## 7 Example of Using CPM to Determine the Resistance of a Continuous I-Beam
### Welded Section
#### 7.2.4 Design Ultimate Resistance  
Table 3 shows the design values of the ultimate resistance for the reliable cross-sections: span \(M^{CP}_{eff,p^{\prime}}\) support \(M^{CP}_{eff,s}\), and the design ultimate loads (\(q_{eff}\)) determined from them according to CPM. The symbol \(q^{CP}_{eff,s}\) indicates the ultimate load value determined from the condition of reaching \(M^{CP}_{eff,s}\) in the support cross-section, and the symbol \(q^{CP}_{eff,p}\) from the conditions needed to achieve \(M^{CP}_{eff,p}\) in the span cross-section. (Note: in Table 3, the values of the moments \(M^{CP}_{eff,s}\) and \(M^{CP}_{eff,p}\) are given \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{1}{c}{**t**} & **s/p** & **ABAQUS** & **CPM** & **CUFSM** & [19] & [22] & **EC3** \\ \hline \multirow{4}{*}{**5**} & 1 & & 2 & 3 & 4 & 5 & 6 & 7 \\  & s & 291.98 & 288.70 & & & & & \\  & p & 247.16 & 243.20 & & & & & \\  & s & 418.32 & 415.73 & & & & & \\  & P & 355.00 & 350.21 & & & & & \\
**7** & s & 566.31 & 565.86 & & & & & \\  & p & 481.86 & 476.67 & & & & & \\
**8** & s & 735.48 & 739.08 & & & & \\  & p & 627.53 & 622.60 & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of critical stresses determined by several methods for the I-300\(\times\)tx250\(\times\)t section and four wall thickness variants. as absolute values). The ultimate resistance of the beam was determined by the minimum load \(q_{eff,min}^{CP}=min\bigg{\{}q_{eff,p}^{CP},q_{eff,s}^{CP}\bigg{\}}\). Furthermore, in Table 3, the symbol \(M_{c,Rd}^{EC3}\) indicates the design cross-section resistance determined according to EC3 [1; 3], and the symbol \(q_{eff,min}^{EC3}\) indicates the minimum value of ultimate load determined on the basis of \(M_{c,Rd}^{EC3}\). Comparison of the results displayed in Table 3 showed that for a wall thickness of t = 5, 6, and 7 mm, the resistance \(M_{eff,s}^{CP}\) was approx. 4.5% higher than \(M_{eff,p}^{CP}\). This was due to different stress distributions and different lengths of the span (\(l_{p}\)) and support (\(l_{s}\)) segment. Despite the fact that \(M_{eff,p}^{CP}<M_{eff,s}^{CP}\), the design ultimate resistance of the beam is determined by the support zone, because \(u=1.351>M_{eff,s}^{CP}/M_{eff,p}^{CP}=1.045\). However, for wall thickness t = 8 mm, the difference between \(M_{eff,s}^{CP}\) and \(M_{eff,p}^{CP}\) was only 0.4%. This resulted from the fact that the cross-section loaded in this way, calculated according to CPM, can be classified as a Class 3 cross-section, for which the value of the reduction factor \(\rho\) is close to one. In contrast, the same cross-section calculated according to EC3 (due to low \(\sigma_{cr,0}\)) is still treated as a Class 4 cross-section (in this case, \(\rho=0.79\)). The percentage increase of \(M_{eff,s}^{CP}\) in relation to \(M_{c,Rd}^{EC3}\) was about 23% for t = 5, 6, and 7 mm, and about 18% for t = 8 mm. The same relationship occurred for the corresponding ultimate loads \(q_{eff}^{min}\). Figure 9 shows the effective cross-sections of the I-300\(\times\)5\(\times\)250\(\times\)5 section determined in the span and support cross-sections according to CPM (Figure 9a,b) and EC3 (Figure 9c,d). For example, the sum of the effective widths \(2b_{eff}\) of the compressed bottom flange of the support cross-section (which determines the resistance of the beam) determined according to CPM was about 39% larger than that determined according to EC3 (see Figure 9b,d). The shift of the neutral axis of the effective cross-section according to CPM in relation to the gross cross-section was e = 12.2 mm and was 51% smaller than e = 24.9 mm calculated according to EC3. This effect additionally affected the differentiation of the effective modulus \(W_{eff}\) of the reliable cross-sections and, consequently, the differences in the design ultimate resistance of the beam determined according to CPM and EC3.