Solid Mechanics | 30. Castigliano’s First Theorem by Abraham | Learn Smarter
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30. Castigliano’s First Theorem

The chapter discusses energy methods in solid mechanics, specifically focusing on Castigliano’s First Theorem which relates energy stored in deformed bodies to generalized forces. Different forms of energy stored in beams due to axial extension, bending, torsion, and shear loads are derived systematically. Verification of reciprocal relations is provided, along with practical examples illustrating these concepts in solving problems related to beams under various loading conditions.

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Sections

  • 1

    Castigliano’s First Theorem

    Castigliano’s First Theorem relates the energy stored in a structure to the displacements caused by applied forces.

    Learning Practice

  • 2

    Deriving Expression For Energy Stored In A Beam In Terms Of Internal Contact Force And Moment Which Acts In The Beam’s Cross-Section

    This section derives expressions for energy stored in a beam due to internal forces and moments acting within its cross-section.

    Learning Practice

  • 2.1

    Axial Extensional Energy

    This section discusses the concept of axial extensional energy in beams, detailing how energy is stored and its significance in structural mechanics.

    Learning Practice

  • 2.2

    Bending Energy

    This section explains the concept of bending energy in beams and its derivation using Castigliano's theorem.

    Learning Practice

  • 2.3

    Torsional Energy

    This section discusses the calculation and significance of torsional energy in beams, focusing on the energy associated with torsion and how it can be expressed mathematically.

    Learning Practice

  • 2.4

    Shear Energy Due To Transverse Load

    This section discusses the shear energy generated in beams due to transverse loads, detailing its mathematical formulation and implications.

    Learning Practice

  • 2.5

    Total Energy Stored In The Beam

    This section discusses the total energy stored in a beam due to various types of loading conditions, employing Castigliano's theorem to facilitate calculations.

    Learning Practice

  • 3

    Verification Of Reciprocal Relation

    This section discusses the verification of the reciprocal relation in beam theory, showcasing its utility in simplifying complex problems.

    Learning Practice

  • 4

    Example 1

    Learning

  • 5

    Example 2

    This section discusses how to apply energy methods to determine reactions from supports in structural problems.

    Learning Practice

  • 6

    Example 3

    This section delves into an analytic approach to solving a complex beam deformation problem using energy methods, specifically focusing on a ring under external forces.

    Learning Practice

References

ch30.pdf

Class Notes

Memorization

What we have learnt

  • Energy in deformable bodies...
  • Castigliano's First Theorem...
  • Different deformation modes...

Final Test

Revision Tests