In structural engineering, for a simply supported beam carrying a central load, where does the bending moment reach its maximum value and where is it least?

Introduction / Context:
A simply supported beam is one of the most common structural members studied in strength of materials. Understanding bending moment distribution under various load cases is critical for safe design.

Given Data / Assumptions:

• Simply supported beam with span L.
• Single concentrated load W at mid-span.
• Supports are idealized as pin and roller.

Concept / Approach:
For symmetric loading, reactions at both supports equal W/2. The bending moment at a section x from one support is M(x) = (W/2) * x for 0 ≤ x ≤ L/2. This increases linearly up to the mid-span. Maximum occurs at mid-span, then decreases symmetrically.

Step-by-Step Solution:
Reaction at each support = W/2.Bending moment at mid-span = (W/2) * (L/2) = WL/4.Moment at supports = 0.Thus, maximum bending moment occurs at the centre, minimum at the supports.

Verification / Alternative check:
Plotting the bending moment diagram shows a triangular distribution with apex at mid-span, confirming the result.

Why Other Options Are Wrong:

• Least at centre: incorrect, it is maximum there.
• Maximum at supports: false, bending moment is zero at supports.
• Least at supports: true, but incomplete, as the maximum is at centre.

Common Pitfalls:

• Confusing shear force diagram with bending moment diagram.
• Forgetting symmetry of load distribution.

Final Answer:
maximum at the centre