# Flexural Strength and Stiffness of Donut-Type Voided Slab
## 4 Test Results and Discussion
### Flexural Stiffness of Donut-Type Voided Slab  
In order to evaluate the flexural stiffness of the donut-type voided slabs, the flexural stiffness and moment of inertia were compared. For flexural stiffness, the secant stiffness at the cracking load and the stiffness between the cracking and yield points are compared. The theoretical moment of inertia of the gross concrete section, cracked section and effective moment of inertia derived from Equations (5)-(9) were compared with the experimental moment of inertia derived from Equation (10). The moment of inertia was calculated for the minimum section and donut section and compared with the test results. The moment of inertia at the minimum and donut sections can be easily calculated by subtracting \(I_{gM}\) and \(I_{gD}\) from \(I_{g}\) of 'Solid', respectively (refer to Figure 13). \[I_{g}=\frac{bh^{3}}{12}-N(I_{gM}\text{ or }I_{gD}) \tag{5}\]  
\[I_{gM}=(\frac{b^{\prime}h^{\prime 3}}{12}+\frac{\pi r^{4}}{4}) \tag{6}\]  
\[I_{gD}=\frac{\pi r^{4}}{4}+\frac{r^{\prime}h^{\prime 3}}{6}+\left(\frac{\pi r ^{\prime 4}}{4}+\pi r^{\prime 2}\bigg{(}\frac{h^{\prime\prime}}{2}+\frac{4r^{ \prime}}{3\pi}\bigg{)}^{2}\right) \tag{7}\]  
\[I_{cr}=\frac{b(kd^{3})}{3}+nA_{s}(d-kd)^{2} \tag{8}\]  
\[I_{c}=(\frac{M_{cr}}{M_{a}})^{3}I_{g}+[1-(\frac{M_{cr}}{M_{a}})^{3}]I_{cr} \tag{9}\]  
\[I_{test}=\frac{Pl_{a}}{48E_{c}\delta}(3l^{2}-4l_{a}^{2}) \tag{10}\]  
* \(I_{g}\): moment of inertia of gross concrete section, neglecting reinforcement
* \(I_{gM}\): moment of inertia of voided area at the minimum section (refer to Figure 13)
* \(I_{gD}\): moment of inertia of voided area at the donut section (refer to Figure 13)
* \(I_{cr}\): moment of inertia of cracked section transformed to concrete
* \(I_{c}\): effective moment of inertia for computation of deflection
* \(I_{test}\): moment of inertia calculated by inverse operation from test results
* \(b\): width of slab
* \(b^{\prime}\): width of plane part of void shaper (refer to Figure 13)
* \(h\): height of slab
* \(h^{\prime}\): height of void shaper (refer to Figure 13)
* \(h^{\prime\prime}\): height of straight part of center hole (refer to Figure 13)
* \(k\): factor used in calculating flexural capacity of a section
* \(l\): length of clear span of specimen
* \(l_{a}\): length between loading and support point
* \(M_{a}\): maximum moment at the stage for which deflections are being computed* \(n\): the number of reinforcements in a layer
* \(N\): the number of voids which are included in the cross-section for calculation
* \(P\): load at the stage for which effective moment of inertia is computed
* \(r\): radius curvature of outer side of void (refer to Figure 13)
* \(r^{\prime}\): radius curvature of upper part of center hole (refer to Figure 13)
* \(E_{c}\): modulus of elasticity of concrete
* \(\delta\): deflection at the stage for which effective moment of inertia is computed  
Table 9 shows the comparison between the test results and theoretical value of stiffness. In all the specimens, \(S_{2,Test}\) was lower than \(S_{2,ACI}\) by 23-28%. It is because Equation (9) generally overestimates the effective moment of inertia of members having a lower steel ratio-like slab [16]. Based on the test results, flexural stiffness was compared between the donut-type voided slabs and solid slab. Although a gap of 20%-37% was obtained in the initial flexural stiffness before cracking (\(S_{1,Test}\)), it might be caused by experimental errors, because slab deflection before cracking was very small. The flexural stiffness between cracking and the yield point was lower in the donut-type voided specimens by 7.7-8.7% than the 'Solid'. It is deduced to have been caused by the reduction of the cross-section due to the voids. However, the donut-type specimens show a higher stiffness between the cracking and yield point than 'OF-V-S-R' by 7%. Therefore, it is deduced that the center hole in the void shaper improves the flexural stiffness.