# Buckling Resistance of Tapered Steel Columns
## 3 Parametric Analysis
### Limitations, Varied Parameters, and Methods of Displaying Results  
To make the results of various methods comparable, it was necessary to determine the range of variations of the variables and set certain limitations so that the results would be valid. Accordingly, in the parametric analysis, the following statements were adopted, which were aligned with the notes from [45]. * The change in the height of the web is linear along the length of the member. * There is no change in the thickness of the cross-section parts or the width of the flange along the length of the member. * The analyzed members are homogeneous (the web and flange are of the same steel quality). * If there are class 4 cross-sections within the length of the element, according to [22], their effective area is larger than the area of the smallest cross-section. The limitations were tied to the calculation methods' capabilities, with no method defining the change in the flange's width or how to incorporate that change. As noted in [45], most automation production equipment requires a constant flange width throughout the member's length, making this limitation somewhat technological. The chosen parameters varied to allow comparability between the methods and to examine the parameters that designers can manipulate when using a non-uniform member. Thus, the design-influenced variables affecting the buckling resistance calculation (slanderness based on the smallest cross-section; ratio of smallest to largest cross-section heights) were chosen for the parametric analysis. The tapering ratio change was an essential variable, as suggested in all the methodologies in Section 2. The non-dimensional slenderness based on the smallest cross-section was also varied, as it was the only commonality between the different calculation methodologies. It allowed for the calculation of a specific member length based on the smallest cross-section's characteristics, which was then applicable to all the calculation methods using the equation derived from the Euler's buckling critical force, as shown in Equation (13). \[L=\overline{\lambda}_{0,min}\cdot\pi\cdot i_{0}\cdot\sqrt{\frac{E}{f_{y}}} \tag{13}\]where \(\lambda_{0,min}\) is the non-dimensional slenderness based on the smallest cross-section and \(i_{0}\) is the radius of inertia of the smallest cross-section. The unalterable parameters throughout the analysis included the initial cross-section shape, steel quality, and production type, as their changes would only linearly translate the resistance/non-dimensional slenderness curve along the resistance axis. For the analysis, the smallest cross-section for all the members was chosen as HEB 300, based on the analyses in [26], the steel quality S235 according to [22], and a welding production process (plated structural elements). The non-dimensional slenderness based on the smallest cross-section \(\lambda_{0,min}\) was varied in steps up to a maximum value depending on the tapering ratio \(\gamma\). The tapering ratios ranged from one for a uniform member up to eight, with variation steps increasing progressively. For the tapering ratios from one to 1.75, the slenderness was varied from zero to three, with a step of 0.20. For the tapering ratios from two to eight, the slenderness range increased by one for each increment (e.g., for the tapering ratio \(\gamma=2\), the slenderness range was from zero to four), and the step was kept so that within the range from zero to four it was 0.20, and after that 0.50. The member lengths were determined using Equation (14) based on the varied non-dimensional slenderness, as shown in Table 3.  
The total number of calculations and a summarized view of the varied variables (and their combinations) are shown in Table 4. The variations and individual steps were chosen based on the area of interest and the required accuracy. The tapering ratios were more densely observed from one to three, representing the most likely range in the typical applications, assuming the height of the members was just a few meters.