## 5 Discussion  
The previous section shows that the longitudinal elongation due to base isolation (BI) needs to consider the role of soil-structure interaction (SSI). The study herein assumes that the three subsystems (soil, isolators and structure) are modelled as linear for expeditious assessments of the performance of the 3 DOF-system. The results show that there two important key parameters: the isolation ratio (n) and the squared ratio between the soil frequency and that structural one (\(a\)), that represents the effects of SSI.  
Parameter \(a\) was used herein to consider the relationship between the fundamental structural period and the shear wave velocity into a single parameter. The mutual interaction between BI and SSI was shown to depend on n when foundation soil is sufficiently rigid, and the structure is sufficiently deformable (high values of \(a\)). However, when soil deformability increases and the structure becomes particularly rigid (for example, in case of low-rise buildings), period elongation depends on \(a\). Therefore, \(a\) needs to be considered within the formulation.  
Formulas (12) may be condensed in a more compact way as:  
\[\mathcal{F}:\frac{T_{1}}{T_{S}}=1.388\cdot(n)^{0.965}\cdot\left(\frac{0.725 \cdot(n)^{-0.897}}{\sqrt{\alpha}}\right)^{\delta} \tag{13}\]  
where:  
For \(\alpha>\Omega\): \(\delta=0\);  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Vs = 1000** & **T1/Ts (6)** & **T1/Ts (10)** & **Error** & **T1/Ts (6)** & **T1/Ts (10)** & **Error** & **T1/Ts (6)** & **T1/Ts (10)** & **Error** \\ \hline \multicolumn{1}{c}{Vs= 200} & & & & & & & & \\ \hline B1 & 2.715 & 2.709 & 0.23\% & 3.998 & 4.006 & \(-0.19\%\) & 5.298 & 5.287 & 0.19\% \\ B2 & 2.716 & 2.709 & 0.27\% & 3.999 & 4.006 & \(-0.17\%\) & 5.298 & 5.287 & 0.20\% \\ B3 & 2.717 & 2.709 & 0.32\% & 3.999 & 4.006 & \(-0.15\%\) & 5.299 & 5.287 & 0.21\% \\ B4 & 2.720 & 2.709 & 0.42\% & 4.001 & 4.006 & \(-0.11\%\) & 5.300 & 5.287 & 0.24\% \\ B5 & 2.723 & 2.709 & 0.54\% & 4.003 & 4.006 & \(-0.07\%\) & 5.301 & 5.287 & 0.26\% \\ B6 & 2.727 & 2.709 & 0.68\% & 4.005 & 4.006 & \(-0.02\%\) & 5.302 & 5.287 & 0.28\% \\ B7 & 2.734 & 2.709 & 0.93\% & 4.008 & 4.006 & 0.05\% & 5.304 & 5.287 & 0.32\% \\ B8 & 2.751 & 2.709 & 1.58\% & 4.013 & 4.006 & 0.19\% & 5.308 & 5.287 & 0.39\% \\ B9 & 2.845 & 2.709 & 5.03\% & 4.029 & 4.006 & 0.58\% & 5.316 & 5.287 & 0.54\% \\ B10 & 3.438 & **3.494** & \(-1.60\%\) & 4.096 & 4.006 & 2.25\% & 5.338 & 5.287 & 0.97\% \\ \hline \multicolumn{1}{c}{Vs = 150} & & & & & & & & & \\ \hline B1 & 2.716 & 2.709 & 0.26\% & 3.999 & 4.006 & \(-0.177\%\) & 5.298 & 5.287 & 0.20\% \\ B2 & 2.718 & 2.709 & 0.34\% & 4.000 & 4.006 & \(-0.142\%\) & 5.299 & 5.287 & 0.22\% \\ B3 & 2.720 & 2.709 & 0.44\% & 4.001 & 4.006 & \(-0.106\%\) & 5.300 & 5.287 & 0.24\% \\ B4 & 2.727 & 2.709 & 0.67\% & 4.005 & 4.006 & \(-0.027\%\) & 5.302 & 5.287 & 0.28\% \\ B5 & 2.735 & 2.709 & 0.98\% & 4.008 & 4.006 & 0.062\% & 5.305 & 5.287 & 0.33\% \\ B6 & 2.747 & 2.709 & 1.42\% & 4.012 & 4.006 & 0.164\% & 5.307 & 5.287 & 0.37\% \\ B7 & 2.778 & 2.709 & 2.55\% & 4.019 & 4.006 & 0.343\% & 5.311 & 5.287 & 0.45\% \\ B8 & 2.910 & 2.802 & 3.85\% & 4.036 & 4.006 & 0.748\% & 5.319 & 5.287 & 0.60\% \\ B9 & 3.471 & **3.531** & \(-1.69\%\) & 4.101 & 4.006 & 2.392\% & 5.340 & 5.287 & 1.00\% \\ B10 & 4.527 & **4.659** & \(-2.82\%\) & **4.646** & **4.789** & \(-2.972\%\) & 5.432 & 5.287 & 2.73\% \\ \hline \multicolumn{1}{c}{Vs = 130} & & & & & & & & \\ \hline B1 & 2.716 & 2.709 & 0.29\% & 3.999 & 4.006 & \(-0.18\%\) & 5.298 & 5.287 & 0.21\% \\ B2 & 2.720 & 2.709 & 0.40\% & 4.001 & 4.006 & \(-0.14\%\) & 5.300 & 5.287 & 0.23\% \\ B3 & 2.723 & 2.709 & 0.54\% & 4.0 For \(\alpha<\Omega\): \(\delta=1\);  
It is worth noting that (13) can be implemented inside code prescriptions as a reference formula for practitioners.