# Buckling Resistance of Tapered Steel Columns
## 4 Comparison of the Results and Discussion
### Normalized Resistance and the Taper Ratio Relationship
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(\lambda_{0}\) & \(\gamma\) & \(\chi_{0,SV}\) & \(\chi_{0,SV}\) & \(\chi_{0,SV}\) & \(\chi_{0,SV}\) & \(\chi_{0,SV}\) & \(\chi_{0,SV}\) & \(\chi_{0,SV}\) & \(\chi_{0,SV}\) & \(\chi_{0,SV}\) \\ \hline
1 & 2 & 3 & 4 & 5 & 6 & 4/6 & 4/6 \\ \hline  & 1.00 & 0.22 & 0.00 & 142.82 & 0.00 & - \\  & 1.25 & 0.27 & 0.23 & 146.95 & 0.03 & 7.88 \\  & 1.50 & 0.32 & 0.45 & 151.07 & 0.06 & 7.76 \\  & 1.75 & 0.37 & 0.66 & 155.195 & 0.09 & 7.63 \\  & 2.00 & 0.41 & 0.87 & 159.32 & 0.12 & 7.50 \\
0.8 & 2.50 & 0.50 & 1.24 & 167.57 & 0.17 & 7.17 \\  & 3.00 & 0.57 & 1.56 & 175.82 & 0.23 & 6.77 \\  & 4.00 & 0.67 & 2.04 & 192.32 & 0.35 & 5.88 \\  & 5.00 & 0.74 & 2.34 & 208.82 & 0.46 & 5.06 \\  & 6.00 & 0.79 & 2.55 & 225.32 & 0.58 & 4.41 \\  & 7.00 & 0.82 & 2.72 & 241.82 & 0.69 & 3.93 \\  & 8.00 & 0.85 & 2.86 & 258.32 & 0.81 & 3.54 \\ \hline \hline \end{tabular} \(\chi_{0,SV}\)—mean value of the normalized resistance; \(A\)—cross-sectional area at L/2.  
\end{table} 
Table 9: Average normalized resistance \(\chi_{0,SV}\) depending on the tapering ratio \(\gamma\) and the ratio of the resistance increase to the mass increase—\(\lambda_{0}\) = 0.8; HEB 300; S235. The tapering ratio's impact on the resistance was most pronounced for the members with a high slenderness despite a smaller resistance range than those with a medium slenderness (Table 12). A member with the highest tapering ratio (\(\gamma\) = 8.0) displayed a resistance increase of 857% compared to a uniform member. In the members with a high slenderness, both the increase in the resistance per unit of the tapering ratio and the ratio of the resistance increase to the mass increase rose. Notably, a member with a tapering ratio of \(\gamma\) = 8.0 showed a resistance increase 10.59 times greater than its mass increase in relation to a uniform member. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \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]—x\({}_{\text{M}}\) & 0.27 & 0.32 & 0.37 & 0.41 & 0.45 & 0.51 & 0.57 & 0.64 & 0.70 & 0.74 & 0.78 & 0.82 \\ \hline \(\chi_{\text{g}}/\chi_{\text{M}}\) & 0.78 & 0.79 & 0.82 & 0.85 & 0.90 & 0.97 & 1.02 & 1.08 & 1.11 & 1.12 & 1.10 & 1.09 \\ \(\chi_{\text{S}}/\chi_{\text{M}}\) & 0.78 & 0.83 & 0.88 & 0.92 & 0.94 & 0.98 & 1.01 & 1.06 & 1.09 & 1.10 & 1.09 & 1.08 \\ \(\chi_{\text{Se}}/\chi_{\text{M}}\) & 0.78 & 0.81 & 0.84 & 0.87 & 0.90 & 0.94 & 0.98 & 1.02 & 1.04 & 1.05 & 1.03 & 1.02 \\ \(\chi_{\text{L,mod}}/\chi_{\text{M}}\) & 0.78 & 0.76 & 0.78 & 0.81 & 0.85 & 0.94 & 1.01 & 1.07 & 1.07 & 1.06 & 1.06 & 1.06 \\ \hline Mean value & 0.78 & 0.80 & 0.83 & 0.86 & 0.90 & 0.96 & 1.00 & 1.06 & 1.08 & 1.08 & 1.07 & 1.06 \\ \hline Coefficient of variation & 0.003 & 0.037 & 0.052 & 0.052 & 0.042 & 0.021 & 0.019 & 0.027 & 0.029 & 0.031 & 0.030 & 0.028 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Normalized resistances compared to the normalized resistances, according to the methodology conceived by [PERSON] et al. [24]—\(\lambda_{0}\) = 2.0; HEB 300; S235. Figure 16: Normalized resistances (\(\chi_{0}\)) for the varied tapering ratios (\(\gamma\)) and a non-dimensional slenderness of \(\lambda_{0}\) = 5.0—[8,21,22,26,32,33,44]. Compared to [PERSON] et al. 's reference method [26], the other methods predicted lower resistances up to a taper ratio of \(\gamma\) = 3.0 and higher (Table 13). The deviation from the reference increased with the tapering ratio, with the slightest deviations for the methodology by [PERSON] et al. [33].