# Seismic Assessment and Strengthening of a Load-Bearing Masonry Structure Considering SSI Effects
## 4 Type of Analysis Performed
### Modal Analysis  
In the modal analysis, the periods and participating mass ratios were calculated for each model. According to seismic codes [23; 29], the cumulative effective modal mass of the modes considered in the spectral analysis must amount to at least 75% of the total mass of the structure, as specified in [29], and 90% as required by [23]. For the purposes of the present study, 200 modes were considered for each model (Tables 7-10):  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**StepNum** & **Period** & **SumUX** & **SumUY** & **SumUZ** \\ \hline
**Unitless** & **Sec** & **Unitless** & **Unitless** & **Unitless** \\ \hline
1 & 0.287175 & 0.00\% & 0.00\% & 0.44\% \\
2 & 0.272051 & 0.02\% & 5.90\% & 0.45\% \\
3 & 0.194507 & 91.14\% & 5.90\% & 0.45\% \\
4 & 0.166549 & 91.16\% & 93.56\% & 0.46\% \\
17 & 0.102015 & 93.04\% & 94.23\% & 84.39\% \\
200 & 0.031548 & 97.52\% & 97.82\% & 87.40\% \\ \hline \hline \end{tabular}
\end{table}
Table 10: M2 ssi—modal participating mass ratios.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**StepNum** & **Period** & **SumUX** & **SumUY** & **SumUZ** \\ \hline
**Unitless** & **Sec** & **Unitless** & **Unitless** & **Unitless** \\ \hline
1 & 0.18602 & 0.00\% & 5.20\% & 0.00\% \\
2 & 0.13667 & 12.09\% & 5.20\% & 0.00\% \\
41 & 0.05178 & 87.83\% & 87.10\% & 21.49\% \\
133 & 0.03032 & 90.03\% & 88.84\% & 84.80\% \\
190 & 0.02547 & 90.49\% & 90.00\% & 86.03\% \\
200 & 0.02468 & 90.56\% & 90.13\% & 86.17\% \\ \hline \hline \end{tabular}
\end{table}
Table 7: M1 fix—modal participating mass ratios.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**StepNum** & **Period** & **SumUX** & **SumUY** & **SumUZ** \\ \hline
**Unitless** & **Sec** & **Unitless** & **Unitless** & **Unitless** \\ \hline
1 & 0.287152 & 0.00\% & 0.00\% & 0.42\% \\
3 & 0.156688 & 5.01\% & 3.73\% & 0.43\% \\
5 & 0.150461 & 73.52\% & 3.74\% & 0.99\% \\
8 & 0.122583 & 75.61\% & 57.18\% & 1.00\% \\
59 & 0.05362 & 90.05\% & 89.25\% & 77.62\% \\
81 & 0.045667 & 90.48\% & 90.03\% & 83.13\% \\
200 & 0.029105 & 92.11\% & 91.54\% & 87.77\% \\ \hline \hline \end{tabular}
\end{table}
Table 10: M2 ssi—modal participating mass ratios.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**StepNum** & **Period** & **SumUX** & **SumUY** & **SumUZ** \\ \hline
**Unitless** & **Sec** & **Unitless** & **Unitless** & **Unitless** \\ \hline
1 & 0.287152 & 0.00\% & 0.00\% & 0.42\% \\
3 & 0.156688 & 5.01\% & 3.73\% & 0.43\% \\
5 & 0.150461 & 73.52\% & 3.74\% & 0.99\% \\
8 & 0.122583 & 75.61\% & 57.18\% & 1.00\% \\
59 & 0.05362 & 90.05\% & 89.25\% & 77.62\% \\
81 & 0.045667 & 90.48\% & 90.03\% & 83.13\% \\
200 & 0.029105 & 92.11\% & 91.54\% & 87.77\% \\ \hline \hline \end{tabular}
\end{table}
Table 9: M2 fix—modal participating mass ratios.  
The following results are extracted from Tables 7-10:  
* Compared to M1 fix and M1 ssi, the fundamental period increases by 54.37% for M2 fix and 51.49% for M2 ssi, respectively.
* The fundamental period of M1 ssi is 1.90% higher than that of M1 fix, while M2 ssi exhibits only a 0.01% increase relative to M2 fix, indicating no significant change in the fundamental natural period in either case.
* The ratios T\({}_{1}\)M1 ssi/T\({}_{1}\)M1 fix = 1.02 and T\({}_{1}\)M2 ssi/T\({}_{1}\)M2 fix = 1.00 are both \(\leq\) 1.08; therefore, the structures can be analyzed as if founded on rigid soil [42]. Nevertheless, the effect of SSI was still examined in order to evaluate its influence on the maximum stress values.