# Buckling Resistance of Tapered Steel Columns
## 4 Comparison of the Results and Discussion
### Normalized Resistance and the Taper Ratio Relationship  
Viewing the relationship between the normalized resistance and taper ratio for various non-dimensional slenderness levels helped to understand the congruence or divergence of the different methods in predicting the buckling resistance of non-uniform members. Specifically, the results for a slenderness of \(\lambda_{0}=0.8\), \(\lambda_{0}=2.0\), and \(\lambda_{0}=5.0\) (representing areas of small, medium, and large slenderness) provided essential insights. Figure 14 and Table 5 show the buckling resistances for small slenderness, similar across all methodologies except for [PERSON] et al. [21], which was excluded from the average and statistical parameter calculations as an outlier. The [PERSON] et al. [8] modified procedure results were considered to avoid artificially reducing the coefficient of variation for the smaller tapering ratios. The most significant deviations were at a tapering ratio \(\gamma=5.0\), with a coefficient of variation of 0.018, suggesting high conformity. Slight deviations were mainly due to increased resistance from the [PERSON] et al. [26] methodology for the tapering ratios over \(\gamma=1.75\). Analyzing the average resistances shown in Table 6, it was clear that the smaller tapering ratios experienced the most resistance increase per tapering ratio unit. For instance, a member with a tapering ratio of \(\gamma=2.0\) saw a 20% resistance increase compared to its uniform counterpart, which gradually diminished as the taper ratio escalated. At a taper ratio of \(\gamma=8.0\), a member's resistance was 37% higher. Comparing this resistance elevation to a corresponding mass increase showed the economic preference for smaller tapering ratios, especially for members with a low slenderness. A member with a \(\gamma=1.25\) taper ratio underwent a resistance increase 2.6 times its mass increase compared to a uniform member. This ratio fell below one as the taper ratio surpassed \(\gamma=3.0\), raising questions about the economic and technical feasibility of further increasing the taper ratio. These findings were corroborated by the average resistance values shown in Table 5, where the resistance increase began to stagnate beyond a \(\gamma=3.0\) tapering ratio, more so beyond \(\gamma=4.0\). A non-uniform member with a tapering ratio of \(\gamma=4.0\) presented 32%  
seen with the methodology proposed by [PERSON] et al. [33] at the tapering ratio of \(\gamma\) = 5.0. The most significant variances were observed at the taper ratio of \(\gamma\) = 1.50. Generally, the average resistance ratio was less than 1.0 across the tapering ratios, indicating that the [PERSON] et al. [26] methodology provided the highest resistance for less slender members. Analyzing Figure 15, which displays the normalized resistances versus the tapering ratios for \(\lambda_{0}\) = 2.0, the resistances according to methodologies by [PERSON] et al. [21] and [PERSON] et al. [8] deviated at higher tapering ratios. Excluding these, the resistances from the other methods were relatively close for all the tapering ratios. Therefore, the resistance from these deviating methodologies was excluded from the statistical analysis (Table 8). There was a more significant dispersion in the resistance results for the members with medium non-dimensional slenderness compared to those with low slenderness. As shown in Table 8, as the tapering ratio increased, the dispersion of the results, measured by the coefficient of variation, decreased, reaching its lowest at \(\gamma\) = 3.0. Compared to a uniform member, the most substantial resistance increase per unit tapering ratio occurred at smaller tapering ratios, paralleling the trend with the members of low slenderness (Table 9). Furthermore, the tapering ratio's impact on the resistance was considerably more significant in the members with a low slenderness. A member with \(\gamma\) = 8.0 offered 286% more resistance than a uniform one but has 81% more mass.