# 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
Figure 5: Modeling of a typical floor mass for vertical excitation. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Frames** & **T\({}_{5}\) (s)** & **T\({}_{21}\) (s)** & **T\({}_{33}\) (s)** & \(\varepsilon_{5}\) (\%)** & \(\varepsilon_{21}\) (\%) & \(\varepsilon_{33}\) (\%) \\ \hline MI & & & & & & \\ \hline \(\mathrm{MI}\) & 0.193 & 0.089 & 0.064 & 76.6 & 5.5 & 3.2 \\ \hline \(\mathrm{MI}\) & & & & & & \\ \hline  & **T\({}_{4}\) (s)** & **T\({}_{18}\) (s)** & **T\({}_{27}\) (s)** & \(\varepsilon_{4}\) (\%)** & \(\varepsilon_{18}\) (\%) & \(\varepsilon_{27}\) (\%) \\ \hline HI & & & & & & \\ \hline \(\mathrm{H\ddot{I}}\) & 0.160 & 0.078 & 0.066 & 78.9 & 3.1 & 2.5 \\ \hline \(\mathrm{H\ddot{I}}\) & & & & & & \\ \hline \end{tabular}
\end{table} 
Table 5: Modal participating mass ratios, \(\varepsilon_{\
u}\) for vertical excitation and natural periods, \(\mathrm{T_{i}}\). where \(\rm A_{sw}\), s and \(\rm f_{ywd}\) are the stirrups' cross-sectional area, spacing and design yield strength, respectively, \(\rm d\) is the effective depth of the member cross-section and \(\rm\theta\) is the crack angle. \(\rm\theta\) was taken to be equal to \(\rm 21.8^{\circ}\) for columns and DCM beams and equal to \(\rm 45^{\circ}\) for DCH beams; 3. Columns' normalized axial force (\(\rm v_{d}\)) limitation: \[\rm v_{d}=\frac{N_{d}}{A_{c}f_{cd}}\leq \begin{cases}0.65,\text{ for DCM}\\ 0.55,\text{ for DCH}\end{cases} ,\] (2) where \(\rm N_{d}\) is the axial force (the compressive force is positive), \(\rm A_{c}\) is the column cross-sectional area and \(\rm f_{cd}\) is the concrete compressional strength; 4. The confinement of columns. This requirement applies to all critical regions of DCH columns (64 cross-sections checked) and to the base of DCM columns (4 cross-sections checked). The mechanical volumetric ratio of confining hoops \(\rm\omega_{wd}\) shall satisfy Equation (3): \[\rm\omega_{wd}\geq \begin{cases}0.12,&\text{ at the base of DCH columns}\\ 0.08,&\text{ at other critical regions}\end{cases} ,\] (3) In addition, the following relation shall be satisfied: \[\rm\alpha\omega_{wd}\geq 30\mu_{\rm\varphi}v_{d}\varepsilon_{sy,d}\frac{b_{c} }{b_{o}}-0.035,\] (4) where \(\rm\alpha\) is the effectiveness factor, \(\rm v_{d}\) is the maximum normalized axial force (the compression is positive), \(\varepsilon_{sy,d}\) is the reinforcement yield strain, \(\rm b_{c}\) is the width of the cross-section, \(\rm b_{o}\) is the concrete core width and \(\mu_{\rm\varphi}\) is the required value of the curvature ductility factor: \[\mu_{\rm\varphi}= \begin{cases}2q_{o}-1,&\text{ if }T_{1}\geq T_{C}\\ 1+\left(2q_{o}-1\right)\frac{T_{C}}{T_{1}},&\text{ if }T_{1}<T_{C}\end{cases} ,\] (5) where \(q_{o}=q\) is the behavior factor, \(T_{1}\) is the first natural period of the frame and \(T_{c}\) is the characteristic period of the horizontal design spectrum; 5. The anchorage of the longitudinal bars of beams passing through joints. The following relations shall be satisfied: \[\rm\frac{d_{bl}}{h_{c}}\leq \begin{cases}\frac{7.5t_{cm}}{\gamma_{Rd}f_{vd}} \frac{1+0.8v_{d}}{1+0.75k_{\rm D}\rho^{\prime}/\rho_{max}},&\text{ for interior joints}\\ \frac{7.5t_{cm}}{\gamma_{Rd}f_{vd}}(1+0.8v_{d}),&\text{ for exterior joints} \end{cases} ,\] (6) where \(\rm d_{bl}\) is the diameter of steel bars, \(\rm h_{c}\) is the column width, \(f_{ctm}\) is the mean concrete tensile strength, \(\rm f_{yd}\) is the reinforcement yield strength, \(\rm\gamma_{Rd}\) is the model uncertainty factor (1.2 or 1.0 for DCH or DCM, respectively), \(\rm v_{d}\) is the minimum column normalized axial force (the compression is positive), \(\rm k_{D}=1\) or \(2/3\) for DCH or DCM, respectively, \(\rm\rho^{\prime}\) is the beam compression reinforcement ratio and \(\rm\rho_{max}\) is the maximum reinforcement ratio under tension; 6. The ratio of the column flexural resistance to the beam flexural resistance at beam-column joints.