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An asymptotic higher-order theory for rectangular beams.

E Nolde1, A V Pichugin1, J Kaplunov2

  • 1Department of Mathematics, CEDPS, Brunel University London, Uxbridge UB8 3PH, UK.

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Summary
This summary is machine-generated.

A new consistent governing equation for rectangular beams is derived using asymptotic integration. This new equation accurately predicts behavior across various aspect ratios, validating its consistency with 3D elasticity and offering insights into shear correction factors.

Keywords:
Timoshenko beam theoryasymptoticsbeam theorylong waveslow frequencyshear correction factor

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Area of Science:

  • Solid Mechanics
  • Continuum Mechanics
  • Structural Engineering

Background:

  • Classical beam theories often lack accuracy for specific geometries and boundary conditions.
  • Three-dimensional elasticity provides a more complete description but is computationally intensive.
  • Need for consistent lower-dimensional models that capture essential physics.

Purpose of the Study:

  • Derive a consistent governing equation for rectangular beams using direct asymptotic integration.
  • Ensure the new equation replicates the long-wave, low-frequency behavior of 3D elasticity.
  • Validate the equation's performance and its implications for classical theories and shear correction factors.

Main Methods:

  • Direct asymptotic integration of the full 3D elasticity problem.
  • Derivation of a consistent governing equation for rectangular cross-section beams.
  • Comparison of the new equation's predictions with finite-element computations.

Main Results:

  • A novel, consistent governing equation for rectangular beams was successfully derived.
  • The equation accurately reproduces the behavior of 3D elasticity for long-wave, low-frequency scenarios.
  • Recovered limiting behaviors for beams with large/small aspect ratios, confirming consistency.

Conclusions:

  • The derived beam equation offers a consistent and accurate model for rectangular beams.
  • Highlights the importance of refined boundary conditions in classical plate theories.
  • Provides guidance on selecting appropriate shear correction factors in Timoshenko's beam theory.