Architectural structures frequently use hollow steel tubes (HSS) to save weight while maintaining high stiffness. Calculate the Moment of Inertia ($I_x$) about the centroidal axis for a hollow rectangular steel tube with outer dimensions of $200 \text{ mm}$ wide by $300 \text{ mm}$ deep, and a uniform wall thickness of $20 \text{ mm}$.

Calculate the Polar Moment of Inertia ($J$) for a solid circular concrete column with a diameter of $400 \text{ mm}$.

Calculate the total horizontal Moment of Inertia ($I_x$) of a concrete T-beam exactly about its horizontal centroidal axis. The T-beam explicitly consists of a top flange ($b = 100 \text{ mm}, h = 20 \text{ mm}$) and a vertical bottom web ($b = 20 \text{ mm}, h = 80 \text{ mm}$). Assume the exact global neutral axis (global centroid) has already been accurately calculated and is securely located at $\bar{Y} = 67.78 \text{ mm}$ vertically from the absolute bottom edge.

Calculate the exact Radius of Gyration ($r_x$) about the horizontal centroidal axis for a perfectly solid rectangular timber column that is $100 \text{ mm}$ wide ($b$) and $200 \text{ mm}$ deep ($h$).

For the exact same $100 \text{ mm} \times 200 \text{ mm}$ solid timber column, calculate the Radius of Gyration about the vertical centroidal axis ($r_y$). Which direction will the column physically buckle if loaded from the top?

A steel structural column is designed as a hollow pipe. The outer diameter is $200 \text{ mm}$ and the inner diameter is $180 \text{ mm}$ (wall thickness is $10 \text{ mm}$). Calculate its Radius of Gyration ($r$) to evaluate its buckling resistance.

Why do structural engineers almost exclusively use Wide-Flange (I-beam) shapes for massive horizontal floor spans instead of perfectly solid rectangular steel blocks of the exact same weight?

An architect designs a spectacular spiral staircase around a single, massive central support column. The stairs cantilever entirely off this one vertical post. Why is the Polar Moment of Inertia ($J$) the most critical mathematical property for this specific central column?