## 3 Layout and Design of the Test Structures
The damping coefficients (\(C_{H0}\), \(C_{V0}\)) of the single HDRB are then obtained assuming different equivalent viscous damping ratios in the horizontal (\(\xi_{H}\) = 10%) and vertical (\(\xi_{V}\) = 5%) direction by  
\[C_{H0}=\frac{4\cdot\pi\cdot\xi_{H}\cdot m_{tot}}{n_{b}\cdot T_{1H}},\qquad C_{V0}= \frac{4\cdot\pi\cdot\xi_{V}\cdot m_{tot}}{n_{b}\cdot T_{1V}}=\frac{2\cdot\xi_{V} \cdot m_{tot}}{n_{b}\cdot\left(\frac{m_{tot}}{K_{V0}}\right)^{0.5}} \tag{29}\]  
Next, the effective geometric dimensions of the HDRB, like diameter \(D\), total thickness \(t_{e}\) of the elastomeric layers and thickness \(t_{i}\) of the single elastomeric layers, are evaluated by an iterative procedure including: a design displacement \(S_{d}\) = 27.8 cm, at the collapse prevention (CP) limit state of the Italian seismic code [6]; a shear modulus \(G\) = 0.035 kN/cm\({}^{2}\); a volumetric compression modulus \(E_{b}\) = 200 kN/cm\({}^{2}\). This procedure is repeated until the effective horizontal and vertical stiffnesses (\(K_{H}\), \(K_{V0}\)) match the initial stiffnesses (\(K_{H0}\), \(K_{V0}\)), and the ultimate limit state verifications regarding the maximum shear strains are fulfilled  
\[\gamma_{tot}=\gamma_{s}+\gamma_{c}+\gamma_{a}\leq 5,\qquad\gamma_{s}\leq 2 \tag{30}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Beams**} & \multicolumn{3}{c}{**Columns**} \\ \hline
**Floors** & **Deep** & **Flat** & **Corner** & **Perimeter** & **Interior** \\ \hline
1 & 30 \(\times\) 55 & 70 \(\times\) 25 & 30 \(\times\) 50 & 30 \(\times\) 50 & 30 \(\times\) 50 \\
2 & 30 \(\times\) 50 & 60 \(\times\) 25 & 30 \(\times\) 40 & 30 \(\times\) 40 & 30 \(\times\) 40 \\
3 & 30 \(\times\) 45 & 50 \(\times\) 25 & 30 \(\times\) 30 & 30 \(\times\) 30 & 30 \(\times\) 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Cross sections of beams and columns of the B7 structure (dimensions in cm). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Structures** & **m1** & **m2** & **m3** & **m4** & **m5** & **m6** & **m7** & **m8** & **m9** & **m10** \\ \hline B3 & 218.55 & 210.80 & 144.95 & - & - & - & - & - & - \\ B7 & 239.45 & 237.72 & 232.82 & 226.19 & 218.55 & 210.80 & 144.95 & - & - & - \\ B10 & 262.90 & 261.26 & 237.72 & 253.72 & 253.72 & 232.82 & 226.19 & 218.55 & 210.80 & 144.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Floor masses of the B3, B7 and B10 test structures (dimensions in kNs\({}^{2}\)/m). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{**Beams**} & \multicolumn{3}{c}{**Columns**} \\ \hline
**Floors** & **Deep** & **Flat** & **Corner** & **Perimeter** & **Interior** \\ \hline
1–2 & 30 \(\times\) 70 & 80 \(\times\) 25 & 30 \(\times\) 70 & 30 \(\times\) 70 & 30 \(\times\) 70 \\
3 & 30 \(\times\) 65 & 80 \(\times\) 25 & 30 \(\times\) 65 & 30 \(\times\) 65 & 30 \(\times\) 65 \\
4 & 30 \(\times\) 60 & 80 \(\times\) 25 & 30 \(\times\) 60 Here \(\gamma_{tot}\), \(\gamma_{s}\), \(\gamma_{c}\) and \(\gamma_{\alpha}\) represent the total shear strain of the elastomer and its shear strains due to the seismic displacement, axial compression, and angular rotation, respectively. In particular, the thickness of the single elastomeric layer (\(t_{i}\)), the total thickness (\(t_{e}\)) of the elastomeric layers and the diameter \(D\) of the bearing can be obtained by using the expressions  
\[\gamma_{s}=1.2.\frac{S_{d}}{t_{e}}\leq 2\qquad\Longrightarrow\qquad t_{e}=1.2.\frac{S_{d}}{2}\qquad\Longrightarrow\qquad D=\sqrt{\frac{4}{\pi}.\frac{K_{H0 }\cdot t_{e}}{G}} \tag{31}\]  
\[\alpha_{K0}=\frac{K_{H0}}{K_{V0}}=G\cdot\left(\frac{1}{6\cdot G\cdot S_{1}^{2} }+\frac{4}{3\cdot E_{h}}\right)^{-1}\Longrightarrow\;\;S_{1}=\frac{\sqrt{E_{h }\cdot\sqrt{\pi_{K0}}}}{\sqrt{6\cdot E_{h}\cdot G^{2}-8\cdot A_{K0}}\cdot G} \;\;\Longrightarrow\;t_{i}=\frac{D}{4\cdot S_{1}} \tag{32}\]  
Two further verifications are added to complete the design of the bearings.