# 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
The Newmark direct integration algorithm was implemented for the parameters \(\beta=0.25\) and \(\gamma=0.50\), while Rayleigh dumping was assumed. The time step used for the integration was taken to be adequately small to ensure the accuracy of the analysis. In Figures 14-19, the plastic hinges developed in the frames due to both H and H+V excitations are shown. The color and the shape of each plastic hinge corresponds to the exceedance of one of the three damage limit states defined in Eurocode 8: the Damage Limitation limit state (green circular), Significant Damage limit state (orange triangular) and Near Collapse limit state (red rectangular). The classification of each plastic hinge as the relevant limit state was based on the acceptance criteria specified in Eurocode 8, Part 3 [35]. The acceptance criteria are expressed as a percentage of the ultimate plastic chord rotation of each member: 0% for Damage Limitation, 75% for Significant Damage and 100% for the Near Collapse limit state. The ultimate plastic chord rotation capacity was calculated in accordance with Eurocode 8. It is worth noting that buildings designed to meet the provisions of Eurocode 8 are expected to experience significant structural damage but without losing their structural integrity. In other words, (green circular) plastic hinges not exceeding the Significant Damage limit state are anticipated and accepted. On the contrary, orange triangular and red rectangular plastic hinges are not supposed to develop. Concerning the frames designed herein, it seems that this fundamental requirement was, in general, achieved for H excitation. The only exceptions were the frames MIII and HIII for the Imperial Valley excitation (Figures 16a and 19a). However, this is not surprising since the scaling method applied for this excitation led to a response spectrum with considerably larger ordinates in relation to the design seismic action prescribed by Eurocode 8 (see Figure 13). Figure 14: Plastic hinge formation for (**a**) H excitation and (**b**) H+V excitation. Frame MI. Figure 15: Plastic hinge formation for (**a**) H excitation and (**b**) H+V excitation. Frame MII. Figure 16: Plastic hinge formation for (**a**) H excitation and (**b**) H+V excitation. Frame MIII. Figure 17: Plastic hinge formation for (**a**) H excitation and (**b**) H+V excitation. Frame HI. Figure 18: Plastic hinge formation for (**a**) H excitation and (**b**) H+V excitation. Frame HII. Figure 19: Plastic hinge formation for (**a**) H excitation and (**b**) H+V excitation. Frame HIII. From Figures 14-19, the following observations can be made:  
* The vertical component led to an increased number of locations where plastic deformations occurred;
* Plastic hinges may develop not only at the ends of beams but at the mid-spans too. Obviously, this should be considered during the detailing of the beams for local ductility;
* One of the main intentions of the design was to preclude soft storey plastic mechanisms. This was supposed to be achieved through the control of the flexural resistance of all the members framing a joint (see Section 3.1(f) above). However, the whole concept was compromised since the vertical component led to the development of several plastic hinges in the columns;
* The vertical component may unfavorably affect the achievement of a limit state;
* The upper stories seemed to be more vulnerable to the vertical excitation;
* Vertical ground motion was more impactful for higher seismic intensities. The aforementioned derivations are supported by all three records applied, and it is expected that they would not change even if more records were used. An important issue arising due to the vertical component was the amplification of the shear forces of the beams, resulting from the additional vertical inertial forces which developed at the spans. Indicatively, Figure 20 illustrates the absolute maximum shear force at three points (left end, mid-point, right end) of beam B12 in the six frames. In the same graphs, the corresponding shear resistances are also plotted.