A rectangular bar with width b and height h is used as a cantilever. If the loading acts in a plane parallel to the side b (i.e., bending about the weak axis), what is the section modulus?

Introduction / Context:
The section modulus Z determines the bending stress for a given bending moment via σ = M / Z. For rectangles, Z depends on which axis the section bends about (strong vs weak axis). Correctly identifying the bending axis is crucial for safe design.

Given Data / Assumptions:

• Rectangle: width b, height h.
• Loading plane is parallel to side b → bending about the weak axis (through the centroid, parallel to side b), hence the moment of inertia uses b^2.
• Linear elastic bending.

Concept / Approach:
Section modulus Z = I / c, where I is the second moment of area about the bending axis and c is the distance from centroid to the extreme fiber along the bending direction.

Step-by-Step Solution:
For bending about the weak axis: I_weak = h * b^3 / 12.Extreme fiber distance c = b/2.Z = I / c = (h * b^3 / 12) / (b/2) = h * b^2 / 6.

Verification / Alternative check:
Compare with strong-axis bending where I_strong = b * h^3 / 12 and Z_strong = b * h^2 / 6; the provided loading plane distinguishes the weak-axis case.

Why Other Options Are Wrong:

• Z = b * h^2 / 6: strong-axis result, not applicable here.
• Z = b^2 * h / 12 and Z = h^2 * b / 12: these are moments of inertia divided by incorrect c or simply incorrect expressions.

Common Pitfalls:

• Confusing which side corresponds to c and which formula for I to use.

Final Answer:
Z = h * b^2 / 6