## 1 Introduction  
### Spatial Arch Bridges and Funicularity  
The structural behavior of spatial arch bridges, which is a concept introduced for the first time in Reference [1], extends from the essentially vertical in-plane behavior of the \"classical\" arch bridges to three-dimensional behavior [1, 2]. This structural type was pioneered by the [PERSON], Landquart, Bohlbach, and Schwandbach deck-stiffened spatial arch bridges built between 1924 and 1933 and designed by [PERSON] who was clearly ahead of his time [3, 4]. This spatial behavior emerges because the arch is subjected to out-of-plane loads. For example, when the arch springings and the deck abutments do not coincide (as in the Hulme Arch in Manchester, see Figure 1 and Reference [5]), when the deck is curved in a plane (as in the Gateshead Footbridge, see Figure 2 and Reference [6]) or when the arch is rotated about either a longitudinal position (as in the York Millennium Bridge (see Figure 3 and Reference [7] or in the La Devesa footbridge [8]) or a vertical axis (as in the Juscelino Kubitschek Bridge [9]).  
Figure 1: Hulme Arch (Photo: [[https://www.flickr.com/photos/raver_mikey](https://www.flickr.com/photos/raver_mikey)]([https://www.flickr.com/photos/raver_mikey](https://www.flickr.com/photos/raver_mikey)) Creative Commons).  
Figure 3: York Millennium footbridge (Photo: [PERSON] Creative Commons).  
Figure 2: Gateshead Millennium footbridge (Photo: [PERSON] Creative Commons).  
In a \"classical\" vertical planar arch bridge subjected only to in-plane loads, its funicular geometry (which is the geometry that results in an equilibrium state free from bending stresses, i.e., simply under axial forces) is contained within a vertical plane. In these (and other) configurations and for a given load case (usually the permanent loads), the three-dimensional anti-funicular geometry of the arch becomes a warped curve, i.e., it is not contained within a plane such as the arch of the well-known Ripshorst footbridge [10] designed by [PERSON] (Figure 4).  
Despite the complex structural behavior and aesthetic potential of this type of bridge, relatively little research has been carried out focused on spatial arch bridges thus far [1; 11; 12; 13]. This lack of interest for understanding the structural response of this type of bridge is particularly striking when the aesthetic quality of a design, mainly in bridges located in urban environments, becomes more frequently, a desired, or even the primary, objective [14; 15]. Similarly, although many research teams are currently carrying out studies that comprehensively explore the full potential of form-finding methods such as the Force Density Method (FDM), Graphic Statics, Thrust Network Analysis or Dynamic Relaxation, the number of studies about anti-funicular geometries in spatial arch bridges is remarkably low. To mention but a few, the work carried out by the teams led by [PERSON] (at the MIT [16]), [PERSON] (at the Form-Finding Lab at Princeton University [17]), [PERSON] (at the Block Research Group at the ETH [18]), and [PERSON] at SBP [19] are of particular interest. Some of the alternative methods to find the funicular form of arch bridges are the Local Eccentricities Method (LEM) and the Global Reaction Method (GRM), which were both developed by [PERSON] and [PERSON] [1; 20]; the application of Graphic Statics [21] and the formulation obtained by [PERSON] et al. [22] by applying the Rankine Theorem. A variation of FDM has been used for form-finding of arches by [PERSON] and [PERSON] [23]. All of these methods are relatively recent. In fact, [PERSON] obtained the warped geometry of the Ripshorst footbridge as the inverted form of a hanging cable [10].  
Figure 4: Ripshorst footbridge (Photo courtesy of [PERSON] structurae.net).