16.8 Summary of beam equations

• Thus: \[\begin{align} \frac{d^2}{d z^2} \left[ E {I_{yy}}{u {}_{,zz}} + E {I_{xy}}{v {}_{,zz}} \right] =& \, p_x \\ \frac{d^2}{d z^2} \left[ E {I_{xy}}{u {}_{,zz}} + E {I_{xx}}{v {}_{,zz}} \right] =& \, p_y \\ \end{align}\]

• If symmetry exists across the \(x\) or \(y\) axis, the bending equations are uncoupled. \[\begin{align} \frac{d^2}{d z^2} \left[ E {I_{yy}}{u {}_{,zz}} \right] =& \; p_x \\ \frac{d^2}{d z^2} \left[ E {I_{xx}}{v {}_{,zz}} \right] =& \; p_y \\ M_x =& \, - E{I_{xx}}{v {}_{,zz}} \\ M_y =& \, + E{I_{yy}}{u {}_{,zz}} \\ \end{align}\]

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