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Purpose <p>To derive simple, approximate analytical formulas for critical buckling loads of uniform columns under combined end and linearly distributed axial compression for common boundary conditions.</p> Methods <p>The end and distributed compression are perturbed in different orders, followed by time and space coordinate transformations, which leads to the most concise form of dimensionless equation. Buckling loads were determined from the condition where the fundamental frequency approaches zero.</p> Results <p>The universal pre-solution data (<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42417_2025_1866_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(\omega _i-l-g\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ω</mi> <mi>i</mi> </msub> <mo>-</mo> <mi>l</mi> <mo>-</mo> <mi>g</mi> </mrow> </math></EquationSource> </InlineEquation>) were obtained, which avert repeatedly solving frequency transcendental equations for each beam with different physical parameters. Simple, approximate analytical buckling load equations were derived for the four boundary conditions, expressing the critical load as the classical Euler load minus a linear correction term for the distributed load. Good agreement was found with FEM and literature results.</p> Conclusion <p>Simple, physically meaningful, and practical approximate formulas for buckling loads under combined compression were successfully derived. They extend Euler’s formulas, offer satisfactory accuracy for design, and simplify the calculation compared to existing complex solutions.</p>

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Approximate Analytical Frequencies and Buckling Loads Equations for Columns Under Both End and Distributed Compression

  • 期刊论文

  • 发表日期：

    2025年5月15日

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  • Ceshi Sun,
  • Chuan Zeng,
  • Junqiang Lin,
  • Gang Zheng,
  • Wenxiu Cai

摘要

Purpose

To derive simple, approximate analytical formulas for critical buckling loads of uniform columns under combined end and linearly distributed axial compression for common boundary conditions.

Methods

The end and distributed compression are perturbed in different orders, followed by time and space coordinate transformations, which leads to the most concise form of dimensionless equation. Buckling loads were determined from the condition where the fundamental frequency approaches zero.

Results

The universal pre-solution data ( \(\omega _i-l-g\) ω i - l - g ) were obtained, which avert repeatedly solving frequency transcendental equations for each beam with different physical parameters. Simple, approximate analytical buckling load equations were derived for the four boundary conditions, expressing the critical load as the classical Euler load minus a linear correction term for the distributed load. Good agreement was found with FEM and literature results.

Conclusion

Simple, physically meaningful, and practical approximate formulas for buckling loads under combined compression were successfully derived. They extend Euler’s formulas, offer satisfactory accuracy for design, and simplify the calculation compared to existing complex solutions.