# 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
While acknowledging the non-uniform, height-dependent nature of actual wind, this simplification facilitates generalization and  
Figure 2: Conceptualization of the high-rise buildings under examination. Refined models (top row) and simplified models (bottom row) for regular (**a**), tilted (**b**), and twisted (**c**) geometries. interpretation of the outrigger behavior through direct inter-model comparisons. The implementation of a detailed, floor-specific wind profile, often characterized by exponential variations with height, would introduce a level of complexity incompatible with the simplified analytical framework employed herein, which lacks a discrete representation of individual floor diaphragms. Furthermore, the assumption of uniformly distributed lateral loads constitutes a recognized and established methodology in the preliminary structural analysis of tall buildings and in foundational studies concerning the behavior of their lateral load-resisting systems (e.g., [6; 7; 11; 12; 54], among many others), enabling the isolation and focused examination of key structural parameters. Although load distribution might affect optimal outrigger placement [35], research has demonstrated that uniform loading tends towards slightly higher optimal locations for outriggers, and these minor level variations minimally impact drift reduction [19; 35; 43; 55]. Notably, [PERSON] et al. [56] found uniform loading to yield the most conservative inter-story drift for one and two-outrigger configurations, supporting its use for a robust preliminary evaluation. Three geometric configurations were analyzed for the reference building: a regular geometry (labeled as the R model), a tilted geometry (T model) with an angle \(\alpha\) relative to the vertical axis, and a twisted geometry (W model) characterized by an angle \(\theta\) with respect to the baseline. All simplified analytical models share the same height and square plan dimensions. As depicted in Figure 2a,b, two-dimensional simplified models adequately represented the regular and tilted configurations. However, the twisted geometry required a three-dimensional model to capture its inherent behavior (Figure 2c). In this latter case, four twisted columns, connected to the central core by four horizontal outrigger arms, were explicitly modeled. Referring to Figure 2, the following parameters are specified. The inner core is defined by a total height (H = 200 m) and flexural stiffness (K\({}_{\text{c}}\)), derived from its moment of inertia (I\({}_{\text{c}}\) = 5.34 \(\times\) 10\({}^{11}\) cm\({}^{4}\)) and the concrete elastic modulus (E\({}_{\text{c}}\) = 37.3 GPa). Each equivalent column, located at a distance b/2 (20 m) from the core centerline, exhibits a flexural stiffness calibrated to represent the contribution of the perimeter columns in each geometric configuration. Specifically, the equivalent column's flexural stiffness is set to one-half of the aggregate flexural stiffness (K\({}_{\text{p}}\)) of the actual perimeter columns for the regular (R) and tilted (T) models, and one-quarter for the twisted (W) model. The vertical location of the outrigger system is defined by the variable x, which represents the distance from the building's base. The equivalent outrigger beams have a flexural stiffness (K\({}_{\text{o}}\)) determined as a fraction of the core's stiffness K\({}_{\text{c}}\) through a non-dimensional characteristic parameter (k). The parameterization directly addresses a fundamental principle in outrigger design: the system's efficiency is critically dependent on the relative flexural stiffness of the core (e.g., [39; 42]). As extensively demonstrated in the literature, in fact, the extent of drift and core moment reduction achieved by outriggers is a complex function of these relative stiffnesses and the vertical placement of the outriggers along the building's height ([23; 27], among others).