## 7 Example of Using CPM to Determine the Resistance of a Continuous I-Beam
### Welded Section
#### 7.2.2 Finite Element and Finite Strip Analysis  
The critical stresses listed in Table 1 (\(\sigma_{cr,s}\), \(\sigma_{cr,p}\)) were verified via the finite element method (FEM) using ABAQUS software [35], and the finite strip method (FSM) using CUFSM software [18]. The FEM enables the analysis of local buckling of the extreme span of a continuous beam by taking into account both the effect of mutual elastic restraint of cross-section's components plates, as well as the effect of longitudinal stress variation. However, the commonly used finite strip method (FSM) allows the effect of mutual elastic restraint of cross-section's component plates to be considered, but with the assumption of a constant stress distribution over the length of the member. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Wall Thickness (mm)** & **t=** & **8** & **7** & **6** & **5** \\ \hline Senderness & \(\lambda\)=b/t= & 15.63 & 17.86 & 20.83 & 25.00 \\ Euler stress (N/mm\({}^{2}\)) & Flange \(\sigma_{\text{E,i}}\)= & 778.24 & 595.84 & 437.76 & 304.00 \\  & Web \(\sigma_{\text{E,i}}\)= & 135.11 & 103.44 & 76.00 & 52.78 \\ Buckling coefficient & for \(l_{p}\): k\({}^{\ast}\)= & 0.800 & 0.800 & 0.800 & 0.800 \\  & for \(l_{s}\): k\({}^{\ast}\)= & 0.950 & 0.950 & 0.950 & 0.950 \\ Critical stress acc. CPM (N/mm\({}^{2}\)) & for \(l_{p}\): \(\sigma_{\text{cr,p}}\)= & 622.60 & 476.67 & 350.21 & 243.20 \\  & for \(l_{s}\): \(\sigma_{\text{cr,s}}\)= & 739.08 & 565.86 & 415.73 & 288.70 \\ Critical stress acc. EC3 (N/mm\({}^{2}\)) & \(\sigma_{\text{cr,p}}\)= & 334.64 & 256.21 & 188.24 & 130.72 \\  & for \(l_{p}\): \(M_{cr,p}^{L}\)= & 448.36 & 300.35 & 189.13 & 109.45 \\ Local critical resistance (kNm) & for \(l_{s}\): \(M_{cr,s}^{L}\)= & 532.24 & 356.54 & 224.52 & 129.93 \\  & \(M_{cr}^{\text{EC3}}\)= & 240.99 & 161.44 & 101.66 & 58.83 \\ Critical load acc. CPM (kN/m) & for \(l_{p}\): \(q_{cr,s}^{CP}\)= & 359.72 & 240.97 & 151.74 & 87.81 \\  & for \(l_{s}\): \(q_{cr,s}^{CP}\)= & 316.02 & 211.70 & 133.31 & 77.14 \\ Critical load acc. EC3 (kN/m) & \(q_{cr}^{\text{EC3}}\)= & 143.09 & 95.85 & 60.36 & 34.93 \\ Percentage increase of resistance (\%) & \(q_{cr}^{CP}/q_{cr}^{\text{EC3}}\)= & 120.86 & 120.86 & 120.86 & 120.86 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of critical results according to critical plate method (CPM) and European Eurocode 3 (EC3) for the I-300x\(\times\)x250 xt section and four wall thickness variants. For the numerical simulations in ABAQUS v. 6.12, the first (extreme) span of a continuous beam was modeled. Due to the small wall thickness (compared to other dimensions), the cross-section's geometry was simplified to so-called centerline. S4R shell elements (four nodes with six degrees of freedom in the node) were used, and the dimensions of the finite element were assumed to be \(12.5\times 12.5\) mm. Such division made it possible to obtain technically sufficient accuracy while reducing calculation time. The transverse load was applied in the form of uniform pressure (p = 0.1 N/mm\({}^{2}\)) on the upper flange with a width of \(b\) = 250 mm, which corresponded to a uniform load of q = 25 kN/m. Conversely, the loading with the \"corresponding\" support moment of the value of M\({}_{\text{s}}\) = 0.105 \(\times\) qL\({}^{2}\) = 42 kNm (where 0.105 is the coefficient from [PERSON]'s tables) was performed by means of loading the support cross-section with a stress block. The boundary conditions in the support cross-sections were assumed to be continuous constraints perpendicular to each of the component plate. An additional tie securing the beam against displacement along the longitudinal axis was applied in the gravity axis of one of the support sections. In Figure 6b, a general view of FEM model of the I-300\(\times\)5\(\times\)250\(\times\)5 cross-section in ABAQUS software is presented. Figure 6 shows the support ties used, and Figure 6c shows the method of loading with the support moment M\({}_{\text{s}}\). The critical load calculations were performed using the \"buckling\" procedure.