This Moment of Inertia of a Beam Section equation computes the moment of inertia about the x- axis, which is defined at the height of the y-centroid of an I-beam's cross-section. The y-centroid is the center of mass of the beam's cross-sectional sections and we visualize the x-axis passing horizontally through the y-centroid of the cross-section of the beam.
The centroid is used to compute the moment of inertia of the beam, which represents a body's tendency to resist angular acceleration. In this case the moment of inertia is about the x-axis through the y-centroid and represents the beams resistance to rotations about the x-axis.
We split the cross-section into into three segments, each segment having a nice rectangular symmetry. We then calculate the area and y-centroid of each of the three segments and compute the entire centroid as:
`bary = sum(A_i*y_i)/(sum A_i)`
The three segments are shown in the figure below, where `A_i = w_i *D_i`:
And the y-centroids of the segments are given as:
Knowing the y-centroid of the beam's cross section computed above, we then compute the moment of inertia of each of the three segments sing the equation:
`barI_i = ("base"_i * "height"_i^3)/12`
Then we find the distances, d_i, of the segment's centroid from the x-axis (the y-centroid we computed for the beam cross-section earlier).
`d_i = |y_i - bary|`
The moment of inertia of the beam in cross-section is then given by:
`I_"(beam section)" = sum(barI_i + A_i * d_i^2)`