# Flexural Strength and Stiffness of Donut-Type Voided Slab
## 2 Optimization of Void Shape
### Finite Element Analysis Model of Voided Slab  
Numerical simulations using nonlinear finite element analysis methods were conducted to derive the optimal void shape by using the FE analysis program named 'LUSAS' [13]. Generally, a 2D model was often used to conduct numerical simulations of the slabs by using finite element method when slabs were uniform in both longitudinal and transverse axes. However, in the case of voided slabs, it was impossible to use 2D model because of the irregular section geometry along the longitudinal and transverse axis due to voids. To consider complex void shapes precisely, a 3D model was used to generate concrete web parts between voids, and tetrahedral elements that had four nodes were used to generate FE meshes of the voided slabs that had extraordinary shapes of voids inside.  
To conduct a nonlinear finite element analysis, two material models were used. The bilinear model, which assumed that the rebar behavior will be totally elastoplastic in the tensile and compressive loading conditions, was used for the rebar, and the LUSAS concrete model 94 [13], which can consider multi-cracks and strength softening of concrete, was used for concrete.  
The properties of slab model were idealized as 8.9 m in length and 300 mm in width to conduct a FE analysis. It was enough to conduct a FE analysis with a line module model of slab to compare the effects of void shapes on flexural capacities, because voids are located with a uniform gap toward the width and length directions of the slab, and this way is more  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Item** & **Sphere** & **Mushroom** & **Ellipse** & \begin{tabular}{c} **Rect.** \\ **Donut** \\ **(D = 50)** \\ \end{tabular} & \begin{tabular}{c} **Rect.** \\ **Donut** \\ **(D = 30)** \\ \end{tabular} & \begin{tabular}{c} **Roct.** \\ **Rect** \\ **(R = 70)** \\ \end{tabular} & \
\begin{tabular}{c} **Round** \\ **Rect** \\ **(R = 50)** \\ \end{tabular} &
\begin{tabular}{c} **Round** \\ **Rect** \\ **Square** \\ \end{tabular} \\ \hline Shape & & & & & & & & \\ \hline Base shape & Sphere & Elliptical Sphere & & & & & \\ \hline Radius & mm & 70 & 50 & 70 & 70 & 70 & 50 & 0 \\ of edge & & & & & & & & \\ \hline Hole & mm & - & - & - & 50 & 30 & - & - & - \\ Diameter & & & & & & & & \\ \hline Volume & cm\({}^{3}\) & 1436 & 5625 & 6300 & 7380 & 7650 & 7785 & 8910 & 10,125 \\ \hline Width & mm & 140 & 270 & 270 & 270 & 270 & 270 & 270 & 270 \\ \hline Height & mm & 140 & 140 & 140 & 140 & 140 & 140 & 140 & 140 \\ \hline Weight & \% & 20.0 & 25.0 & 28.0 & 32.8 & 34.0 & 34.6 & 39.6 & 45.0 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Properties of the void shaper for the FE analysis model.  
time-efficient than using whole slab model. To simulate the real state of continuous slabs for building, the support conditions were set to be the fixed condition, and a distributed load was imposed. Loads, imposed on slabs, increased until the slab was destroyed in order of the self-weight of the slab, dead load and live load. Further information of the FE model is shown in Table 3.  
The convergency test was conducted by mesh size to validate the FE model in terms of the accuracy of results and the time efficiency. The five tetrahedral elements with different mesh sizes of 10 mm, 20 mm, 30 mm, 50 mm and 80 mm were evaluated. The results converged into the theoretical ultimate load-bearing capacity of 32 kN/m\({}^{2}\) with less than a 5% difference when the mesh size was less than 30 mm, as shown in Figure 3. Therefore, a tetrahedral element with 30-mm mesh size was used in the FE analysis for voided slabs.