**A.3 FORM calculation steps**

The calculation steps for this method are listed below [10].

1.*Formulate the LSF/performance function in terms of the original random variables, xi:*

$$Z(\mathbf{X}) = Z(\boldsymbol{\pi}\_1, \boldsymbol{\pi}\_2, \dots, \dots, \dots, \boldsymbol{\pi}\_n)$$

*Perspective Chapter: Probabilistic Modeling of Failure – Nonlinear Approximation DOI: http://dx.doi.org/10.5772/intechopen.109266*

2.*Assume the initial MPPF/design point as the given mean of each variable.*


$$\frac{\partial \mathbb{Z}(\mathbf{U}^\*)}{\partial u\_i} \tag{35}$$

7.*Compute the standard deviation of the LSF using the following Equation:*

$$
\sigma\_x = \sqrt{\sum\_{i=1}^n \left(\frac{\partial Z(U^\*)}{\partial u\_i}\right)^2} \tag{36}
$$

8.*Compute the initial reliability index β:*

$$
\beta = \frac{\mu\_Z}{\sigma\_Z} \tag{37}
$$

9.*Compute the directional cosines α<sup>i</sup> using the following Equation:*

$$a\_i = -\frac{\left(\frac{\partial Z(U^\*)}{\partial u\_i}\right)}{\sqrt{\sum\_{i=1}^n \left(\frac{\partial Z(U^\*)}{\partial u\_i}\right)^2}}\tag{38}$$

10.*Determine the new MPPF/design point in U space using the following Equation and in the X space using Eq. (33):*

$$
u\_i^\* = \beta a\_i \tag{39}$$


$$\beta = \frac{Z(\mathbf{U}^\*) - \sum\_{i=1}^n \frac{dZ(\mathbf{U}^\*)}{\vartheta u\_i} \left(u\_i^\*\right)}{\sqrt{\sum\_{i=1}^n \left(\frac{dZ(\mathbf{U}^\*)}{\vartheta u\_i}\right)^2}} \tag{40}$$

14.*Repeat steps 9 through 13 until β converges. A tolerance, ε, of 0.001 is usually used:*

$$
\varepsilon = \frac{|\beta\_i - \beta\_{i-1}|}{\beta\_{i-1}} < 0.001 \tag{41}
$$

15.*Use Eq. (31) to calculate the probability of failure*.
