described in the previous chapter for indices *D<sup>i</sup>* . For example, for *P<sup>f</sup>* = 0.3, indices *C*1, *C*2, and *C*<sup>12</sup> (load action) have slightly smaller values and indices *C*3, *C*4, *C*5, *C*34, *C*35, *C*45, and *C*<sup>345</sup> (resistance) have slightly higher values, compared to the perfect symmetry. For the other indices, there is a mix of both influences.

In the interval *P<sup>f</sup>* ∈[0.1, 0.9], the first-, fourth-, and fifth-order indices generally have higher values than the second- and third-order indices.

In civil engineering, the target values of *P<sup>f</sup>* for reliability classes RC1, RC2, and RC3 taken from [4] are 8.5 <sup>×</sup> <sup>10</sup>−<sup>6</sup> , 7.2 <sup>×</sup> <sup>10</sup>−<sup>5</sup> , and 4.8 <sup>×</sup> <sup>10</sup>−<sup>4</sup> (also see [19]). Figure 11b shows the contribution of all 31 indices for target value *<sup>P</sup><sup>f</sup>* <sup>=</sup> 7.2 <sup>×</sup> <sup>10</sup>−<sup>5</sup> . First-order indices are represented minimally, where P *S<sup>i</sup>* = 0.017. On the contrary, the representation of higher-order indices is significant, especially those related to *fy*, *F*1, and *F*<sup>2</sup> (see Figure 11b).

In Figure 11, *f<sup>y</sup>* occurs in all significant parts of the graph, but the same is true for *F*<sup>1</sup> or *F*2. Determining the order of importance of input variables using 31 indices can be difficult. The use of total indices *CTi* is more practical. Input variables are ranked based on *CTi* as *fy*, *F*1, *F*2, *t*, and *b* (see Figure 12). This is the same ranking as was found using index *D<sup>i</sup>* (Figure 7b).

Figure 12b shows the total sensitivity indices for small *P<sup>f</sup>* , which are relevant for the design of building structures. Figure 13 shows the local extremes of some sensitivity indices in the interval of small *P<sup>f</sup>* . Interestingly, the sensitivity indices of small *P<sup>f</sup>* have plots that are not obvious (cannot be *fy*, *F1*, and *F2* (see Figure 11b).

extrapolated) from the plots in the interval *P<sup>f</sup>* ∈ [0.1, 0.9]. Similar local extremes as in Figure 13 were not observed for *D<sup>i</sup>* in Figure 7. small *Pf*. Interestingly, the sensitivity indices of small *Pf* have plots that are not obvious (cannot be extrapolated) from the plots in the interval *P<sup>f</sup>* ∈ [0.1, 0.9]. Similar local extremes as in Figure 13 were not observed for *D<sup>i</sup>* in Figure 7.

Figure 12). This is the same ranking as was found using index *Di* (Figure 7b).

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 13 of 20

On the contrary, the representation of higher-order indices is significant, especially those related to

In Figure 11, *fy* occurs in all significant parts of the graph, but the same is true for *F<sup>1</sup>* or *F2*. Determining the order of importance of input variables using 31 indices can be difficult. The use of total indices *CTi* is more practical. Input variables are ranked based on *CTi* as *fy*, *F*1, *F*2, *t*, and *b* (see

#### *5.3. Global ROSA—Contrast Q Indices 5.3. Global ROSA—Contrast Q Indices*

In the case study, contrast *Q* indices were estimated using the Latin Hypercube Sampling (LHS) method [43,44], according to the procedure in [35]. Indices *Q<sup>i</sup>* were estimated from Equation (17) using double-nested-loop computation. In the outer loop, E[·] was computed using one thousand runs of the LHS method. In the nested loop, conditional contrast values were computed using four million runs of the LHS method. The unconditional contrast value in the denominator was computed using four million runs of the LHS method. Higher-order indices were estimated similarly. In the case study, contrast *Q* indices were estimated using the Latin Hypercube Sampling (LHS) method [43,44], according to the procedure in [35]. Indices *Qi* were estimated from Equation (17) using double-nested-loop computation. In the outer loop, E[·] was computed using one thousand runs of the LHS method. In the nested loop, conditional contrast values were computed using four million runs of the LHS method. The unconditional contrast value in the denominator was computed using four million runs of the LHS method. Higher-order indices were estimated similarly.

The target value *<sup>P</sup><sup>f</sup>* <sup>=</sup> 7.2 <sup>×</sup> <sup>10</sup>−<sup>5</sup> is considered according to [4]. In Equation (7), the design value of resistance *R<sup>d</sup>* is considered as the 0.0036-quantile and the design load value *F<sup>d</sup>* is considered as the 0.9963-quantile (see Figure 6). Sensitivity analysis is performed for *R* with a three-parameter lognormal pdf when no or one variable in Equation (26) is fixed; otherwise, a Gaussian pdf is used in the stochastic model. The target value *Pf* = 7.2 × 10−5 is considered according to [4]. In Equation (7), the design value of resistance *R<sup>d</sup>* is considered as the 0.0036-quantile and the design load value *F<sup>d</sup>* is considered as the 0.9963-quantile (see Figure 6). Sensitivity analysis is performed for *R* with a three-parameter lognormal pdf when no or one variable in Equation (26) is fixed; otherwise, a Gaussian pdf is used in the stochastic model.

It can be noted that standard design quantiles *F<sup>d</sup>* = *R<sup>d</sup>* = 321.01 kN computed using Equation (7) consider *F* and *R* with a Gaussian pdf. However, the design resistance value computed using a three-parameter lognormal pdf (stochastic model) is 325.00 kN. The small difference is because the skewness *a<sup>R</sup>* = 0.111 was neglected in Equation (7). It can be noted that standard design quantiles *Fd* = *Rd* = 321.01 kN computed using Equation (7) consider *F* and *R* with a Gaussian pdf. However, the design resistance value computed using a threeparameter lognormal pdf (stochastic model) is 325.00 kN. The small difference is because the skewness *aR* = 0.111 was neglected in Equation (7).

The SA results of the 0.9963-quantile of *F* are depicted in Figure 14a. Input random variables for *<sup>F</sup>* are considered according to Table 1, where the value of <sup>µ</sup>*<sup>P</sup>* for *<sup>P</sup><sup>f</sup>* <sup>=</sup> 7.2 <sup>×</sup> <sup>10</sup>−<sup>5</sup> is µ*<sup>P</sup>* = −79.592 kN. Input random variables for *R* are considered according to Table 2. The results of SA of the 0.0036-quantile of *R* are depicted in Figure 14b. The SA results of the 0.9963-quantile of *F* are depicted in Figure 14a. Input random variables for *F* are considered according to Table 1, where the value of *μP* for *Pf* = 7.2 × 10−5 is *μP* = −79.592 kN. Input random variables for *R* are considered according to Table 2. The results of SA of the 0.0036-quantile of *R* are depicted in Figure 14b.

**Figure 14.** Contrast *Q* indices: (**a**) 0.9963-quantile of *F* and (**b**) 0.0036-quantile of *R*. **Figure 14.** Contrast *Q* indices: (**a**) 0.9963-quantile of *F* and (**b**) 0.0036-quantile of *R*.

By computing total indices *QT*1 = 0.71, *QT*2 = 0.70 and *QT*3 = 0.86, *QT*4 = 0.59, and *QT*5 = 0.13, the order of importance of input variables can be determined as *F*1 and *F*2 and *fy*, *t*, and *b*. Variables *F* and *R* have the same weight in Equation (2) and therefore, the order of importance of all five input variables can be determined as *fy*, *F*1, *F*2, *t*, and *b*, based on the estimates of all *QTi*. By computing total indices *QT*<sup>1</sup> = 0.71, *QT*<sup>2</sup> = 0.70 and *QT*<sup>3</sup> = 0.86, *QT*<sup>4</sup> = 0.59, and *QT*<sup>5</sup> = 0.13, the order of importance of input variables can be determined as *F*<sup>1</sup> and *F*<sup>2</sup> and *fy*, *t*, and *b*. Variables *F* and *R* have the same weight in Equation (2) and therefore, the order of importance of all five input variables can be determined as *fy*, *F*1, *F*2, *t*, and *b*, based on the estimates of all *QTi*.

This is a typical example of how the ranking of input parameters based on total indices can give reliable results. The results are satisfactory, although ROSA is not evaluated directly using *Pf*; it is "only" based on the SA of design quantiles *Rd* and *Fd*. This is a typical example of how the ranking of input parameters based on total indices can give reliable results. The results are satisfactory, although ROSA is not evaluated directly using *P<sup>f</sup>* ; it is "only" based on the SA of design quantiles *R<sup>d</sup>* and *Fd*.

In the presented study, the results for other values of the α-quantile are the same as in Figure 14. In practice, this means that the change in µ*<sup>P</sup>* (generally a change in µ*F*) is not reflected in the results of contrast *Q* indices.
