# Numerical Study on the Bending Resistance of Lightweight Built-Up Steel-Concrete Composite Beams
## 2 Materials and Methods
### Analytical Approaches for the Bending Resistance of Composite Beams
#### 2.1.2 Analytical Bending Resistance of Composite LWT-FLOOR Beams
Moreover, the used bolt diameter (12 mm) is smaller than specified in the European standard [9] (16 mm-25 mm). However, it was assumed that these equations are also valid for bolts used as shear studs. In addition, according to EN 1994-1-1 [9], to calculate the resistance of the bolt when the ribs of the metal sheet are positioned perpendicular to the direction of the composite beam, it is necessary to apply the reduction coefficient, \(\mathrm{k_{t}}\). \[\mathrm{k_{t}}=0.7/\mathrm{sqrt(n_{r})}\times\mathrm{b_{0}}/\mathrm{h_{p}} \times(\mathrm{h_{sc}}/\mathrm{h_{p}}-1)\leq\mathrm{k_{t,max}}, \tag{5}\]  
where \(\mathrm{b_{0}}\) is a mean width of a concrete rib, \(\mathrm{h_{sc}}\) is the overall nominal height of a connector, \(\mathrm{n_{r}}\) is the number of stud connectors in one rib, \(\mathrm{h_{p}}\) is the overall depth of the profiled steel sheet (excluding embossments), and \(\mathrm{k_{t,max}}\) is the limit value, in accordance with the findings in [9]. EN 1994-1-1 [9] requires the minimum height of the shear connectors to be:  
\[\mathrm{h_{sc}}=\mathrm{h_{p}}+2\times\mathrm{d}, \tag{6}\]  
where \(\mathrm{d}\) is the diameter of the shear connector. The shear connector resistance is finally calculated as:  
\[\mathrm{P_{t,Rk}}=\mathrm{k_{t}}\times\mathrm{P_{Rk}}, \tag{7}\]  
where \(\mathrm{P_{Rk}}\) is the design value of the shear resistance of a single connector. Table 1 shows the calculated bending resistances of composite CFS-concrete beams using the investigated steel beam heights in cases where shear connectors are arranged in pairs or are staggered, with and without the reduction coefficient, \(\mathrm{k_{t}}\). The third considered approach concerns the non-linear bending resistance, which is calculated in accordance with [9, 32, 33]. First, the elastic bending resistance of the cross-section must be found, which can be calculated by converting the concrete slab into a steel cross-section. To calculate the ideal area of the composite cross-section, the ideal area of the concrete slab cross-section must be found, as well as the distance between the center of gravity of the composite section and the concrete slab. The ideal area of the concrete slab is calculated by considering the thickness of the concrete slab above the center of gravity of the composite section. Then, it is possible to calculate the ideal moment of inertia for \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Bolt Arrangement**} & \multicolumn{2}{c}{**Bending Resistance [kNm]**} & \multicolumn{2}{c}{**Degree of Shear Connection [–]**} \\ \cline{2-5}  & **without k\({}_{\mathrm{t}}\)** & **with k\({}_{\mathrm{t}}\)** & **without k\({}_{\mathrm{t}}\)** & **with k\({}_{\mathrm{t}}\)** \\ \hline \multirow{4}{*}{In pairs} & \multicolumn{4}{c}{h = 400 mm} \\  & 257.7 & 181.7 & 1.0 & 0.64 \\ \cline{1-1} \cline{2-5}  & 210.0 & 142.7 & 0.78 & 0.46 \\ \cline{1-1} \cline{2-5}  & \multicolumn{4}{c}{h = 500 mm} \\ \cline{1-1} \cline{2-5}  & 299.7 & 215.3 & 1.0 & 0.64 \\ \cline{1-1} \cline{2-5} Staggered position & 246.8 & 171.9 & 0.78 & 0.46 \\ \cline{1-1} \cline{2-5}  & \multicolumn{4}{c}{h = 600 mm} \\ \cline{1-1} \cline{2-5}  & 341.7 & 249.2 & 1.0 & 0.64 \\ \cline{1-1} \cline{2-5}  & 283.7 & 201.6 & 0.78 & 0.46 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Bending resistances for steel beam heights of 400 mm, 500 mm, and 600 mm. the concrete slab, and finally, for the composite section. The stiffness of the composite cross-section is possible to calculate using Equation (8):  
\[\rm EI_{L}=E_{a}\times I_{i,L}, \tag{8}\]  
where \(\rm E_{a}\) is the modulus of the elasticity of steel and \(\rm I_{i,L}\) is the ideal second moment of the area. Non-linear bending resistance can be considered for propped and unpropped systems.