# Flexural Strength and Stiffness of Donut-Type Voided Slab
## 2 Optimization of Void Shape
### Finite Element Analysis Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Division** & **Solid** & **Sphere** & **Mushroom** & **Ellipse** & \begin{tabular}{c} **Rect** \\ **Donuts** \\ **(D = 50)** \\ \end{tabular} & \begin{tabular}{c} **Rect** \\ **Donuts** \\ **(D = 30)** \\ \end{tabular} & \begin{tabular}{c} **Round** \\ **Rect** \\ **(R = 70)** \\ \end{tabular} & \begin{tabular}{c} **Round** \\ **Rect** \\ **(R = 50)** \\ \end{tabular} & **Square** \\ \hline \begin{tabular}{c} Self-weight \\ (kN/m\({}^{2}\)) \\ \end{tabular} & 5.89 & 5.48 & 4.41 & 4.24 & 3.96 & 3.88 & 3.85 & 3.56 & 3.3 \\ \hline \begin{tabular}{c} Design Load \\ (kN/m\({}^{2}\)) \\ \end{tabular} & 9.70 & 9.29 & 8.22 & 8.05 & 7.77 & 7.69 & 7.66 & 7.37 & 7.11 \\ \hline \begin{tabular}{c} Ultimate Load \\ (kN/m\({}^{2}\)) \\ \end{tabular} & 34.01 & 33.8 & 33.78 & 33.83 & 33.94 & 33.81 & 33.81 & 32.08 & 28.50 \\ \hline \begin{tabular}{c} Residual Load \\ (kN/m\({}^{2}\)) \\ \end{tabular} & 24.31 & 24.51 & 25.2 & 25.78 & 26.17 & 26.12 & 26.15 & 24.71 & 21.39 \\ \hline \begin{tabular}{c} Deflection at D.L \\ (mm) \\ \end{tabular} & 2.80 & 2.80 & 2.52 & 2.58 & 2.49 & 2.47 & 2.46 & 2.41 & 2.34 \\ \hline \begin{tabular}{c} Deflection at U.L \\ (mm) \\ \end{tabular} & 22.42 & 23.32 & 25.2 & 25.78 & 24.03 & 25.55 & 26.5 & 25.42 & 22.3 \\ \hline \begin{tabular}{c} Initial \\ Stiffness \\ \end{tabular} & 3.45 & 3.30 & 3.25 & 3.10 & 3.10 & 3.10 & 3.10 & 3.04 & 3.02 \\ \hline \begin{tabular}{c} Failure \\ Stiffness \\ \end{tabular} & 1.52 & 1.45 & 1.34 & 1.31 & 1.41 & 1.32 & 1.28 & 1.26 & 1.28 \\ \hline \
\begin{tabular}{c} Transition Stiffness \\ \end{tabular} & 1.24 & 1.20 & 1.13 & 1.11 & 1.22 & 1.13 & 1.09 & 1.08 & 1.07 \\ \hline
\begin{tabular}{c} Failure mode \\ at U.L \\ \end{tabular} & Y & Y & Y & Y & Y & Y & C & C \\ \hline \hline \end{tabular} Note: \(\gamma\)’ is the failure with the yield of rebars, and ‘C’ is the failure with the concrete crushing at the edge of void; ‘UL’ is the load at the failure, and ‘DL’ is the design load.  
\end{table} 
Table 4: The results of the FE analysis. Figure 5: \(Cont\). Based on FE analysis results, the effect of each void shape parameter was investigated. The base shape of void was closely related with the volume of void, not the section properties of voided slab. Therefore, it was indicated that the base shape of void mainly influenced the load-bearing capacity of voided slab, as shown in Figure 5a. The curvature radius of edge was closely related with the stress concentration at the edge of void, not the section properties of the voided slab. The FE analysis also showed a concrete crushing failure at the edge of void before the yield of bottom rebars owing to stress concentration when the curvature radius of edge decreased less than 70 mm. Therefore, it was indicated that the curvature radius of edge of void influenced the load-bearing capacity and failure mode of the voided slab, as shown in Figure 5b. The hole diameter was related with the flexural stiffness of the voided slab. Among the flexural stiffness, the hole diameter mainly influenced the transition stiffness. Therefore, it was indicated that the hole diameter influences the flexural stiffness after the crack occurred, as shown in Figure 5c.