## Verification by Simulated Switching Loadings  
### Simulation of Load Cases  
The stationary case has already been checked in [13]. Therefore, attention here is focused on the switching case. Three types of switching loadings, which may represent practical cases, are considered:  
* Load A: sequence of three states of equal length \(T_{S,i}=100\) s but different mean \(m_{i}=0,\ 1\) and standard deviation \(s_{i}=1,\ 2\) (see Figure 3). This load case was chosen because of its similarity to the measured loads in Section 6;
* Load B: same \(m_{i}\) and \(s_{i}\) as Load A but with different lengths \(T_{S,i}\) with values: \(T_{S,1}=50\) s (state 1), \(T_{S,2}=175\) s (state 2), \(T_{S,3}=75\) s (state 3);
* Load C: sequence of four states arranged in a random sequence. Since states 1 and 2 appear twice (see Figure 4a), the analysis requires a state reordering (the reordered sequence, with full state lengths \(T_{S,i}\), is shown in Figure 4b). States have the following characteristics: \(T_{S,1}=50\) s, \(m_{1}=0\), \(s_{1}=1\) (state 1); \(T_{S,2}=175\) s, \(m_{2}=1\), \(s_{2}=1\) (state 2); \(T_{S,3}=75\) s, \(m_{3}=1\) and \(s_{3}=2\) (state 3); \(T_{S,A}=100\) s, \(m_{4}=0\), \(s_{4}=2\) (state 4). Each switching load \(z(t)\) is obtained by first simulating a sample loading separately for each stationary state and then by arranging the simulated loadings in the desired order. The previous simulation procedure was repeated so that a total of \(N=2\cdot 10^{5}\) realizations \(z_{i}(t)\), \(i=1,2,\ldots,N\) were simulated for each load type A, B, C. This sample size \(N\) is comparable to similar studies [11]. Fatigue damage values \(D_{i}(T)\), \(i=1,2,\ldots,N\) were calculated for all realizations by assuming an S-N slope \(k=3\). A sample mean damage \(\overline{D}(T)\) is computed as in Equation (2); since \(N\) is very large, the approximation \(E[D(T)]\ \cong\overline{D}(T)\) holds true with enough accuracy. Figure 2: Example of state reordering: stress-time history (**a**) before and (**b**) after reordering. Besides the sample of large size \(N\), a smaller set of \(N_{val}=20\) realizations--called the validation set--is also generated for each load type A, B, C. This validation set will become fundamental with measured loadings, see Section 6. In fact, whereas it is effortless to numerically simulate large samples of load realizations to be used for approximating \(E[D(T)]\) accurately as above, it is instead almost impossible to collect a huge number of measured records, and the \"small\" validation set then becomes the only means to approximate \(E[D(T)]\cong\overline{D}(T)\). On the other hand, when the sample size \(N\) reduces, it may be presumed that \(\overline{D}(T)\) can occasionally be less close to \(E[D(T)]\) because of the increased sampling variability. Introducing the validation set also in numerical simulations has the purpose to assess the accuracy of the approximation \(E[D(T)]\cong\overline{D}(T)\) when \(\overline{D}(T)\) is computed over a much smaller sample (\(20\ll 2\cdot 10^{5}\)). Figure 4: Load C: (**a**) with replicated states and (**b**) after reordering (values of \(m_{i}\) and \(s_{i}\) in bottom subplots). Figure 3: Load A: three states of same duration \(T_{S,i}\); the values of \(m_{i}\) and \(s_{i}\) are in bottom subplots. The damage values \(D_{val,i}(T)\), \(i=2,3,\ldots,N_{val}\) from the validation set are used to compute a sample mean and sample variance:  
\[\overline{D}_{val}(T)=\frac{1}{N_{val}}\sum_{i=1}^{N_{val}}D_{val,i}(T)\,,\, \hat{\rho}_{\overline{D}_{val}}^{2}=\frac{1}{N_{val}-1}\sum_{i=1}^{N_{val}} \bigl{[}D_{val,i}(T)-\overline{D}_{val}(T)\bigr{]}^{2} \tag{20}\]  
Obtained results show that the approximation \(E[D(T)]\;\cong\;\overline{D}_{val}(T)\) has only a 2% difference lower than the approximation in which the sample mean damage is computed on the larger set 2\(\cdot\)10\({}^{5}\).