# 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 following relation shall be satisfied at all joints except those of the upper floor: \[\rm\sum M_{Rc}\geq 1.3\rm\sum M_{Rb},\] (7)where \(\sum\)M\({}_{\rm{Rc}}\) and \(\sum\)M\({}_{\rm{Rb}}\) are the sums of the resistance moments of the columns and the beams, respectively, framing the joint; (g) The verification of diagonal struts' compressive resistance at the joints. The following relations shall be satisfied: \[\mathrm{V_{jhd}}\leq\left\{ \begin{array}{ll}\mathrm{\eta f_{cd}\sqrt{1-\frac{ \mathrm{v_{d}}}{\mathrm{\eta}}}\mathrm{b_{j}h_{c}},&\qquad\text{for interior joints}\\ 0.8\mathrm{\eta f_{cd}\sqrt{1-\frac{\mathrm{v_{d}}}{\mathrm{\eta}}}\mathrm{b_{ j}h_{c}},&\qquad\text{for exterior joints}\end{array} \right\},\] (8) where \(\mathrm{V_{jhd}}\) is the shear force acting horizontally on the concrete core, \(\mathrm{\eta}=0.6\) (\(1-\mathrm{f_{ck}}/250\)), \(\mathrm{v_{d}}\) is the maximum value of the column's normalized axial force above the joint (the compression is positive) and \(\mathrm{b_{j}}\) is the effective joint width. This provision applies for DCH joints only; (h) The confinement of joints. The total cross-sectional area A\({}_{\rm{sh}}\) of horizontal hoops shall be provided: \[\frac{\mathrm{A_{sh}f_{ywd}}}{\mathrm{b_{j}h_{jw}}}\geq\frac{\left(\frac{ \mathrm{v_{jhd}}}{\mathrm{b_{jh_{c}}}}\right)^{2}}{\mathrm{f_{cd}+v_{d}f_{cd }}}-\mathrm{f_{ctd}},\] (9) where \(\mathrm{h_{jw}}\) is the distance between the bottom and top longitudinal reinforcing bars of the beams, \(\mathrm{h_{jc}}\) is the distance between column longitudinal reinforcing bars of opposite faces, \(\mathrm{f_{ctd}}\) is the concrete tensile strength and \(\mathrm{v_{d}}\) is the minimum column normalized axial force above the joint (the compression is positive). This provision applies for DCH joints only; (i) Vertical joint reinforcement. A minimum vertical reinforcement, A\({}_{\rm{sv,i}}\), shall be provided: \[\mathrm{A_{sv,i}}\geq\frac{2}{3}\mathrm{A_{sh}\frac{h_{jc}}{h_{jw}}}\] (10) where \(\mathrm{A_{sv,i}}\) is the cross-sectional area of the intermediate bars placed in the relevant column faces. This provision applies for DCH joints only. Regarding the aforementioned code provisions (a) to (i), the number of inadequacies that occurred due to H+V excitation out of the total number of checks conducted is given in Table 6. Other code provisions like those related to column buckling, the limitation of storey drifts, etc., were not remarkably influenced by the vertical motion, and they were not critical. \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 Limitation of columns’ & 4/32 & 8/32 & 10/32 & 0/32 & 2/32 & 8/32 \\ normalized axial force & 0/32 & 0/32 & 0/32 & 0/32 & 0/32 & 0/32 \\ \hline Verification of columns’ & 8/32 & 4/32 & 1/32 & 14/32 & 18/32 & 12/32 \\ shear resistance & 2/4 & 4/4 & 4/4 & 36/64 & 40/64 & 48/64 \\ critical regions & 0/24 & 18/24 & 20/24 & 2/24 & 14/24 & 22/24 \\ \hline Verification of beams’ & 0/24 & 18/24 & 20/24 & 2/24 & 14/24 & 22/24 \\ bending resistance & 0/24 & 18/24 & 20/24 & 2/24 & 14/24 & 22/24 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Number of inadequacies/number of checks. Apparently, the vertical component's impact with the greatest importance was the inducement of significant variation in the columns' normalized axial forces, v\({}_{d}\), especially for the interior ones. As an example, the variation in v\({}_{d}\) for columns C11 and C12 of all the frames under H and H+V excitations is shown in Figure 6. On the one hand, the vertical oscillation increased the maximum value of v\({}_{d}\) (by up to about 49%), even exceeding the prescribed limit (see subclause (c) above). In addition, the confinement requirements at the critical regions of the columns were increased too (Equation (4)), whereas the diagonal struts' compressive resistance at beam-column joints was reduced (Equation (8)).