## 2 Calculation of Distortional Buckling Loads  
### Numerical Method  
In this work, the distortional buckling loads according to the numerical method in EN 1993-1-3, are calculated using the semi-analytical form of Generalized Beam Theory [18], as implemented in the GBTUL program [11], which is freely available from [[https://sites.fct.unl.pt/gbt/](https://sites.fct.unl.pt/gbt/)]([https://sites.fct.unl.pt/gbt/](https://sites.fct.unl.pt/gbt/)). This constitutes an accurate and computationally efficient method to (i) retrieve the so-called \"signature curve\" pioneered by [PERSON] [7]--the buckling load as a function of the buckling mode half-wavelength, as required by EN 1993-1-3--and (ii) identify the nature of the buckling mode at the relevant half-wavelengths (local, distortional or global). It is worth noting that, even though GBT-based analytical formulas for distortional buckling have been proposed [19; 20; 21], they are not used in this work, due to their inherent simplifications.  
For many cross-section geometries, the minimum distortional buckling load corresponds to a local minimum of the signature curve, but this is not always the case. For this reason, the determination of the minimum distortional buckling load \(\lambda_{d}\) for either axial force (\(N_{d}\)), major axis bending (\(M_{y,d}\)) or minor axis bending (\(M_{z,d}\) causing compression in the lips) was performed in the following steps (see Figure 1):  
1. Definition of the cross-section geometry, material parameters and GBT cross-section discretization, which allows for calculating the so-called \"GBT cross-section deformation modes\" (global, distortional and local-plate). To ensure obtaining accurate results, the cross-section discretization adopted involves five intermediate nodes in the web and three intermediate nodes in each flange.  
2. Calculation of the signature curve for the loading considered using (i) all deformation modes (the true signature curve) and (ii) only the distortional modes \(5+6\) (the pure distortional curve). The pure distortional curve is used to obtain \(L_{d}\), the half-wavelength corresponding to the minimum of the curve, which serves as an initial estimate for \(L_{min}\), the true distortional minimum (which generally has contributions from the local-plate modes). 3. The minimum distortional buckling load \(\lambda_{d}\) is searched on the true signature curve, within a reasonable interval centered on \(L_{d}\). If the minimum does not correspond to a stationary point, which can occur for certain geometries, the value corresponding to \(L_{d}\) is adopted. The procedure is executed in batch processing, using a MATLAB [22] script.  
1. Cross-section geometry, material parameters, GBT discretization and deformation modes  
2. Signature curves for the loading (\(N\), \(M_{y}\) or \(M_{z}\)) using (i) all modes and (ii) only modes \(5+6\)  
The GBT semi-analytical solution for distortional buckling in a single deformation mode allows obtaining directly the minimum distortional buckling load and the corresponding half-wavelength, through [8]  
\[\lambda_{d} =\frac{D_{d}+2\sqrt{C_{d}B_{d}}}{X_{d}}, \tag{1}\] \[L_{min} =\pi\sqrt[4]{\frac{C_{d}}{B_{d}}}, \tag{2}\]  
where \(\lambda_{d}\) is the minimum distortional buckling load, \(B_{d}\), \(C_{d}\), \(D_{d}\) and \(X_{d}\) are cross-section stiffness properties associated with distortion--\(B_{d}\) relates to plate-like transverse wall bending, \(C_{d}\) to warping (membrane and bending), \(D_{d}\) to plate-like torsion and Poisson  
Figure 1: Procedure adopted for the numerical calculation of the distortional buckling loading using GBTUL.  
couplings, and \(X_{d}\) to the geometric non-linear effects associated with the loading considered. These cross-section stiffness properties can be obtained from the GBTUL output.