# Parametric Analysis of Outrigger Systems for High-Rise Buildings with Different Geometric Shapes
## 2 Simplified Approach for Outrigger Systems
### Results and Discussion  
Linear static and modal analyses were performed for each simplified model. The results were compared in terms of lateral displacement, core bending moment, and fundamental period of vibration. Such Engineering Demand Parameters (EDPs) are frequently adopted to evaluate the global performance of outrigger systems, and thereby their effectiveness. The fundamental periods for each model, both with and without outriggers, are presented in Table 2, Table 3, and Table 4 for the regular, tilted, and twisted configurations, respectively. As previously stated, models without outriggers were simulated with very low flexural stiffness. Consistent with the existing literature, the results demonstrate that the addition of outriggers increases the overall lateral stiffness, which, in turn, yields a reduction in the fundamental period for all the analyzed structures. This effect is more pronounced with increasing flexural stiffness of the equivalent beams representing the outriggers. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**k** & **x = 1/3** & **x = 1/2** & **x = 2/3** & **x = 1/3 and x = 2/3** \\
**(\%)** & **(s)** & **(s)** & **(s)** & **(s)** \\ \hline
0.005 & 2.90 & 2.90 & 2.90 & 2.90 \\
0.1 & 2.88 & 2.77 & 2.93 & 2.61 \\
0.5 & 2.85 & 2.66 & 2.81 & 2.41 \\
1.0 & 2.74 & 2.51 & 2.64 & 2.20 \\
3.0 & 2.60 & 2.32 & 2.45 & 2.03 \\
6.0 & 2.45 & 2.15 & 2.30 & 1.91 \\
11.5 & 2.40 & 2.10 & 2.28 & 1.89 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fundamental period of the regular model (R). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**k** & **x = 1/3** & **x = 1/2** & **x = 2/3** & **x = 1/3 and x = 2/3** \\
**(\%)** & **(s)** & **(s)** & **(s)** & **(s)** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 3: Fundamental period of the tilted model (T) for different tilt angles (\(\alpha\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**k** & **x = 1/3** & **x = 1/2** & **x = 2/3** & **x = 1/3 and x = 2/3** \\
**(\%)** & **(s)** & **(s)** & **(s)** & **(s)** \\ \hline \hline \multicolumn{5}{c}{\(\alpha=4^{\circ}\)} \\ \hline
0.005 & 2.95 & 2.95 & 2.95 & 2.95 \\
0.1 & 2.90 & 2.73 & 2.87 & 2.56 \\
0.5 & 2.81 & 2.60 & 2.70 & 2.30 \\
1.0 & 2.70 & 2.43 & 2.51 & 2.15 \\
3.0 & 2.53 & 2.25 & 2.39 & 2.00 \\
6.0 & 2.48 & 2.10 & 2.35 & 1.98 \\
11.5 & 2.46 & 2.09 & 2.33 & 1.96 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fundamental period of the twisted model (W) for different twist angles (\(\theta\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**k** & **x = 1/3** & **x = 1/2** & **x = 2/3** & **x = 1/3 and x = 2/3** \\
**(\%)** & **(s)** & **(s)** & **(s)** & **(s)** \\ \hline \hline \multicolumn{5}{c}{\(\alpha=1^{\circ}\)} \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._with [PERSON] and [PERSON] [59], who attributed this effect to discontinuities in stiffness, mass, and geometry along the building height. Notably, the introduction of outriggers generally contributes in mitigating such irregularities, especially when two outriggers are adopted (last column of the tables). For example, assuming a core-to-outrigger flexural stiffness ratio of 11.5% for the regular model results in the following reductions in fundamental periods: 17% with an outrigger at one-third of the building height, 28% at mid-height, 21% at two-thirds height, and 35% when two outriggers are considered. Similar reduction percentages are observed for the tilted model with minimal differences across the tilt angles (i.e., 18% at one-third height, 28% at one-half height, 23% at two-thirds height, and 36% at one-third and two-thirds locations). These values progressively decrease in twisted models, where the twist angle is a determining factor. Specifically, fundamental period reductions of approximately 14% (at one-third height), 21% (at mid-height), 8% (at two-thirds height), 22% (at one-third height and two-thirds) are observed for twist angles of 1\({}^{\circ}\) and 2\({}^{\circ}\) per floor.