Overhangs on a uniformly loaded simply supported beam A simply supported beam carries a uniformly distributed load across its span and has two equal overhangs. To make the maximum bending moment as small as possible, what should be the ratio (overhang length) / (total beam length)?

Introduction / Context:
Overhangs can be used to reduce peak bending moments under uniform loads by shifting the location and magnitude of positive and negative moments. There exists an optimal overhang ratio that minimizes the maximum absolute bending moment in the member.

Given Data / Assumptions:

• Simply supported interior span with identical overhangs at both ends.
• Uniformly distributed load of constant intensity over the entire physical length.
• Prismatic beam; linear elastic behavior.

Concept / Approach:
The optimal condition is found by equating the magnitude of the maximum positive moment in the span to the magnitude of the maximum negative moment over the supports, then solving for the ratio a/L, where a is each overhang and L is the total physical length.

Step-by-Step Solution:
Let w be the load per unit length, a the overhang on each end, and L the total length.Write expressions for the reactions and bending moments as functions of a/L.Set the peak negative support moment equal in magnitude to the peak positive span moment.Solve the resulting equation for a/L; the well-known solution is approximately 0.207 (about 20.7% of total length).

Verification / Alternative check:
Design handbooks and classical beam tables list this optimum ratio near 0.207. A quick energy or influence-line based derivation confirms the same optimum.

Why Other Options Are Wrong:
Values 0.307, 0.407, 0.508 and 0.107 either overestimate or underestimate the required overhang, yielding unbalanced positive and negative peaks and thus a larger maximum bending moment.

Common Pitfalls:
Confusing the interior clear span with total length; forgetting that both overhangs must be equal; applying point-load formulas instead of uniform-load relations.

Final Answer:
0.207