As part of a project, I'm calculating deflection of a cantilever beam, something like the image below where the beam bends due to a load.
To do so requires its moment of inertia, and I'm thinking of using the beam's rectangular cross section. However, all the sources I've looked at only describe moment of inertia about the x or y axes and rotating about the center of the object–not one end of the beam and the z axis as is shown with the cantilever beam.
So, would the moment of inertia in this case still be $I=\frac{1}{12}bh^3$ as is the case for all the other rectangular cross section moment formulas I've seen, or would it be different and why?
Best Answer
Note that the $I(x)$ term in the beam deflection formula is the area moment of inertia of a cross-section of the beam about an axis perpendicular to the plan of the cross-section. $I(x)$ may be a function of the distance $x$ along the beam - although in your example its is not, as the cross-section of the beam is the same at all points along it.
The perpendicular axis theorem tells us that the area moment of inertia of a two-dimensional shape about an axis perpendicular to its plane is the sum of its moments of inertia about two perpendicular axes within its plane. For a rectangle with height $h$ and breadth $b$ this gives
$\displaystyle I_z = I_x + I_y = \frac 1 {12} bh^3 + \frac 1 {12} b^3h = \frac 1 {12} bh(h^2+b^2)$