**4. Case Study of the Ultimate Limit State**

Probability-based reliability analysis considers a stochastic model of an ultimate limit state of a bar under tension (see Figure 5a). The structural member is safe when the sum of loads is less than the relevant resistance.

33.94 kN.

width *b* [40]:

the relevant resistance.

*Q*(*Y*|*Xi*) and mean value *E*(*Y*|*Xi*).

**4. Case Study of the Ultimate Limit State** 

numerical illustrations of contrast *Q* indices are presented in [38,39].

In general, SSA is relevant to the mean value of *Y*, while the SA of the quantile (QSA) is relevant to the *α*-quantile of *Y*. However, in many cases, there is a strong similarity between the conclusions of QSA and SSA if all or at least the total sensitivity indices are examined. It can be shown in a simple example of *Y* = *X*1 + *X*2 that corr(*Q*(*Y*|*Xi*), *E*(*Y*|*Xi*)) ≈ 1, where *Q*(*Y*|*Xi*) is the conditional *α*-quantile and *E*(*Y*|*Xi*) is the conditional mean value. Changing *Xi* causes synchronous changes in the *α*-quantile

Although contrasts are of a different type, similarities between the results of QSA and SSA have been observed in the task of SA of the resistance of a building load-bearing element [35]. Other

Probability-based reliability analysis considers a stochastic model of an ultimate limit state of a

**Figure 5.** Static model: (**a**) Bar under tension and (**b**) probability density functions of *R*, *F*1, and *F*<sup>2</sup> for *μ<sup>p</sup>* = 0. **Figure 5.** Static model: (**a**) Bar under tension and (**b**) probability density functions of *R*, *F*<sup>1</sup> , and *F*<sup>2</sup> for µ*p* = 0.

The bar is loaded by two statistically independent forces *F1* and *F2*, both of which have a Gaussian pdf (see Figure 5b and Table 1). Parameter *μP* changes the mean value of the axial load of the bar, while the standard deviation of *F* is constant. The resulting force *F* = *F<sup>1</sup>* + *F2* has a Gaussian pdf with a mean value of *<sup>F</sup>* = *<sup>F</sup>*<sup>1</sup> + *F*2 = 309.56 kN + *μP* and standard deviation *F* = ( 2 *F*1 + 2 *F*2 ) 0.5 = The bar is loaded by two statistically independent forces *F*<sup>1</sup> and *F*2, both of which have a Gaussian pdf (see Figure 5b and Table 1). Parameter µ*<sup>P</sup>* changes the mean value of the axial load of the bar, while the standard deviation of *F* is constant. The resulting force *F* = *F*<sup>1</sup> + *F*<sup>2</sup> has a Gaussian pdf with a mean value of µ*<sup>F</sup>* = µ*F*<sup>1</sup> + µ*F*<sup>2</sup> = 309.56 kN + µ*<sup>P</sup>* and standard deviation σ*<sup>F</sup>* = (σ 2 *F*1 + σ 2 *F*2 ) 0.5 = 33.94 kN.

**Table 1.** The input random variables on the load action side.


Load Action 2 *F<sup>2</sup>* 68.16 + 0.5·*μ<sup>P</sup>* 23.86 kN The stochastic computational model for the evaluation of the static resistance *R* is a function of three statistically independent random variables: The yield strength *fy*; plate thickness *t*; and plate The stochastic computational model for the evaluation of the static resistance *R* is a function of three statistically independent random variables: The yield strength *fy*; plate thickness *t*; and plate width *b* [40]:

$$R = f\_y \cdot t \cdot b\_\prime \tag{26}$$

*y btfR* , (26) where *t*·*b* is the cross-sectional area. The resistance *R* is a function of material and geometric characteristics *fy*, *t*, and *b*, whose random variabilities are considered according to the results of experimental research [41,42]. Random variables *fy*, *t*, and *b* are statistically independent and are introduced with Gaussian pdfs (see Table 2).

**Table 2.** The input random variables on the resistance side.


The arithmetic mean µ*R*, standard deviation σ*R*, and standard skewness *a<sup>R</sup>* of resistance *R* can be expressed using equations (see [40]), based on arithmetic means µ*fy*, µ*<sup>t</sup>* , and µ*<sup>b</sup>* and standard deviations σ*fy*, σ*<sup>t</sup>* , and σ*<sup>b</sup>* presented in Table 2.

The mean value of *R* can be written as

$$
\mu\_{\mathbb{R}} = \mu\_{fy} \cdot \mu\_{t} \cdot \mu\_{\mathbb{b}}.\tag{27}
$$

The standard deviation of *R* can be written as

$$
\sigma\_{\mathcal{R}} = \sqrt{\mu\_{fg}^2 \cdot \left(\mu\_t^2 \cdot \sigma\_b^2 + \sigma\_t^2 \cdot \left(\mu\_b^2 + \sigma\_b^2\right)\right) + \mu\_t^2 \cdot \sigma\_{fg}^2 \cdot \left(\mu\_b^2 + \sigma\_b^2\right) + \sigma\_{fg}^2 \cdot \sigma\_t^2 \cdot \left(\mu\_b^2 + \sigma\_b^2\right)}}\tag{28}
$$

The standard skewness of *R* can be written as

$$a\_{\mathbb{R}} = 6 \cdot \frac{\mu\_{\mathbb{R}}}{\sigma\_{\mathbb{R}}^3} \cdot \left(\mu\_{fg}^2 \cdot \sigma\_t^2 \cdot \sigma\_b^2 + \sigma\_{fg}^2 \cdot \mu\_t^2 \cdot \sigma\_b^2 + \sigma\_{fg}^2 \cdot \sigma\_t^2 \cdot \mu\_b^2 + 4 \cdot \sigma\_{fg}^2 \cdot \sigma\_t^2 \cdot \sigma\_b^2\right). \tag{29}$$

For example, for input random variables from Table 2, we can write µ*<sup>R</sup>* = 412.68 kN, σ*<sup>R</sup>* = 34.057 kN, and *a<sup>R</sup>* = 0.111.

Goodness-of-fit and comparison tests [40] have shown that probabilities down to 1 <sup>×</sup> <sup>10</sup>−<sup>19</sup> are estimated relatively accurately using the approximation of probability density *R* by a three-parameter lognormal pdf with parameters µ*R*, σ*R*, and *aR*. This approximation is also suitable when one variable in Equation (26) is fixed. Fixing two variables leads to *R* with a Gaussian pdf with parameters µ*<sup>R</sup>* and σ*R*.

In SA, the failure probability *P<sup>f</sup>* = *P*(*Z* < 0) = *P*(*R* < *F*) can be computed using distributions *F* (Gaussian) and *R* (three-parameter lognormal or Gaussian) as the integral:

$$P\_f = \int\_{-\infty}^{\infty} \Phi\_R(y)\varphi\_F(y) \mathrm{d}y \,\tag{30}$$

where ϕF(*y*) is the pdf of load action, ΦR(*y*) is the distribution function of resistance, and *y* denotes a general point of the force (the observed variable) with the unit of Newton. The integration in Equation (30) is performed in the case study numerically using Simpson's rule, with more than ten thousand integration steps over the interval [µ*<sup>Z</sup>* − 10σ*Z*, µ*<sup>Z</sup>* + 10σ*Z*].
