# Flexural Strength and Stiffness of Donut-Type Voided Slab
## 2 Optimization of Void Shape
### Finite Element Analysis Results and Discussion  
To evaluate the structural capacity of voided slabs, the load-bearing capacity and the flexural stiffness were compared. A design load, an ultimate load and a residual load were compared in terms of a load-bearing capacity, and an initial stiffness, a failure stiffness and a transition stiffness were compared in terms of a flexural stiffness (refer to Figure 4). The design load is the sum of the dead load, live load and self-weight of the slab. The ultimate  
Figure 3: The results of the convergency test. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{Rebar} & Transverse direction & Top & D10 \(\times\) 2 \\ \cline{2-4}  & Longitudinal direction & Bottom & D13 \(\times\) 2 \\ \hline \multirow{2}{*}{Mesh} & Types of Mesh & Tetrahedral & \\ \cline{2-4}  & Mesh size & 30 mm & \\ \hline \multirow{2}{*}{Size} & Width & 300 mm & \\ \cline{2-4}  & Height & 250 mm & \\ \cline{2-4}  & Length & 8900 mm & \\ \hline \multirow{2}{*}{Load} & Self-weight & 3.2–5.9 kN/m\({}^{2}\) & \\ \cline{2-4}  & Dead load & 1.81 kN/m\({}^{2}\) & \\ \cline{2-4}  & Live load & 1.96 kN/m\({}^{2}\) & \\ \hline \multirow{2}{*}{Material} & \(f_{ck}\) & 24 MPa & \\ \cline{2-4}  & \(f_{y}\) & 400 MPa & \\ \hline \multicolumn{2}{c}{Boundary condition} & Fixed end \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Properties of the FE analysis model. load is a load at failure. The residual load is a difference between the design load and the ultimate load. The initial stiffness is a slope of a line through the origin and the point on the load-deflection curve at the design load. The failure stiffness is a slope of a line through the origin and the point on the load-deflection curve at the ultimate load. The transition stiffness is a slope of a line through the point on the load-deflection curve at the design load and the point on the load-deflection curve at the ultimate load. Table 4 and Figure 5 show the results of the FE analysis of about eight cases of voided slabs and solid slab. All the slab models were found to be on the safety side under the design load and showed a general flexural behavior with some flexural cracks until the design load. The voided slabs except 'Square' and 'Round Rect (R = 50 mm)' showed flexural failure with the yield of bottom rebars at the ultimate load. On the other hand, the voided slabs with 'Square' and 'Round Rect (R = 50 mm)' failed with concrete crushing at the edge of void due to stress concentration at the ultimate load before the yield of rebars. The design load was varied by the void shapes with its volume. The solid slab showed the largest design load of 9.70 kN/m\({}^{2}\). In the case of voided slab, the voided slab with 'Sphere' showed the largest design load of 9.29 kN/m\({}^{2}\), and the voided slab with 'Square' showed the smallest design load of 7.11 kN/m\({}^{2}\). Voided slabs showed the decrease in the design load by 4-27% compared to that of the solid slab. The ultimate load decreased in the cases of void shape such as 'Round Rect (R = 50 mm)' and 'Square'. The solid slab showed the ultimate load of 34.01 kN/m\({}^{2}\). The voided slabs, except 'Round Rect (R = 50 mm)' and 'Square', showed the similar ultimate load compared to the solid slab. The voided slabs with 'Round Rect (R = 50 mm)' and 'Square' showed the ultimate loads of 32.08 kN/m\({}^{2}\) and 28.50 kN/m\({}^{2}\), respectively. These were 94% and 84% of that of the sold slab, respectively. The residual load was varied by the void shapes in contrast with the ultimate load. The solid slab showed the residual load of 24.31 kN/m\({}^{2}\). In the case of voided slab, the voided slab with 'Rect Donuts (D = 50 mm)' showed the largest load-bearing capacity of 26.17 kN/m\({}^{2}\) and the voided slab with 'Square' showed the smallest load-bearing capacity of 21.39 kN/m\({}^{2}\). The voided slabs except in the cases of void shape such as 'Round Rect (R = 50 mm)' and 'Square' showed the increase in the residual load by up to 8% compared to that of the solid slab.