## 3 Examples  
### Cross-Section under Flexure  
A cross-section under a factored flexural moment \(M_{x,n}=400\) kNm is studied. The strength of materials are \(f_{c^{\prime}}=30\) MPa for concrete and \(f_{y}=500\) MPa for steel. The clear cover of reinforcement is 40 mm and the stirrup  
tuned for this example, as can be seen in [29]. The values adopted in this work are \(\beta_{0}=1\), \(m=2.0\), \(\gamma_{0}=10\), \(\alpha_{0}=1\), \(NF=300\), \(k_{max}=1000\) and \(t_{c}=0.1\). The best solution obtained from 50 runs using MPFA with eight subpopulations is considered to be the optimal design solution. The optimal design solutions obtained considering ACI318 and EC2 standards are shown in Table 1.  
The average time of 50 runs, considering ACI318 standard and using MPFA with different numbers of subpopulations, was obtained. Although there is a difference in the average time depending on the computer used, the average time was considered to work out the metric \"speed-up\", which refers to how much a parallel FA is faster than a sequential FA [34]. The speed-up is shown in Figure 3. \"Efficiency\", which is the ratio of speed-up and the number of processors used to solve the example, measures how efficiently the MPFA uses the parallel resources [34]. Efficiency is shown in Figure 4. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **ACI318** & **EC2** \\ \hline \(z\)**(mm)** & 152.3 & 136.3 \\ \(\theta\)**(rad)** & 0 & 0 \\ \(b\)**(mm)** & 300 & 325 \\ \(h\)**(mm)** & 585 & 605 \\ \(\varphi^{\prime}x\)**(mm)** & 10 & 10 \\ \(n^{\prime}x\) & 2 & 4 \\ \(\varphi_{x}\)**(mm)** & 32 & 20 \\ \(n_{x}\) & 3 & 6 \\
**Cost (\(\ell\)/m)** & 85.54 & 86.19 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimal design solutions obtained using the MPFA. Figure 3: Cross-section under flexure. Speed-up. The parallelization produces satisfactory results despite its simple implementation (Figure 3). However, the efficiency quickly drops with more than four processors (Figure 4). The computer that was used to run the calculations is a workstation equipped with an Intel Xeon CPU E5-1620 v2 3.70 GHz with eight cores. \"Accuracy\", which is the mean value of several independent runs when compared to a reference value, measures how efficiently the MPFA uses the parallel resources [34]. \"Robustness\" is linked to the scatter of obtained results [34] (for example, standard deviation) for several independent runs of MPFA. The mean cost, accuracy (reference value of Cost = 85.54, from Table 1), and robustness (standard deviation) from 50 runs considering ACI318 standard and using MPFA with a different number of subpopulations were obtained. These metrics are shown in Tables 2-4, considering only LSRD, only GSRD, and both LSRD and GSRD, respectively. For comparison purposes, the average values of the performance metrics were also obtained, and they are shown in the last row of Tables 2-4. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Subpopulations** & **Mean Cost** & **Accuracy** & **Robustness** \\ \hline
1 & 85.81 & 0.9969 & 0.51 \\
2 & 85.83 & 0.9966 & 0.52 \\
3 & 85.78 & 0.9972 & 0.76 \\
4 & 85.68 & 0.9984 & 0.35 \\
5 & 85.86 & 0.9963 & 0.81 \\
6 & 85.68 & 0.9983 & 0.36 \\
7 & 85.76 & 0.9974 & 0.47 \\
8 & 85.98 & 0.9949 & 0.95 \\
9 & 86.26 & 0.9917 & 1.32 \\
10 & 85.94 & 0.9953 & 0.99 \\
11 & 86.52 & 0.9887 & 1.49 \\
12 & 86.35 & 0.9906 & 1.72 \\
13 & 86.84 & 0.9851 & 1.90 \\
14 & 87.05 & 0.9827 & 1.94 \\
15 & 88.02 & 0.9719 & 2.40 \\
16 & 87.98 & 0.9723 & 2.68 \\ \hline
**Averaged value** & 86.33 & 0.9909 & 1.20 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Cross-section under flexure. Local, small, random displacement (LSRD). Performance metrics. Figure 4: Cross-section under flexure. Efficiency. The method is able to obtain designs close to the global optimum as shown in Tables 2-4. When the method considers both LSRD and GSRD (Table 4), the minimum average value of the mean costs, the maximum average value of the accuracies, and the minimum average value of the standard deviations are obtained.