# 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
It can be observed that, even though the secant stiffness was slightly less than the tangential stiffness, \(K_{i}\) and the degradation ratio \(\beta_{K}\) were almost identical. Figure 13 shows the degradation ratios in stiffness as the size of the opening increases. The ratios of the stiffness decline almost linearly with a gentle slope from M0 to M8. When the opening area exceeds 22% (M8), the decreasing trends are also regarded as being linear, but they become steeper. From the similar trends in the results of linear and nonlinear analyses, it can be concluded that the loss of area due to multiple openings is a more influential factor on the stiffness of outrigger walls than the material degradation. The maximum vertical reaction, \(R_{i}\), in the nonlinear analysis can be defined as the strength of the outrigger walls. The ratio \(\beta_{R}\) refers to the ratio of the vertical reaction at the ultimate state of the \(i\)-th models \(R_{i}\) to \(R_{0}\) of the M0 model. The maximum vertical reactions and degradation ratio in strength from the nonlinear analysis are shown in Table 4 and Figure 14. The reduction in strength is a function of the opening ratio, which is similar to the results of stiffness. However, the outrigger wall began to experience larger losses in strength when the opening area reached 24% (M9) in which the shear strength reduced to 77%. Comparing the reduction of stiffness and strength in model M0 to model M8, where the sizes of openings were less than 22% of the overall area of the outrigger wall, the gentle reductions in stiffness and strength were similar, while the sharp reduction occurred earlier in the stiffness (M8) than in strength (M9). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{\(A_{op}\)/\(A_{out}\)} & \multicolumn{2}{c}{**Linear Analysis**} & \multicolumn{2}{c}{**Nonlinear Analysis**} \\ \cline{3-6}  & & \(K_{i}\) (MN) & \(\beta_{K}\) & \(K_{si}\) (MN) & \(\beta_{K}\) \\ \hline M0 & 0.00 & 2124 & 1.00 & 1817 & 1.00 \\ M1 & 0.03 & 1973 & 0.93 & 1690 & 0.93 \\ M2 & 0.05  
Figure 14: Degradation of strength of the outrigger wall according to the size of the openings. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{\(A_{\text{op}}/A_{\text{out}}\)} & \multicolumn{2}{c}{**Nonlinear Analysis**} \\ \cline{3-4}  & & \(R_{i}\) **(kN)** & \(\beta_{R}\) \\ \hline M0 & 0.00 & 39,932 & 1.00 \\ M1 & 0.03 & 38,193 & 0.96 \\ M2 & 0.05 & 36,798 & 0.92 \\ M3 & 0.08 & 35,938 & 0.90 \\ M4 & 0.11 & 35,604 & 0.89 \\ M5 & 0.14 & 34,635 & 0.87 \\ M6 & 0.16 & 34,150 & 0.86 \\ M7 & 0.19 & 33,314 & 0.83 \\ M8 & 0.22 & 31,729 & 0.79 \\ M9 & 0.24 & 30,815 & 0.77 \\ M10 & 0.27 & 27,853 & 0.70 \\ M11 & 0.29 & 19,083 & 0.48 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Evaluation of strength of the outrigger wall. Figure 13: Degradation of stiffness of the outrigger wall according to the size of openings. From the analysis results, it can be concluded that the reinforced concrete outrigger walls can accommodate four openings without significantly decreasing the stiffness and strength of the outrigger walls if the openings do not severely interfere with the critical load path.