Two prismatic beams of equal cross-sectional area—one with a circular cross-section and the other with a square cross-section—are subjected to the same bending conditions. Considering section modulus (economy in bending), which beam is more economical for resisting bending?

Introduction:
Economy in bending for a given cross-sectional area is judged by the section modulus, Z = I / y_max. A larger Z yields lower bending stress for the same bending moment. This question compares circular and square cross-sections of equal area to determine which is more efficient in bending.

Given Data / Assumptions:

• Equal cross-sectional areas: A_square = A_circle.
• Homogeneous, isotropic, linearly elastic material.
• Same loading and span so bending moment demand is identical.

Concept / Approach:
For a square of side a: I = a^4 / 12 and y_max = a / 2, hence Z_square = a^3 / 6. For a circle of diameter d: I = (π * d^4) / 64 and y_max = d / 2, hence Z_circle = (π * d^3) / 32. With equal areas, a^2 = (π * d^2) / 4, so a = (d * √π) / 2. Substituting shows Z_square > Z_circle, i.e., the square section is more economical in bending.

Step-by-Step Solution:
1) Area equality: a^2 = π d^2 / 4 ⇒ a = (d √π) / 2.2) Z_square = a^3 / 6 = ((d √π) / 2)^3 / 6.3) Z_circle = (π d^3) / 32.4) Ratio Z_square / Z_circle > 1 for all positive d, confirming the square is more efficient.

Verification / Alternative check:
Numerical substitution (e.g., d = 1) yields Z_square ≈ 1.182 * Z_circle, validating the theoretical comparison.

Why Other Options Are Wrong:
Circular more economical: contradicted by computed Z.

Equally economical / none / cannot be compared: incorrect because Z permits a direct, objective comparison when areas are equal.

Common Pitfalls:
Comparing moments of inertia alone without dividing by y_max; ignoring the equal-area constraint.

Final Answer:
square beam is more economical