# Buckling Resistance of Tapered Steel Columns
###### Abstract  
Tapered steel members are widely used in structural and architectural engineering for their efficiency and adaptability, allowing for optimal material usage tailored to specific load levels. However, their complex stability characteristics have hindered their representation in modern design standards like EN 1993 and AISC 360. Existing buckling solutions are limited, and practical research is lacking. This paper comprehensively examines the buckling resistance calculation methodologies for members with variable cross-sections, addressing discrepancies across the methodologies, particularly regarding the tapering ratio. A parametric analysis and numerical simulations were conducted to evaluate each methodology's applicability, emphasizing the need to balance computational simplicity and accuracy. The study primarily focused on tapered steel beams, considering the different tapering ratios and loading conditions. The findings provide valuable insights into the buckling behavior in tapered members and the practical implications for real-world structural designs. By examining the available analytical methods for calculating the buckling resistance of tapered elements, a better understanding of how to accommodate the non-uniformity of a member was gained, enabling an overview of the variance in the determined resistances and an assessment of the method's applicability. Moreover, specific calculation methodologies were found to have shortcomings that require modifications for a more accurate parametric analysis. This research contributes to the field by bridging the gaps in modern design standards and enhancing the understanding of buckling in tapered steel members. v 2023 13 11  
The early work on columns with varying cross-sections is linked to [PERSON] in 1770 and later [PERSON]. Research on the elastic buckling strength of tapered columns intensified in the 20 th century due to [PERSON] [2; 3] and [PERSON] [4], leading to successive approximations for determining critical loads combined with finite difference expressions. This resulted in efficient solutions for determining the column buckling strength. Significant work on steel frames using tapered I-members began with the Column Research Council and Welding Research Council in 1966 [1] as a continuation of the research at Columbia University led by [PERSON] [5; 6]. The experimental data from [PERSON] et al. [7] led to the practical design procedures for tapered columns by [PERSON] et al. [8], using an equivalent uniform member to determine the buckling resistance. This procedure was adopted into the AISC [9] specifications but was removed in 2005 to encourage further development of the guidelines for tapered I-shaped members. A similar approach was studied by [PERSON] et al. [10] based on the concept of a prismatic column with the same length but with an equivalent moment of inertia. Experimental studies of I-shaped tapered members grew in subsequent years, with crucial contributions from [PERSON] et al. [11] and [PERSON] et al. [12; 13], but only for compact members. [PERSON] et al. [14] executed research to evaluate the design provisions made by Star Building Systems and the procedure by [PERSON] et al. [15] concluded that no design provisions accurately predicted the frame resistance for all the loading combinations. The pilot study conducted by [PERSON] et al. [16] evaluated the AISC [17] stability checks for tapered members, proposing several methods for determining the elastic buckling resistance. Recent research based on the finite element method (FEM) has shown that using uniform finite elements to analyze members with variable cross-sections (by dividing the member into smaller areas) can lead to significant errors when torsion influences the behavior, as found in the works of [PERSON] et al. [18; 19] as well as [PERSON] et al. [20]. [PERSON] et al. [21] were among the first to address the lack of design guidelines for tapered members in EN 1993 [22], proposing a procedure based on the existing provisions for uniform columns.