## 3 Application to the Case Study
### LASA Equilibrium Solution  
As shown in Section 2.1, a discrete number of Linear Arches can be considered within the stair in order to model, by means of uniaxial stress along that arches, the transmission of the given vertical load to the supporting structures. Obviously, the number and the geometry of the family of Linear Arches should take into account the way in which the steps are in contact with each other. There are cases where the steps, made of monolithic stone blocks, are in contact only along an internal rib (for example, the stair of San Domingo de Bonaval analyzed in [21]), and thus a single linear arch must be considered. If this is not the case, a number of Linear Arches can be accounted for, taking into account that the class of compressive equilibrium solutions enlarges as more and more arches are assumed. A balance between the two competing demands for richness of solutions and reasonable computational effort must be found. The choice of a convenient inter-spacing is obviously related to the minimum transverse dimension of the step. For a non-monolithic stair, say a tile vault made of bricks, this width is also dependent on the size of the bricks and of their arrangement. For the analysis shown below, referred to a stair made of monolithic steps, four spiral Linear Arches (\(n_{l}=4\)) spaced apart 0.19 m from each other have been considered (see Figure 6b). The steps of the stair are made of stone with a specific weight \(\rho=25.5\)\(\text{ kN}/\text{m}^{3}\), so we consider a distributed load \(q=8.0\)\(\text{ kN}/\text{m}^{2}\), taking into account the self-weight. The geometry of the four linear arches \(\Gamma_{i}^{*}\), of projected radius \(r_{i}\) and total projected length \(L_{i}\), can be described as follows  
\[\Gamma_{i}^{*}=\bigg{\{}\mathbf{x}_{i}^{*}=r_{i}\cos\frac{s_{i}}{r_{i}} \hat{\mathbf{e}}_{1}+r_{i}\sin\frac{s_{i}}{r_{i}}\hat{\mathbf{e}}_{2}+f_{i}( s_{i})\hat{\mathbf{e}}_{3},\;s_{i}\in[0,L_{i}]\bigg{\}}, \tag{14}\]  
where the functions \(f_{i}(s_{i})\) are assumed linear, and described by the expression:  
\[f_{i}(s_{i})=4H\bigg{(}1-\frac{s_{i}}{L_{i}}\bigg{)}. \tag{15}\]  
In order to represent the results, it is convenient to introduce the non-dimensional variable \(\alpha=s_{i}/r_{i}\) to express both the geometric and force variables. Writing Equation (11) in terms of \(\alpha\)  
\[S_{i}(\alpha)=\frac{1}{f_{i}^{\prime}}Q\alpha r_{i}+S^{0}, \tag{16}\] \begin{table}
\begin{tabular}{c c c} \hline  & type: & circular \\  & rotation: & laevrotatory \\
**PLAN** & staircase diameter: & 1.83 m \\  & hole (eye) diameter: & 0.30 m \\  & step angle: & 31\({}^{\circ}\) \\  & number of steps: & 36 \\ \hline  & central hole solution: & radial \\  & total length (\(L_{step}\)): & 0.76 m \\  & usable length: & 0.61 m \\
**STEP** & minimum width: & 0.16 m \\  & maximum width: & 0.49 m \\  & step height: & 0.41 m \\  & medium height (\(b_{m}\)): & 0.21 m \\ \hline \end{tabular}
\end{table} 
Table 1: Spiral staircase in the tower of Nisida: synthesis of the main metric and dimensional parameters. where \(Q=\frac{qL_{stop}}{4}\) is the load for unit projected length carried by each linear arch. The maximum thrust value occurs at line 4 at the bottom end of the line and its value is \(S_{max}=35.5\) kN (see Figure 6b). To obtain the normal forces \(N_{i}(\alpha)\) we project back the thrusts \(S_{i}(\alpha)\) on the lines  
\[N_{i}(\alpha)=S_{i}(\alpha)\sqrt{1+f_{i}(\alpha)^{2}}, \tag{17}\]  
In Figure 6c the plots of \(N_{i}\) as a function of \(\alpha\) is reported. The maximum normal force value, that is the maximum value of the thrust transmitted by one step to the one below, occurs in line 4 at the bottom of the stair, and its value is \(N_{max}=45.6\) kN.