Section modulus of a beam: The section modulus Z used in flexural design is expressed in terms of the second moment of area I and the distance to the extreme fiber y. What is Z?

Introduction / Context:
Section modulus Z is a geometric property that links bending moment to maximum normal stress at the outer fiber of a beam. It is widely used to size beams in structural and machine design.

Given Data / Assumptions:

• Linear elastic behavior and Bernoulli–Euler beam theory.
• Maximum stress occurs at the extreme fiber located a distance y from the neutral axis.

Concept / Approach:
The flexure formula is σ_max = M * y / I. Rearranging gives M / Z = σ_max where Z = I / y. Thus for a given allowable stress, the required section modulus is M / σ_allow, translating directly into geometry selection.

Step-by-Step Solution:

Verification / Alternative check:
For standard shapes (rectangular, circular, I-sections), published Z values equal I / y_max, confirming the definition. Example: for a rectangle b × h, I = b h^3 / 12 and y = h/2, giving Z = b h^2 / 6, a well-known result.

Why Other Options Are Wrong:

• I * y and y / I have incorrect dimensions.
• M / I is unrelated to Z; it appears in curvature κ = M / (E I).
• I / y^2 does not match flexure relations for stress at the extreme fiber.

Common Pitfalls:
Using the wrong y (not the extreme fiber); confusing elastic section modulus with plastic section modulus (Zp), which is different and used in plastic design.

Final Answer:
I/y