Beam basics: bending moment, section modulus, and deflection

One-line orientation

Beam sizing is a short chain — load gives a bending moment, the moment over the section modulus gives a bending stress, and a separate deflection check keeps it serviceable — and the exam tests the logic, not heavy arithmetic.

Key points

• Bending moment (uniform load, simple span): M = wL² / 8, where w is the uniformly distributed load and L is the span. This is the maximum internal bending force and what the beam must be sized for.
• Bending stress: fb = M / S — the moment divided by the section modulus. For a fixed moment, increasing S lowers the stress.
• Section modulus: S = I / c, the moment of inertia I over the distance c from the neutral axis to the extreme fiber. Higher S = greater bending resistance; S guides the choice of shape and size.
• Depth dominates. Because depth enters the moment of inertia strongly, a deeper section produces a much larger S than a wider one of the same area — depth is the efficient way to gain bending capacity.
• Deflection is a separate, serviceability check. Adequate strength does not guarantee adequate stiffness; the beam must also stay within a deflection limit so it doesn’t sag or bounce noticeably and damage finishes.

Confusions / comparison

Quantity	Relationship	What it tells you
Bending moment M	M = wL² / 8 (uniform load, simple span)	The internal force to size for; grows with the square of span
Bending stress fb	fb = M / S	Stress in the material; lower when S is larger
Section modulus S	S = I / c	Bending efficiency of the shape; driven mostly by depth
Deflection	Serviceability limit (separate check)	Stiffness, not strength — controls sag/vibration

Related

→ Column buckling & K-values (this topic): the compression counterpart to bending behavior · End connections (this topic): a simple span assumes pinned ends; fixed ends change the moment diagram · Structural system selection (this topic): how required span steers the member and material choice.