# The Influence of Vertical Ground Motion on the Design of Common R/C Frames
## 3 Evaluation of Frames for Vertical Ground Motion  
### Evaluation Through Spectral Analysis  
The modal response spectrum analysis method was applied considering that horizontal and vertical ground motions act simultaneously ('H+V' excitation). The vertical seismic component was represented by the Eurocode 8 vertical design spectrum for the Peak Ground Acceleration (PGA) equal to 90 percent of the horizontal one and a behavior factor equal to 1.5 regardless of the ductility class of the frames (Figure 4). The directional combination of seismic actions was conducted applying the SRSS rule. The mass was simulated as illustrated in Figure 5. The reliability of such a modeling approach was demonstrated in a previous publication [31] which indicated that the resulting critical \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Frames** & **T\({}_{1}\) (s)** & **T\({}_{2}\) (s)** & **T\({}_{3}\) (s)** & \(\varepsilon_{1}\) (\%)** & \(\varepsilon_{2}\) (\%) & \(\varepsilon_{3}\) (\%) \\ \hline MI & & & & & \\ \hline MIII & 1.317 & 0.479 & 0.289 & 77.6 & 12.6 & 4.8 \\ \hline MIII & & & & & \\ \hline HI & & & & & \\ \hline HII & 1.091 & 0.372 & 0.213 & 79.0 & 12.3 & 4.1 \\ \hline HII & & & & & & \\ \hline HII & & & & & \\ \hline HII & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Modal participating mass ratios, \(\varepsilon_{i}\), for horizontal excitation and natural periods, T\({}_{i}\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Columns** & \multicolumn{6}{c}{**Frames**} \\ \cline{2-6}  & **MI** & **MI** & **MIII** & **HI** & **HI** & **HIII** & **HIII** \\ \hline C51-C54 & 35/35 & 35/35 & 35/35 & 40/40 & 40/40 & 40/40 \\ \hline C52-C53 & 40/40 & 40/40 & 40/40 & 50/50 & 50/50 & 50/50 \\ \hline C61-C64 & 35/35 & 35/35 & 35/35 & 40/40 & 40/40 & 40/40 \\ \hline C62-C63 & 40/40 & 40/40 & 40/40 & 50/50 & 50/50 & 50/50 \\ \hline C71-C74 & 30/30 & 30/30 & 30/30 & 40/40 & 40/40 & 40/40 \\ \hline C72-C73 & 35/35 & 35/35 & 35/35 & 50/50 & 50/50 & 50/50 \\ \hline C81-C84 & 25/25 & 25/25 & 25/25 & 40/40 & 40/40 & 40/40 \\ \hline C82-C83 & 35/35 & 35/35 & 35/35 & 50/50 & 50/50 & 50/50 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._ \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Beams** & \multicolumn{6}{c}{**Frames**} \\ \cline{2-6}  & **MI** & **MI** & **MIII** & **HI** & **HII** & **HIII** \\ \hline B11-B12-B13 & 30/75 & 30/75 & 30/75 & 30/75 & 30/75 & 30/75 \\ \hline B21-B22-B23 & 30/75 & 30/75 & 30/75 & 30/75 & 30/75 & 30/75 \\ \hline B31-B32-B33 & 30/70 & 30/70 & 30/70 & 30/75 & 30/75 & 30/75 \\ \hline B41-B42-B43 & 30/70 & 30/70 & 30/70 & 30/75 & 30/75 & 30/75 \\ \hline B51-B52-B53 & 30/70 & 30/70 & 30/70 & 30/70 & 30/70 & 30/70 \\ \hline B61-B62-B63 & 30/70 & 30/70 & 30/70 & 30/70 & 30/70 & 30/70 \\ \hline B71-B72-B73 & 30/70 & 30/70 & 30/70 & 30/70 & 30/70 & 30/70 \\ \hline B81-B82-B83 & 30/70 & 30/70 & 30/70 & 30/70 & 30/70 & 30/70 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Cross-sectional dimensions of beams (cm/cm). response quantities were almost equal to the exact values (mean error of up to 1-2%). As a consequence, the natural vibration modes of the structural models ran into the hundreds. The response spectrum analysis was conducted considering the first fifty of them. This number ensured that over 90 percent of the total mass was activated for both horizontal and vertical excitations. The natural periods and the modal participating mass ratios of the first three modes dominating the vertical response are given in Table 5.  
The following Eurocode provisions related to member design were verified considering the response quantities produced by H+V excitation:  
1. Members' flexural resistance with or without axial force;
2. Members' shear resistance, which is governed by the transverse reinforcement: \[\mathrm{V_{Rd,s}}=\frac{\mathrm{A_{sw}}}{8}0.9\mathrm{d}\mathrm{f_{ywd}} \mathrm{cot}\theta,\] (1)  
Figure 4: Vertical pseudo-acceleration design spectra for the three seismicity zones of Greece (\(\gamma_{1}=1\), S = 1, q = 1.5).