## 4 Interaction of Beam with Slab
As for the diagrams from Robot, given in Figure 19, the jumps between values are observed due to its discretization into FEs with specific lengths. Figure 22 displays a contour plot of bending moments \(m_{xx}\) for the slab. Together with this distribution, the values of \(m_{xx}\) along the width \(L_{y}\) at midpoint 0 are distinguished. They are needed to determine \(I_{mxx}^{\text{s}}\) by the trapezoidal rule. After this integration, the total value of the moment \(M_{y}^{\Sigma}\) for the Midas computations, using type II and the finer mesh, equals:  
Figure 16.2  
An alternative approach to operating with the offset in the model is to preserve the beam-slab configuration in the basic form, as defined in option I (see Figure 18), and then appropriately modify the value of the moment of inertia for the beam. Its initial value is  
Figure 16.3  
Figure 16.4 Diagram of bending moments \(M_{y}\) [kNm] for beam, results in Midas, connection type II via offset, finer mesh. Figure 16.5 Middle panel cut of slab’s bending moments \(m_{xx}\) [kNm/m] perpendicular to beam with corresponding integrals \(I_{mxx}^{\text{s}}\) [kNm], results in Robot, connection type II via offset, finer mesh (**left**), coarser mesh (**right**). Adopting the parallel axis theorem, the moments of inertia for connection type II with \(z_{\Pi}=12.5\) cm and type III with \(z_{\Pi\Pi}=27.5\) cm are calculated, respectively, as follows:  
Figure 16.6  
Figure 16.7  
In Midas, such adjusted geometric characteristics have to be directly input into the beam data. The Robot software tool requires the entry of the so-called section's moment of inertia coefficient depending on the distance \(z\). For configurations II and III, these entries are determined by the ratio:  
Figure 16.8  
The coefficient \(C\) is originally given to account for the cracking of a section, and in that case, it is usually reduced for the beam analysis. However, here it is regarded to indicate the position of the beam related to the slab as for the offset option. Note that both modified values of the moment of inertia, directly as in Midas or by means of \(C\) as in Robot, are also transferred to the next software modules for dimensioning. The total bending moment \(M_{y}^{\Sigma}\) is still calculated using Equation (20), but according to connection type I, where \(z=z_{1}=0\) cm, the second term vanishes. This approach, where the beam's moment of inertia is updated, is further denoted by suffix \"-m\". The bending moment \(M_{y}^{\Sigma}\) and its particular components are determined at the beam mid-span for each connection type, either by using the offset option or by considering the moment of inertia. Table 7 shows the results for Robot, while Table 8 shows the results for Midas. The following columns are specified: the connection type, the distance \(z\), mesh density, the moment \(M_{y}\), and the axial force \(F_{x}\) read from the diagram; their combination \(M_{y}+F_{x}\,z\); the integral \(I_{mxx}^{\Sigma}\) from the panel cut, the total value \(M_{y}^{\Sigma}\), and additionally the deflection \(w_{0}\) at point 0 (Figure 17). The suffix \"-o\" or \"-m\" refers to how the position of the beam relative to the slab is taken into account. It is evident in the cases with non-zero offset (\"-o\") that the force \(F_{x}\) has to be included in calculating \(M_{y}^{\Sigma}\). Generally, the values of this total bending moment correspond to the theoretical \(M_{y}^{0}=4.5\) kNm. The results for the coarser mesh are less accurate, especially in Midas, where the integrals \(I_{mxx}^{\Sigma}\) are estimated using the trapezoidal rule along the panel cut, which is based on four FEs. It should be  
Figure 16.9 Contour plot of bending moments \(m_{xx}\) [kNm/m] for slab with values required to calculate integral \(I_{mxx}^{\Sigma}\), results in Midas, connection type II via offset, finer mesh.