**2. Probability-Based Assessment of Structural Reliability**

Let the reliability of building structures be a one-dimensional random variable Z:

$$Z = \mathcal{g}(X) = \mathcal{g}(X\_1, X\_2, \dots, X\_M), \tag{1}$$

where *X*1, *X*2, . . . , *X<sup>M</sup>* are random variables employed for its computation. The classical theory of structural reliability [27] expresses Equation (1) as a limit state using two statistically independent random variables, the load effect (action *F*), and the load-carrying capacity of the structure (resistance *R*).

$$Z = R - F \ge 0\tag{2}$$

The variable that unambiguously quantifies reliability or unreliability is the probability that inequality (2) will not be satisfied. If *Z* is normally distributed, reliability index β is given as

$$
\beta = \frac{\mu\_{\rm Z}}{\sigma\_{\rm Z}},
\tag{3}
$$

where µ*<sup>Z</sup>* is the mean value of *Z* and σ*<sup>Z</sup>* is its standard deviation. By modifying Equation (3), we can express µ*<sup>Z</sup>* −β·σ*<sup>Z</sup>* = 0. The failure probability *P<sup>f</sup>* can then be expressed as

$$P\_{\rm I} = P(Z < 0) = P(Z < \mu\_Z - \beta \cdot \sigma\_Z) = \Phi\_{\rm U}(-\beta),\tag{4}$$

where Φ*U*(·) is the cumulative distribution function of the normalized Gaussian probability density function (pdf). Reliability is defined as *P<sup>s</sup>* = (1 − *P<sup>f</sup>* ). For other distributions of *Z*, β is merely a conventional measure of reliability. Equation (3) can be modified for normally distributed *Z*, *F*, and *R* as

$$\beta = \frac{\mu\_{\rm Z}}{\sigma\_{\rm Z}} = \frac{\mu\_{\rm R} - \mu\_{\rm F}}{\sqrt{\sigma\_{\rm R}^2 + \sigma\_{\rm F}^2}} = \frac{\mu\_{\rm R} - \mu\_{\rm F}}{\frac{\sigma\_{\rm R}^2}{\sqrt{\sigma\_{\rm R}^2 + \sigma\_{\rm F}^2}} + \frac{\sigma\_{\rm F}^2}{\sqrt{\sigma\_{\rm R}^2 + \sigma\_{\rm F}^2}}} = \frac{\mu\_{\rm R} - \mu\_{\rm F}}{\alpha\_{\rm R} \cdot \sigma\_{\rm R} + \alpha\_{\rm F} \cdot \sigma\_{\rm F}},\tag{5}$$

where α*<sup>F</sup>* and α*<sup>R</sup>* are values of the first-order reliability method (FORM) sensitivity factors.

$$\alpha\_{\mathbb{R}} = \frac{\sigma\_{\mathbb{R}}}{\sqrt{\sigma\_{\mathbb{R}}^2 + \sigma\_{\mathbb{F}}^2}}, \ \alpha\_{\mathbb{F}} = \frac{\sigma\_{\mathbb{F}}}{\sqrt{\sigma\_{\mathbb{R}}^2 + \sigma\_{\mathbb{F}}^2}}, \text{with } |\alpha| \le 1 \tag{6}$$

It can be noted that Sobol's first-order indices are equal to the squares of α*<sup>F</sup>* and α*R*: *S<sup>F</sup>* = α 2 *F* and *S<sup>R</sup>* = α 2 *R* , respectively [19]. By applying α*<sup>F</sup>* and α*<sup>R</sup>* according to Equation (6), Equation (5) can be written with formally separated random variables as

$$
\mu\_F + \alpha\_F \cdot \beta \cdot \sigma\_F = \mu\_R - \alpha\_R \cdot \beta \cdot \sigma\_R.\tag{7}
$$

Equation (7) is a function of the four statistical characteristics of µ*F*, σ*F*, µ*R*, and σ*R*, from which β, α*F*, and α*<sup>R</sup>* are computed. The left side in Equation (7) is the design load *F<sup>d</sup>* (upper quantile) and the right side is the design resistance *R<sup>d</sup>* (lower quantile).

Standard [4] verifies the reliability by comparing the obtained reliability index β with the target reliability index β*d*, according to the equation β ≥ β*d*, which transforms Equation (7) into the design condition of reliability:

$$
\mu\_F + \alpha\_F \cdot \beta\_d \cdot \sigma\_F \le \mu\_R - \alpha\_R \cdot \beta\_d \cdot \sigma\_{R\prime} \tag{8}
$$

where α*<sup>F</sup>* and α*<sup>R</sup>* may be considered as 0.7 and 0.8, respectively [4].
