## 3 Formulation and Flowchart
### Formulation
\[N_{ij}\frac{y_{j}-y_{i}}{L_{ij}}+N_{ik}\frac{y_{k}-y_{i}}{L_{ik}}+N_{il}\frac{y _{l}-y_{i}}{L_{il}}+N_{im}\frac{y_{m}-y_{i}}{L_{im}}+F_{yi}=0 \tag{2}\]2  
\[N_{ij}\frac{y_{j}-y_{i}}{L_{ij}}+N_{ik}\frac{y_{k}-y_{i}}{L_{ik}}+N_{il}\frac{y _{l}-y_{i}}{L_{il}}+N_{im}\frac{y_{m}-y_{i}}{L_{im}}+F_{yi}=0 \tag{2}\]3  
\[N_{ij}\frac{y_{j}-y_{i}}{L_{ij}}+N_{ik}\frac{y_{k}-y_{i}}{L_{ik}}+N_{il}\frac{y _{l}-y_{i}}{L_{il}}+N_{im}\frac{y_{m}-y_{i}}{L_{im}}+F_{yi}=0 \tag{2}\]4  
Since both sets of equations (composed of Equations (8), (9), and (11) and Equations (12)-(14)) of equilibrium are uncoupled in the \(x\)-\(y\) and \(x\)-\(z\) planes, the location of the nodes (\(x_{i}\), \(y_{i}\), \(z_{i}\)) of the three-dimensional anti-funicular geometry can be obtained by solving two independent systems successively in which each one defines a form contained within a coordinate plane. The spatial procedure is, thus, reduced to two planar procedures which are, nevertheless, related to each other because the values of \(N_{x,i}\), \(x_{i}\), and \(F_{xi}\) must be the same for both of them. The method will be iterative because the loads that act upon the arch such as the self-weight of the arch, \(g_{A}\), or the axial forces at the hangers, \(T_{i}\), depend on the geometry of the arch (i.e., the node coordinates), which are the solutions for the system of equations. The effect of \(g_{A}\) can be easily taken into account by assigning an additional load \(Fz_{i}\) to the given point \(i\) equal to the weight of the tributary length of the point (i.e., half the weight of the two segments of arch adjacent to the point \(i\)). A similar procedure can be used to transform a distributed external load into a set of concentrated loads. A strong variation of the load distribution can then be accurately reproduced by simply defining a finer spacing of \(x_{i}\). To express the horizontal equilibrium of the nodes, \(n\) more equations should be written. However, this is not necessary in our case if the value of \(N_{x}\) is known at only one given node (usually the nodes 1 or \(n\), which correspond to the springing points) because, then, \(N_{x,i}\) at any point can be evaluated by Equation (15) as shown in Figure 7 and subsequently introduced directly into the systems of equations defined by Equations (8), (9), and (11) and Equations (12)-(14). If \(H_{1}\) is given at the node 1. \[N_{ij}\frac{y_{j}-y_{i}}{L_{ij}}+N_{ik}\frac{y_{k}-y_{i}}{L_{ik}}+N_{il}\frac{y _{l}-y_{i}}{L_{il}}+N_{im}\frac{y_{m}-y_{i}}{L_{im}}+F_{yi}=0 \tag{2}\]5  
\[N_{ij}\frac{y_{j}-y_{i}}{L_{ij}}+N_{ik}\frac{y_{k}-y_{i}}{L_{ik}}+N_{il}\frac{y _{l}-y_{i}}{L_{il}}+N_{im}\frac{y_{m}-y_{i}}{L_{im}}+F_{yi}=0 \tag{2}\]6 Use of FDM for arch form-finding. Vertical projection. If \(\Delta x_{i}=1\), \(q_{i}=N_{x,i}\) (i.e., the force density is equal to the horizontal component of the axial force at the arch). If, additionally, \(Fx_{i}=0\), \(q_{i}=H_{1}\), i.e., \(q_{i}\) is constant and equal to the longitudinal component of the reaction at the springing points, \(H_{1}\) (or \(H_{n}\)), which happens, for example, in a vertical plane arch linked to the deck by vertical hangers and subjected only to vertical loads. When a positive or negative value is given to \(H_{1}\), a cable composed of tensioned segments or an arch under compression is obtained, respectively. Strictly speaking, the axial forces \(N_{i}\) can change its sign depending on the values of \(Fx_{i}\) and a compression-only shape might transform into a tension-only shape. However, these structural configurations are not frequent in engineering. For a given set of external loads, there is one unique anti-funicular shape for each value of \(H\) (i.e., there is an infinite number of anti-funicular geometries for a given set of loads). In a 2-D process, the arch can be forced to pass through a given point (\(x_{C}\), \(z_{C}\)) by adjusting the value of \(H\).