For a simply supported rectangular beam of span L carrying a central load W, determine the length of the elasto–plastic zone of the plastic hinge that forms at midspan (answer in terms of L).

Introduction / Context:
As bending moment increases under a central point load, yielding starts at the extreme fibers near midspan and spreads until a plastic hinge forms. The length of the yielded (elasto–plastic) region depends on the shape factor of the section and the moment diagram around midspan.

Given Data / Assumptions:

• Simply supported rectangular beam, span L.
• Central load W → triangular bending-moment diagram peaking at midspan.
• Rectangular section with shape factor S = Mp / My = 1.5.
• Linear elastic-perfectly plastic behavior; small deflection.

Concept / Approach:
Let z measure distance from midspan. For a central point load, near midspan M(z) decreases linearly with slope equal to the shear V = W/2. The elasto–plastic zone is where M(z) ≥ My. At collapse, M(0) = Mp = S*My. The half-length of the yielded zone z_y satisfies M(z_y) = My, which allows solving for z_y in terms of S and L, and then doubling to get the full zone length.

Step-by-Step Solution:
1) Maximum moment at midspan: M_max = W*L/4.2) At collapse for a rectangle: M_max = Mp = S*My with S = 1.5.3) Near midspan, M(z) = M_max - (W/2)*|z| (linear drop with shear W/2).4) Define z_y by M(z_y) = My ⇒ S*My - (W/2)*z_y = My ⇒ (S - 1)*My = (W/2)*z_y.5) Also, My = M_max / S = (W*L/4)/S.6) Substitute to get z_y = 2*(S - 1)*My / W = 2*(S - 1)*(W*L/4S)/W = (S - 1)*L/(2S).7) Full elasto–plastic length = 2*z_y = (S - 1)*L/S = (1.5 - 1)*L/1.5 = L/3.

Verification / Alternative check:
Dimensional check shows a length proportional to L. Using S = 1.5 reproduces classic results for rectangular sections under a central point load.

Why Other Options Are Wrong:

• L/6, L/4: Underestimate the spread implied by S = 1.5.
• L/2: Overestimates; would imply too large a yielded region for a rectangular section.

Common Pitfalls:

• Confusing the plastic hinge length with the entire constant-moment region (there is none here).
• Forgetting the role of the shape factor in determining spread of yield.

Final Answer:
L / 3.