# Parametric Analysis of Outrigger Systems for High-Rise Buildings with Different Geometric Shapes
## 2 Simplified Approach for Outrigger Systems
### Results and Discussion
Minimal lateral stiffness gains are reported for twist angles of 3\({}^{\circ}\) and 4\({}^{\circ}\) per floor, with reductions of approximately 6%, 6%, 5%, and 7%, respectively. Figure 5 graphically illustrates the normalized fundamental periods of the analyzed structures, expressed as a ratio of the fundamental period of the outrigger-equipped structure to that of the equivalent structure without outriggers (T\({}_{1,0}\)/T\({}_{1}\)). Consistent trends are observed in the bending moment and lateral top displacement results. Such EDPs are widely recognized as crucial metrics for assessing outrigger system effectiveness. Bending moments at the core base inform the overall demand on the main lateral load-resisting element, and their reduction is a primary objective of outrigger systems. Lateral top displacements directly quantify the building's deformation induced  
Figure 5: Normalized fundamental periods for the analyzed structures. by lateral loads. Excessive sway can cause serviceability issues (e.g., occupant's discomfort) and even instability. Thus, minimizing displacement is another essential task. Top displacements and bending moments for each analyzed typology are illustrated in Figure 6 and Figure 7, respectively, as a function of the outrigger-to-core stiffness ratio. Core base bending moments are normalized by the corresponding value for the no-outrigger case (\(\mathrm{M_{b,o}}/\mathrm{M_{b}}\)) to clearly show the impact of the outrigger system on moment demand. Similarly, roof displacements are normalized (\(\mathrm{d_{roof,o}}/\mathrm{d_{roof}}\)). At the peak analyzed core-to-outrigger flexural stiffness ratio (11.5%), maximum reductions in roof displacements are achieved compared to the baseline non-outrigger configuration. Reduction percentages range from 40% to 60%, varying with outrigger placement along the building height. Two-outrigger configurations yield the highest reductions, while outriggers at one-third height yield the lowest, with intermediate reductions of 50% observed at mid-height and two-thirds height. Similar results are observed for the core bending moments, with reductions slightly lower ranging from 40% (one-third height) to 45% (two-outriggers). However, the mid-height and two-thirds height locations exhibit, in general, the least favorable reductions among the analyzed configurations. Twisted structures confirm the trends, exhibiting significantly diminished gains in both displacements and core moments as the twist angle increases from 0 = 1\({}^{\circ}\) to 0 = 4\({}^{\circ}\). Specifically, in this latter case, reductions are approximately 20% for both displacements and core moments at one-third height and with two outriggers. Very limited improvements are observed at mid-height (5%) and two-thirds height (1%). Overall, the presented results are in good agreement, demonstrating that enhanced lateral stiffness from outriggers translates to reduced top displacement and core bending moment demands compared to the non-outrigger condition. This effect intensifies with increasing outrigger stiffness, aligning with the fundamental period findings. Figure 6: Normalized roof displacements for the analyzed structures. To shed some light on the distinctive profiles, a detailed analysis of the impact of the core-to-outrigger stiffness ratio on the identified EDPs is presented in Figure 8 for the examined structures. As already observed by [PERSON] [51], the rate of improvement diminishes beyond a certain outrigger flexural stiffness threshold, where further increments yield only marginal performance gains. However, structural geometry also plays a critical role in this regard. In the plots for regular and tilted models, a sharp decrease in normalized bending moments and lateral displacements is shown up to a relative stiffness ratio of 3% (approximately a 25% reduction for both EDPs and all geometries). Between 3% and 6%, small improvements are observed (relative difference of approximately 10%).