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?
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ASame as A
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BDouble of A
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CFour times that of A
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DSix times that of A
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EEight times that of A
Answer
Correct Answer: Four times that of A
Explanation
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:
Z_A = b * d^2 / 6.Z_B = b * (2d)^2 / 6 = b * 4d^2 / 6 = 4 * (b * d^2 / 6) = 4 * Z_A.Hence elastic strength_B / elastic strength_A = Z_B / Z_A = 4.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