Japan Design Guide

Basic Cross-Section Properties

Important Section Properties
	Rectangle	Circle	Triangle

Area (A) 断面積 (A)	$$bh$$	$$\frac{\pi d^2}{4}$$	$$\frac{bh}{2}$$
Moment of Inertia (I) 断面２次モーメント (I)	$$\frac{bh^3}{12}$$	$$\frac{\pi d^4}{64}$$	$$\frac{bh^3}{36}$$
Radius of Gyration (r) 断面２次半径 (i)	$$\frac{h}{\sqrt{12}}$$	$$\frac{d}{4}$$
Section Modulus (S) 断面係数 (Z)	$$\frac{bh^2}{6}$$	$$\frac{\pi d^3}{32}$$
Plastic Modulus (Z) 塑性断面係数 (Zp)	$$\frac{bh^2}{4}$$	$$\frac{d^3}{6}$$

Notes

The above properties for rectangles and circles are important building blocks that designers should know by heart. At a minimum, the Area and Moment of Inertia formulas should be memorized; other properties can easily be derived as needed.
For reference, comparing a square with side lengths equal to the diameter of a circle, the area of that square is 12% larger, and the moment of inertia is 14% larger. It's helpful to have a "gut feel" for these types of relationships.
The properties shown for the triangle case are for reference only. These are not commonly used and can be looked up as needed.

Quick Cases: Moment of Inertias

Equivalent Cases
Case	Notes
	A straight-forward case: this is also very useful for circles as well.
	This case is similar to the previous, without the contribution of "webs".
	The "wide-flange" or "i-beam" relationship is critical for everyone to know.
	A variant of the "donut" case above, less one web, the "channel" case is important as a way of understanding the next case.
	The "zee" shape can be solved by likening it to the "channel" case above.

Parallel Axis Theorem

It's important to be able to use the parallel axis theorem to be able to rapidly analyze complex cross-sections by breaking them into smaller, simplified shapes.

Below is the standard formula for calculating an object's moment of inertia about an arbitrary axis. It's determined as the sum of two components: (1) the object's moment of inertia about it's centroidinal axis AND (2) the product of the area of that object, and the square of the distance measured from that centroid to the arbitrary axis:

$$I=I_o+Ay^2$$

Make sure you are familiar with this formula as an important tool for cross-section analysis.