Two beams of equal length carry the same maximum bending moment under the same permissible bending stress. One beam has a circular cross-section and the other a square cross-section. What is the ratio of their flexural weights (circular to square)?

Introduction / Context:
Flexural (bending) design often compares different cross-sections that must carry the same maximum bending moment at the same permissible bending stress. The required section modulus governs the size, and therefore the relative weight (area for unit length) follows from the geometry needed to achieve that section modulus.

Given Data / Assumptions:

• Equal length beams subjected to the same maximum bending moment M.
• Same permissible bending stress, so required section modulus Z is the same.
• Cross-sections: circle of diameter d, and square of side a.
• Flexural weight proportional to cross-sectional area (for unit length and same material density).

Concept / Approach:
Use the section modulus Z of each shape and equate them to find the relationship between a and d. Then compute the ratio of areas A_circular / A_square, which is the ratio of flexural weights for the same bending capacity.

Step-by-Step Solution:
Z_circle = (π * d^3) / 32Z_square = a^3 / 6Set Z_circle = Z_square → (π * d^3) / 32 = a^3 / 6Solve for a in terms of d: a^3 = (3π/16) * d^3 → a/d = (3π/16)^(1/3)Areas: A_circle = π d^2 / 4 ; A_square = a^2 = ((3π/16)^(2/3)) * d^2Flexural weight ratio (circle/square) = A_circle / A_square = (π/4) / ( (3π/16)^(2/3) ) ≈ 1.118

Verification / Alternative check:
The circular section is known to be slightly less area-efficient in flexure than the optimally oriented rectangular/square section for the same Z, so the circular beam needing marginally more area (≈ 11.8%) is reasonable.

Why Other Options Are Wrong:

• 1.338, 1.228, 1.108: Do not match the computed ratio from equating section moduli and areas.
• 0.894: Would imply the circle is lighter than the square for same bending capacity, which contradicts the derivation.

Common Pitfalls:

• Confusing section modulus with moment of inertia or forgetting division by extreme fiber distance.
• Equating areas instead of section modulus when bending capacity is the criterion.

Final Answer:
1.118