## 3 The Limit Analysis under Horizontal Longitudinal Loads
The collapse mechanism and the corresponding acceleration can be found using the same iteration procedure shown in the case of the vertical travelling load and based on the principle of the virtual works. Consider a given mechanism and the corresponding diagrams of the virtual displacement components (Figure 5). If \(\eta(\hat{z})\) are the vertical components of the virtual displacements, the virtual work of dead loads is:  
\[L_{w}=\gamma_{w}bL^{3}\cdot\int_{0}^{1}\left[\hat{w}_{w}(\hat{z})\eta(\hat{z}) +\hat{w}_{b}(\hat{z})\eta(\hat{z})\right]d\hat{z}=\gamma_{w}bL^{3}\cdot\hat{L} _{w} \tag{9}\]  
Figure 4: Structural model of a stone voussoir arch under longitudinal seismic acceleration. Figure 5: Collapse mechanism and virtual displacements. Analogously, if \(\zeta(\hat{z})\) are the horizontal components of the virtual displacements, the virtual work of the reference seismic load can be written as:  
\[\sigma=\frac{f_{u}}{\gamma_{w}L} \tag{3}\]0  
If \(\Delta\varphi_{i}\) are the relative rotations at the \(n\) hinges and \(\hat{d}_{i}=d_{i}/L\), the internal work is:  
\[\sigma=\frac{f_{u}}{\gamma_{w}L} \tag{3}\]1  
It is worth noting that \(\hat{d}_{i}\) decreases when \(\sigma\) increases. With \(2\sigma\hat{d}_{i}\) being a finite quantity equal to the axial force, \(2\sigma\hat{d}_{i}^{2}\to 0\) when \(\sigma\rightarrow\infty\). In the presence of a horizontal longitudinal seismic acceleration \(a\), expressed as a fraction of the gravity acceleration \(g\), for an assigned mechanism, the equilibrium equation is  
\[\sigma=\frac{f_{u}}{\gamma_{w}L} \tag{3}\]2  
and allows for the determination of a kinematic admissible seismic acceleration:  
\[\sigma=\frac{f_{u}}{\gamma_{w}L} \tag{3}\]3  
The upper bound theorem ensures that the collapse acceleration is the minimum of all the kinematic admissible ones, while the lower bound theorem ensures that the collapse acceleration is the maximum of all the statically admissible ones. Considering both these theorems, an iteration procedure can be used, which can be started by assigning a first mechanism and considering the corresponding diagrams of the virtual displacements (Figure 5). From Equation (13), a kinematically admissible seismic acceleration can be deduced and so the seismic load that corresponds to the assigned mechanism. With the seismic loads being known, it is easy to find out the external reactions and then the line of thrust passing through the yielded zones, i.e., the internal forces acting at each section. The acceleration value is the collapse one only if, for all the sections, the points that represent the stress state are not outside the cross-section limit domain, but they are on the limit domain at a sufficient number of sections to turn the arch into a mechanism. If these conditions are not satisfied, the procedure must be continued, and the hinges must be moved to the sections where there are the maximum distances between the couples \((\hat{N},\hat{e})\) and the limit domain, i.e., where the compression stresses are at a maximum, changing the hinge cross-sections and axial force eccentricities. In the investigation reported in the following, a high number of voussoirs were considered. So, the obtained seismic accelerations that determines the onset of motion are certainly not lower than the actual ones. The latter, in fact, are related to the actual number and size of the voussoirs, which limit the possible mechanisms. Furthermore, by limiting the analysis to a plane model, the width \(b=1\) was assumed.