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The geometry of structural equilibrium.

Allan McRobie1

  • 1Cambridge University Engineering Department , Trumpington Street, Cambridge CB2 1PZ, UK.

Royal Society Open Science
|April 14, 2017
PubMed
Summary

This study introduces a geometric method for analyzing stress in 3D frames, generalizing Rankine

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Area of Science:

  • Structural mechanics
  • Geometric analysis
  • Computational engineering

Background:

  • Traditional methods for structural analysis, like Rankine's reciprocal diagrams, have limitations in three-dimensional applications.
  • A need exists for a comprehensive geometric approach to understand stress resultants in complex 3D frames.
  • Historical context from Maxwell, Rankine, and Klein highlights the evolution of structural equilibrium descriptions.

Purpose of the Study:

  • To present a novel geometric description for structural equilibrium in general three-dimensional frames.
  • To develop a graphic analysis procedure for determining all six stress resultants (axial forces, shear forces, torsion, and bending moments).
  • To extend and resolve limitations in existing methods, particularly Rankine's approach for 3D trusses.

Main Methods:

  • Embedding dual abstract 4-polytopes within dual four-dimensional vector spaces.
  • Utilizing the oriented area of generalized polygons to represent stress resultants.
  • Applying four-dimensional Clifford algebra for calculation of stress resultants.

Main Results:

  • A complete geometric procedure for the graphic analysis of stress resultants in 3D frames.
  • Identification of all six components of stress resultants through geometric properties in 4D space.
  • Demonstration of the method's applicability to engineering structures, including gridshell roofs.

Conclusions:

  • The proposed geometric description provides a powerful and comprehensive tool for 3D frame analysis and design.
  • This method overcomes the incompleteness issues found in previous descriptions of 3D structural equilibrium.
  • The approach offers a natural generalization of reciprocal diagrams and facilitates practical structural engineering applications.
Keywords:
Clifford algebraMaxwell reciprocal diagramsRankine reciprocal diagramsgraphic staticsthree-dimensional frames

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