In plastic analysis, the shape factor for a solid circular section (ratio of plastic moment capacity to elastic moment capacity) is approximately 1.697.

Introduction / Context:
The shape factor is central to plastic analysis of beams. It quantifies how much additional moment capacity is available beyond first yield due to redistribution of stresses from an elastic to a fully plastic distribution. It depends only on cross-sectional shape, not on material strength.

Given Data / Assumptions:

• Section: solid circular cross-section.
• Material behaves ideally elastoplastic at the section level.
• No effects of strain hardening or residual stresses considered in the definition.

Concept / Approach:
Shape factor S_f = M_p / M_y, where M_p is the fully plastic moment and M_y is the elastic yield moment. For a solid circle, the plastic neutral axis passes through the centroid and divides the circle into two equal areas under uniform yield stress, leading to a known closed-form factor ≈ 1.697.

Step-by-Step Solution:
Compute elastic section modulus: Z = I / y_maxCompute plastic section modulus: Z_p from equal area blocks at yield stressM_y = f_y * Z; M_p = f_y * Z_pShape factor S_f = M_p / M_y = Z_p / Z ≈ 1.697 for a solid circle

Verification / Alternative check:
Reference values: rectangle ≈ 1.5, solid circle ≈ 1.697, diamond (square at 45°) ≈ 2.0. The circular value sits logically between rectangle and diamond due to distribution of area away from the neutral axis.

Why Other Options Are Wrong:

• 1.50: matches a solid rectangle, not a circle.
• 1.60: close but not the standard value for a circle.
• None of these: incorrect because 1.697 is correct.

Common Pitfalls:

• Confusing elastic section modulus with plastic section modulus.
• Using hollow circular values for a solid circular section.

Final Answer:
1.697