## 3 Bending of Slab Supported by Column
When the larger parameter \(W\) is used, the moments are underestimated. However, it should be noted that in this case, the reduction option activates starting from the coarsest mesh with \(8\times 8\) FEs. For smaller \(W=50\) cm, it becomes effective for \(32\times 32\) FEs or denser meshes. Meshes with 8 and 16 FEs along the edge are too coarse to trigger a moment reduction for this distance. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & **No** & \multicolumn{2}{c}{**Reinforcement: Reduction via Parameter \(W\)**} \\
**Division** & **FE Size** & **Reinforcement** & \(W=0\)**cm** & \(W=50\)**cm** & \(W=150\)**cm** \\ \(N\times N\) & \(\begin{array}{c}h_{\text{FE}}\\ \text{\bf[cm]}\end{array}\) & \(\begin{array}{c}\text{\bf max}\ m_{xx}\\ \text{\bf[kNm/m]}\end{array}\) & \(\begin{array}{c}\text{\bf max}\ m_{xx}\\ \text{\bf[kNm/m]}\end{array}\) & \(\begin{array}{c}\text{\bf max}\ m_{xx}\\ \text{\bf[kNm/m]}\end{array}\) \\ \hline \(8\times 8\) & 62 & 8.88 & 11.99 & 11.99 & 10.32 \\ \(16\times 16\) & 31 & 12.22 & 16.50 & 16.50 & 11.39 \\ \(32\times 32\) & 15.5 & 15.60 & 17.11 & 11.99 & 9.41 \\ \(64\times 64\) & 7.75 & 18.93 & 17.89 & 12.64 & 9.30 \\ \(128\times 128\) & 3.875 & 22.21 & 18.40 & 12.83 & 9.29 \\ \(256\times 256\) & 1.9375 & 25.48 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Influence of discretization on moment reduction in slab above column—results from Midas. Figure 14: Idea of determining reduction zone in Robot dependent on column area (detailed explanation in the text). Table 5 indicates the effect of the reduction width on the bending moment value in Midas. Now, only one mesh with 64 FEs along the edge is employed while \(W\) grows. As expected, similarly to the dependence on the column size in Robot as shown in Table 3, the value of the reduced bending moment for the slab decreases as parameter \(W\) increases. The most suitable choice seems to be distance \(W=50\) cm or 75 cm. For both cases, \(m_{xx}\) after reduction is close to 12 kNm/m, agreeing reasonably well with the case of \(b=30\) cm in Robot. The reduced moment \(m_{xx}=9.30\) kNm/m for \(W=150\) cm is clearly underestimated, which corresponds to the exaggeratedly large reduction area; see Figure 15.  
The reduction in Midas works as a weighted average method. The original value of \(m_{xx}\) is recalculated at the central node, selected due to the connection of the slab to the column and in its defined vicinity. First, this vicinity is specified as a square area with a width \(W\). All FEs within this region, i.e., given by the size \(W/2\) in horizontal left and right as well as vertical up and down, are taken into account. It concerns internal and contiguous elements restricted to the square. The bending moments are computed by extrapolation from integration points at each node of a given FE. These values belong to each separate element. The final value at the central node is then averaged based on nodal values but derived from patch FE functions. Moreover, weighting factors, which depend on the distances between the central node and sequentially considered nodes, affect this averaging operation. Thus, the determined bending moment \(m_{xx}\) is assigned to all nodes within the zone defined by the given width \(W\). The last part of this section summarizes the entire analysis discussed above. The abbreviations listed in Table 6 are useful for understanding Figure 16. This figure concisely highlights the key aspects of this study. Each case in Figure 16 represents the diagram of moments \(m_{xx}\) at the center of the slab as a function of FE density from the coarsest to the finest mesh. The abscissa axis changes in the logarithmic scale. The cases R-no and M-no refer to the original results in Robot and Midas, respectively, i.e., without any reduction in the post-processing stage. They are marked by red lines with empty triangles, straight and inverted, respectively. It is noticed that both diagrams grow almost linearly, but the R-no for Robot starts from a larger value.