# Buckling Resistance of Tapered Steel Columns
## 4 Comparison of the Results and Discussion
### Normalized Resistance and the Taper Ratio Relationship
Economically, the best resistance increase relative to the mass increase ratio was found in the members with a taper ratio of \(\gamma\) = 1.25. It was intriguing that for all the tapering ratios, the rise in the resistance substantially exceeded the mass increase relative to a uniform member, a phenomenon not witnessed with members of a low slenderness. However, this pattern aligned with the \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \(\gamma\) & **1.00** & **1.25** & **1.5** & **1.75** & **2.00** & **2.50** & **3.00** & **4.00** & **5.00** & **6.00** & **7.00** & **8.00** \\ \hline [PERSON] et al. [26]\(\rightarrow\)xm & 0.72 & 0.78 & 0.82 & 0.85 & 0.88 & 0.92 & 0.94 & 0.97 & 1.00 & 1.00 & 1.00 & 1.00 \\ \(\chi_{\mathrm{S}}/\chi_{\mathrm{M}}\) & 1.01 & 1.00 & 0.99 & 0.99 & 0.98 & 0.98 & 0.98 & 0.97 & 0.98 & 0.99 & 1.00 \\ \(\chi_{\mathrm{S}}/\chi_{\mathrm{M}}\) & 1.01 & 1.01 & 1.01 & 1.00 & 0.99 & 0.98 & 0.98 & 0.97 & 0.97 & 0.98 & 0.99 & 1.00 \\ \(\chi_{\mathrm{Se}}/\chi_{\mathrm{M}}\) & 1.01 & 1.01 & 1.00 & 0.99 & 0.98 & 0.97 & 0.97 & 0.96 & 0.97 & 0.98 & 0.98 \\ \(\chi_{\mathrm{L,mod}}/\chi_{\mathrm{M}}\) & 1.01 & 0.99 & 0.98 & 0.98 & 0.97 & 0.98 & 0.98 & 0.97 & 0.96 & 0.97 & 0.98 & 0.99 \\ \hline Mean value & 1.01 & 1.00 & 0.99 & 0.99 & 0.99 & 0.98 & 0.98 & 0.97 & 0.96 & 0.98 & 0.99 & 0.99 \\ Coefficient of variation & 0.001 & 0.011 & 0.013 & 0.011 & 0.008 & 0.004 & 0.003 & 0.005 & 0.006 & 0.007 & 0.007 & 0.007 \\ \hline \end{tabular}
\end{table} 
Table 7: Normalized resistances compared to the normalized resistances, according to the methodology conceived by [PERSON] et al. [24]—\(\lambda_{0}\) = 0.8; HEB 300; $235.$more significant influence of the tapering ratio on the buckling resistance identified for the members with a low slenderness. Table 10 shows the ratios of the calculated normalized buckling resistances using various methodologies and those according to the methodology by [PERSON] et al. [26] for the varied taper ratios and a slenderness of \(\lambda_{0}\) = 2.0. It was evident that all the methodologies predicted a lower resistance for the members up to a tapering ratio of \(\gamma\) = 3.0, while for the ratios above \(\gamma\) = 3.0, the reference methodology anticipated higher resistances. The resistance ratios were uniform across all the tapering ratios, as indicated by the maximum coefficient of variation of 0.085 for a member with a taper ratio of \(\gamma\) = 1.50. The obtained normalized resistances for the members of a high slenderness, with \(\lambda_{0}\) = 5.0, across the varied tapering ratios are presented in Figure 16. The results deviated significantly at larger taper ratios, particularly for the method proposed by [PERSON] et al. [8]. Thus, the modified results, denoted as [PERSON] et al.--mod, were used for the comparison and statistical calculations. The method proposed by [PERSON] et al. [26] gave the smallest buckling resistance values, and as the tapering ratio increased, the results became more dispersed. With the increase in the tapering ratio, both the buckling resistance and the coefficient of variation increased (Table 11), with the latter being the smallest for the tapering ratio of \(\gamma\) = 3.0 (0.027) and the largest for \(\gamma\) = 8.0 (0.206). Discrepancies were most prominent for the method by [PERSON] et al. [26], suggesting potential calibration issues with high tapering ratios and slenderness.