# The Behaviour of Half-Slabs and Hollow-Core Slab in Four-Edge Supported Conditions
## 4 Computational Analysis  
### Simplified Analytical Method  
Engineering calculations for the deflection of slabs were made in accordance with the standards [52; 53; 54] and compared with the test results to highlight differences in the behaviour of individual slabs. Two models were analysed: the beam and the slab model under non-uniformly distributed short- and long-term loading sequences K-11 and L-12, presented in Figure 12. In the first case, the analysis included a single span simply supported beam (Figure 21a) on which the uniformly distributed short- and long-term load \(\mathrm{p}\,=\,7.7\mathrm{kN}/\mathrm{m}^{2}\) was exerted. The calculated results, presented in Table 6, differ from the measured values. The most convergent values equal 36% of the deflections noticed during the tests for the RP model. A convergence of only 5% was determined for the LGR model and was 25% in the case of the prestressed HC slab. Similar results were obtained for the calculations of long-term loading, and they are presented in Table 7. The convergence level was similar for the RP and HC models, and it was higher by 24% for the LGR model. The analysis of the slab model covered the rectangular slab under four-edge-supported conditions (Figure 21b), which was non-uniformly loaded on the surface under the loading sequences K-11 and L-12 (cf. Figure 12). The deflection, a function of the maximum bending moment at the midspan of the slab, was calculated using the (Marcus) simplified method of elastic meshes. Considering assumptions of this method, the actual non-uniform loading of the slab was replaced with the uniform loading \(\overline{\mathrm{p}}\). For that purpose, the load exerted on the slabs was distributed into the symmetric and non-symmetric load (applying the rule shown in Figure 21c). The resulting uniform loading of the slab was equal to \(\overline{\mathrm{p}}=3.0+1.7=4.7\ \mathrm{kN}/\mathrm{m}^{2}\). Under the loading \(\overline{\mathrm{p}}\), the deflection of the slab midpoint can be  
Figure 21: Slab models used in calculations: (**a**) the beam model; (**b**) the slab model; (**c**) the principle of establishing the equivalent loading of the slab \(\overline{\mathrm{p}}\) under non-symmetric loads. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Day of Loading**} & \multicolumn{6}{c}{**Maximum Displacement of Slabs (mm)**} \\ \cline{2-9}  & \multicolumn{2}{c}{**LGR**} & \multicolumn{2}{c}{**LGF**} & \multicolumn{2}{c}{**RP**} & \multicolumn{2}{c}{**HC**} \\ \cline{2-9}  & **a** & **a/l** & **a** & **a/l** & **a** & **a/l** & **a** & **a/l** \\ \hline
0 & 1.93 & 1/3140 & 2.67 & 1/2270 & 3.9 & 1/1554 & 3.57 & 1/1697 \\
100 & 7.0 & 1/866 & 15.0 & 1/404 & 8.0 & 1/758 & 7.0 & 1/866 \\
200 & 9.0 & 1/673 & 17.0 & 1/356 & 10.0 & 1/606 & 7.0 & 1/866 \\
300 & 12.0 & 1/505 & 18.0 & 1/337 & 11.0 & 1/551 & 10.0 & 1/606 \\
365 & 13.0 & 1/466 & 19.0 & 1/319 & 12.0 & 1/505 & 10.0 & 1/606 \\ \hline \hline \end{tabular} a—maximum displacement, l—span length.  
\end{table} 
Table 5: Maximum displacements of individual models on consecutive days of loading. determined from the relationships \(\overline{\overline{a}}_{\text{cal}}\big{(}\text{M}_{\text{P}}\big{)}=\overline{a}_{ \text{cal}1}+\overline{a}_{\text{cal}2}\), which depend on the value of the maximum bending moment. Symbols \(\overline{a}_{\text{cal}1}\) and \(\overline{a}_{\text{cal}2}\) denote the displacement from the load of 1.7 \(\text{kN}/\text{m}^{2}\) and 3.0 \(\text{kN}/\text{m}^{2}\). The maximum moment for the rectangular slab under the uniform loading was \(\text{M}_{\text{P}}=0.0365\overline{\text{p}}^{2}=0.172\text{l}^{2}\) (where 1 mean calculation span), and for the simply supported beam, it was equal to \(\text{M}_{\text{P}}=0.125\overline{\text{p}}\text{l}^{2}=0.96\text{l}^{2}\), and the coefficient for determining the slab moment proportionally to deflections of the beam was equal to 0.172/0.963 = 0.178.