# Modified, Reliability-Based Robust Geotechnical Design Method, in accordance with Eurocode 7
## 2 Materials and Methods
### A Modification of the RGD Procedure
#### 2.2.2 Simplification of the RGD Procedure by Substituting the Robustness Measure
[PERSON] and [PERSON] [14] suggest PEM with seven estimating points, used in the outer loop of the RGD procedure, whose accuracy is significantly greater than the earlier instances of the method using 2, 3 and 5 estimating points. Figure 3: A graphic illustration of the relation of \(\beta\), \(P_{f}\) for different values of system response. The main shortcoming of PEM with seven estimating points is the fact that the random variables are transformed into standard normal space using the Rosenblatt transformation. In the example of correlated random variables, the transformation of random variables requires having total probabilistic information data, which is almost never the case in engineering. By substituting the Rosenblatt transformation with the Nataf transformation in PEM, we enable the consideration of correlated variables with the knowledge of the correlation matrix, their marginal distributions, mean values and standard deviations. The data on correlations between geotechnical parameters, their statistical distribution and standard deviation are available in literature, and the mean values can be determined from the geotechnical investigation results. The modification of PEM by introducing the Nataf transformation was suggested by [PERSON] et al. (2016), and they named the modified method IPEM. The Nataf transformation and its inverse are expressed in the following equations:  
\[T_{N}:x=F_{X}^{-1}[\Phi(L_{0}u)], \tag{5}\]  
\[T_{N}^{-1}:u=L_{0}^{-1}\Big{\{}\phi^{-1}[F_{X}(x)]\Big{\}}, \tag{6}\]  
where \(F_{X}(. )\) is the marginal cumulative probability density function (CDF) of \(X\), \(\Phi(. )\) is the standard normal CDF of \(X\), \(L_{0}\) is the lower triangle matrix yielded from the Cholesky decomposition of the correlation matrix \(R_{0}=\left[\rho_{0,ij}\right]\). The procedure of determining the correlation matrix \(R_{0}\) is composed of a series of complex function integrations ([PERSON] and others, 2017), but its approximation is possible, based on a set of semiempirical equations suggested by [PERSON] and [PERSON] (2018). \(R_{0}\) is approximated based on the known correlations of \(\rho_{ij}\) and the ratio of \(F\) in the following way:  
\[F=\frac{\rho_{0,ij}}{\rho_{ij}}, \tag{7}\]  
The value of the ratio of \(F\) is given for different sets of marginal distributions, divided into two groups. According to the distribution pair, tables are given which provide the values/equations of \(F\). The IPEM procedure is conducted using simpler mathematical operations. The inverse Nataf transformation, which is a part of IPEM, is required only at the estimating points, and can be done simply, e.g., by using the built-in function of Microsoft Excel, or the Python  
Figure 4: The relation \(\beta-\beta_{\beta}\) for different foundation widths and variation coefficients of the soil’s internal friction angle. program language used with the \"SciPy\" package. Since the accuracy of the calculations of \(\beta\) is highly dependent on the method used [19], the accuracy of IPEM for ULS and SLS of shallow foundation was examined. The Monte Carlo method was chosen as the control method, and the results are given in Tables 3 and 4. The average error in the estimation of the reliability index using the IPEM method for ULS and SLS of shallow foundations, for different variation coefficients of the internal friction angle, is approximately 1%--which is negligible.