In plastic analysis, identify the shape factor ( Mp / My ) for a rectangular cross-section (keeping symbols and values unchanged).

Introduction / Context:
The shape factor S (also written as Mp / My) quantifies the reserve strength beyond first yield in plastic analysis. It depends only on cross-sectional shape, not on material properties, and is central to plastic hinge formation and collapse load calculations.

Given Data / Assumptions:

• Section is rectangular, subjected to bending.
• My = yield moment (based on elastic stress distribution up to sigma_y).
• Mp = plastic moment (fully plastic stress block).

Concept / Approach:
For a rectangle, the elastic section modulus (Z) and plastic section modulus (Zp) have a constant ratio: S = Zp / Z = Mp / My. Classical plastic analysis yields S = 1.5 for a rectangular section because the plastic neutral axis divides the section into two equal areas in full plasticity, doubling the lever arm relative to the elastic centroidal distribution by a factor leading to 1.5 overall.

Step-by-Step Solution:
1) My = sigma_y * Z (elastic).2) Mp = sigma_y * Zp (plastic).3) Shape factor S = Mp / My = Zp / Z.4) For a rectangle, Zp / Z = 1.5 → S = 1.5.

Verification / Alternative check:
Handbooks and steel design texts tabulate S = 1.5 for rectangles, approximately 1.1 for I-sections (compact flanges/webs vary), and 1.7 for diamonds, confirming the standard value for rectangles.

Why Other Options Are Wrong:

• 1.4, 1.6, 1.7: Belong to other shapes or bounds; not the canonical value for a pure rectangular section.

Common Pitfalls:

• Confusing shape factor with safety factor or with section modulus itself.
• Using a noncompact or perforated section where local buckling alters plastic capacity.

Final Answer:
1.5.