Bending strength for equal-area beams under the same bending moment A square beam and a circular beam have the same cross-sectional area and are subjected to the same bending moment. Based on section modulus and resulting extreme fiber stress, which statement about economy/strength is correct (assuming identical material and allowable stress)?
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Acircular beam is more economical
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Bsquare beam is more economical
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Cboth beams are equally strong in bending
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Dboth beams are equally economical regardless of shape
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Enone of these
Answer
Correct Answer: square beam is more economical
Explanation
Introduction / Context:When two beams are made of the same material and carry the same bending moment, the bending stress at the extreme fiber is inversely proportional to the section modulus. For a fixed cross-sectional area, the shape that yields a larger section modulus produces lower bending stress and is therefore stronger and more economical in bending design.
Given Data / Assumptions:
- Same material and same allowable stress for both beams.
- Square beam side = a; circular beam diameter = d.
- Equal area condition: a^2 = (π d^2)/4.
- Pure bending; linear elastic behavior; no shear effects considered.
Concept / Approach:
Bending stress relation: σ = M / Z, where Z = I / c is the section modulus. For equal area, the beam with the larger Z has smaller σ for the same moment M, hence is stronger/more economical. Compute Z for square and circle under the equal-area constraint and compare magnitudes.
Step-by-Step Solution:
Equal area: a^2 = (π d^2)/4 ⇒ d = 2a/√π.Square: I = a^4/12, c = a/2 ⇒ Z_square = (a^4/12)/(a/2) = a^3/6.Circle: I = π d^4/64, c = d/2 ⇒ Z_circle = (π d^4/64)/(d/2) = (π d^3)/32.Insert d = 2a/√π ⇒ Z_circle = (π (8a^3/π^{3/2}))/32 = a^3/(4√π) ≈ 0.141 a^3.Z_square = a^3/6 ≈ 0.167 a^3 > Z_circle. Therefore square produces lower stress for the same M.Verification / Alternative check:
Numerical example with a = 1 unit gives Z_square ≈ 0.167 and Z_circle ≈ 0.141. Since σ ∝ 1/Z, the square shows about 18% lower extreme fiber stress, confirming superior bending efficiency for equal area.
Why Other Options Are Wrong:
- Circular beam is more economical: False; its section modulus is smaller for equal area.
- Both equally strong/economical: False; Z differs noticeably.
- None of these: Incorrect because one shape is clearly better in bending for equal area.
Common Pitfalls:
- Comparing second moment I alone; the correct comparison uses section modulus Z = I/c.
- Forgetting to enforce equal area before comparing Z values.
Final Answer:
square beam is more economical.