## 7 Example of Using CPM to Determine the Resistance of a Continuous I-Beam
### I-Section Consisting of Two Cold-Formed Channel Sections (2C)  
For compound sections (type 2C), detailed calculations were made for five-span beams with a constant span length for variants of \(L=4\), 5, 6, and 7 m made of S355-grade steel. For comparison purposes, the analyzed dimensions of 2C sections (Figure 10) with the contour of Sk250\(\times\)250\(\times\)t box sections (for t = 2, 3, 4, and 5 mm) presented in Reference [6] were used. In this case, the elastic gross section bending moduli \(W_{el}\) for Sk and 2C sections were equal. Table 4 presents the results of the analysis of the effect of the span length on the \"local\" critical moment value \(M_{cr}^{L}\) and the resistance \(M_{eff}\) calculated according to CPM and EC3 [1,3] for the 2C-250\(\times\)250\(\times\)4 (2C-h\(\times\)b\(\times\)t) section. For the support segment decisive of the beam resistance, the difference between the critical load value \(q_{cr}\) obtained according to CPM method in relation to EC3 ranged from + 158% for \(L=4\)\(m\) up to + 141% for \(L=7\)\(m\). The difference between the load ultimate value \(q_{eff}\) ranged from + 22% for \(L=4\)\(m\) up to +20% for \(L=7\)\(m\). Figure 10: Compared cross-sections of continuous beams: (**a**) box cross-section according to Reference [6], (**b**) compound cross-section type 2C. Figure 9: Effective cross-sections of the I-300\(\times\)5\(\times\)250\(\times\)5 section according to CPM and EC3 methods. (**a**) span cross-section acc. CPM, (**b**) support cross-section acc. CPM, (**c**) span cross-section acc. EC3, (**d**) support cross-section acc. EC3. Note: for comparison, Table 4 also includes the values of \(q_{cr}\) and \(q_{eff}\) determined according to CPM for the case where the resistance of the beam is decided by the span section. This would be the case, for example, in protection against local buckling of the support segment. Table 5 compares the \"local\" critical resistances (\(M_{cr}^{L}\), \(q_{cr}\)) of beams with box Sk250\(\times\)250\(\times\)4 sections according to Reference [6] and 2C-250\(\times\)250\(\times\)4 compound I-sections depending on the span length of the continuous beam (\(L\) = 4, 5, 6, and 7 m). Comparison of the values presented in Table 5 shows that for 2C sections, where critical resistance is determined by cantilever walls, \"local\" critical moments \(M_{cr}^{L}\) determined according to CPM were lower on average by about 25-26% in relation to corresponding box sections [6], and the reduction \(M_{cr}^{EC3}\) was as much as 54%. The same relationship occurred when comparing the respective critical loads \(q_{cr}\). Meanwhile, the relation \(q_{cr}^{CP}/q_{cr}^{EC3}\) for 2C sections in relation to Sk sections increased by as much as 62%. This effect resulted from a much larger reserve of the critical resistance of the elastically restrained cantilever plate (which determines the resistance of the 2C section) in relation to the internal plate of the compression flange, which in turn determines the resistance of the box section. In fact, this is the difference between the critical stresses for the plates (cantilever and internal) that are simply supported in relation to the same plates that are elastically restrained at one or two edges, respectively. For comparison, the relation between the buckling coefficient for the compressed cantilever plate when fully restrained (\(k_{u}=1.25\)[7; 19]) and simply supported (\(k_{p}=0.43\)[3]) was \(k_{u}/k_{p}=1.25/0.43=2.91\). For an internal plate, on the other hand, the same relation was much smaller and was \(k_{u}/k_{p}\)= 6.97/4