8.5

8.5.1 Bending moments about \(x\)

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\[\begin{align} {\sigma_{zz}}=& \, E \left({\varepsilon_{zz 0}}- x {u {}_{,zz}} - y {v {}_{,zz}}\right) \end{align}\]

  • The resultant bending moment about the centroid is: \[M_x = \int_A y \cdot {\sigma_{zz}}\, dA\]

  • \[M_x = E \left( {\varepsilon_{zz 0}}\cdot \int_A y \, dA - {u {}_{,zz}} \cdot \int_A x y \, dA - {v {}_{,zz}} \cdot \int_A y^2 \, dA\right)\]

  • Recall:
  • \(\int_A y \, dA = 0\) : By choice of coordinate system
  • \(\int_A x y \, dA = {I_{xy}}\) : Area product of inertia
  • \(\int_A y^2 \, dA = {I_{xx}}\) : Area moment of inertia about \(x\)-axis
  • Note: \({I_{xy}}= 0\) if there is symmetry about the \(x\) or \(y\) axis. (If there is symmetry about another axis, the computation can be transformed to be about the principal axis of the cross section.)
  • If \(E=E(x,y)\), then it must remain inside the integral and we do not have a clean separation between material and cross-section properties. (See ME6520 Composite Materials)

8.5.2 Bending moments about \(y\)

\[\begin{align} {\sigma_{zz}}=& E \left({\varepsilon_{zz 0}}- x {u {}_{,zz}} - y {v {}_{,zz}}\right) \end{align}\]

  • The resultant bending moment about the centroid is: \[M_y = - \int_A x \cdot {\sigma_{zz}}\, dA\]

\[M_y = - E \left( {\varepsilon_{zz 0}}\cdot \int_A x \, dA - {u {}_{,zz}} \cdot \int_A x^2 \, dA - {v {}_{,zz}} \cdot \int_A x y \, dA\right)\]

  • Recall:
  • \(\int_A x \, dA = 0\) : By choice of coordinate system
  • \(\int_A x y \, dA = {I_{xy}}\) : Area product of inertia
  • \(\int_A x^2 \, dA = {I_{yy}}\) : Area moment of inertia about \(y\)-axis
  • Note: \({I_{xy}}= 0\) if there is symmetry about the \(x\) or \(y\) axis (If there is symmetry about another axis, the computation can be transformed to be about the principal axis of the cross section.)

\[\begin{align} M_x =& \, - E{I_{xy}}{u {}_{,zz}} - E{I_{xx}}{v {}_{,zz}} \\ M_y =& \, + E{I_{yy}}{u {}_{,zz}} + E{I_{xy}}{v {}_{,zz}} \\ \frac{P}{A} =& \, E {\varepsilon_{zz 0}}= {\sigma_{zz}}^{\mathrm{ave}} \\ \end{align}\]

  • These equations are uncoupled due to the centroidal \({z}\)-axis
  • Note also we can invert and solve