Moment of Inertia Visualizer
Two panels side by side. Left panel shows the cross-section shape drawn to scale with the neutral axis (NA) marked in red, a bending stress distribution triangle on the right edge, and a color gradient showing how material further from NA contributes more to bending resistance.
Right panel shows Ixx vs Iyy comparison bars, efficiency (I/A) bars, and for composite shapes, a parallel axis theorem breakdown showing how much comes from I local vs Ad².
6 cross-section shapes:
Simple shapes:
Rectangle — Ixx = bh³/12. Doubling height increases Ixx by 8× (cubic). The most important insight.
Solid Circle — Ixx = Iyy = πd⁴/64. Equal in all directions. Good for shafts.
Hollow Tube — Ixx = π(D⁴-d⁴)/64. Removes low-stress center material. 90% of solid I at 50% weight.
Composite shapes (parallel axis theorem):
T-beam — flange + web. Parallel axis theorem: I = Σ(Ilocal + Ad²). The d² term from the flange dominates.
I-beam — two flanges + web. Maximum efficiency. Both flanges far from NA give huge Ad² contributions.
C-Channel — asymmetric. Centroid shifts toward web, causing shear center offset.
Key slider experiments::
Rectangle: drag height from 60 to 120. Watch Ixx jump 8× while area only doubles. Height cubed is the lesson.
I-beam vs Rectangle: set both to same overall dimensions. I-beam Ixx is much higher at much less area. The Ixx/A efficiency ratio tells the story.
Hollow tube: increase inner diameter. Watch I barely drop while area shrinks fast. The center material was doing almost nothing.