## 3 Case Study
### Design Requirements and the Definition of Boundary Constraints and Load Conditions  
Load conditions, to be applied, are stress-strain conditions derived from the static analysis of the central body and the effects due to the assembly of arms and wings on the inner frame, as described in Figure 3b. Wings and arms introduce a bending moment in the lateral plane of the statue that passes through its left knee (P1 in Figure 4a). From the design requirements point of view, the central body stress must be maintained in the same conditions as before the restoration, characterized by the presence of a filling of resin able to minimize relative rotation between the inner frame and the statue. This filling has been removed for material preservation requirements, hence the necessity of a new inner frame. To define the worst reference condition derived from the filling removal, a preliminary FEA on the central body was done. Figure 4c shows von Mises equivalent stress superimposed to the magnified plot of the relative displacements from the undeformed shape (grey mesh), as derived from the gravity load, without imposing any filling inside. The gravity load of the central body defines a bending effect, in the frontal plane, that overstresses the base of the neck, where a visible crack is present. This defines a major constraint on the compliance of the structure that must reduce the final displacement of the head (OR2.1, 2.2 and 3.1). \begin{table}
\begin{tabular}{l l l} \hline \hline \multicolumn{1}{c}{**Design Requirements (DR)**} & \multicolumn{2}{c}{**TO Optimization Requirements (OR)**} \\ \hline
1 Preserve structural integrity of the ancient material & \(\rightarrow\) &
\begin{tabular}{l} 1.1 Maximize the safety margin for local stress (at least equal to 2) \\ 1.2 Avoid interfaces on thin or weak areas \\ \end{tabular} \\ \hline
2 Limit relative displacement at the neck & \(\rightarrow\) &
\begin{tabular}{l} 2.1 Reduce the head displacement \\ 2.2 Minimize bending compliance in the front and lateral planes \\ of the statue \\ \end{tabular} \\ \hline
3 Reduce stress at the right part of the hip & \(\rightarrow\) &
\begin{tabular}{l} 3.1 Minimize bending compliance in the front and lateral planes \\ of the statue \\ \end{tabular} \\ \hline
4 Do not apply loads on the base of the skirt & \(\rightarrow\) & 4.1 Suspended directly on the inner frame \\ \hline
5 Avoid contact among wings and back of the central body & \(\rightarrow\) & 5.1 Suspended directly on the inner frame \\ \hline
6 Connect the statue as a rigid body to the anti-seismic basement & \(\rightarrow\) & 6.1 Transfer loads to the ground for the static balance \\  & \(\rightarrow\) & 6.2 Reduce relative compliance between central body and arms \\  & & and wings, making the system as a rigid body \\ \hline
7 Guarantee the position of the arms and the wings in the respect of the central body & \(\rightarrow\) & 7.1 Wings and arms must be connected directly to the central \\  & \(\rightarrow\) & beam through upper openings of the central body \\ \hline
8 Allow assembly and disassembly through the limited areas of the openings i.e., at the base, around the arms and wings connection areas & \(\rightarrow\) & 8.1 Reduction of the design space at the minimum section of the statue to guarantee maneuvering \\  & \(\rightarrow\) & 8.2 Possible manufacturing constraints related to the adoption of \\  & & a wireframe structures made of beams and mold/machined \\  & & parts at the interfaces with bronze \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design Requirements declared by the restorers and their translation to optimization requirements. Areas useful for suspending the central body with a simple contact interface are the left leg (above the knee) and the right shoulder (P2 and P1 in Figure 4a,b).