## 6 Verification by Loadings Measured in a Mountain Bike
### Experimental Verification of Confidence Intervals: Stationary Case  
In this section, the confidence intervals in Equation (3) (Case M) and (9) (Case S) are applied to the measured stationary records. In Case M, the analysis considered groups of \(N\) records (with \(N\) ranging from 2 to 10) selected from the 10 measured initially; in every group, each of the 10 records appeared only once. The \(N\) records have damages \(D_{i}(T)\), \(i=1,2,\ldots N\), used in Equation (2). In Case S, the individual record was subdivided into \(N_{B}\) disjoint blocks (with \(N_{B}\) ranging from 2 to 10). The block damages \(D_{B,i}(T_{B})\), \(i=1,2,\ldots N_{B}\) are used in Equations (5) and (9). Finally, the validation set is used to approximate the  
Figure 10: Run test applied to a measured stress-time history classified as non-stationary: (**a**) \(T_{s}=5\) s and (**b**) \(T_{s}=10\) s.  
Figure 9: Run test applied to a measured stress-time history classified as stationary: (**a**) \(T_{s}=5\) s and (**b**) \(T_{s}=10\) s.  
expected damage as \([D(T)]\cong\overline{D}_{val}(T)\), where \(\overline{D}_{val}(T)\) is computed by Equation (20) with \(D_{val,i}\) from \(N_{val}=30\) measured records in the validation set. Equation (20) also provides the sample variance of validation damage, \(\vartheta_{D_{val}}^{2}\). Damage computation considered an S-N slope \(k=3\). The main statistics are summarized in Table 3. The first three columns (Case M) list the sample mean \(\overline{D}(T)\) and sample standard deviation \(\hat{\sigma}_{D}\) of damage as a function of \(N\). The last three columns (Case S) list the total damage calculated as \(D(T)\cong N_{B}\cdot\overline{D}_{B}(T_{B})\) and the standard deviation as \(\sigma_{D}^{2}\cong N_{B}\cdot\vartheta_{D_{B}}^{2}\), as a function of \(N_{B}\). Damage values are normalized to \(\overline{D}_{val}(T)\), standard deviations to \(\hat{\sigma}_{D_{val}}\). In Case M, the sample mean damage \(\overline{D}(T)\) varies with \(N\) but not significantly. Apart from two exceptions, the trend of \(\hat{\sigma}_{D}\) is to decrease as \(N\) increases, the lowest value being for the largest sample of records (\(N=10\)). In Case S, this trend is not observed, and \(\hat{\sigma}_{D}\) remains almost independent of \(N_{B}\). Interestingly, at the largest \(N_{B}\), \(\hat{\sigma}_{D}\) is about 50% greater than its value for \(N=10\) in Case M. Compared to Case M, in Case S, the damage \(D(T)\) varies less with \(N_{B}\), a result somehow predictable. In fact, the small variations in the values of \(D(T)\) are caused by the cycles lost after block subdivision, the number of which is indeed negligible. These findings are in line with those observed in numerical simulations [13]. The values of \(\overline{D}_{val}(T)\) and \(\hat{\sigma}_{D_{val}}\) are comparable to those of Case M, especially for the highest number \(N=10\) that is the closest to \(N_{val}=30\) of the validation set. This similarity is a further proof that all load records were measured under similar conditions. On the other hand, the standard deviation \(\hat{\sigma}_{D_{val}}\) is slightly lower (about 0.3%) than the standard deviation \(\hat{\sigma}_{D}\) for \(N=10\) in Case M. This result confirms that an increase in sample size reduces the statistical variability (variance) of sample mean damage. If the number of load records were infinite, \(\hat{\sigma}_{D_{val}}\) would be zero and \(\overline{D}_{val}(T)\) would approach \(E[D(T)]\). The values in Table 3 are used to compute the 95% confidence intervals (normalized to the expected damage) displayed in Figure 11 as a function of \(N\) (Case M) or \(N_{B}\) (Case S). In each figure, the horizontal continuous line represents the expected damage approximated as \(E[D(T)]\cong\overline{D}_{val}(T)\). In Figure 11a, markers are the sample means \(\overline{D}(T)\). As \(N\) increases, \(\overline{D}(T)\) tends to approach \(E[D(T)]\), while the confidence interval becomes narrower--trends that confirm what was already observed in simulations.