E-04 Section Properties

live equation

I_strong = bd³/12 = 4.0 × 12.0³ / 12 = 576.0 in⁴ I_weak = db³/12 = 12.0 × 4.0³ / 12 = 64.0 in⁴

the tall orientation is 9.0× stiffer — same material, same weight.

I = Measures resistance to bending: I = bd³/12 for a rectangle

b = Width of the rectangular cross-section

d = Depth of the cross-section

S = Elastic bending capacity: S = I/c

explained

Moment of inertia (I) measures how a cross-section resists bending. Depth matters far more than width because it enters as the cube — a beam on its side is dramatically weaker than the same beam standing tall, even though the area is identical.

key concepts

overview Depth is cubed — orientation changes everything

Moment of inertia measures how far material is from the neutral axis — and it rewards depth far more than width because d is cubed. A 4×12 beam has 9× the I of a 12×4 beam (same material, same weight), just because it's oriented tall instead of flat. This is the geometry behind every I-beam ever made: put the flanges as far from the neutral axis as possible, connect them with a thin web to hold position, and you have an efficient section that maximizes I while minimizing weight.

geometry is destiny Shape matters more than amount of material

A cross-section's shape determines everything about how it resists load. Two beams with identical area but different shapes will have wildly different bending capacity. An I-shape concentrates material far from the neutral axis (in the flanges), maximizing the moment of inertia I. A solid square of the same area has far less I because material near the center contributes almost nothing to bending resistance. Shape is more important than amount.

the properties that matter A, I, S, Z, and r — all pure geometry

The key properties: Area (A) — governs axial capacity. Moment of inertia (I) — governs bending stiffness and deflection, units in⁴. Section modulus (S = I/c) — governs elastic bending stress. Plastic section modulus (Z) — governs ultimate bending capacity. Radius of gyration (r = √(I/A)) — governs buckling resistance. These are all just geometry — they don't depend on the material, only on the shape.

the three numbers I, S, and r — what engineers actually use

I, S, and r aren't arbitrary — each answers a specific question about how the section performs under bending, stress, or buckling.

Moment of Inertia

I = bd³/12

Measures how much the section resists bending — specifically, how the area is distributed relative to the neutral axis. Used in deflection calculations (EI stiffness) and bending stress (M = σI/c).

units: in⁴

Section Modulus

S = I / c = bd²/6

Shortcut for bending design: σ = M/S. c is the distance from the neutral axis to the extreme fiber. For a symmetric section, c = d/2. Larger S → lower bending stress for the same moment.

units: in³

Radius of Gyration

r = √(I / A)

Used in column buckling design. Represents the "effective" distance from the centroid where all area is concentrated. Columns buckle about the axis with the smallest r — which is why wide-flange columns have two r values.

units: in

Test Your Understanding

1. Set b = 6 in and d = 12 in. What is the I_strong value shown by the tool?

216.0 in&sup4; 864.0 in&sup4; 432.0 in&sup4;

2. In I = bd³/12, why does depth (d) have a much larger effect on I than width (b)?

Because d is cubed while b is only linear — doubling d increases I by 8× Because d is squared while b is constant Because d and b contribute equally but d is always larger

3. Two beams have the same cross-sectional area but different shapes. Which statement is true?

They will have the same moment of inertia since A is equal The heavier beam will always be stiffer Shape matters more than amount — concentrating material far from the neutral axis increases I

depth is cubed. orientation is everything.

drag the section. watch I change.

E-04 Section Properties