Plastic analysis — which cross-section has the largest shape factor? Among the following cross-section shapes subjected to bending about their centroidal axis, which one typically exhibits the largest shape factor (plastic section modulus / elastic section modulus)?

Introduction / Context:
The shape factor, defined as Zp/Z (plastic section modulus divided by elastic section modulus), measures a section’s reserve strength beyond first yield in bending. A larger shape factor indicates greater plastic redistribution capacity before full plastic collapse, which is central to plastic design of beams and frames.

Given Data / Assumptions:

• Bending is about a centroidal principal axis.
• Material is ideally elastic–perfectly plastic.
• Sections considered are common shapes used in practice.

Concept / Approach:
The shape factor depends on how the area is distributed with respect to the neutral axis. Shapes concentrating more area away from the neutral axis tend to have higher elastic modulus but may not maximize plastic capacity. Among standard solid shapes, typical shape factors are approximately: rectangular ≈ 1.5, solid circular ≈ 1.7, I-sections ≈ 1.1–1.2 (depending on proportions). A square turned through 45° (diamond) develops a higher plastic-to-elastic ratio, yielding one of the largest shape factors among simple solid shapes, commonly taken as about 2.0 for idealized plastic analysis.

Step-by-Step Solution:
Compare known shape factors: I (≈1.1–1.2) < rectangular (≈1.5) < circular (≈1.7) < diamond (≈2.0).Identify the maximum among listed shapes → diamond.Select “Diamond (square turned through 45°)”.

Verification / Alternative check:
Plastic design handbooks enumerate these typical values; the diamond’s shape factor exceeds that of rectangle and circle for bending about a symmetry axis.

Why Other Options Are Wrong:
Rectangular and solid circular: Lower shape factors (≈1.5, ≈1.7).
I-section: Typically around 1.1–1.2 due to flange–web distribution.
Thin-walled tube: Depending on wall thickness may be competitive, but for common comparisons the diamond’s idealized factor is larger.

Common Pitfalls:

• Confusing section efficiency in elastic design (high Z) with plastic reserve (high Zp/Z).
• Comparing about a non-principal axis, which changes numerical values.

Final Answer:
Diamond (square turned through 45°)