# Elasto-Plastic Short Exoskeleton to Improve the Dynamic and Seismic Performance of Frame Structures
## 4 Parametric Analysis  
Parametric analyses are performed to evaluate the dynamic and the seismic effectiveness of the coupling. To simulate cases close to reality, two structures with characteristics reported in Table 1 are used as references. For each structure, all storeys have the same mass, height, and stiffness. The characteristics reported in Table 1 are used in the reducing order technique adopted in [36; 37; 38] to derive the mechanical characteristics of equivalent 2-DOF models of the structures. It is worth observing that the values of \(k_{1}\) and \(k_{2}\) change with the yielding system connection level, even if the equivalent 2-DOF models refer to the same building (see the second and the third rows of Table 2). In Table 2, \(\xi\) shows the modal damping ratio adopted for the two oscillation modes of the stand-alone 2-DOF equivalent model. The value \(\xi=0.05\) is commonly used in various building codes for structural frame structures, independently of the material of the structure, and it is the value used in this paper. The parameters that are varied in the analyses are the following:  
* Mechanical characteristics of the 2-DOF system \(m_{i}\) and \(k_{i}\) (\(i=1,2\)), representing real M-DOF frame structures;
* Post- and pre-yielding stiffness ratio \[\psi=\frac{k_{\text{plast}}}{k_{\text{exo}}};\] (4) \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Storeys** & **Storey Area** & **Storey Mass \(m_{\text{s}}\)** & **Storey Height** & **Main Period** \\ \hline
3 & 100 m\({}^{2}\) & \(120.6\times 10^{3}\) kg & 3 m & 0.390 s \\
6 & 300 m\({}^{2}\) & \(361.8\times 10^{3}\) kg & 3 m & 0.655 s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric and mechanical characteristics of the two reference multi-degree-of-freedom frames (M-DOF). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Storeys** & **Connection** & \(k_{1}\) **[N/m]** & \(k_{2}\) **[N/m]** & \(m_{1}\) **[kg]** & \(m_{2}\) **[kg]** & \(\xi\) \\ \hline
3 & 1 & \(2.19219\times 10^{8}\) & \(0.93951\times 10^{8}\) & \(120.6\times 10^{3}\) & \(241.2\times 10^{3}\) & 0.05 \\
6 & 1 & \(1.02972\times 10^{9}\) & \(1.99300\times 10^{8}\) & \(361.8\times 10^{3}\) & \(1809.0\times 10^{3}\) & 0.05 \\
6 & 2 & \(9.30067\times 10^{8}\) & \(1.99300\times 10^{8}\) & \(723.6\times 10^{3}\) & \(1447.2\times 10^{3}\) & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mechanical characteristics of the 2-DOF equivalent systems. Ratio between pre-yielding stiffness of the yielding system and stiffness of the substructure  
\[\mu=\frac{k_{\text{exo}}}{k_{1}}; \tag{5}\]  
Ratio between yielding force and weight of the yielding system \(w_{\text{exo}}\) (obtained multiplying the mass \(m_{\text{exo}}\) by the gravitational acceleration \(g\))  
\[\eta=\frac{F_{y}}{w_{\text{exo}}}=\frac{k_{\text{exo}}u_{y}}{m_{\text{exo}}g}; \tag{6}\]  
Ratio between mass of the yielding system and total mass of the structure  
\[\gamma=\frac{m_{\text{exo}}}{m_{1}+m_{2}}. \tag{7}\]  
The comparison between the displacement \(u_{1}\) and the drift \(\Delta u=u_{2}-u_{1}\) of the coupled system and those of the stand-alone structure provides an indication of the efficiency of the coupling. The above-mentioned comparison is performed by means of two gain indexes  
\[\alpha_{1}=\frac{\max\lvert u_{1}(t)\rvert}{\max\lvert\bar{u}_{1}(t)\rvert}, \ \ \alpha_{2}=\frac{\max\lvert u_{2}(t)-u_{1}(t)\rvert}{\max\lvert\bar{u}_{2}(t)- \bar{u}_{1}(t)\rvert}=\frac{\max\lvert\Delta u(t)\rvert}{\max\lvert\Delta\bar {u}(t)\rvert} \tag{8}\]  
where the displacements \(\bar{u}_{1}\) and \(\bar{u}_{2}\) are those of the 2-DOF model of the stand-alone structure. According to Equation (8), the effectiveness of the coupling increases as the values of \(\alpha_{1}\) and \(\alpha_{2}\) become smaller and smaller compared to unity. The parametric analysis is performed by evaluating the gain indexes \(\alpha_{1}\) and \(\alpha_{2}\) for each set of variable parameters and a single base motion and plotting such values in a specific parameters plane, thus creating gain maps.