# Buckling Resistance of Tapered Steel Columns
## 4 Comparison of the Results and Discussion
### Relation of Normalized Resistance and Reduced Slenderness  
Figure 11a presents the buckling resistance calculations of a uniform member using various methods. Except for the [PERSON] et al. [25] method, all matched the buckling curve \(b\), suggesting a correct application. The differences with [PERSON] et al. [26] might have been due to unique imperfection factors and potential non-calibration for uniform members. The results indicated that EN 1993-1-1 [22] and the other methods yielded conservative results for a slenderness greater than \(\lambda_{0}\) = 0.8 and uniform members. As the tapering ratio mildly increased from \(\gamma\) = 1.0 to \(\gamma\) = 1.25, the buckling resistance results from the various methodologies converged closer to the ones proposed by [PERSON] et al. [26] (Figure 11b). Except for [PERSON] et al. [19], all the methodologies closely matched up to the non-dimensional slenderness of \(\lambda_{0}\) = 1.0, beyond which slight discrepancies appeared. The resistance increases were noted compared to a uniform member, particularly in the mid-slenderness area. These trends were accentuated with a further rise in the tapering ratio to \(\gamma\) = 1.50 (Figure 12a). Compared to the numerical models from ANSYS, it was noticeable that the methods of [PERSON] et al. [8], [PERSON] et al. [32], and [PERSON] et al. [33] gave lower values for the reduction factor, as well as the methods according to [PERSON] [44] and [PERSON] et al. [26] for the slenderness ratios less than 1.0, while the method according to [PERSON] et al. [21] gave higher reduction factors for the slenderness of 1.0 and less. For the tapering ratios of 1.5 and 2.0, a similar trend was observed. The methods of [PERSON] et al. gave lower buckling resistance values compared to the numerical models [8], [PERSON] et al. [32], and [PERSON] et al. [33], as well as the methods according to [PERSON] [44] and [PERSON] et al. [26] for the slenderness up to 1.2 for a tapering ratio of 1.5 and the slenderness of 1.6 in the  
Figure 11: The relationship between the normalized resistance (\(\chi_{0}\)) and non-dimensional slenderness of the smallest cross-section (\(\lambda_{0}\)) for the various calculation methods [8; 21; 22; 26; 32; 33; 44]—tapering ratio (**a**) \(\gamma\) = 1.0 and (**b**) 1.25.  
case of a tapering ratio of 2. According to [PERSON] et al. [20], the results overestimated the buckling resistance up to a slenderness of 1.2 for a tapering ratio of 1.5 and 1.4 for a tapering ratio of 2.0. With a tapering ratio of \(\gamma=4.0\) (Figure 12(a)), there was a slight redistribution of the obtained resistances, according to the various calculation methods. [PERSON] et al. [21] stood out with divergent results up to \(\lambda_{0}=2.8\). Excluding this methodology, the results were densely grouped. Compared to lower tapering ratios, the resistances by [PERSON] et al. [26] were the lowest across the most varied non-dimensional slenderness ranges (except up to \(\lambda_{0}=1.2\)). The calculations of [PERSON] et al. [33] were almost identical, as were [PERSON] [44], [PERSON] et al. [32], and [PERSON] et al. [8]. The most significant deviation of the obtained resistances from a uniform element of the same length was in the mid-non-dimensional slenderness range. For a tapering ratio of \(\gamma=6.0\), the previous trends for lower ratios persisted with a slightly higher calculation result dispersion between the methodologies (Figure 13(b)). Apart from the [PERSON] et al. [21] known divergence, the methodology proposed by [PERSON] et al. [26] stood out with the lowest resistances. The results according to [PERSON] et al. [33] and [PERSON] et al. [8] were relatively close, and for a higher non-dim  
Figure 12: The relationship between the normalized resistance (\(\chi_{0}\)) and non-dimensional slenderness of the smallest cross-section (\(\lambda_{0}\)) for the various calculation methods [8,21,22,26,32,33,44]—tapering ratio (**a**) \(\gamma=1.50\) and (**b**) 2.0.