## 5 Algorithm of the CPM Method for Thin-Walled I-Cross-Sections under Bending
Identifying CP for a \"simple\" cross-section based on the condition \[\sigma_{cr}^{0}=min\{\sigma_{cr,i}^{0}\}\] (7) \[\sigma_{\sigma,i}^{0}=k_{i}^{0}\sigma_{E,i}\] (8) where \(\sigma_{E,i}\)--Euler stress for the i-th plate according to the formula: \(\sigma_{E,i}=190,000\left(t_{i}/b_{i}\right)^{2}\). Note: for \(h<h_{0}\) acc. to Equation (6), the critical plate (CP) is the cantilever wall of the compression flange and the rigid plate (RP) is the web;
3. Adoption of the initial CP edge restraint coefficient value (for the so-called zero step), e.g., \(\kappa_{0}=0.4\);
4. Estimation of the critical length (\(l_{cr}\)) for a single half-wave of CP buckling [26] according to the formula \[l_{cr}=b_{s}\Big{(}\frac{2.02-0.37\kappa}{\kappa^{0.25}}\Big{)}\] (9)
5. Determination of the coefficient \(\eta\)[31], depending on the static scheme and the way of forcing (loading) RP by buckling CP (Figure 2) for the critical length according to Equation (9) [10] \[\eta=\sqrt{33.4+50.7(b_{r}/l_{cr})^{2}}-2.78\] (10) where \(b_{r}\)--RP width;
6. Determination of \(\sigma_{cr}^{L}\) according to the formula \[\sigma_{cr}^{L}=k^{*}\sigma_{E,s}\] (11) where \(k^{*}\)--buckling coefficient according to a more accurate calculation model. The \(k^{*}\) coefficient can be determined on the basis of Reference [26] from the following formulas: \[\bullet\] for linear stress distribution (Figure 3a), \[ \begin{array}{c}k^{\star}(\kappa,m,\gamma)=k_{\infty}(\kappa)+\left[0.765m-0.31m^{2}+0.227m^{3}+\right.\\ \left.\left(3.201m-0.307m^{2}-3.724m^{3}+2.842m^{4}\right)\kappa^{2}+\right.\\ \left.\left(-3.887m-9.205m^{2}+21.528m^{3}-12.306m^{4}\right)\kappa^{3}+ \right.\\ \left.\left(-1.132m+22.933m^{2}-37.767m^{3}+19.091m^{4}\right)\kappa^{4}+ \right.\\ \left.\left(2.559m-14.341m^{2}+20.975m^{3}-10.023m^{4}\right)\kappa^{5}\right] /\gamma_{s}^{(0.59+0.16m)}\end{array} \] (12)  
Figure 2: Division of I-cross-section into critical plate (CP) and restraining plate (RP) for \(h<h_{0}\). for non-linear stress distribution (according to \(2^{\circ}\) parabola, Figure 3b), \[ \begin{split} k^{\prime}(\kappa,m,\gamma)=k_{\infty}(\kappa)+\left[1.096m-0.808m^{2}+0.363m^{3}+\right.\\ \left.\left(20.064m-63.78m^{2}+74.842m^{3}-30.565m^{4}\right)\kappa^{2 }+\right.\\ \left.\left(-58.826m+195.88m^{2}-232.857m^{3}+95.684m^{4}\right) \kappa^{3}+\right.\\ \left.\left(61.802m-211.851m^{2}+254.031m^{3}-104.82m^{4}\right) \kappa^{4}+\right.\\ \left.\left(-22.086m+77.761m^{2}-93.977m^{3}+38.925m^{4}\right) \kappa^{5}\right]/\gamma_{s}^{(1,06+0.04m)}\end{split} \] (13) where \(m=1-\sigma_{1}/\sigma_{0}\)--longitudinal stress distribution coefficient (Figure 3), \(\gamma_{s}=l_{s}/b_{s}\). The plate buckling coefficient \(k_{\infty}(\kappa)\) of an elasticly restrained and infinitely long cantilever plate can be determined according to Reference [26], using the formula \[ \begin{split} k_{\infty}(\kappa)=0.425+2.893\kappa-19.433\kappa^ {2}+83.849\kappa^{3}-195.943\kappa^{4}+250.971\kappa^{5}\\ -165.321\kappa^{6}+43.833\kappa^{7}\end{split} \] (14) On the other hand, Reference [28] presents a simplified formula for \(k_{\infty}(\kappa)\) within the range \(0.05\leq\kappa\leq 1\) in the form of \[k_{\infty}(\kappa)=0.49+0.974\kappa-0.822\kappa^{2}+0.632\kappa^{3}\] (15)
7. Estimation of the critical stress \(\sigma_{\sigma,r}\) for a RP bent in its plane (width \(b_{r}\)) [32] for one half-wave of CP buckling length \(l_{cr}\) according to Point 4. \[\sigma_{\sigma,r}=\frac{Et_{r}^{2}}{l_{cr}^{2}b_{r}^{4}}\big{(}11.32l_{cr}^{ 4}+1.97b_{r}^{4}+12.06l_{cr}^{2}b_{r}^{2}\big{)}\] (16)
8. Determination of rotational spring stiffness \(C_{\theta}\) according to Equation (17) and the restraint coefficient \(\kappa_{i+1}\) according to Equation (3) for the first (\(i=1\)) and subsequent iteration steps.