# Influence of Multiple Openings on Reinforced Concrete Outrigger Walls in a Tall Building
## 3 Evaluation of the Influence of Multiple Openings
### Results of Finite Element Analysis  
The absolute maximum principal stress distribution calculated by the nonlinear finite element analysis is shown in Figure 10. In the outrigger wall without web openings (M0), the results show that the compressive strut was formed from the loading plate, and the bottom of the outrigger wall and tensile zone was formed at the top of the outrigger wall. In the outrigger wall with web openings (M8), compressive strut was more clearly formed at the diagonal line. Additionally, the bottom strut was formed separately from the diagonal strut. It was also observed that the cross-section of the diagonal strut was slightly defected by the openings. The maximum plastic strain distribution at the ultimate state is shown in Figure 11. The maximum plastic strain represents the tensile strain in concrete, and the distribution shows the crack pattern of the analysis models. In the M0 model, the widest crack developed in the flexural critical section at which the outrigger wall meets the core wall. Several flexural cracks developed along the top reinforcement. The diagonal crack initiated at the loading plate and developed along the diagonal reinforcement, which was subjected to compression, was also noticeable. In model M1, many cracks developed along the top and diagonal reinforcements, which were subjected to tension. It was also noticed that wide local cracks were developed at the bottom left corner of the openings. In M8, the crack initiated at the loading plate and propagated along the top reinforcement, which formed a large arc along the corners of openings. In M11, in which the compressive struts interfered by the openings, the ultimate state developed by the local failure at the corners of the top and right openings. Figure 10: Absolute maximum principal stress distribution of (**a**) M0 and (**b**) M8. Figure 9: Material model of reinforcement subjected to tension and compression. Figure 12 shows the relation between the applied vertical displacement and total vertical reactions measured at the pinned nodes from the nonlinear analysis. It can be clearly observed that the outrigger walls experienced a progressive decrease in both stiffness and strength as the size of the opening increased. However, the ultimate displacement of all the models, except M11, were around 60 mm and the initial yielding was developed at about 10 mm. Figure 11: Plastic strain distribution at the ultimate state of (**a**) M0, (**b**) M1, (**c**) M8, and (**d**) M11. Figure 12: Displacement-reaction curves of nonlinear analysis. In order to quantify the degradation in stiffness due to the size of openings, the initial tangential slope, \(K_{ti}\), calculated from the load-displacement curves shown in Figure 12 and compared it with the slope \(K_{i}\) from the linear analysis. In order to evaluate the stiffness of outrigger walls, all the analysis models were constructed as the same as the nonlinear analysis, with an exception of the plastic modeling removed in the linear analysis. The slope of the linear analysis \(K_{i}\) is defined as the vertical force developing a unit deflection at the loading point of the \(i\)-th model in the linear analysis. The term \(K\) can be defined as the tangential stiffness of the outrigger walls. The initial tangential slopes \(K_{ti}\) from the nonlinear analysis and stiffness \(K_{i}\) from the linear analysis were identical. To include the effect of initial cracking in the nonlinear analysis, the secant stiffness, \(K_{si}\), were defined as the slope from the origin to the point with vertical displacement of 10 mm and are shown in Table 3 with the stiffness \(K_{i}\) from the linear analysis. As mentioned before, \(\beta_{K}\) is defined as the ratio of \(K_{i}\) to \(K_{0}\) or \(K_{si}\) to the outrigger wall without openings. \(\beta_{K}\) is regarded as an index to show the degradation of stiffness of the outrigger walls.