Beam with equal overhangs under full-length UDL — midspan bending moment condition A simply supported beam with equal overhangs a on both sides (total length l + 2a) carries a uniformly distributed load over its entire length. The bending moment at the middle of the span will be zero if:

Introduction / Context:
Beams with overhangs subjected to a uniformly distributed load (UDL) develop hogging moments over the supports and sagging moments within the span. For certain overhang-to-span ratios, the sagging at midspan can be exactly cancelled by the overhang effects, making the midspan bending moment zero. This is a classic contraflexure balance problem.

Given Data / Assumptions:

• Equal overhangs each of length a; clear span between supports is l.
• UDL of constant intensity along entire (l + 2a).
• Linear elastic behaviour; small deflections.

Concept / Approach:

By symmetry, reactions are equal. The midspan moment is influenced by span loading (sagging) and by negative moments transmitted from the loaded overhangs (hogging at supports). Setting the resultant midspan moment to zero yields a relationship between l and a.

Step-by-Step Solution (outline):

Verification / Alternative check:

Standard beam tables confirm that for equal overhangs under continuous UDL, the midspan moment changes sign through zero at l = 2a. For l < 2a, the overhang effect dominates (midspan hogging), and for l > 2a, sagging persists at midspan.

Why Other Options Are Wrong:

• l = 4a, l > a, l > 3a: do not satisfy the precise balance condition.
• l < 2a: implies reversal but not zero at midspan; the equality is required for zero specifically at the middle.

Common Pitfalls:

• Mistaking the condition for zero support moment rather than zero midspan moment.

Final Answer:

l = 2a.