# Buckling Resistance of Tapered Steel Columns
## 2 Available Stability Design Procedures for Tapered Members
### [PERSON] [44]  
[PERSON] [44] described the energy method for calculating non-uniform compressive elements with a regular change in cross-section. In the appendix of his work, [PERSON] [44] provided a display of a factor for adjusting the critical buckling force, factor \(m\), which depends on the ratio of moments of inertia of the smallest and largest cross-sections of the non-uniform element.  
According to [PERSON], following the proposed procedures, the basic expression for the critical force takes the form of Equation (7).  
\[N_{cr,taper}=\frac{m\cdot E\cdot I_{y,max}}{L^{2}} \tag{7}\]  
where \(m\) is the factor for adjusting the critical buckling force, \(I_{y,max}\) is the moment of inertia of the largest cross-section, and \(L\) is the buckling length of the member around the corresponding axis. After calculating the critical buckling force according to Equation (7), a further procedure for calculating the buckling resistance is performed using the properties of the largest cross-section of the non-uniform member. Figure 6 provides a summarized representation of the calculation process, according to this methodology.  
The values of the factor \(m\) depending on the ratio of moments of inertia, as proposed by [PERSON] [44], are shown in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \(I_{y,min}/I_{y,max}\) & 0.010 & 0.050 & 0.100 & 0.200 & 0.300 & 0.400 & 0.500 & 0.600 & 0.700 & 0.800 & 0.900 & 1.000 \\ \hline \(\alpha_{n}\) & 0.563 & 0.629 & 0.676 & 0.740 & 0.788 & 0.829 & 0.864 & 0.895 & 0.924 & 0.951 & 0.976 & 1.000 \\ \hline \end{tabular}
\end{table}
Table 1: Values of the calibration factor \(\alpha_{n}\)[25].  
The fact that [PERSON] (2003) offered values for the factor \(m\) only to the value of the ratio of the moments of inertia of 0.1 dramatically limits the application of this method. Namely, with the increase in the tapering ratio (\(h_{max}\)/\(h_{min}\)), there is a pronounced increase in the moment of inertia.  
The paper (2003) did not discuss calculating load-bearing capacity after determining the critical buckling force. The methodology (Figure 6) assumes that the highest cross-section determines the reduction factor \(\chi\). The resistance reduction using the calculated reduction factor is then performed based on the smallest cross-section in the non-uniform element. In addition to the fact mentioned in Section 2.4 that the area of the smallest cross-section certainly limits the resistance, this intervention maintains its simplicity. Namely, at higher tapering ratios, the sections of greater heights enter the cross-section class 4. However, their effective area is slightly larger than the area of the smallest cross-section, making resistance gains negligible at the expense of a more complex calculation. In addition, considering that the effective area only extends to the area of slenderness within which it applies, the reduction factor is equal to one or the curve reduction factor/non-dimensional slenderness slightly translates along the axis of non-dimensional slenderness.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Cantilever** & \(\frac{l_{y,min}}{l_{y,max}}\) & 0.000 & 0.100 & 0.200 & 0.300 & 0.400 & 0.500 & 0.600 & 0.700 & 0.800 & 0.900 & 1.000 \\ \cline{2-13}  & \(m\) & - & 1.202 & 1.505 & 1.710 & 1.870 & 2.002 & 2.116 & 2.217 & 2.308 & 2.391 & \(\pi^{2}/4\) \\ \hline
**Pin-** & \(l_{y,min}\)/\(l_{y,max}\) & 0.000 & 0.100 & 0.200 & 0.300 & 0.400 & 0.500 & 0.600 & 0.700 & 0.800 & 0.900 & 1.000 \\ \cline{2-13}
**supported** **element** & \(m\) & - & 4.808 & 6.020 & 6.840 & 7.480 & 8.008 & 8.464 & 8.868 & 9.232 & 9.564 & \(\pi^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Values of factor m (2003).  
Figure 6: The process of integrating the buckling calculation into EN 1993-1-1, according to the methodology proposed by [PERSON] (2003).