# Parametric Analysis of Outrigger Systems for High-Rise Buildings with Different Geometric Shapes
## 3 Refined Modeling for Outrigger Systems
### Results and Discussion  
An investigation into the behavior of the refined structures, with and without outriggers, is initially performed. A subscript \"\(o\)\" is appended to denote the outrigger condition in the results (i.e., R\({}_{0}\) for the regular building, T4\({}_{0}\) for the tilted building, and W2\({}_{0}\) for the twisted building). Natural periods and modal participating mass ratios of the first twelve vibration modes are detailed in Tables 7-9 for the three refined models. Notably, irregular structures require a greater number of modes to achieve 85% mass participation, reflecting a more complex dynamic behavior compared to the regular structure. It is crucial to recognize that a direct comparison of the first mode natural periods between the simplified (Tables 2-4) and refined models (Tables 7-9) is inappropriate due to their inherent differences in structural mass. Specifically, the refined models incorporate mass derived from floor framing, slabs, and gravity loads to enable a complete assessment of their dynamic characteristics. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & & **R** & & & **R\({}_{0}\)** & \\ \hline \multirow{2}{*}{**Mode**} & **Period** & **UX** & **UY** & **Period** & **UX** & **UY** \\  & **(s)** & **(\%)** & **(\%)** & **(s)** & **(\%)** & **(\%)** \\ \hline
1 & 4.98 & 67\% & 3\% & 3.77 & 66\% & 9\% \\
2 & 4.98 & 70\% & 70\% & 3.77 & 75\% & 75\% \\
3 & 3.55 & 70\% & 70\% & 3.19 & 75\% & 75\% \\
4 & 1.35 & 78\% & 75\% & 1.32 & 82\% & 77\% \\
5 & 1.35 & 83\% & 83\% & 1.32 & 84\% & 84\% \\
6 & 1.14 & 83\% & 83\% & 1.07 & 84\% & 84\% \\
7 & 0.65 & 83\% & 83\% & 0.57 & 84\% & 84\% \\
8 & 0.62 & 88\% & 83\% & 0.49 & 89\% & 84\% \\
9 & 0.62 & 88\% & 88\% & 0.49 & 89\% & 89\% \\
10 & 0.43 & 88\% & 88\% & 0.42 & 89\% & 89\% \\
11 & 0.35 & 91\% & 88\% & 0.35 & 91\% & 89\% \\
12 & 0.35 & 91\% & 91\% & 0.35 & 92\% & 92\% \\ \hline \end{tabular}
\end{table} 
Table 7: Modal periods and cumulative participation mass ratios for refined regular model with and without outrigger. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & & **T\({}_{4}\)** & & & **T\({}_{4,0}\)** & \\ \hline \multirow{2}{*}{**Mode**} & **Period** & **UX** & **UY** & **Period** & **UX** & **UY** \\  & **(s)** & **(\%)** & **(\%)** & **(s)** & **(\%)** & **(\%)** \\ \hline
1 & 4.74 & 39\% & 29\% & 3.85 & 1\% & 67\% \\
2 & 4.71 & 70\% & 69\% & 3.56 & 75\% & 68\% \\  
The introduction of outriggers consistently reduces natural periods, indicating an increase in structural stiffness. Such a stiffening effect is crucial for maintaining the top and inter-story drifts within acceptable limits, ensuring serviceability and preventing structural damage under wind loads. For wind-induced lateral displacement control, the maximum allowable top drift is set to H/500, and the maximum allowable inter-story drift is set to h/300, where H and h denote the total building height and the inter-story height, respectively. Figure 11 presents a comparative analysis of outrigger efficacy in controlling the top displacements (d\({}_{\text{root}}\)) and inter-story drifts (IDR) within the refined models, showcasing configurations with and without outrigger systems. Figure 12 presents the same comparison for core base bending moments. The refined model without outriggers exhibits a maximum lateral deflection at the top story and an IDR exceeding the maximum allowable lateral top displacement of 0.4 m and the allowable inter-story drift of 1.3%, respectively. The inclusion of an outrigger with a belt-truss demonstrably reduces both maximum wind-induced lateral deflection at the top story and inter-story drift values in all other outrigger-equipped models. Importantly, these EDPs now fall within the design-acceptable thresholds, confirming the outrigger's effectiveness in mitigating lateral displacement demands under wind load. Displacement reductions are observed in both regular and irregular building geometries at the outrigger level.