Plastic Hinge Development in a Simply Supported I-Section A simply supported I-section beam of span L carries a central load W. In plastic analysis, what is the approximate length of the elasto-plastic zone associated with the plastic hinge that forms at mid-span under increasing load?

Introduction / Context:
When a prismatic I-section beam under bending approaches plastic collapse, a plastic hinge forms at the critical section. The hinge region is not a mathematical point; it spreads over a finite length where fibers transition from elastic to fully plastic — the elasto-plastic zone. Designers often use practical approximations for its length.

Given Data / Assumptions:

• Simply supported beam, span L.
• Central concentrated load W creating maximum moment at mid-span.
• Prismatic I-section of overall depth D.
• Typical steel behavior with strain hardening neglected in first-order approximation.

Concept / Approach:
The plastic hinge length l_p is commonly estimated using simple rules of thumb tied to the member depth. For rolled I-sections in bending, an accepted approximation in plastic design is l_p ≈ D, representing the region over which curvature is high and stresses redistribute from elastic to plastic states.

Step-by-Step Solution:
At mid-span, moment is maximum and a hinge initiates.Curvature localizes over a zone scaling with section depth D.Adopting standard plastic design practice: l_p ≈ D.

Verification / Alternative check:
Refined models relate l_p to material strain capacity and curvature distribution, yielding values between about 0.5 * D and 2 * D. The commonly adopted design approximation is l_p ≈ D for steel I-sections, balancing safety and realism.

Why Other Options Are Wrong:

• 0.5 * D and 2 * D: lower/upper bounds sometimes cited but not the usual design default.
• 1.5 * D: larger than the standard approximation for typical rolled sections.
• Depends only on yield stress: hinge length also depends on section geometry; depth is influential.

Common Pitfalls:
Treating the hinge as a zero-length point or ignoring section geometry can misestimate rotation capacity in plastic analysis.

Final Answer:
D