## 5 Nonlinear Time History Analyses
### Results Obtained for a Reference Masonry Pier  
Figures 12-14 report the IDRs of masonry pier P1 (IDR\({}_{\rm P1}\)) obtained by varying the vertical load and subjecting the case-study structure to earthquakes #1, #2, and #3. Each figure reports the IDRs versus time curves of P1 for the VL-a (a), VL-b (b), and VL-c (c). The red lines indicate the 2% IDR limit (drift limit representative of the failure condition), while the black and grey lines represent the results with and without the vertical seismic component. Figure 11: Normalized GMPs of the H component (**a**) and the V component (**b**) of the selected ground motion records. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Ground Motion** & \multicolumn{2}{c}{**\#1**} & \multicolumn{2}{c}{**\#2**} & \multicolumn{2}{c}{**\#3**} \\ \cline{2-7}
**Parameters** & **H** & **V** & **H** & **V** & **H** & **V** \\ \hline PGA (m/s\({}^{2}\)) & 0.872 & 0.640 & 1.653 & 1.558 & 1.853 & 1.369 \\ \hline PGV (m/s) & 0.058 & 0.053 & 0.136 & 0.073 & 0.089 & 0.114 \\ \hline AI (m/s) & 0.086 & 0.047 & 0.308 & 0.243 & 0.521 & 0.360 \\ \hline SED (m\({}^{2}\)/s) & 0.0026 & 0.0051 & 0.0253 & 0.0072 & 0.0189 & 0.0151 \\ \hline ASI (m/s) & 0.404 & 0.335 & 1.400 & 1.014 & 1.499 & 1.254 \\ \hline HI (m) & 0.166 & 0.137 & 0.378 & 0.251 & 0.350 & 0.314 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Ground motion parameters of the H and V components of #1, #2, and #3 ground motion records. Figure 12: IDRs obtained subjecting the structure with VL-a (**a**), VL-b (**b**), and VL-c (**c**) to earthquake #1. The red lines indicate the IDR limit of 2%. Figure 13: IDRs obtained subjecting the structure with VL-a (**a**), VL-b (**b**), and VL-c (**c**) to earthquake #2. The red lines indicate the IDR limit of 2%. The figures show that the vertical component of the earthquakes tends to increase the IDR of the pier, especially for high vertical loads. The differences between results with or without the V component may depend on GMPs. This is shown, for example, by the comparison of IDR time histories in Figure 13c, with significant differences between the grey and black curves and IDR time histories in Figure 12c, with minor differences between the two curves, even though the two earthquakes have similar PGAs and the same loading condition. Earthquake #3, characterized by a lower PGA than earthquakes #1 and #2, generates significantly lower IDRs (Figure 11), which never exceed the 1% value. Figures 15-17 show the IDR-\(V_{b}\) curve obtained on masonry pier P1 and the damage pattern when the structure is subjected to earthquakes #1, #2, and #3. Each figure reports these results with and without the V component and for the different loading conditions (VL-a (a), VL-b (b) 10%, and VL-c (c)). The obtained results confirm that the V component of the earthquakes increases the EDPs, and this increase becomes more significant as the vertical load and the value of the IDR increase. As with the shear-displacement curves, the damage scenario indicates more extensive damage when the vertical seismic component is considered, particularly for high vertical loads. Note that, although earthquake #3 generates a limited damage scenario (Figure 17) compared to those obtained from earthquakes #1 (Figure 15) and #2 (Figure 16), there is still a widespread crack pattern when the V component of the earthquake is taken into account. Figure 14: IDRs obtained subjecting the structure with VL-a (**a**), VL-b (**b**), and VL-c (**c**) to earthquake #3. The red lines indicate the IDR limit of 2%. Figure 15: IDR-\(V_{b}\) curves and crack patterns obtained for earthquake #1. TH results for the H component are in grey, TH results for H + V components are in black, the PO curve for positive displacements is the continuous red line, and the PO curve for negative displacements is the dashed red line. Loading conditions: VL-a (**a**), VL-b (**b**), and VL-c (**c**). Figure 16: IDR-\(V_{b}\) curves and crack patterns obtained for earthquake #2.