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In this chapter, we will explore an alternative approach to solving boundary value problems in elasticity: energy methods, specifically variational methods. The power of the principles of minimum potential energy and complementary energy is effectively demonstrated through illustrative examples of beam bending and Saint-Venant torsion, showcasing a self-consistent approach to simplifying complex three-dimensional elasticity problems into more manageable two-dimensional or even one-dimensional models. Their direct applications to numerical solutions are illustrated via the Rayleigh-Ritz method.

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Variational Methods for Infinitesimal Deformation Problems

  • 章节

  • 发表日期：

    2025年

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  • Junqian Zhang,
  • Yicheng Song,
  • Bo Lu

摘要

In this chapter, we will explore an alternative approach to solving boundary value problems in elasticity: energy methods, specifically variational methods. The power of the principles of minimum potential energy and complementary energy is effectively demonstrated through illustrative examples of beam bending and Saint-Venant torsion, showcasing a self-consistent approach to simplifying complex three-dimensional elasticity problems into more manageable two-dimensional or even one-dimensional models. Their direct applications to numerical solutions are illustrated via the Rayleigh-Ritz method.