## 4 Parametric Studies
### Influence of \(\beta_{T}\) on the Applicability of the Proposed Method  
Figure 13 shows the boundary lines, which indicate the case when the conventional and proposed methods have the same expected cost, considering the parameters \(\beta_{T}\) and \(k_{b}\). The lower left area of each boundary line indicates the applicable area of the proposed method. Note that \(k_{f}\) differs in each case because it is the function of \(\beta_{T}\) and \(k_{b}\), as shown in Equation (16). The target reliability index, \(\beta_{T}\), 1.5, 3.0, and 4.0 imply serviceability limit state (SLS), lower level of ultimate limit state (ULS), and higher level of ULS, respectively, with reference to [44].  
As \(k_{b}\) increases, the boundary line moves to the upper right and the applicability widens, which is explained in Figure 10.  
The slope of the boundary line changes with \(\beta_{T}\). When \(\beta_{T}\) is greater than 1.5, the line seems vertical, and the applicability of the proposed method is primarily associated with \(\text{COV}_{\text{R};\text{P}}\). Conversely, when \(\beta_{T}\) is as small as 1.5, both \(k_{c}\) and \(\text{COV}_{\text{R};\text{P}}\) are important. In other words, both the CM costs and the uncertainty of the estimation using piling data need to be reduced. Moreover, even if \(\text{COV}_{\text{R};\text{P}}\) is extremely low, \(k_{c}\) must be less than a certain value (Figure 13a).  
Influence of \(\text{COV}_{\text{Q}}\) and \(\text{COV}_{\text{R};\text{C}}\) on the Applicability  
For the remaining parameters, a lower \(\text{COV}_{\text{Q}}\) and higher \(\text{COV}_{\text{R};\text{C}}\) naturally expand the applicability of the proposed method.  
Loads generally consist of dead loads, impact loads, and seismic loads, and the variability depends on their proportions. Variable loads are typically modeled with the extreme distribution. This has a heavier tail than the lognormal distribution, which would underestimate the probability of failure, especially at a high target reliability index. However, a lognormal distribution was also used for the higher \(\text{COV}_{\text{Q}}\) case to focus on the differences in the amount of variation in this paper.  
Figure 12: Relationship between the parameters of the proposed method (\(\text{COV}_{\text{R};\text{P}}\) and \(k_{b}\)) and \(P_{c}\) when \(k_{b}=1.0\) and \(k_{f}=80\).  
Figure 14 illustrates the boundary lines when the \(\text{COV}_{\text{Q}}\) increases to 20%. The boundary lines move to the lower left compared to Figure 13. This is because the value of updating the capacity with the piling data would decrease, and consequently, the applicability of the proposed method would also decrease. Note that a distribution with a heavier tail would further limit the applicability of the proposed method.  
Although the \(\text{COV}_{\text{Rc}}\) was assumed to be 0.35 for soils whose characteristics are well known, it should be larger for unfamiliar rocks with high spatial variability. Figure 15 illustrates the boundary lines if the \(\text{COV}_{\text{Rc}}\) increases to 50% with reference to [65]. The boundary lines move to the upper right, contrary to those in Figure 14, and the applicability of the proposed method increases significantly. When \(\beta_{\text{T}}\) is 4.0, the visible area in Figure 15c can be mostly applied to the proposed method.  
Hence, the proposed method seems less effective in cases in which seismic loads are dominant in determining the structural parameters or in cases in which the reliability of the design using the conventional method improves. Conversely, the scope of the application of the method increases when dead loads are dominant or when the information about the ground and piling methods is insufficient.