## 5 Case studies: Examples of Application  
To illustrate the capabilities of the simplified FDM described above, the funicular geometry has been obtained for four case studies, A to D. The study has been carried out mainly to show the effect of different geometrical configurations and, consequently, the same reference bridge with equal mechanical parameters has been considered for all cases. The reference model (Figure 11) is an all-steel arch bridge spanning \(L=100\) m. The cross section of the arch is a circular hollow section (CHS) with \(\phi=1000\) mm and thickness \(t=30\) mm. The cross section of the deck is a 4000 \(\times\) 1000 mm rectangular hollow section (RHS) with a thickness \(t_{f}=t_{w}=15\) mm. The springing points of the arch are clamped while the deck is simply supported at the abutments. Apart from the self-weight of the arch and the deck, \(g_{A}\) and \(g_{D}\), respectively, the bridge is assumed to be subjected to a dead load acting downwards, \(g_{DL}\), of \(-12\) kN/m.  
The models A and B (Figure 12) correspond to symmetrical curved decks with horizontal sag of 10 m. The arch of the model A is also symmetrical and the coordinates of the springing points are (\(\pm 50\), \(6\), \(0\)) while the arch of the model B is non-symmetrical and the coordinates of its springing points are (\(-50\), \(5\), \(-3\)) and (\(50\), \(2\), \(8\)). The decks C and D (Figure 13) are symmetrical with respect to the origin of coordinates, which also illustrates the aesthetical potential of this type of structure. Deck C is straight while deck D is an S-shaped deck composed of two half-waves with a horizontal sag of 10 m among each one. An S-shaped deck similar to this one has been proposed by [PERSON] [14; 19].  
Figure 11: Reference bridge: cross-section and loads considered.  
Figure 14 shows the bending moments at the bridge defined in the case study A. The FDM is based on cable structures, a type of structure that is usually assumed to be simply supported at the springing points and made out of a material of infinite axial stiffness. Because the arch of the case study A is, in fact, made of steel and clamped at the springing points, bending moments due to axial stiffness are not considered.  
Figure 12: Case studies A and B. Anti-funicular geometry (m): Elevation, plan, and perspective views.  
Figure 13: Case studies C and D. Anti-funicular geometry (m): Elevation, plan, and perspective views.  
shortening of the arch appear and the bending moments at the arch are not equal to zero. For the axial force at the crown of the arch shown in Figure 14, which is equal to \(-3202\) kN, the eccentricity of \(M_{y}\) and \(M_{z}\) are, respectively, \(30\) and \(16\) mm at the crown for a \(1000\) mm-diameter pipe.  
Although these results can be considered excellent because their accuracy is smaller than the erection tolerance, the results can be refined even further by using jacks at the springing points to set the bending moments \(M_{y}\) and \(M_{z}\) at the springings and the \(M_{y}\) moment at the crown to zero. The results from this process are shown in Figure 14. For this case, the axial forces at the arch N at the crown and at the springings are equal to \(-3225\) kN and to \(-3627\) kN, respectively. Therefore, after refining the results, the maximum eccentricity of the resultant of the axial forces at any point of the arch is roughly \(2/3125\), which is \(0.6\) mm. This value is much smaller than the one obtained before using jacks.  
For the GRM and LEM, these accurate results are directly obtained as a part of the process itself since both methods use jacks for setting the bending moments at the selected degrees of freedom of the chosen points to zero before correcting the geometry, as detailed in Reference [1] or Reference [20].