# Parametric Analysis of Outrigger Systems for High-Rise Buildings with Different Geometric Shapes
## 2 Simplified Approach for Outrigger Systems
### Parametrization of the Core-Column-Outrigger System
To capture the sensitivity of the structural response to variations in outrigger-to-core stiffness, seven distinct cases, encompassing seven discrete values of the k parameter, were analyzed. The selection of these parameter values was informed by a comprehensive survey of the existing literature on outrigger system modeling and analysis (e.g., [54; 57; 58]). While acknowledging that inherent real-world complexities, such as assembly tolerances and connection stresses/deformations, are approximated within this simplified analytical framework--a common and necessary step in preliminary parametric studies to isolate key behavioral trends--the chosen range of k values aims to capture a spectrum of realistic and theoretically significant outrigger stiffness ratios encountered in practice and explored in prior research. For regular and tilted models, this stiffness is distributed across two arms (k\(\cdot\) K\({}_{\rm c}\)/2), whereas for twisted structures it is distributed across four arms (k\(\cdot\) K\({}_{\rm c}\)/4). Detailed stiffness parameters for the outriggers are presented in Table 1. In the simplified models, the stiffness variation was simulated by adjusting the equivalent outrigger cross-section using S450 circular hollow tubes (E\({}_{\rm s}\) = 210 GPa). Models without outriggers were simulated using the first cross-section typology, characterized by a near-zero k value. The geometric attributes of the tilted and twisted configurations were parametrically varied, as illustrated in Figure 3. The tilt angle (\(\alpha\)) ranged from 1\({}^{\circ}\) to 4\({}^{\circ}\), a limit established to maintain the core within the building's footprint. This corresponded to a floor-to-floor lateral shift (d) of 0.07 m (for \(\alpha\) = 1\({}^{\circ}\)), 0.14 m (\(\alpha\) = 2\({}^{\circ}\)), 0.21 m (\(\alpha\) = 3\({}^{\circ}\)), and 0.28 m (\(\alpha\) = 4\({}^{\circ}\)), yielding a total lateral offset D (i.e., the distance between the exterior facade and the core perimeter wall) of 3.5 m, 7 m, 10.5 m, and 14 m, respectively. The floor twist rate (\(\theta\)) also varied from 1\({}^{\circ}\) to 4\({}^{\circ}\), resulting in four models with global twist angles (\(\Theta\)) of 50\({}^{\circ}\), 100\({}^{\circ}\), 150\({}^{\circ}\), and 200\({}^{\circ}\), respectively. The selection of the 1\({}^{\circ}\) to 4\({}^{\circ}\) range for both tilt and twist angles is intended to cover the lower spectrum of geometric deviations observed in real-world non-orthogonal high-rise buildings. Several iconic constructed examples demonstrate the application of these ranges, including the Shanghai Tower by [PERSON] (\(\theta\)\(\approx\) 1\({}^{\circ}\) twist), the Infinity Tower by SOM (\(\theta\)\(\approx\) 1.2\({}^{\circ}\) twist), and the conceptual Chicago Spire by [PERSON] (\(\theta\)\(\approx\) 2\({}^{\circ}\) twist). For tilt, it is observed that some prominent examples like the Gate of Europe (\(\alpha\) = 15\({}^{\circ}\)), Veer Towers (5\({}^{\circ}\)), and Altair Towers (13.8\({}^{\circ}\)) exhibit larger angles than those assumed in this study. However, this range allows for a focused parametric investigation of the early-stage effects of tilt and twist on the outrigger performance and overall building behavior as these geometric complexities are introduced. This is also supported by the work of [PERSON] [37], who explored similar bounds for twisted and tilted structures, providing a valuable point of comparison. For each building configuration (regular, tilted, and twisted), four outrigger locations (x) were analyzed: one-half, one-third, and two-thirds of the building height for single outriggers, and one-third and two-thirds of the height for two-outrigger systems. The analyzed locations are depicted in Figure 4. Finally, a total of 28(R) + 112(T) + 112(W) = 252 structural models were generated to systematically explore the combined influence of building geometry and outrigger placement.