*2.3. Sobol Sensitivity Indices—Sobol Indices*

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*S*

*S*

*i*

*i*

Sobol variance-based sensitivity analysis is the most frequently used SA method [2,3]. Sobol SA is based on the decomposition of the variance of the model output. Sobol SA estimates the degree of variance that each parameter contributes to the model output, including interaction effects. The first-order *S<sup>i</sup>* index can be written as SA estimates the degree of variance that each parameter contributes to the model output, including interaction effects. The first-order *Si* index can be written as *XRVERV* 

*i*

$$S\_{\bar{i}} = \frac{V(\mathbb{R}) - E(V(\mathbb{R}|X\_{\bar{i}}))}{V(\mathbb{R})} \,\,\,\,\tag{18}$$

where *E*(·) is considered across all *X<sup>i</sup>* . The total effect index *STi*, which measures first and higher-order effects (interactions) of variable *X<sup>i</sup> ,* is another popular variance-based measure [1] higher-order effects (interactions) of variable *Xi,* is another popular variance-based meas- *XRVERV* <sup>~</sup>

$$S\_{\bar{i}} = \frac{V(R) - E(V(R|X\_{\sim \bar{i}}))}{V(R)},\tag{19}$$

where the second term in the numerator contains the conditional variance evaluated for input random variable *X<sup>i</sup>* and fixed variables (*X*1, *X*2, . . . , *Xi–*1, *Xi+*1, . . . , *XM*). where the second term in the numerator contains the conditional variance evaluated for input random variable *Xi* and fixed variables (*X*1, *X*2, …, *Xi–*1, *Xi+*1, …, *XM*).

Sobol SA is dependent only on the variance. The similarity of the results of Sobol SA and quantile-oriented SA can be sought in connection with the degree of the influence of the variance on the quantile. Sobol SA is dependent only on the variance. The similarity of the results of Sobol SA and quantile-oriented SA can be sought in connection with the degree of the influence of the variance on the quantile.

#### **3. Resistance of Steel Member under Compression 3. Resistance of Steel Member under Compression**

Most forms of civil engineering structures are designed using European unified design rules [75]—Eurocodes. The limit state is the structural condition past which it no longer satisfies the pertinent design criteria [76]. Limit state design requires the structure to satisfy two fundamental conditions: the ultimate limit state (strength and stability) and the serviceability limit state (deflection, cracking, vibration). Most forms of civil engineering structures are designed using European unified design rules [75]—Eurocodes. The limit state is the structural condition past which it no longer satisfies the pertinent design criteria [76]. Limit state design requires the structure to satisfy two fundamental conditions: the ultimate limit state (strength and stability) and the serviceability limit state (deflection, cracking, vibration).

The aim of the case study presented in this article is to analyse the static resistance (load-carrying capacity) of a slender steel member, which is limited by the strength of the material and stability. The resistance *R* is a random variable that depends on material and geometrical characteristics, which are generally random variables. A structure is considered to satisfy the ultimate limit state criterion if the random realization of the external load action is less than the low (design) quantile of load-carrying capacity *R<sup>d</sup>* . Standard [59] enables the determination of design value *R<sup>d</sup>* as 0.1 percentile [77–81]. The aim of the case study presented in this article is to analyse the static resistance (load-carrying capacity) of a slender steel member, which is limited by the strength of the material and stability. The resistance *R* is a random variable that depends on material and geometrical characteristics, which are generally random variables. A structure is considered to satisfy the ultimate limit state criterion if the random realization of the external load action is less than the low (design) quantile of load-carrying capacity *Rd*. Standard [59] enables the determination of design value *Rd* as 0.1 percentile [77–81].

The stochastic model of ultimate limit state of a hot-rolled steel member under longitudinal compression load action *F* is shown in Figure 4a. Biaxially symmetrical cross section HEA 180 of steel grade S235 is considered; see Figure 4b. The stochastic model of ultimate limit state of a hot-rolled steel member under longitudinal compression load action *F* is shown in Figure 4a. Biaxially symmetrical cross section HEA 180 of steel grade S235 is considered; see Figure 4b.

**Figure 4.** Static model: (**a**) steel member under compression; (**b**) Cross-section HEA 180. **Figure 4.** Static model: (**a**) steel member under compression; (**b**) Cross-section HEA 180.

The resistance of the steel structural member shown in Figure 4a was derived in [82] using the equation *e* = *e*0/(1 − *F*/*Fcr*), where *Fcr* is Euler's critical load. Increasing the external load action *F* increases the compressive stress *σx* until the yield strength *fy* is attained in the middle of the span in the lower (extremely compressed) part of the cross-section; see The resistance of the steel structural member shown in Figure 4a was derived in [82] using the equation *e* = *e*0/(1 − *F*/*Fcr*), where *Fcr* is Euler's critical load. Increasing the external load action *F* increases the compressive stress *σ<sup>x</sup>* until the yield strength *f<sup>y</sup>* is attained in the middle of the span in the lower (extremely compressed) part of the cross-

Figure 4a. Hooke's law with Young's modulus *E* is considered. The dependence of *σ<sup>x</sup>* on

section; see Figure 4a. Hooke's law with Young's modulus *E* is considered. The dependence of *σ<sup>x</sup>* on *F* is non-linear if *e*<sup>0</sup> > 0, where *F* < *Fcr*. The elastic resistance *R* (unit Newton) is the maximum load action *F*; a higher value of force *F* would cause overstressing and structural failure. The resistance *R* can be computed using the response function [82]

$$R = \frac{Q \cdot A + F\_{cr} \cdot \mathcal{W}\_z - \sqrt{Q^2 \cdot A^2 + 2 \cdot A \cdot F\_{cr} \cdot \mathcal{W}\_z \cdot \left(|\mathbf{e}\_0| \cdot F\_{cr} - f\_y \cdot \mathcal{W}\_z\right) + F\_{cr}^2 \cdot \mathcal{W}\_z^2}}{2 \cdot \mathcal{W}\_z},\tag{20}$$

where

$$Q = F\_{cr} \cdot |e\_0| + f\_y \cdot \mathcal{W}\_{z\prime} \tag{21}$$

$$A = 2 \cdot b \cdot t\_2 + t\_1 \cdot (h - 2t\_2),\tag{22}$$

$$F\_{\rm cr} = \pi^2 E I\_z / L^2 \,\text{.}\tag{23}$$

$$I\_z = 2 \cdot t\_2 \cdot b^3 / 12 + (h - 2t\_2) \cdot t\_1^3 / 12,\tag{24}$$

$$\mathbf{W}\_{\mathbf{z}} = \mathbf{2} \cdot \mathbf{l}\_{\mathbf{z}} / \mathbf{b}\_{\mathbf{z}} \tag{25}$$

where *e*<sup>0</sup> is the amplitude of initial axis curvature, *L* is the member length, *h* is the crosssectional height, *b* is the cross-sectional width, *t*<sup>1</sup> is the web thickness and *t*<sup>2</sup> is the flange thickness. These variables are used to further compute the following variables: *A* is crosssectional area and *I<sup>z</sup>* is second moment of area around axis *z*.

It can be noted that *e*<sup>0</sup> is the amplitude of pure geometrical imperfection with an idealized shape according to the elastic critical buckling mode [83]. Amplitude *e*<sup>0</sup> is not an equivalent geometrical imperfection [84–86], which would replace the influence of other imperfections, such as the residual stress. In Equation (20), the influence of residual stress is neglected.

The member length *L* is a deterministic parameter. Equation (20) is a non-linear, non-monotonic function for *R* > 0 that has the typical elastic resistance properties of a compressed member with initial material and geometrical imperfections with the exception of residual stress. Although *R* is a vector quantity, the direction is still horizontal; see Figure 4a) and only the magnitude is a random variable. Thus, in this article, the resistance *R* is examined as a scalar model output.

The material and geometrical characteristics of hot-rolled steel beams have been studied experimentally [87,88]. Studies [82,89–91] have confirmed that the variance of *t*<sup>1</sup> and *h* have a minimal influence on *R*. Therefore, these variables can be considered as deterministic with values *t*<sup>1</sup> = 6 mm and *h* = 171 mm. The input random variables are listed in Table 1. All random variables are statistically independent of each other.



All random variables have Gauss pdf, but with the condition *f<sup>y</sup>* > 0, *E* > 0, *t*<sup>2</sup> > 0 and *b* > 0. However, negative realizations of random variables *fy*, *E*, *t*<sup>2</sup> and *b* practically never occur if the LHS method [92,93] is used with no more than tens of millions of runs. Theoretically, if *f<sup>y</sup>* → 0 then *R* → 0 (due to no stress), if *E* → 0 then *R* → 0 (due to zero stiffness), if *e*<sup>0</sup> → 0

then *R* → *Fcr* or *R* → *fy*·*A* (pure buckling for high *L* or simple compression for low *L*), if *L* → 0 then *R* → *fy*·*A* (simple compression). the member with non-dimensional slenderness [94] . The common non-dimensional slenderness of a strut in an efficient structural system is around one, but struts usu-

The member length *L* is a deterministic parameter that changes step-by-step as *L* = 0.001, 0.424, 0.849, …, 6.366 m. The step value is *L*0/10, where *L*0 = 4.244 m is the length of

#### **4. Results of Sensitivity Analysis** ally do not have non-dimensional slenderness higher than two [95]. The slenderness is

**4. Results of Sensitivity Analysis**

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The member length *L* is a deterministic parameter that changes step-by-step as *L* = 0.001, 0.424, 0.849, . . . , 6.366 m. The step value is *L*0/10, where *L*<sup>0</sup> = 4.244 m is the length of the member with non-dimensional slenderness [94] *λ*= 1.0. The common nondimensional slenderness of a strut in an efficient structural system is around one, but struts usually do not have non-dimensional slenderness higher than two [95]. The slenderness is directly proportional to the length. It is possible, for the presented case study, to write the transformation *L* =*λ*· *L*0, which makes it easier to understand the lengths. directly proportional to the length. It is possible, for the presented case study, to write the transformation *L* = · *L*0, which makes it easier to understand the lengths. All three types of SA are based on double-nested-loop algorithms. Estimation of indices was software-based by implementing randomized Latin Hypercube Samplingbased Monte Carlo simulation (LHS) algorithms [92,93], which have been tuned for sensitivity assessments [78,80]. Using LHS runs, the outer loop is repeated 2000 times to esti-

All three types of SA are based on double-nested-loop algorithms. Estimation of indices was software-based by implementing randomized Latin Hypercube Sampling-based Monte Carlo simulation (LHS) algorithms [92,93], which have been tuned for sensitivity assessments [78,80]. Using LHS runs, the outer loop is repeated 2000 times to estimate the arithmetic mean *E*(·) of the samples (*l*, *l* <sup>2</sup> or variance), which are estimated using an inner loop algorithm. The inner loop is repeated 4 million times (4 million LHS runs) to compute statistics (*l*, *l* <sup>2</sup> or variance) with some random realizations fixed by the outer loop. mate the arithmetic mean *E*(·) of the samples (*l*, *l* 2 or variance), which are estimated using an inner loop algorithm. The inner loop is repeated 4 million times (4 million LHS runs) to compute statistics (*l*, *l* 2 or variance) with some random realizations fixed by the outer loop. The subject of interest for the two quantile-oriented SA is the 0.001-quantile *θ*\* of *R*. The estimate *l* quantifies the population distribution around the 0.001-quantile *θ*\*, where

The subject of interest for the two quantile-oriented SA is the 0.001-quantile *θ*\* of *R*. The estimate *l* quantifies the population distribution around the 0.001-quantile *θ*\*, where 0.001-quantile *θ*\* is estimated as the 4000th smallest value in the set of four million LHS runs ordered from smallest to largest [78,80]. 0.001-quantile *θ*\* is estimated as the 4000th smallest value in the set of four million LHS runs ordered from smallest to largest [78,80]. The estimates of the unconditional characteristics in the denominators of the indices are computed using four million runs of the LHS method. Higher-order indices are esti-

The estimates of the unconditional characteristics in the denominators of the indices are computed using four million runs of the LHS method. Higher-order indices are estimated similarly. mated similarly. The same set of (pseudo-) random numbers is used in each member length *L,* hereby

The same set of (pseudo-) random numbers is used in each member length *L*, hereby ensuring that sampling and numerical errors do not swamp the statistics being sought [96,97]. ensuring that sampling and numerical errors do not swamp the statistics being sought [96,97].

Figures 5–8 show contrast *Q* indices, *K* indices and Sobol indices for four selected member lengths corresponding to non-dimensional slenderness values *λ* = 0, 0.5, 1, 1.5. The outer coloured ring displays 31 sensitivity indices, and the inner white-grey pie chart shows the representation of member lengths of first-order indices (white area of the chart) and higher-order indices (grey areas). Figures 5–8 show contrast *Q* indices, *K* indices and Sobol indices for four selected member lengths corresponding to non-dimensional slenderness values = 0, 0.5, 1, 1.5. The outer coloured ring displays 31 sensitivity indices, and the inner white-grey pie chart shows the representation of member lengths of first-order indices (white area of the chart) and higher-order indices (grey areas).

= 0).

**Figure 5. Figure 5.** Comparison of three types of Comparison of three types of sensitivity analysis (SA) for sensitivity analysis (SA) for *L* = 0 m ( *L* = 0 m ( *λ* = 0). 

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**Figure 6.** Comparison of three types of SA for *L* = 2.122 m ( = 0.5). **Figure 6.** Comparison of three types of SA for *L* = 2.122 m (*λ* = 0.5). **Figure 6.** Comparison of three types of SA for *L* = 2.122 m ( = 0.5). **Figure 6.** Comparison of three types of SA for *L* = 2.122 m ( = 0.5).

**Figure 7.** Comparison of three types of SA for *L* = 4.244 m ( = 1.0). **Figure 7.** Comparison of three types of SA for *L* = 4.244 m ( = 1.0). **Figure 7.** Comparison of three types of SA for *L* = 4.244 m (*λ* = 1.0). **Figure 7.** Comparison of three types of SA for *L* = 4.244 m ( = 1.0).

**Figure 8.** Comparison of three types of SA for *L* = 6.366 m ( = 1.5). **Figure 8.** Comparison of three types of SA for *L* = 6.366 m ( = 1.5). **Figure 8.** Comparison of three types of SA for *L* = 6.366 m ( = 1.5). **Figure 8.** Comparison of three types of SA for *L* = 6.366 m (*λ* = 1.5).

The results in Figures 5–8 show that Sobol indices have the largest proportion of firstorder sensitivity indices; see the white area in the inner circles in Figures 5–8. Small The results in Figures 5–8 show that Sobol indices have the largest proportion of firstorder sensitivity indices; see the white area in the inner circles in Figures 5–8. Small The results in Figures 5–8 show that Sobol indices have the largest proportion of firstorder sensitivity indices; see the white area in the inner circles in Figures 5–8. Small The results in Figures 5–8 show that Sobol indices have the largest proportion of first-order sensitivity indices; see the white area in the inner circles in Figures 5–8. Small

higher-order indices make Sobol first-order indices transparent without the need to eval-

higher-order indices make Sobol first-order indices transparent without the need to eval-

The new quantile-oriented *K* indices also have a relatively small proportion of higherorder indices (grey areas in the inner circles), and thus approach Sobol indices with their properties. *Q* indices have the lowest proportion of first-order sensitivity indices and a

The new quantile-oriented *K* indices also have a relatively small proportion of higherorder indices (grey areas in the inner circles), and thus approach Sobol indices with their properties. *Q* indices have the lowest proportion of first-order sensitivity indices and a

order indices (grey areas in the inner circles), and thus approach Sobol indices with their properties. *Q* indices have the lowest proportion of first-order sensitivity indices and a

higher-order indices make Sobol first-order indices transparent without the need to evaluate total indices. Unfortunately, Sobol indices are not quantile-oriented. *Symmetry* **2021**, *13*, x FOR PEER REVIEW 11 of 22 *Symmetry* **2021**, *13*, x FOR PEER REVIEW 11 of 22

> The new quantile-oriented *K* indices also have a relatively small proportion of higherorder indices (grey areas in the inner circles), and thus approach Sobol indices with their properties. *Q* indices have the lowest proportion of first-order sensitivity indices and a high proportion of higher-order indices (interaction effects), which makes the results less comprehensible, and the evaluation of total indices is then necessary. high proportion of higher-order indices (interaction effects), which makes the results less comprehensible, and the evaluation of total indices is then necessary. high proportion of higher-order indices (interaction effects), which makes the results less comprehensible, and the evaluation of total indices is then necessary.

> Figures 9 and 10 display the plots of all thirty-one *Q* indices vs. member length *L*. A finer step is used in places where the curves change course faster. Figures 9 and 10 display the plots of all thirty-one *Q* indices vs. member length *L*. A finer step is used in places where the curves change course faster. Figures 9 and 10 display the plots of all thirty-one *Q* indices vs. member length *L*. A finer step is used in places where the curves change course faster.

**Figure 9.** *Q* indices: (**a**) first-order sensitivity indices; (**b**) second-order sensitivity indices. **Figure 9.** *Q* indices: (**a**) first-order sensitivity indices; (**b**) second-order sensitivity indices. **Figure 9.** *Q* indices: (**a**) first-order sensitivity indices; (**b**) second-order sensitivity indices.

**Figure 10.** *Q* indices: (**a**) third-order sensitivity indices; (**b**) fourth- and fifth-order sensitivity indices. **Figure 10.** *Q* indices: (**a**) third-order sensitivity indices; (**b**) fourth- and fifth-order sensitivity indices. **Figure 10.** *Q* indices: (**a**) third-order sensitivity indices; (**b**) fourth- and fifth-order sensitivity indices.

As for *Q* indices, the clear influence of individual variables on the 0.001-quantile of *R* is evident only after the evaluation of total indices, see Figure 11. The yield strength *f<sup>y</sup>* is dominant for low values of *L* (low slenderness), imperfection *e*0 is dominant for intermediate lengths *L* (intermediate slenderness), Young's modulus and flange thickness gain dominance in the case of long members (high slenderness). As for *Q* indices, the clear influence of individual variables on the 0.001-quantile of *R* is evident only after the evaluation of total indices, see Figure 11. The yield strength *f<sup>y</sup>* is dominant for low values of *L* (low slenderness), imperfection *e*0 is dominant for intermediate lengths *L* (intermediate slenderness), Young's modulus and flange thickness gain dominance in the case of long members (high slenderness). As for *Q* indices, the clear influence of individual variables on the 0.001-quantile of *R* is evident only after the evaluation of total indices, see Figure 11. The yield strength *f<sup>y</sup>* is dominant for low values of *L* (low slenderness), imperfection *e*<sup>0</sup> is dominant for intermediate lengths *L* (intermediate slenderness), Young's modulus and flange thickness gain dominance in the case of long members (high slenderness).

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**Figure 11.** *Q* indices: total indices. **Figure 11.** *Q* indices: total indices. **Figure 11.** *Q* indices: total indices. **Figure 11.** *Q* indices: total indices.

Figures 12 and 13 display the plots of all thirty-one *K* indices vs. member length, with the proportion of higher-order sensitivity indices being relatively small. The total *KT* indices shown in Figure 14 provide very similar (but not the same) information as the firstorder *Ki* indices depicted in Figure 12. Figures 12 and 13 display the plots of all thirty-one *K* indices vs. member length, with the proportion of higher-order sensitivity indices being relatively small. The total *K<sup>T</sup>* indices shown in Figure 14 provide very similar (but not the same) information as the first-order *K<sup>i</sup>* indices depicted in Figure 12. Figures 12 and 13 display the plots of all thirty-one *K* indices vs. member length, with the proportion of higher-order sensitivity indices being relatively small. The total *KT* indices shown in Figure 14 provide very similar (but not the same) information as the firstorder *Ki* indices depicted in Figure 12. Figures 12 and 13 display the plots of all thirty-one *K* indices vs. member length, with the proportion of higher-order sensitivity indices being relatively small. The total *KT* indices shown in Figure 14 provide very similar (but not the same) information as the firstorder *Ki* indices depicted in Figure 12.

**Figure 12.** *K* indices: (**a**) first-order sensitivity indices; (**b**) second-order sensitivity indices. **Figure 12.** *K* indices: (**a**) first-order sensitivity indices; (**b**) second-order sensitivity indices. **Figure 12.** *K* indices: (**a**) first-order sensitivity indices; (**b**) second-order sensitivity indices. **Figure 12.** *K* indices: (**a**) first-order sensitivity indices; (**b**) second-order sensitivity indices.

**Figure 13.** *K* indices: (**a**) third-order sensitivity indices; (**b**) fourth- and fifth-order sensitivity indices. **Figure 13.** *K* indices: (**a**) third-order sensitivity indices; (**b**) fourth- and fifth-order sensitivity indices. **Figure 13.** *K* indices: (**a**) third-order sensitivity indices; (**b**) fourth- and fifth-order sensitivity indices. **Figure 13.** *K* indices: (**a**) third-order sensitivity indices; (**b**) fourth- and fifth-order sensitivity indices.

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**Figure 14.** *K* indices: total indices. **Figure 14.** *K* indices: total indices. **Figure 14.** *K* indices: total indices.

Figure 15a shows Sobol first-order sensitivity indices. Sobol higher-order sensitivity indices are not shown because they are practically zero. The total indices shown in Figure 15b provide practically the same information as Sobol first-order sensitivity indices *Si*. Figure 15a shows Sobol first-order sensitivity indices. Sobol higher-order sensitivity indices are not shown because they are practically zero. The total indices shown in Figure 15b provide practically the same information as Sobol first-order sensitivity indices *S<sup>i</sup>* . Figure 15a shows Sobol first-order sensitivity indices. Sobol higher-order sensitivity indices are not shown because they are practically zero. The total indices shown in Figure 15b provide practically the same information as Sobol first-order sensitivity indices *Si*.

**Figure 15.** Sobol indices: (**a**) first-order sensitivity indices; (**b**) total indices. **Figure 15.** Sobol indices: (**a**) first-order sensitivity indices; (**b**) total indices. **Figure 15.** Sobol indices: (**a**) first-order sensitivity indices; (**b**) total indices.

Examples of percentage differences between first-order indices are as follows. The yield strength has the greatest influence for *L* = 0 m, with *Q*1 being 37% smaller than *S*1, and *K*1 being 3% smaller than *S*1. Imperfection *e*0 has the greatest influence for *L* ≈ 3.8 m, with *Q*3 being 46% smaller than *S*3, and *K*3 being 16% smaller than *S*3. *Ki* indices are closer to *Si* indices (compared to *Qi* indices). Examples of percentage differences between first-order indices are as follows. The yield strength has the greatest influence for *L* = 0 m, with *Q*1 being 37% smaller than *S*1, and *K*1 being 3% smaller than *S*1. Imperfection *e*0 has the greatest influence for *L* ≈ 3.8 m, with *Q*3 being 46% smaller than *S*3, and *K*3 being 16% smaller than *S*3. *Ki* indices are closer to *Si* indices (compared to *Qi* indices). Examples of percentage differences between first-order indices are as follows. The yield strength has the greatest influence for *L* = 0 m, with *Q*<sup>1</sup> being 37% smaller than *S*1, and *K*<sup>1</sup> being 3% smaller than *S*1. Imperfection *e*<sup>0</sup> has the greatest influence for *L* ≈ 3.8 m, with *Q*<sup>3</sup> being 46% smaller than *S*3, and *K*<sup>3</sup> being 16% smaller than *S*3. *K<sup>i</sup>* indices are closer to *S<sup>i</sup>* indices (compared to *Q<sup>i</sup>* indices).

The percentage differences between total indices are as follows. The yield strength has the greatest influence for *L* = 0 m, with *QT*1 being 22% greater than *ST*1, and *KT*1 being 7% greater than *ST*1. Imperfection *e*0 has the greatest influence for *L* ≈ 3.8 m, with *QT*3 being 13% greater than *ST*3 and *KT*3 being 3% smaller than *ST*3. *KTi* indices are closer to *STi* indices (compared to *QTi* indices). The percentage differences between total indices are as follows. The yield strength has the greatest influence for *L* = 0 m, with *QT*1 being 22% greater than *ST*1, and *KT*1 being 7% greater than *ST*1. Imperfection *e*0 has the greatest influence for *L* ≈ 3.8 m, with *QT*3 being 13% greater than *ST*3 and *KT*3 being 3% smaller than *ST*3. *KTi* indices are closer to *STi* indices (compared to *QTi* indices). The percentage differences between total indices are as follows. The yield strength has the greatest influence for *L* = 0 m, with *QT*<sup>1</sup> being 22% greater than *ST*1, and *KT*<sup>1</sup> being 7% greater than *ST*1. Imperfection *e*<sup>0</sup> has the greatest influence for *L* ≈ 3.8 m, with *QT*<sup>3</sup> being 13% greater than *ST*<sup>3</sup> and *KT*<sup>3</sup> being 3% smaller than *ST*3. *KTi* indices are closer to *STi* indices (compared to *QTi* indices).

A comparison of the results of all three types of SA shows that the conclusions are very similar, despite being reached in a different way. The new *K* indices with their properties approach Sobol indices due to quadratic measures of sensitivity using *l* 2 , which behaves similarly to variance 2 *R* . A comparison of the results of all three types of SA shows that the conclusions are very similar, despite being reached in a different way. The new *K* indices with their properties approach Sobol indices due to quadratic measures of sensitivity using *l* 2 , which behaves similarly to variance 2 *R* . A comparison of the results of all three types of SA shows that the conclusions are very similar, despite being reached in a different way. The new *K* indices with their properties approach Sobol indices due to quadratic measures of sensitivity using *l* 2 , which behaves similarly to variance *σ* 2 *R* .

#### **5. Static Dependencies between** *l* **and** *σ<sup>R</sup>* **and Other Connections 5. Static Dependencies between** *l* **and** *<sup>R</sup>* **and Other Connections 5. Static Dependencies between** *l* **and** *<sup>R</sup>* **and Other Connections** Quantile-oriented sensitivity indices are based on the quantile deviation *l* or its

Quantile-oriented sensitivity indices are based on the quantile deviation *l* or its square *l* <sup>2</sup> while Sobol sensitivity indices are based on variance *σ* 2 *R* . The subject of interest of both quantile-oriented SA is the 0.001-quantile of *R*. Quantile-oriented sensitivity indices are based on the quantile deviation *l* or its square *l* 2 while Sobol sensitivity indices are based on variance 2 *R* . The subject of interest of both quantile-oriented SA is the 0.001-quantile of *R*. square *l* 2 while Sobol sensitivity indices are based on variance 2 *R* . The subject of interest of both quantile-oriented SA is the 0.001-quantile of *R*.

To better understand the essence of the computation of sensitivity indices, screening of statistics *l* and *σ<sup>R</sup>* is performed when *X<sup>i</sup>* is fixed. The aim is to identify similarities and differences between *l* and *σR*, rather than to accurately quantify sensitivity using Equations (8), (13) and (18). Samples *σR*|*X<sup>i</sup>* and *l*|*X<sup>i</sup>* are plotted for 400 LHS runs of *X<sup>i</sup>* , otherwise the solution is the same as in the previous chapter. Skewness *aR*|*X<sup>i</sup>* and kurtosis *kR*|*X<sup>i</sup>* are added for selected samples, see Figures 16–21. To better understand the essence of the computation of sensitivity indices, screening of statistics *l* and *<sup>R</sup>* is performed when *Xi* is fixed. The aim is to identify similarities and differences between *l* and *R*, rather than to accurately quantify sensitivity using Equations (8), (13) and (18). Samples *<sup>R</sup>*|*Xi* and *l*|*Xi* are plotted for 400 LHS runs of *Xi*, otherwise the solution is the same as in the previous chapter. Skewness *aR*|*Xi* and kurtosis *kR*|*Xi* are added for selected samples, see Figures 16–21. To better understand the essence of the computation of sensitivity indices, screening of statistics *l* and *<sup>R</sup>* is performed when *Xi* is fixed. The aim is to identify similarities and differences between *l* and *R*, rather than to accurately quantify sensitivity using Equations (8), (13) and (18). Samples *<sup>R</sup>*|*Xi* and *l*|*Xi* are plotted for 400 LHS runs of *Xi*, otherwise the solution is the same as in the previous chapter. Skewness *aR*|*Xi* and kurtosis *kR*|*Xi* are added for selected samples, see Figures 16–21.

**Figure 16.** Samples of *<sup>R</sup>*|*X*<sup>1</sup> and *l*|*X*<sup>1</sup> for *L* = 4.244 m ( = 1.0). **Figure 16.** Samples of *σR*|*X*<sup>1</sup> and *l*|*X*<sup>1</sup> for *L* = 4.244 m (*λ* = 1.0). **Figure 16.** Samples of *<sup>R</sup>*|*X*<sup>1</sup> and *l*|*X*<sup>1</sup> for *L* = 4.244 m ( = 1.0).

**Figure 17.** Samples of skewness *aR*|*X*<sup>3</sup> and kurtosis *kR*|*X*<sup>3</sup> for *L* = 4.244 m ( = 1.0). **Figure 17.** Samples of skewness *aR*|*X*<sup>3</sup> and kurtosis *kR*|*X*<sup>3</sup> for *L* = 4.244 m ( = 1.0). **Figure 17.** Samples of skewness *aR*|*X*<sup>3</sup> and kurtosis *kR*|*X*<sup>3</sup> for *L* = 4.244 m (*λ* = 1.0).

**Figure 18.** Samples of *<sup>R</sup>*|*X*<sup>2</sup> and *l*|*X*<sup>2</sup> for *L* = 4.244 m ( = 1.0). **Figure 18.** Samples of *σR*|*X*<sup>2</sup> and *l*|*X*<sup>2</sup> for *L* = 4.244 m (*λ* = 1.0).

**Figure 19.** Samples of

**Figure 20.** Samples of

*<sup>R</sup>*|*X*<sup>4</sup> and *l*|*X*<sup>4</sup> for *L* = 4.244 m (

half.

*<sup>R</sup>*|*X*<sup>3</sup> and *l*|*X*<sup>3</sup> for *L* = 4.244 m (

= 1.0).

The samples in Figure 19 are symmetric due to the symmetric shape of the probability distribution (Gauss pdf) of input variable *e*0 with a mean value of zero. Only the absolute value of this variable is applied in Equation (20). The output *R* is not monotonically dependent on *e*0. The practical consequence is that in the case of an even number of LHS runs, it is sufficient to compute the nested loop in Equations (8), (13) and (18) only once for the positive value of random realization *e*0, because the solution is the same for a negative value. This reduces the computational cost of estimating indices *Q*3, *K*3 and *S*3 by

= 1.0).

*Symmetry* **2021**, *13*, x FOR PEER REVIEW 15 of 22

*<sup>R</sup>*|*X*<sup>2</sup> and *l*|*X*<sup>2</sup> for *L* = 4.244 m (

*<sup>R</sup>*|*X*<sup>2</sup> and *l*|*X*<sup>2</sup> for *L* = 4.244 m (

= 1.0).

= 1.0).

**Figure 18.** Samples of

**Figure 18.** Samples of

**Figure 19.** Samples of *<sup>R</sup>*|*X*<sup>3</sup> and *l*|*X*<sup>3</sup> for *L* = 4.244 m ( = 1.0). **Figure 19.** Samples of *σR*|*X*<sup>3</sup> and *l*|*X*<sup>3</sup> for *L* = 4.244 m (*λ* = 1.0). ative value. This reduces the computational cost of estimating indices *Q*3, *K*3 and *S*3 by half.

**Figure 20.** Samples of *<sup>R</sup>*|*X*<sup>4</sup> and *l*|*X*<sup>4</sup> for *L* = 4.244 m ( = 1.0). **Figure 20.** Samples of *σR*|*X*<sup>4</sup> and *l*|*X*<sup>4</sup> for *L* = 4.244 m (*λ* = 1.0). *Symmetry* **2021**, *13*, x FOR PEER REVIEW 16 of 22

**Figure 21.** Samples of *<sup>R</sup>*|*X*<sup>5</sup> and *l*|*X*<sup>5</sup> for *L* = 4.244 m ( = 1.0). **Figure 21.** Samples of *σR*|*X*<sup>5</sup> and *l*|*X*<sup>5</sup> for *L* = 4.244 m (*λ* = 1.0).

quantile-oriented sensitivity indices *Qi* and *Ki*.

paring two statistically independent variables.

pendences *l*|*Xi* vs. *σR*|*Xi*, *i* = 2, 3, 4, 5 are approximately linear.

values of changes.

**6. Discussion** 

The smaller the estimated *<sup>R</sup>*|*Xi*, the more fixing of *Xi* reduces the uncertainty of the output in terms of variance, which measures change around *μR*. The smaller the estimated *l*|*Xi*, the more the fixing of *Xi* reduces the uncertainty of the output in terms of parameter *l*, which measures change around *θ*. Imperfection *e*0 (*X*3) has the greatest influence in both cases, see low values on the vertical axes in Figure 19. In all cases, the dependence *l*|*Xi* vs. *σR*|*Xi* is approximately linear with the exception The samples in Figure 19 are symmetric due to the symmetric shape of the probability distribution (Gauss pdf) of input variable *e*<sup>0</sup> with a mean value of zero. Only the absolute value of this variable is applied in Equation (20). The output *R* is not monotonically dependent on *e*0. The practical consequence is that in the case of an even number of LHS runs, it is sufficient to compute the nested loop in Equations (8), (13) and (18) only once for the positive value of random realization *e*0, because the solution is the same for a negative value. This reduces the computational cost of estimating indices *Q*3, *K*<sup>3</sup> and *S*<sup>3</sup> by half.

of the concave course on the right in Figure 16. Pearson correlation coefficient between

(compared to other variables) is due to the conflicting influences of *σR*, *aR* and *kR*. By approximating *R* using the Hermite distribution *R*~*H*(*μR*, *σR*, *aR*, *kR*), the effects of *μR*, *σR*, *a<sup>R</sup>* and *k<sup>R</sup>* on *l* can be observed separately as follows: change in *μR* has no influence on *l*, increasing *σR* increases *l*, increasing *aR* decreases *l*, increasing *kR* increases *l*, assuming small

The influence of *X*1 is interesting. Figure 16, on the left, shows that with increasing *X*1, *σR*|*X*1 has an approximately decreasing plot, with the exception of the beginning on the left. Figure 17 shows that *aR*|*X*1 has an approximately decreasing course, *kR*|*X*1 has an increasing course. At the beginning on the left, increasing *X*1 causes an increase in *σR*|*X*1, *kR*|*X*1 and *aR*|*X*1, which, taken together, increases *l*|*X*1 due to the dominance of the joint effect of *σR*|*X*1, *kR*|*X*1 and *aR*|*X*1. The region where *σR*|*X*1 starts decreasing but *l* still increases is interesting. Although the standard deviation is an important output characteristic, a change in the input variable can have a stronger influence on the quantile through skewness and kurtosis. At the opposite end (right), increasing *X*1 causes a decrease in *σR*|*X*1, a decrease in *aR*|*X*1 and an increase in *kR*|*X*1, which together reduces *l*|*X*1, because the decreasing sole effect of *σR*|*X*1 is dominant. The whole course of *l*|*X*1 vs. *X*1 is shown in Figure 16 in the middle. The example shows the combined effect of standard deviation, skewness and kurtosis on the quantile deviation *l*, which is the core of the computation of

For other variables *X*2, *X*3, *X*4 and *X*5 the range of *σR*|*Xi* is significantly larger than that of *σR*|*X*<sup>1</sup> (99.1 − 90.3 = 8.8 MPa) and *σR*|*Xi* has a crucial influence on *l*|*Xi*. Hence, the de-

Low quantiles represent a significant part of the analysis of reliability of load-bearing structures. SA of design quantiles can be used wherever reliability can be judged by com-

Both types of quantile-oriented sensitivity analysis identified a very similar sensitivity order to Sobol SA. Identical identification of probabilistically insignificant variables can serve to reliably decrease the dimension of random design space by introducing non-

The smaller the estimated *σR*|*X<sup>i</sup>* , the more fixing of *X<sup>i</sup>* reduces the uncertainty of the output in terms of variance, which measures change around *µR*. The smaller the estimated *l*|*X<sup>i</sup>* , the more the fixing of *X<sup>i</sup>* reduces the uncertainty of the output in terms of parameter *l*, which measures change around *θ*. Imperfection *e*<sup>0</sup> (*X*3) has the greatest influence in both cases, see low values on the vertical axes in Figure 19.

In all cases, the dependence *l*|*X<sup>i</sup>* vs. *σR*|*X<sup>i</sup>* is approximately linear with the exception of the concave course on the right in Figure 16. Pearson correlation coefficient between 400 samples *l*|*X<sup>i</sup>* vs. *σR*|*X<sup>i</sup>* is approximately 0.66. The concave course and lower correlation (compared to other variables) is due to the conflicting influences of *σR*, *a<sup>R</sup>* and *kR*. By approximating *R* using the Hermite distribution *R*~*H*(*µR*, *σR*, *aR*, *kR*), the effects of *µR*, *σR*, *a<sup>R</sup>* and *k<sup>R</sup>* on *l* can be observed separately as follows: change in *µ<sup>R</sup>* has no influence on *l*, increasing *σ<sup>R</sup>* increases *l*, increasing *a<sup>R</sup>* decreases *l*, increasing *k<sup>R</sup>* increases *l*, assuming small values of changes.

The influence of *X*<sup>1</sup> is interesting. Figure 16, on the left, shows that with increasing *X*1, *σR*|*X*<sup>1</sup> has an approximately decreasing plot, with the exception of the beginning on the left. Figure 17 shows that *aR*|*X*<sup>1</sup> has an approximately decreasing course, *kR*|*X*<sup>1</sup> has an increasing course. At the beginning on the left, increasing *X*<sup>1</sup> causes an increase in *σR*|*X*1, *kR*|*X*<sup>1</sup> and *aR*|*X*1, which, taken together, increases *l*|*X*<sup>1</sup> due to the dominance of the joint effect of *σR*|*X*1, *kR*|*X*<sup>1</sup> and *aR*|*X*1. The region where *σR*|*X*<sup>1</sup> starts decreasing but *l* still increases is interesting. Although the standard deviation is an important output characteristic, a change in the input variable can have a stronger influence on the quantile through skewness and kurtosis. At the opposite end (right), increasing *X*<sup>1</sup> causes a decrease in *σR*|*X*1, a decrease in *aR*|*X*<sup>1</sup> and an increase in *kR*|*X*1, which together reduces *l*|*X*1, because the decreasing sole effect of *σR*|*X*<sup>1</sup> is dominant. The whole course of *l*|*X*<sup>1</sup> vs. *X*<sup>1</sup> is shown in Figure 16 in the middle. The example shows the combined effect of standard deviation, skewness and kurtosis on the quantile deviation *l*, which is the core of the computation of quantile-oriented sensitivity indices *Q<sup>i</sup>* and *K<sup>i</sup>* .

For other variables *X*2, *X*3, *X*<sup>4</sup> and *X*<sup>5</sup> the range of *σR*|*X<sup>i</sup>* is significantly larger than that of *σR*|*X*<sup>1</sup> (99.1 − 90.3 = 8.8 MPa) and *σR*|*X<sup>i</sup>* has a crucial influence on *l*|*X<sup>i</sup>* . Hence, the dependences *l*|*X<sup>i</sup>* vs. *σR*|*X<sup>i</sup>* , *i* = 2, 3, 4, 5 are approximately linear.
