# Influence of Multiple Openings on Reinforced Concrete Outrigger Walls in a Tall Building
## 2 Design of Reinforced Concrete Outrigger Wall with Multiple Openings
### Effect and Demand of Outrigger Walls  
The internal shear force of the outrigger walls due to lateral loads is derived based on the simplified model of the core wall and outrigger structural system [15]. A simplified model of tall building structures with multiple outriggers is shown in Figure 2. The core wall is connected to the perimeter columns through the outriggers. When lateral loads are applied at the central core wall, the outrigger develops axial forces in the perimeter columns. The limitation of the simplified model is that only flexural deformation of the outrigger is considered, and the length of the outrigger is assumed as the distance from the center of the core to that of the perimeter column. These assumptions are different from the actual behavior of outrigger walls. To enhance the accuracy of this simplified model, an equivalent bending stiffness \(\left(EI\right)_{\mathcal{O}}\) was proposed by considering the shear deformation of beams and the clear span of outrigger walls. Because the outrigger walls belong to the category of deep beams [16], the total deformation is the sum of flexural and shear deformations, and the actual flexural distance is the clear span \(l_{o}\), as shown  
Figure 2: Simplified analysis model with multiple outriggers. The core wall is connected to the perimeter columns through the outriggers. in Figure 3. According to the principle of virtual works, for the outrigger wall with a moment of inertia \(I_{o}\) and cross-sectional area \(A_{o}\), the equivalent bending stiffness \(\left(EI\right)_{o}\) can be obtained by the following equation. \[\left(EI\right)_{o}=\left(1+\frac{c}{2l_{o}}\right)^{3}EI_{o}/\left(1+\frac{12 EI_{o}}{G\kappa_{s}A_{o}L^{2}}\right) \tag{1}\]  
where \(E\) and \(G\) are the modulus of elasticity and shear modulus, respectively. \(\kappa_{s}\) is the shear coefficient to account for the shear deformation, which is 0.83 for a rectangular section. \(c\) is the width of the core wall, and \(L\) is the distance from the center of the core wall to the center of perimeter column, as shown in Figure 3.  
For the core, the rotation \(\theta_{ii}\) at the \(i\)th outrigger level due to the lateral load \(w\) and the restraining moment \(M_{ri}\) can be given by the following equation. \[\theta_{ii}=\frac{1}{\left(EI\right)_{t}}\Big{[}\int_{x_{1}}^{x_{2}}\left( \frac{wx^{2}}{2}-M_{r1}\right)dx+\int_{x_{2}}^{x_{3}}\left(\frac{wx^{2}}{2}-M_ {r1}-M_{r2}\right)dx+\ldots+\int_{x_{n}}^{H}\left(\frac{wx^{2}}{2}-M_{r1}- \ldots-M_{m}\right)dx\Big{]} \tag{2}\]  
where \(\left(EI\right)_{t}\) represents the bending stiffness of the core wall. At the same outrigger level, the rotation of the inboard end of the outrigger \(\theta_{oti}\), where it attaches to the core, is due to the axial forces of perimeter columns and overturning moment \(M_{ri}\). It was noted that the restraining moment for the core and overturning moment for the outrigger walls have the same value and opposite directions. Hence, \(M_{ri}\) is used to represent both. The rotation \(\theta_{oti}\) can be obtained by the following equation. \[\theta_{oti}=\frac{2M_{r1}\left(H-X_{1}\right)}{L^{2}\left(EA\right)_{c}}+ \frac{2M_{r2}\left(H-X_{2}\right)}{L^{2}\left(EA\right)_{c}}+\ldots+\frac{2M_ {ri}\left(H-X_{i}\right)}{L^{2}\left(EA\right)_{c}}+\frac{M_{ri}L}{12\left( EI\right)_{oi}} \tag{3}\]  
where \(\left(EA\right)_{c}\) represents the axial stiffness of perimeter columns. Figure 3: Shear and flexural deformations of the outrigger wall are considered and the clear span rather than center-to-center span is used in the equivalent bending stiffness of the outrigger wall. Because of \(\theta_{ti}=\theta_{otti}\), the restraining moment \(M_{ri}\) can be obtained in the following matrix form. \[\left[ \begin{array}{c}M_{n}\\ M_{n}\\ \vdots\\ M_{ni}\\ \vdots\\ M_{m}\end{array} \right]=\left[