A beam of span L is pin-supported at both ends and subjected to a pure bending couple of magnitude M applied at mid-span. What is the maximum bending moment in the beam?

Introduction / Context:
Beams under applied moments (couples) behave differently from beams under transverse loads. Understanding how a concentrated couple influences reactions, shear, and bending moment is key to correct analysis.

Given Data / Assumptions:

• Simply supported (pin–pin) beam of length L.
• A pure couple of magnitude M applied at mid-span.
• Linear elastic behavior.

Concept / Approach:
A couple has no net force; the end reactions must form an equal and opposite couple to maintain equilibrium. Let left reaction be +R and right reaction be −R so that ΣF = 0 and the couple R*L balances M.

Step-by-Step Solution:
Global moment balance → R * L = M → R = M / L.Left span shear V = +R; right span shear V = −R.Bending moment for 0 < x < L/2: M(x) = R x → at x = L/2: M = (M/L)*(L/2) = M/2.Across the applied couple, internal M has a jump of magnitude M; immediately to the right of mid-span, M = M/2 − M = −M/2.Thus the maximum absolute bending moment in the beam is M/2.

Verification / Alternative check:
Shear is piecewise constant, so M(x) is linear on each side, peaking in magnitude at mid-span with value ±M/2.

Why Other Options Are Wrong:

• M: would require zero reactions or fixed ends; not applicable here.
• M/3, ML/2: do not satisfy equilibrium with pin supports under a mid-span couple.

Common Pitfalls:

• Assuming a couple creates zero internal moments everywhere; it induces linear M with a jump at the application point.

Final Answer:
M/2