# Modified, Reliability-Based Robust Geotechnical Design Method, in accordance with Eurocode 7
## 2 Materials and Methods
### A Modification of the RGD Procedure
#### 2.2.1 Harmonising the RGD Method Results with Eurocode 7  
In the existing RGD method, a feasible region encompasses all designs that meet the pre-defined target failure probability requirement (\(P_{f}^{T}\)), i.e., the associated value of the target reliability index (\(\beta^{T}\)). The connection between \(P_{f}\) and \(\beta\) is mathematically defined, and these can therefore be used interchangeably. In this paper we use \(\beta\) because it is more user-friendly than \(P_{f}\). The associated range of \(P_{f}\) for \(0.5\leq\beta\leq 10\) would be \(0.3\leq P_{f}\leq-7.6\times 10^{-24}\).  
Eurocode 0 ([PERSON] et al., 2012), in Annex C, gives a recommendation on limit values for \(\beta^{T}\), in accordance with the considered limit state, reference period and reliability class. Thus, for a 50-year reference period and reliability class RC2, the following recommendations for \(\beta^{T}\) were given: \(\beta^{T}_{ULS}=3.8\), \(\beta^{T}_{SLS}=1.5\). The same values are used in the foundation optimisation using the original RGD procedure.  
Meeting the criteria for the aimed reliability index value does not necessarily ensure meeting the criteria for ultimate and serviceability limit states defined in EC7. [PERSON] and [PERSON] ([PERSON], 1998) study the reliability of shallow foundation designed according to Eurocode 7, in the case of ultimate limit state (ULS). In their study, they conduct a series of analyses, which consider foundation's reliability indexes according to EC7, and according to the allowable stress design format, with overall Factors of Safety \(FOS=2\), and \(FOS=3\). Different design situations are defined by the variance of the resultant force position (centric and eccentric load), soil type (granular and fine grain soils), relative load sizes (large and small loads), and the correlation \(tan_{\phi^{\prime}}-c^{\prime}\) (with and without correlation). They conduct multi-parameter analyses for all the listed design situations, with the variance of geotechnical parameters: vertical scale of fluctuation (\(\delta_{V}\)), tangent of the angle of internal friction (\(V_{tamp}\)), characteristic values of \(\phi^{\prime}\) (\(\phi_{k:mean}\) and \(\phi^{\prime}_{k:low}\)). The presented results indicate that there are designs that meet the target reliability index, but do not meet the ultimate limit state criterion defined in EC7.  
An example of such designs is shown in Table 2, which lists selected final designs attained through the application of the RGD procedure from the illustrative example given in ([PERSON], 2012). The designs marked grey fail to meet the ULS criterion defined by EC7; therefore, additional assessments of limit states according to EC7 are necessary for the chosen design, along with the RGD procedure.  
By introducing additional constraints into the optimisation process, it is possible to exclude all designs that do not meet the ULS and SLS criteria defined in EC7 from the feasible region. Figure 2 schematically illustrates the design space and the existing and suggested feasible regions in the case of ULS.  
According to the existing RGD procedure, the feasible region is composed of regions 1 and 3 from Figure 2, and is defined as follows:  
\[\mathbb{F}=\Big{\{}D|\beta_{ULS}(D)\geq\beta_{ULS}^{\mathrm{T}},\ \beta_{ SLS}(D)\geq\beta_{SLS}^{\mathrm{T}}\Big{\}}, \tag{2}\]  
Region 3 includes designs that meet the criterion concerning the aimed reliability index value, while at the same time failing to meet the criteria defined by EC7. By introducing additional constraints, region 3 is excluded from the feasible region, whose definition is then altered as follows:  
\[\mathbb{F}=\Big{\{}D|\beta_{ULS}(D)\geq\beta_{ULS}^{\mathrm{T}},\ \beta_{ SLS}(D)\geq\beta_{SLS}^{\mathrm{T}},\ \ ODF_{ULS}(D)\geq 1,\ \ ODF_{SLS}(D)\geq 1\Big{\}}, \tag{3}\]