Comparing elastic strength in bending: Two rectangular beams with same length l and width b, but beam B has depth 2d (double of beam A). How does the elastic strength of B compare to A?

Introduction / Context:
Elastic strength in bending is proportional to the section modulus Z for the given cross-section. For a rectangular section, section modulus strongly depends on depth, so changing depth substantially affects bending capacity.

Given Data / Assumptions:

• Beam A: width b, depth d.
• Beam B: width b, depth 2d.
• Same material, same allowable stress, same length.

Concept / Approach:
For a rectangle, Z = b * d^2 / 6 about the strong axis. Bending stress formula: sigma = M / Z at elastic limit. If Z increases, allowable moment at the same stress increases proportionally, i.e., elastic strength scales with Z.

Step-by-Step Solution:

Verification / Alternative check:
Keeping width constant while doubling depth squares the influence of depth, hence a factor of 4. This result is standard in beam design tables.

Why Other Options Are Wrong:
“Same” or “double” underestimates the square dependence on depth. “Six times” and “eight times” overestimate the effect.

Common Pitfalls:
Confusing second moment of area I (b * d^3 / 12) with section modulus Z (I / c). For a rectangle, Z depends on d^2 due to division by c = d/2.

Final Answer:
Four times that of A