Final Answer:
The moment of inertia of the given U-beam about its centroid is I = (b1h1³ - b2h2³) / 3, where b1, b2 are the widths and h1, h2 are the heights of the beam sections.
Step-by-step explanation:
The moment of inertia of a U-beam about its centroid is calculated by considering the individual contributions of its rectangular sections. The formula used is I = (b1h1³ - b2h2³) / 3, where b1 and b2 are the widths of the top and bottom rectangles respectively, while h1 and h2 are their corresponding heights.
This formula for the moment of inertia of a U-beam is derived from the parallel axis theorem and the formula for the moment of inertia of rectangles about their centroids. By subtracting the moment of inertia of the smaller rectangle (b2h2^3) from the larger one (b1h1^3) and dividing by 3, the resulting value provides the overall moment of inertia for the U-beam about its centroid.
The approach considers the beam as two separate rectangles, simplifying the calculation of its moment of inertia while accurately accounting for the distribution of mass about its centroid.