## 5 Algorithm of the CPM Method for Thin-Walled I-Cross-Sections under Bending  
For the transverse bending of I-beams of a Class 4 cross-section (Figure 2), their resistance is usually determined by local buckling of the compression flange (of width \(b_{f}\cong 2b_{s}\) and thickness \(t_{f}\)) or local buckling of the web (of height \(h_{w}\) and thickness \(t_{w}\)) under bending and shear interaction. Because of their geometric topology, those cross-sections were categorised in Reference [10] as part of the group of so-called \"simple\" cross-sections. The ultimate height \(h_{0}\) of the cross-sections at which the compression flange buckling and web in-plane bending with a slight impact of shear stresses (\(\tau\leq 0.25\tau_{cr}\)) occur almost simultaneously can be determined using Equation (5) [10]  
\[h_{0}=\sqrt{\frac{k_{w}^{0}}{k_{f}^{0}}}\left(\frac{t_{w}b_{s}}{t_{f}}\right) \tag{5}\]  
where \(k_{i}^{0}\)--basic buckling coefficient for the hinged supported i-th plate at given load distribution (\(k_{f}^{0}=0.43\) for axial cantilever plate compression, \(k_{w}^{0}=23.9\) for bending in the plane of the web plate). For \(h<h_{0}\), it is the compression flange that determines local buckling of the cross-section, whereas for \(h>h_{0}\), the weakest wall is the web under in-plane bending. For the mean proportion of shear stresses (\(0.25\tau_{cr}\leq\tau\leq 0.5\tau_{cr}\)) Equaiton (5) can be modified to the form:  
\[h_{0}=\sqrt{\frac{k_{w}^{0}}{k_{f}^{0}}}\left(\frac{t_{w}b_{s}}{t_{f}}\right) \left[1-\left(\frac{\tau}{\tau_{cr}}\right)^{2}\right]^{0.25} \tag{6}\]  
where: \(\tau_{cr}\)--critical stress for the web plate in shear. However, if there is a large proportion of transverse forces in the slender webs (plates supported at four edges) that are part of beam cross-sections, shear stresses \(\tau\) may need to be considered more closely, as their contribution to local loss of web stability may be significant. Appropriate formulas for elastic strain energy and work done by external forces allowing critical stresses to be determined (using the energy method) for internal plates loaded (within their plane) with bending with high shear were derived in Reference [11]. For thin-walled I-sections, for which \(h<h_{0}\) according to Equation (6), the CPM assumptions are as follows: (1) the compression flange of the cross-section consists of two critical plates (CPs) which width of \(b_{s}\leq b_{f}/2\) each, which are supported on a web plate of the height \(h_{w}\), (note: dimensions \(b_{s}\) and \(h_{w}\) can be determined based on the rules given in Reference [10]); (2) a single CP acts as a cantilever plate, with one side elastically restrained against rotation; (3) the CP to RP connection (i.e., the web) is rigid (i.e., on the longitudinal edge of their connection, the conditions of continuity of displacements (rotation angles) and forces (bending moments), are met); (4) the transverse edges of the plates (CP and RP) are simply supported on the segment ends; and (5) the thin-walled bar segment (with the length \(l_{s}\)), as in Reference [6], is defined as follows: (a) for constant longitudinal stress distribution, as the distance between the so-called buckling nodal lines, (b) for longitudinal stress variation, as the distance between transverse stiffeners (diaphrags, ribs, or supports) that maintain a rigid cross-section contour, but not longer than the range of the compression zone in the critical plate [26]. The conditions under which Assumption 3 can be adopted were discussed in Reference [10]. The case of I-sections with high and slender webs (at \(h>h_{0}\) acc. to Equation (6)), where the \"local\" critical resistance is determined by the buckling of the bent and shear web, will be discussed in a separate study. The calculation algorithm is as follows:  
1. Division of the cross-section into component plates according to Figure 2;
2.