Square beam orientation effect: For the same allowable stress, the ratio of the moment of resistance when the square's sides are horizontal to that when its diagonal is horizontal equals what value?

Introduction / Context:
The moment of resistance of a beam section at a given allowable stress is proportional to its section modulus. For a square, rotating the section by 45 degrees (diagonal horizontal) changes the extreme fiber distance even though the centroidal second moment of area about any axis through the center remains the same. This affects flexural capacity.

Given Data / Assumptions:

• Square cross-section of side a.
• Linear elastic bending with the same allowable (design) stress in both orientations.
• Comparison between orientation with sides horizontal versus diagonal horizontal.

Concept / Approach:
Moment of resistance MR ∝ Z, where Z = I / y_max. For a square, I about a centroidal axis is a^4 / 12 regardless of rotation because I_x = I_y and I_xy = 0. However, y_max (distance from neutral axis to extreme fiber) differs by orientation: a/2 with sides horizontal, and a/√2 with diagonal horizontal.

Step-by-Step Solution:

Verification / Alternative check:
Numerical check with a = 100 mm: compute Z for both orientations and take the ratio; the result is approximately 1.414.

Why Other Options Are Wrong:
1 or 1/2 assumes no change in y_max or incorrect I.2 overestimates; 1/√2 reverses the true ratio.

Common Pitfalls:
Assuming I changes with rotation for a square; forgetting that extreme fiber distance changes with orientation, which controls Z.

Final Answer:

√2