Skip to main content
Engineering LibreTexts

2.3: Generalized Forces and Bending Moments in Plates

  • Page ID
    21478
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    In plates there are three in-plane components of the stress tensor \(\sigma_{\alpha \beta}\{\sigma_{xx}, \sigma_{yy}, \sigma_{xy}\}\). Replacing \(\sigma_{xx}\) by \(\sigma_{\alpha \beta}\) or \(\sigma_{z\alpha}\) in Equations (2.2.16-2.2.18) the generalized forces and couples are defined

    \[M_{\alpha \beta} = \int_{-\frac{h}{2}}^{\frac{h}{2}} \sigma_{\alpha \beta}z dz \; [\mathrm{Nm/m}] = [\mathrm{N}] \label{2.3.1}\]

    \[N_{\alpha \beta} = \int_{-\frac{h}{2}}^{\frac{h}{2}} \sigma_{\alpha \beta} dz \; [\mathrm{Nm/m}]\]

    \[V_{\alpha} = \int_{-\frac{h}{2}}^{\frac{h}{2}} \sigma_{z\alpha} dz \; [N/m] \label{2.3.3}\]

    Note that in the plate theory the integration is performed over the thickness of the plate rather than the entire surface. Therefore the dimensions of the quantities defined by Equations \ref{2.3.1}-\ref{2.3.3} are “per unit length”.


    This page titled 2.3: Generalized Forces and Bending Moments in Plates 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.