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7: Energy Methods in Elasticity

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    The energy methods provide a powerful tool for deriving exact and approximate solutions to many structural problems.

    • 7.1: The Concept of Potential Energy
      This page covers kinetic energy, potential energy, and total potential energy in mechanical systems, detailing relevant equations and the effects of height on potential energy. It introduces total potential energy, including elastic energy, and defines static equilibrium conditions.
    • 7.2: Equivalence of the Minimum Potential Energy and Principle of Virtual Work
      This page covers virtual displacement, a key concept in mechanics related to kinematic boundary conditions, and introduces virtual strains linked to stress increments via elasticity laws. It explains that total strain energy consists of elastic strain energy and external work, with equilibrium demanding stationary potential energy.
    • 7.3: Two Formulations for Beams
      This page explores the bending theory of beams, emphasizing total potential energy in beam deflection. It provides formulations for total potential energy related to bending moments and external loads, highlighting the connection between curvature and displacement. Various methods for approximating displacement, such as polynomial representation, Fourier series, and finite element methods, are introduced.
    • 7.4: Fourier Series Expansion and the Ritz Method
      This page covers the analysis of a symmetrically loaded simply supported plate under a central point force, focusing on the total potential energy expressed as a deflection-load integral. It uses a Fourier series expansion to derive the equilibrium deflection shape while adhering to boundary conditions. The initial approximation yields a close solution with a 1.4% error.
    • 7.5: Solution by Taylor expansion
      This page analyzes a cantilever beam under a tip point force, using a truncated power series to model its deflection. It derives a simplified deflection equation from displacement boundary conditions and establishes a total potential energy expression, leading to linear equations for coefficients \(a_2\) and \(a_3\). The resulting deflection function accurately predicts the beam's tip deflection, demonstrating the efficacy of the power series approximation.
    • 7.6: Castigliano Theorem
      This page covers the applications of Castigliano's theorem in statically determined structures, focusing on deriving bending moment distributions and displacement equations through energy formulations. It analyzes cantilever beams and elbows, illustrating energy relationships and deflection predictions for multiple loads.

    This page titled 7: Energy Methods in Elasticity is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Tomasz Wierzbicki (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.