## 3 Results
### Simulation Results
#### 3.2.2 Simulation 2  
We consider a sectioning span of 1 m in both cantons, while maintaining the elevation of the sloping end at 0.1 m. In each calculation, the length of the slope length is varied from 0.5 to 1 m with an increase of 0.05 m.  
The analysis of the results is carried out in the common area and, as a result, we obtain the graphs shown in Figure 11, which represent the average, maximum, and minimum values of the pantograph effort on the contact wire in Newtons, the standard deviation, and the average distance of take-offs. In this case, there are no take-offs in the common area of the catenary, so this graph is omitted.  
The abscissa axis of the graphs represents the length of the slope in meters and the ordinate axis represents the effort (mean, maximum, minimum, and deviation) in Newtons obtained in the integration calculated with the given elevation.  
Analysis of the graphs (Figure 11) shows that the deviation of the stresses in the common area is adequate in cases in which the length of the slope is less than or equal to 0.45 m. The deviation of the stresses in the common area is adequate in cases in which the length of the slope is less than or equal to 0.45 m. From this length, we see how the maximum and the deviation increase as the length of the slope increases until it coincides with the whole section span. Similarly, the minimum stresses decrease as the maximum increases from 0.45 m in slope length, confirming the increase in deviation.  
Figure 10: Graphs of simulation 1: an overlapping span of 1 m, slope length 1 m, and variation in the height of the slope at its end. (**a**) Average value of pantograph effort, (**b**) standard deviation of pantograph effort, (**c**) maximum value of pantograph effort, and (**d**) minimum value of pantograph effort.  
Figure 11: Graphs of simulation 2. Overlapping span of 1 m variation in slope length in the range of 0.5 to 1 m with 0.05 m increments and 0.1 m slope elevation. (**a**) Average value of pantograph effort, (**b**) standard deviation of pantograph effort, (**c**) maximum value of pantograph effort, and (**d**) minimum value of pantograph effort.  
The effort at the passage of the pantograph through the common area has a better behavior in the case of small lengths of the slope (considering an elevation of 0.1 m from the end). This result is congruent with simulation 1 because the case of 0 m of the slope is equivalent to a horizontal overlapping span in which only the elevation obtained in the initial static configuration is applied. We then observe that the length of the overlapping span can significantly influence the results obtained for a small elevation of the beam and that it could be sufficient with the elevation obtained in the initial static configuration. In this case, the length of the overlapping span profile influences the elevation of the beam and, therefore, the results obtained.  
To test this hypothesis, the following two simulations start from the 0 m elevation of the slope and the variation of the overlapping span length. In simulation 3, we consider a configuration equal to that used previously (i.e., normal spans of 10 m), and in simulation 4 we change the length of the normal spans to 5 m to check the influence of the length of the normal spans on the result.