Simply supported beam – central point load For a simply supported beam of span l carrying a point load W at the centre, what is the maximum bending moment?

Introduction / Context:
Bending moment evaluation is fundamental in beam design. A classic case is a simply supported beam with a central point load, which appears in bridges, crane girders, and machine frames. Knowing the closed-form maximum bending moment allows quick sizing and verification.

Given Data / Assumptions:

• Straight, prismatic beam of span l.
• Point load W applied at midspan.
• Supports are simple (pin and roller), so end moments are zero.
• Linearly elastic behavior; small deflections.

Concept / Approach:
For symmetric loading, reactions at supports are equal. The internal bending moment diagram features a peak under the load. The maximum bending moment occurs where shear force crosses zero, which, for symmetry, is at the midspan.

Step-by-Step Solution:
Support reactions: R_A = R_B = W / 2.Section at a distance x from the left: M(x) = R_A * x for x ≤ l/2; beyond midspan, M(x) decreases symmetrically.At x = l/2: M_max = (W / 2) * (l / 2) = W l / 4.

Verification / Alternative check:
Shear force V(x) is W / 2 up to midspan, then −W / 2 after midspan; V changes sign at the centre, confirming the location of M_max. Standard tables also list M_max = W l / 4.

Why Other Options Are Wrong:
W l / 2 and W l are too large; W l^2 / 4 has wrong dimensions; W l / 8 is the maximum reaction times half span error, not the correct bending moment.

Common Pitfalls:
Mixing up total load and reaction; writing W l^2 / 8 (a deflection-like form) by mistake; forgetting symmetry.

Final Answer:
W l / 4