## 7 Example of Using CPM to Determine the Resistance of a Continuous I-Beam
### Welded Section  
#### 7.2.1 Calculation of Critical Resistance According to CPM  
Detailed calculations were carried out for a five-span continuous beam with a span length \(L=4\)\(m\) and a thin-walled I-section welded from sheets of dimensions I-300x\(\mathrm{t}_{w}\)\(\times\)250x\(\mathrm{t}_{f}\) (I-hx\(\mathrm{t}_{w}\)\(\times\)bx\(\mathrm{t}_{f}\)) for t = \(\mathrm{t}_{f}=\mathrm{t}_{w}=\) 5, 6, 7, and 8 mm made of S355-grade steel. It should be noted that in this case, adopting a smaller web thickness, i.e., \(\mathrm{t}_{w}<\mathrm{t}_{f}\), led to a significant reduction in the elastic restraint of the compressed flange in the plate of the web. This resulted in a much greater reduction in the cross-sectional resistance than that resulting from a reduction in the elastic section modulus caused solely by a reduction in the thickness of \(\mathrm{t}_{w}\).  
Table 1 shows the results of the calculation of \"local\" critical moments for the ruling cross-sections: span \(M_{cr,p}^{L}\), support \(M_{cr,s}^{L}\), and the critical loads \(q_{cr}\) determined from them. The symbol \(q_{cr,s}^{CP}\) indicates  
Figure 5: Calculation model of the first (end) span of a uniformly loaded continuous beam.  
the critical load value determined from the condition of reaching \(M_{cr,s}^{L}\) in the support cross-section, and the symbol \(q_{cr,p}^{CP}\) indicates that determined from the condition of reaching \(M_{cr,p}^{L}\) in the span cross-section. (Note: in Table 1, the values of the moments \(M_{cr,s}^{L}\) and \(M_{cr,p}^{L}\) are given as absolute values). The critical resistance of the beam is determined by the minimum load \(q_{cr,min}^{CP}=min\left\{q_{cr,p}^{CP},q_{cr,s}^{CP}\right\}\).  
Furthermore, in Table 1, the symbol \(M_{cr}^{\text{EC3}}\) indicates the critical cross-sectional moment determined according to \(\sigma_{cr}\) calculated according to EC3 [3], and the symbol \(q_{cr,min}^{\text{EC3}}\) stands for the minimum value of critical load determined on the basis of \(M_{cr}^{\text{EC3}}\). Due to the simplified calculation model used in the standard, \(M_{cr}^{\text{EC3}}\) does not depend on the longitudinal stress distribution or on the degree of mutual elastic restraint of the cross-section component walls. It therefore determines the critical resistance of both the span cross-section (\(p\)) and the support cross-section (\(s\)).  
Note: in the example under consideration, determined on the basis of the \(q_{cr,min}^{CP}\) parameter \(h_{0}\) according to Equation (6), \(h_{0}=880\) mm and was almost three times greater than the cross-section height \(h=300\) mm. Of course, the determination of the parameter \(h_{0}\) according to Equation (6) was iterative because the shear stresses \(\tau\) and \(\tau_{cr}\) depend on the cross-section height, among other things. This check calculation allows the correct identification of the critical plate in a given cross-section loading condition to be confirmed.  
A comparison of the results in Table 1 shows that \(M_{cr,s}^{L}\) was approx. 18.7% higher than \(M_{cr,p}^{L}\). This was due to different stress distributions and different reliable lengths of the span (\(l_{p}\)) and support (\(l_{s}\)) segments. Despite the fact that \(M_{cr,p}^{L}<M_{cr,s}^{L}\), the critical resistance of the beam was determined by the support zone, because: \(u=1.351>M_{cr,s}^{L}/M_{cr,p}^{L}=1.187\). The percentage increase of \(M_{cr,s}^{L}\) in relation to \(M_{cr}^{\text{EC3}}\) was about 121%. The same relationship occurred for the corresponding critical loads \(q_{cr}^{min}\).