# The Influence of Vertical Ground Motion on the Design of Common R/C Frames
## 3 Evaluation of Frames for Vertical Ground Motion
### Evaluation Through Time History Analysis
In this particular case, the mean horizontal spectrum presented in Figure 13 experienced quite large ordinates in the period range from 0 s to about 0.2 s, which is out of the critical range of 0.2T\({}_{\rm 1 min}\) to 2T\({}_{\rm 1 max}\). On the contrary, the \(\rm r_{i}\) values for the'modal' periods that dominated the horizontal response were not excessively large (the quotation marks indicate that the term'modal' is not strictly valid for a non-linear range of behavior). However, as explained below, even if the \(\rm r_{i}\) values did not exceed 1.2-1.3, this may have caused an overestimation of the frame's response to the horizontal component. Although this quite interesting issue constitutes a weakness of the scaling procedures, it is not of great importance for this particular study, because the latter aims to identify the consequences of omitting the vertical seismic component regardless of the specific seismic intensity level. Two series of inelastic dynamic analyses were conducted in SAP2000v23: for H excitation (for the horizontal component only) and H+V excitation (horizontal and vertical components concurrently). The structural models presented in Sections 2 and 3 were modified in order to account for potential inelastic behavior at the critical regions as well  
Figure 13: Horizontal pseudo-acceleration response spectra for seismic hazard level zone I. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Ground Motion**} & \multirow{2}{*}{**sf1**} & \multirow{2}{*}{**sf2**} & \multicolumn{4}{c}{**sf3**} & \multicolumn{4}{c}{**Product sf1**-**sf2**-**sf3**} \\ \cline{3-10}  & & & **I** & **II** & **III** & **I** & **II** & **III** \\ \hline Imperial Valley & 0.667 & & & & & 1.308 & 1.962 & 2.943 \\ \hline Landers & 0.203 & 1.961 & 1.00 & 1.50 & 2.25 & 0.398 & 0.597 & 0.896 \\ \hline N. Palm Springs & 0.727 & & & & & 1.426 & 2.138 & 3.208 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Scale factors for the three seismicity zones (I, II, III). as at the mid-spans of beams. For this purpose, a concentrated plasticity model in accordance with Eurocode 8 [35] was applied. In particular, simplified bilinear elastic-perfectly plastic moment-rotation diagrams were used to describe the flexural behavior of the plastic hinges. The yield and the ultimate chord rotations were determined by applying the relevant equations from Eurocode 8, Part 3 [35]. The yield moment of each plastic hinge was calculated using the mean values of the material properties prescribed by Eurocode 2 [29] for the concrete class C20/25 and reinforcing steel B500C without any safety factors. For the concrete, a parabola-rectangle stress-strain diagram under compression was adopted. The strain upon reaching the maximum strength and the ultimate strain were \(\varepsilon_{c2}=2\%\) and \(\varepsilon_{\text{cu2}}=3.5\%\), respectively. For the reinforcing steel, an idealized bilinear elastic-perfectly plastic stress-strain diagram under compression and tension was adopted. The yield strain was \(\varepsilon_{\text{sy}}=2.174\%\), while no specific limit for the ultimate strain was specified. Concerning the shear behavior of structural members, no specific constitutive law was included in the models. In other words, it was assumed that the shear strength of the members was unlimited. This choice was based on the fact that the frames were designed to meet the requirements of Eurocodes 2 and 8, including the capacity design rule for the shear strength verification. As a consequence, the shear failure of structural members under typical seismic loading was, in general, precluded. Indeed, this was confirmed through a shear verification conducted after the analyses. As will also be discussed subsequently, despite the considerable increase in beams' shear forces due to the action of the vertical seismic component, no exceedances of members' shear strength occurred. This means that including a shear behavior law in the analysis models from the beginning would not alter the results at all.