8.4 Beam loads
8.4.1 Axial load in a beam
- The axial load in a beam can be expressed as: \[P = \int_A {\sigma_{zz}}\, dA\] \[dA = dx \cdot dy\]
- Therefore: \[\begin{align} P =& \, E \int_A \left({\varepsilon_{zz 0}}- x {u {}_{,zz}} - y {v {}_{,zz}}\right) \, dA\\ =& \, E \left({\varepsilon_{zz 0}}\int_A \, dA - {u {}_{,zz}} \int_A x \, dA - {v {}_{,zz}} \int_A y \, dA \right) \end{align}\]
8.4.2 Definitions
Lets define some terms: \[\begin{align} \int_A \, dA =& \, A \\ \int_A x \, dA =& \, x_c A \\ \int_A y \, dA =& \, y_c A \\ \end{align}\]
Now we have: \[\begin{align} P =& \, E \left(A {\varepsilon_{zz 0}}- x_c A {u {}_{,zz}} - y_c A {v {}_{,zz}} \right) \\ \end{align}\]
8.4.3 Choice of coordinate system
We choose our coordinate system so that: \[\begin{align} x_c = 0 \\ y_c = 0 \\ \end{align}\]
Which leads to: \[\begin{align} P =& \, E A \, {\varepsilon_{zz 0}}\\ \end{align}\]
This means that the axial force in the beam is due to the elongation of the centroidal axis and is not related to the bending of the beam.