## 3 The Limit Analysis under Horizontal Longitudinal Loads  
According to the safe theorem, an arch is safe if an equilibrium solution can be found, in which the couples \((\hat{N},\hat{e})\) are always inside the limit domain. The failure of an arch occurs when sufficient hinges form to turn it into a mechanism. Due to the hypotheses on the material and the stress distribution, all the hinges form contemporary. The uniqueness theorem states that the solution exists and is unique, and so is the load factor. On the point of collapse, the internal forces acting at the hinge sections contribute to the stability of the arch, while they do not influence the equilibrium in the case of \(f_{u}\rightarrow\infty\). The coordinates of the arch centre line are given in the non-dimensional form as a ratio of the span \(L\) as follows:  
\[\hat{z}=z/L\quad,\quad\hat{y}(\hat{z})=y(z)/L \tag{2}\]  
The geometrical parameters are also given in the following non-dimensional forms:  
* The ratio \(\hat{f}=f/L\) between the sag (or height) and the span;
* The ratio \(\hat{t}(\hat{z})=t(z)/L\) between the thickness and the span, where the variable thickness is given as a function of its value at the crown \(t_{c}\): \(t(z)=t_{c}\cdot t^{\prime}(z)\) (\(t^{\prime}(z)=1\) at the crown);
* The ratio between the depth of the fill above the crown and the span \(\hat{h}=h/L\). The material compression strength can be represented by the non-dimensional parameter as follows:  
\[\sigma=\frac{f_{u}}{\gamma_{w}L} \tag{3}\]  
where \(\gamma_{w}\) is the weight per unit volume of the structural material. The arch is subject to the dead load \(w\), which is given by the summation of the ring self-weight \(w_{w}\) and the weight \(w_{b}\) of the backfill, whose weight per unit volume is \(\gamma_{b}=\gamma\cdot\gamma_{w}\). The dead loads per unit span can be written as follows (\(\alpha(\hat{z})=\) angle between the tangent to the arch centre line and the horizontal axis \(z\)):  
\[w_{w}(z)=\gamma_{w}\ bL\cdot\hat{w}_{w}(\hat{z})=\gamma_{w}\ bL\cdot\frac{ \hat{t}(\hat{z})}{\cos\alpha(\hat{z})} \tag{4}\]  
\[w_{b}(z)=\gamma_{w}bL\cdot\hat{w}_{b}(\hat{z})=\gamma_{w}bL\cdot\gamma\left( \hat{h}+\hat{f}+\frac{\hat{t}(\hat{z})}{2}-\hat{y}(\hat{z})-\frac{\hat{t}( \hat{z})}{2\ \cos\alpha(\hat{z})}\right) \tag{5}\]  
The seismic action is represented by the product of the horizontal acceleration \(a\), which is expressed as a fraction of the gravity acceleration \(g\), acting at the base of both the abutments and the gravity load \(w_{E}(z)\) associated with the distributed seismic masses as follows:  
\[p_{E}(z)=a\cdot w_{E}(z) \tag{6}\]  
The load \(w_{E}(z)\) certainly contains the ring self-weight and the contribution of the backfill, for which different models can be assumed, as will be shown later. It can be written in non-dimensional form as follows:  
\[w_{E}(z)=\gamma_{w}bL\cdot\hat{w}_{E}(z) \tag{7}\]Then, the distributed seismic load can be written as follows:  
\[p_{E}(z)=\gamma_{w}bL\cdot a\cdot\hat{w}_{E}(z)=\gamma_{w}bL\cdot\hat{p}_{E}(z) \tag{8}\]  
All the loads, vertical and horizontal, are evaluated with reference to the centre line of the arch and are supposed to act on it. The infill located over the maximum height of the masonry arch has no effects on the arch in this model; this is acceptable for bridges where \(h\) is usually low. Vertical accelerations are not considered; however, in this model, a vertical component of the acceleration is equivalent to a different weight of the permanent load. Consider the masonry arch in Figure 4 and suppose that an equilibrium solution under dead loads exists. This means that the points representing the stress states at each cross-section are always within the limit domain \((|\vec{\epsilon}|\leq(1-\hat{N})/2)\). When the seismic loads are put in action and increase from zero to the collapse value, the line of thrust changes, and at least four hinges form. On the point of collapse, the thrust line must pass through the hinge points.