Skip to main content
Engineering LibreTexts

7.3: Two Formulations for Beams

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    In the bending theory of beams, the total potential energy is

    \[\prod = \int_{0}^{l} \frac{1}{2} M\kappa dx − \int_{0}^{l} q(x)w dx \]

    Using the moment curvature relation \(M = EI\kappa\), either \(M\) or \(\kappa\) can be eliminated from Equation (7.2.7), leading to

    \[U = \int_{0}^{l} \frac{1}{2} M\kappa dx = \begin{cases} \int_{0}^{l} \frac{EI}{2}\kappa^2 dx \quad \text{ displacement formulation} \\ \int_{0}^{l} \frac{1}{2 EI} M^2 dx \quad \text{ stress formulation} \end{cases}\]

    In statically determined problems the bending moments can be expressed in terms of the prescribed line load or point load. In the latter case the \(M = M(P)\) and the total potential energy takes the form

    \[\prod = U(P) − P w \]

    The above representation will lead to the Castigliano theorem which will be covered later in this lecture.

    The more general displacement formulation will be covered next. The curvature is proportional to the second derivative of the displacement. The expression of the total potential energy becomes

    \[\prod = \int_{0}^{l} \frac{EI}{2} (w^{\prime\prime})^2 dx − \int_{0}^{l} q(x)w dx \]

    The problem is reduced to express the displacement field in terms of a finite number of free parameters \(w(x, a_i)\) and then use the stationary condition, Equation (7.1.14) to determine these unknown parameters. This could be done in three different ways:

    1. Polinomial representation or Taylor series expansion
    2. Fourier series expansion
    3. Finite element or finite difference method

    Each of the above procedure will be explained separately.

    This page titled 7.3: Two Formulations for Beams 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; a detailed edit history is available upon request.