# Improving the Structural Behavior of Tied-Arch Bridges by Doubling the Set of Hangers
## 3 Structural Behavior of the Tied-Arch Bridge with One Set of Central Hangers  
This section shows the behavior of bridges V1 and N1. The results of this section are compared with the results of the studies carried out in Section 3, Section 4, Section 5, Section 6, Section 7. This section also exemplifies the problems, mentioned above, that can appear in them. Regarding longitudinal bending (Figure 7), the behavior of bridges such as the V1 is well known, and the most adverse load case corresponds to \(q_{3}\) (Figure 6). In the V1 bridge, the bending moment diagram both at the arch and the deck has the well-known S-shape. Global bending is shared by the arch and the deck. The proportion of the global bending moment supported by the arch is proportional to \(EI_{A}/(EI_{A}+EI_{D})\), where \(EI_{A}\) and \(EI_{D}\) correspond to the flexural stiffness of the arch and the deck, respectively. The same applies to the deck (see Menn [19]). Regarding the N1, the reduction of bending achieved with respect to the V1, thanks to the Nielsen-type cable arrangement, is very high, around 70% for the bridge studied, both for the arch and deck. Regarding deflections, and consistently with the results of bending, in Figure 8, it is shown how, under \(q_{3}\), the N1 model has a much smaller deflection at the centerline of the deck than that of the V1 model. At a point located at a quarter of the span (\(x=-L/4\)), the vertical deflection reduction is 72%. Logically, the deflections at the edge, \(\delta_{E}\), for the load \(q_{3}\) are identical to the deflections at the axis of the deck, \(\delta_{D}\), since the distribution of the live load \(q_{3}\) is symmetric with respect to the plane of the arch. Figure 6: Live loads considered: (**a**) pedestrian live loads; (**b**) wind load. However, for the \(q_{2}\) load (Figure 9), at the center of the span (\(x=0\) in Figure 9a), the deflection at the edge of the deck, \(\delta_{E}\), is approximately 2.5 times that of the deck axis, \(\delta_{D}\). This is because the eccentric live load \(q_{2}\) can be resolved into a downward vertical load and a torsional moment, as shown in Figure 10. Subsequently, the deflection at the edge of the deck, \(\delta_{E}\), can be resolved into two addends:  
\[\delta_{E} = \delta_{D}+\theta.\frac{b}{2} \tag{1}\]  
where \(\delta_{D}\) is the deflection on the deck axis and \(\theta\) is the torsional twist of the cross-section (Figure 10b). \(\delta_{D}\) depends only on the global stiffness of the arch-deck vertical structural system, whereas \(\theta\) depends only on the torsional stiffness of the deck. That means that \(\theta\) is independent of the hanger arrangement when it is attached to the axis of the deck. Figure 8: Deflections under \(q_{3}\) at the deck for V1 and N1. Figure 7: Longitudinal bending under \(q_{3}\) at the arch (**a**) and the deck (**b**) for V1 and N1. Therefore, it may happen, if the deck is not torsionally stiff enough, that the most adverse scenario to satisfy the serviceability criteria is not the deflection at quarter spans (\(x=\pm L/4\)) under \(q_{3}\) but the deflection at midspan (\(x=0\)) under \(q_{2}\). In other words, even in a bridge with high vertical rigidity, such as the N1, with the Nielsen-type arrangement, a problem could appear at the edge of the deck due to excessive deflections. To illustrate this, Figure 9b shows how the deflection at the deck edge, \(\delta_{E}\), is extremely increased when the torsional stiffness of the deck cross-section is reduced to 10% of its original value, and, subsequently, the difference \(\delta_{E}\)-\(\delta_{D}\) is 10 times higher than that shown in Figure 9a. Several design strategies can be used to tackle these design problems.