# 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
In a propped system, regarding the criteria of the maximum stresses allowed in steel and concrete sections, two equations can be formulated (Equations (9) and (10)) to calculate the bending resistance; the lower result from the two equations should then be considered for further calculations. \[\rm\sigma_{a}=E_{a}\times M_{\rm RK,1}/EI_{L}\times z_{a}, \tag{9}\]  
\[\rm\sigma_{c}=E_{c}\times M_{\rm RK,2}/EI_{L}\times z_{c}, \tag{10}\]  
where \(\rm M_{\rm RK,1}\) is the maximum allowable moment that will cause stress in the steel section, and \(\rm M_{\rm RK,2}\) is the maximum allowable moment that will cause stress in the concrete section. To choose the right expression for elastic bending resistance, as demonstrated in [9], the value of a normal compressive force in a concrete slab which corresponds with \(\rm M_{\rm EI,RK}\) can be calculated as:  
\[\rm N_{c,el}=k\times M_{el,RK}/I_{i,L}\times(A_{c,L}\times z_{ic,L}), \tag{11}\]  
\[\rm z_{ic,L}=A_{a}\times a_{a}/A_{i,L}. \tag{12}\]  
where \(\rm k\) is the minimum value between \(\rm k_{a}\), \(\rm k_{s}\), and \(\rm k_{c}\). These values are obtained from the edge fibers of the cross-section that has reached its limit in terms of stress. \(\rm A_{c,L}\) is the area of the ideal composite cross-section, \(\rm z_{ic,L}\) is the distance between the centers of gravity of the composite cross-section and the center of gravity of the concrete slab; \(\rm a_{a}\) is the distance between the center of gravity of the cross-section steel part and the center of gravity of the composite cross-section. An unpropped system must be considered during the phases of construction. In phase I, the weight of the steel beam and concrete must be considered, and then the characteristic bending moment of the calculated load is used to calculate the stresses on the steel section. In phase II, the stress in the steel section is assumed to be limited by the yielded stress; this allows the remaining stress to be taken into account. Two bending moments were calculated using the equations that calculate stresses in the steel (Equation (13)) and in the concrete (Equation (14)); the minimum value is associated with \(\rm M_{\rm RK,II}\). \[\rm\sigma_{a,II}=E_{a}\times M_{\rm RK,II,a}/EI_{L}\times z_{a}, \tag{13}\]  
\[\rm\sigma_{c}=E_{c}\times M_{\rm RK,II,c}/EI_{L}\times z_{c}, \tag{14}\]  
The elastic bending resistance of the composite section is then calculated as follows:  
\[\rm M_{el}=M_{\rm RK,I}+k\times M_{\rm RK,II}, \tag{15}\]  
where \(\rm M_{\rm RK,I}\) is the design bending resistance from phase I, and \(\rm k\) is the lowest factor when a stress limit is reached. Following the conditions given in EN-1994-1-1 [9], the characteristic value of the bending resistance of a composite section is calculated using Equations (16) and (17). \[\rm N_{c}\leq N_{c,el};\ M_{\rm RK}=M_{a,RK}+(M_{el,RK}-M_{a,RK})\times N_{c}/ N_{c,el}, \tag{16}\]  
\[\rm EI_{L}=E_{a}\times I_{i,L}, \tag{8}\]0  
Calculated values are shown in \[\rm EI_{L}=E_{a}\times I_{i,L}, \tag{8}\]1  
In Figure 2, a simplified relationship between M and N\({}_{\rm c}\) is shown. In all diagrams in Figure 2, curves for propped (red lines) and unpropped (purple lines) construction are shown. Diagrams are defined in accordance with [9]. For both types of construction, propped and unpropped, one and two bolts in each rib are analyzed. Each line has three marked spots--one spot is used for both degrees of shear connection and is defined as M\({}_{\rm{Rk}}\)/M\({}_{\rm{pl,Rk}}\) and N\({}_{\rm{c}}\)/N\({}_{\rm{c},f}\), and two spots represent M\({}_{\rm{el,Rk}}\)/M\({}_{\rm{pl,Rk}}\) and N\({}_{\rm{c,el}}\)/N\({}_{\rm{c,f}}\); these are related to the degree of shear connection. The calculated characteristic bending resistances align well with the non-linear approach of the propped system; the steel beam height of 400 mm, with bolts in pairs and a staggered arrangement, have resistances of 161.74 kNm and 112.55 kNm, respectively.