# Buckling Resistance of Tapered Steel Columns
## 4 Comparison of the Results and Discussion
### Evaluation of the Applicability of a Specific Methodology  
Assessing the suitability of specific methodologies considered their application potential within the existing regulatory framework (computation sequence per EN 1993-1-1 [22]). This assessment factored in the discrepancies in the computation results compared to the benchmark methodology ([PERSON] et al. [26]). The idea revolved around the balance between introducing new variables required for calculating a member's resistance and maintaining the simplicity and intelligibility of the computational process for engineers. Increased complexity can lead to a higher potential for errors and reduced clarity of the process. Each methodology introduced a unique set of variables for calculating the buckling resistance for non-uniform members, as seen in Section 2's calculation procedures. For example, the method by [PERSON] et al. [8] included just two new variables, the taper ratio \(\gamma\) and the length change factor \(g\), then proceeded according to the standard EN 1993-1-1 methodology. The other methods, like those proposed by [PERSON] et al. [21] and [PERSON] et al. [32], introduced three or four new variables. Along with the [PERSON] et al. [27] method, these were relatively simple, requiring minimal additional computations. Contrarily, the methodologies proposed by [PERSON] et al. [33] and [PERSON] et al. [26] were complex, requiring many new variables and computational steps, including numerical buckling simulations for each case. For instance, the [PERSON] et al. [33] calculation procedure introduced 12 new variables, and after their determination, the conventional computation for a uniform \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\gamma\) & **1.00** & **2.50** & **3.00** & **4.00** & **5.00** & **6.00** & **7.00** & **8.00** \\ \hline [PERSON] et al. [26]—\(\chi_{\text{M}}\) & 0.05 & 0.11 & 0.13 & 0.16 & 0.20 & 0.23 & 0.24 & 0.27 \\ \hline \(\chi_{\text{S}}/\chi_{\text{M}}\) & 0.79 & 0.97 & 1.04 & 1.17 & 1.29 & 1.40 & 1.61 & 1.70 \\ \(\chi_{\text{S}}/\chi_{\text{M}}\) & 0.79 & 0.98 & 1.02 & 1.12 & 1.23 & 1.33 & 1.53 & 1.62 \\ \(\chi_{\text{S}}/\chi_{\text{M}}\) & 0.79 & 0.93 & 0.97 & 1.02 & 1.06 & 1.09 & 1.20 & 1.23 \\ \(\chi_{\text{L,mod}}/\chi_{\text{M}}\) & 0.79 & 0.92 & 1.02 & 1.15 & 1.16 & 1.16 & 1.35 & 1.49 \\ \hline Mean value & 0.79 & 0.95 & 1.01 & 1.11 & 1.18 & 1.24 & 1.42 & 1.51 \\ Coefficient of variation & 0.0004 & 0.0302 & 0.0307 & 0.0571 & 0.0815 & 0.1143 & 0.1277 & 0.1356 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Normalized resistances compared to the normalized resistances, according to the methodology conceived by [PERSON] et al. [24]—\(\lambda_{0}\) = 5.0; HEB 300; S235. member began. [PERSON] et al. [26] introduced nine new variables, one of which needed to be determined by a numerical buckling simulation, increasing the complexity and time required for each case. A summary of the required additional variables for the non-uniform member calculations, compared to the uniform element calculations, is provided in Table 14.  
In order to evaluate the deviation of the results of each methodology from the reference one ([PERSON] et al. [26]), all the possible combinations of variables in the parametric analysis were considered. The ratios of the normalized resistances of each methodology and the reference one for the same non-dimensional slenderness and tapering ratio were examined (Table 15). In the most commonly used taper ratio range, \(\gamma\) = 1.0-\(\gamma\) = 6.0, the most minor deviation from the reference methodology was shown by the methodologies of [PERSON] et al. [8] and [PERSON] et al. [33]. In order to include a specific parameter in the assessment of its applicability, its importance must be assigned in the context of the purpose of the calculation and engineering aspirations.