16.6

16.6.1 Bending moments about \(x\)

Unlabelled Image Missing

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

  • 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)

@Bartel2006 @Bartel2006

@Bartel2006

@Bartel2006

Cross section of the rat femur, showing the neutral axis, which goes through the centroid and has an orientation with respect to the ML axis. Note that for this example, the bending moment is applied only about the AP axis. Also shown are the locations of maximum tensile and compressive bending stress, respectively.

16.6.2 Bending moments about \(y\)

Unlabelled Image Missing

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

  • 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