## 3 Validation of the Simplified Design Method
### Presentation of the Simplified Design Method
The method uses the following equation to compute the design moment resistance \(\mathrm{M_{fi,t,Rd}}\):  
\[\mathrm{M_{fi,t,Rd}}=\sum_{i=1}^{n}\mathrm{A_{i}}\cdot\mathrm{z_{i}}\cdot \mathrm{k_{y,\theta,i}}\cdot\left(\frac{\mathrm{f_{y,i}}}{\gamma_{\mathrm{M,fi,a} }}\right)+\alpha_{\mathrm{slab}}\cdot\sum_{j=1}^{m}\mathrm{A_{j}}\cdot\mathrm{ z_{j}}\cdot\mathrm{k_{c,\theta,j}}\cdot\left(\frac{\mathrm{f_{c,j}}}{\gamma_{\mathrm{M,fi,c}}}\right) \tag{1}\]  
\(\mathrm{z_{i}}\), \(\mathrm{z_{j}}\) is the distance from the plastic neutral axis to the centroid of the area \(\mathrm{A_{i}}\) or \(\mathrm{A_{j}}\)  
\(\mathrm{f_{y,i}}\) is the nominal yield strength;  
\(\mathrm{f_{y}}\), for the steel area \(\mathrm{A_{i}}\), (structural steel or reinforcement) taken as positive on the compression side of the plastic neutral axis and negative on the tension side--the reinforcement can be considered only in tension;  
\(\mathrm{f_{c,j}}\) is the design strength of the concrete area \(\mathrm{A_{j}}\) at 20 \({}^{\circ}\)C;  
\(\mathrm{k_{y,\theta,i}}\) and \(\mathrm{k_{c}}\), \(\mathrm{e_{j}}\) are defined in Tables 3.2 and 3.3 of EN 1994-1-2 [27];  
\(\alpha_{\mathrm{slab}}\) is the coefficient considering the assumption of the rectangular stress block when designing slabs and shallow composite floor beams--\(\alpha_{\mathrm{slab}}=0.85\). The possible contribution of the concrete below the upper flange of the steel section should be neglected in the calculation, and thus only the concrete above the upper flange is considered. The design value of the concrete compression force can be calculated using the following equation:  
\[\mathrm{N_{fi,c,Rd}}=\frac{\mathrm{f_{c}}}{\gamma_{\mathrm{M,fi,c}}}\cdot \mathrm{b_{eff}}\cdot\mathrm{c_{z}}\cdot\mathrm{k_{h}}\leq\mathrm{N_{fi,t,Rd} }=\sum_{i=1}^{n}\mathrm{A_{i}}\cdot\mathrm{k_{y,\theta,i}}\cdot\left(\frac{ \mathrm{f_{y,i}}}{\gamma_{\mathrm{M,fi,a}}}\right) \tag{2}\]  
where:  
\(\mathrm{k_{h}}=0.85\) for \(\mathrm{c_{z}/h}>0.4\)  
\(\mathrm{k_{h}}=1.0\) for all other cases  
The width of the bottom flange is reduced to a value \(\mathrm{b_{fb,eff,fi}}\) using Equation (3), as follows:  
\[\mathrm{b_{fb,eff,fi}}=\mathrm{max}(\mathrm{b_{ft}}\,;\mathrm{b_{fb}}-2\cdot \left(\mathrm{l_{a}}+\mathrm{e_{fb}}\right)\cdot\mathrm{k_{c}}) \tag{3}\]  
where:  
\(\mathrm{k_{c}}=0.5\) for a solid slab that covers 100% of the upper surface of the welded plate  
\(\mathrm{k_{c}}=1.0\) for all other cases  
Based on [PERSON] and [PERSON] [15], the temperature \(\mathrm{\theta_{fb}}\) of the bottom flange can be assumed constant and is calculated using Equation (4):  
\[\mathrm{\Theta_{fb}}=\mathrm{A_{i}}\cdot\mathrm{e_{fb}}^{2}+\mathrm{B_{i}} \cdot\mathrm{e_{fb}}+\mathrm{C_{i}} \tag{4}\]  
Figure 9: (**a**) SFB [21] and (**b**) asymmetric double-T cross-section with notation. where:  
A\({}_{\rm i}\), B\({}_{\rm i}\), C\({}_{\rm i}\) are given in Table 4[15]. The temperature of the profile web \(\theta_{\rm w}\) is assumed constant and is calculated using Equation (5), based on [PERSON] and [PERSON] [15]:  
\[\Theta_{\rm w}=(A_{\rm w}\cdot\ln\ e_{\rm fb}+B_{\rm w})\cdot\rm e(C_{\rm w}\cdot \ln e_{\rm fb}+D_{\rm w})\cdot\frac{B_{\rm w}}{40} \tag{5}\]  
where:  
A\({}_{\rm w}\), B\({}_{\rm w}\), C\({}_{\rm w}\), D\({}_{\rm w}\) are given in Table 4[15]. In this equation, \(e_{\rm fb}\) and h\({}_{\rm w}\) are expressed in [mm]. Within the field of application of slim floor beams, the upper flange can be assumed as non-affected by the fire exposure. For the longitudinal reinforcement placed within the zone determined by h\({}_{\rm s}\) and u\({}_{\rm c}\), u\({}_{\rm w}\), u\({}_{\rm r}\) according to Figure 9, the temperature \(\theta_{\rm r}\) can be calculated using the equation, based on [PERSON] et al.