## 4 Interaction of Beam with Slab
upper flange of beam faces bottom plane of slab—offset \(z=27.5\) cm (**right**). The offset function, rigid links, or connector members form a particular type of frame. As a result, additional axial forces \(F_{x}\) are generated in the beam, along with bending moments \(M_{y}\). These axial forces occur due to the presence of offsets, specified in different ways, and the creation of a frame structure consisting of beam FEs and the connector members linking the beam to the slab as short study. The bending moment \(M_{y}\) and axial force \(F_{x}\) diagrams are presented in Figure 19 for case II(b) as an example. The remaining options (a) and (c) exhibit almost the same distributions of \(M_{y}\) and \(F_{x}\). Additionally, the equilibrium of selected beam nodes has been verified for case (c), thanks to which the formation of the frame and axial forces has been confirmed. This entire analysis proves that the offset function effectively acts as a rigid link between the beam and the slab. As a result, all three aforementioned options are equivalent. Moreover, non-zero axial forces in that case become apparent. The character of both \(M_{y}\) and \(F_{x}\) diagrams is discontinuous, and jumps are governed by the lengths of beam FEs. The presence of axial forces, when the offset option is used, must be considered in calculating the final value of the bending moment for the beam. In further computations, case (a) will be used, and at a later stage of the study, it will be denoted by the suffix \"-o\". The beam in this benchmark, cooperating with the slab, provides the function of bending moments \(M_{y}(x)\) as for a simply supported beam. Therefore, the theoretical value of \(M_{y}\) for the beam can be determined. The extreme point is located at point 0, where \(x=L_{x}/2=3\) m (see Figure 17), and the moment equals:  
\[M_{y}^{0}=0.125\,p_{Z}\,{L_{x}}^{2}=4.50\ [\mathrm{kNm}] \tag{19}\]  
When deriving the total bending moment \(M_{y}^{\Sigma}\) for a beam interacting with the slab, considering the offset effect, three components should be taken into account:  
\[M_{y}^{\Sigma}=M_{y}+F_{x}\,z+I_{mxx}^{\mathrm{s}} \tag{20}\]  
Maximum values of the bending moment \(M_{y}\) and the axial force \(F_{x}\) (absolute value) at midpoint 0 are read from the beam diagrams, as in case II(b), where they are highlighted in the boxes in Figure 19. The integral \(I_{mxx}^{\mathrm{s}}\) is determined based on the panel cut perpendicular to the beam so that slab bending moment \(m_{xx}\) along the \(y\)-axis is summed:  
\[I_{mxx}^{\mathrm{s}}=\int_{0}^{L_{y}}m_{xx}\ \mathrm{d}y \tag{21}\]  
The panel cuts of \(m_{xx}\) along width \(L_{y}\) (in the middle of length \(L_{x}\)) are illustrated in Figure 20 for the computations carried out in Robot using two meshes. In addition, the respective integral values \(I_{mxx}^{\mathrm{s}}\) are shown as well. Figure 20 also depicts the schemes of the model, FE meshes for both densities, supports along width \(L_{y}\), and the position of load \(p_{Z}\) is visible  
Figure 16.0 Diagrams of bending moments \(M_{y}\) [kNm] (top) and axial forces \(F_{x}\) [kN] (bottom) for beam connected with slab via rigid link (marked in red), results in Robot, connection type II, finer mesh. as well. For example, the total value of the bending moment \(M_{y}^{\Sigma}\) at midpoint 0 is then calculated for the finer mesh as  
Figure 16.1  
This arithmetic takes into account the appropriate absolute values based on the results obtained in Robot. In the Midas program, it is not possible to read the value of the integral \(I_{mxx}^{\text{s}}\) directly for a given panel cut. Hence, to assess it, the values of the slab's bending moments \(m_{xx}\) along an indicated intersection can be taken into account. After that, a numerical method such as the trapezoidal rule can be applied to derive \(I_{mxx}^{\text{s}}\). The results for the model in Midas with offset type II and the finer mesh are presented in Figures 21 and 22. Figure 21 visualizes the diagram of \(M_{y}\) for the beam.