# 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
On the other hand, the vertical component decreased the minimum value of v\({}_{d}\) (by up to about 51%), unfavorably affecting the anchorage of the longitudinal reinforcing bars of the beams (subclause (e) above) and increasing the reinforcement requirements at beam-column joints (subclauses (h) and (i) above). Obviously, the axial force variation also reduced the minimum flexural resistance of the columns. Due to various provisions of the Eurocodes (e.g., (c), (e), (f), (g), (h), (i)), the columns of the frames examined here were quite over-dimensioned against bending, so no deficiencies of bending resistance occurred. However, the ratio of the flexural strengths of columns and beams framing the joints (Equation (7)) was disrupted, and the crucial code provision for preventing the formation of a soft storey plastic mechanism was not satisfied in several cases. Furthermore, the axial force variation increased the maximum flexural resistance and the column design shear forces (capacity design rule), potentially leading to the exceedance of the shear resistance. Conclusively, the fluctuation of the columns' normalized axial forces was the main cause of the most inadequacies reported in Table 6.  
Some indicative figures quantifying the aforementioned inadequacies of columns and joints are given subsequently. In Figure 7, the maximum developing shear forces of columns C12 (MI, MII, MIII) and C82 (HI, HII, HII) due to H and H+V excitations are shown. In Figure 7, the shear resistance of the non-critical regions, where the inadequacies occurred, is also plotted. The shear resistance was calculated using Equation (1) for \(\theta=21.8^{\circ}\). Despite the unconservative value of \(\theta\), the shear demand exceeded the resistance of non-critical regions by up to about 12% (HIII). Figure 8 illustrates the confinement demands due to H and H+V excitations at the base of column C12 along with the available confinement in terms of \(\alpha\omega_{\text{wd}}\). Apparently, the vertical seismic component considerably increased the percentage of the confinement demands which exceeded the available values from about 10% to 42%. This was more pronounced for higher seismic hazard levels and for DCH \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Eurocode Provisions**} & \multicolumn{6}{c}{**Frames**} \\ \cline{2-7}  & **MI** & **MII** & **MIII** & **HI** & **HIII** & **HIII** \\ \hline Verification of beams’ shear resistance & 0/24 & 0/24 & 4/24 & 0/24 & 0/24 & 5/24 \\ \hline Verification of joints’ diagonal struts’ compressive resistance & Not applicable & Not applicable & Not applicable & 0/32 & 0/32 & 8/32 \\ \hline Horizontal confinement & Not  applicable & Not applicable & Not applicable & 10/32 & 14/32 & 20/32 \\ \hline Vertical reinforcement & Not applicable & Not applicable & Not applicable & 6/32 & 6/32 & 18/32 \\ \hline Anchorage of longitudinal bars of beams & 2/32 & 4/32 & 14/32 & 6/32 & 8/32 & 12/32 \\ \hline Ratio of column flexural resistance to beam flexural resistance & 4/28 & 12/28 & 18/28 & 0/28 & 0/28 & 4/28 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: _Cont._  
Figure 8: Required and available confinement \(\omega\omega_{\text{wd}}\) at the base of column C12. Figure 10: Required and real width (cm) of column C52. Figure 7: Maximum shear forces and shear resistance (kN) of non-critical regions of columns (**a**) C12 and (**b**) C82. Figure 9: Bending resistance ratio of columns and beams framing joint J72. A considerable number of deficiencies regarding the flexural resistance of beams resulted due to the additional bending moments produced by the vertical seismic component, especially on the upper stories. Indicatively, the maximum and minimum moments at the ends as well as at the mid-span of beam B82 of all the frames for H and H+V excitations are shown in Figure 12. It is obvious that the contribution of the vertical component was quite important since it even led to sign reversals of the mid-span moment.