Simply supported beam with a central point load: Does the maximum bending moment occur at the point of loading (i.e., at midspan)?

Introduction / Context:
For a simply supported beam under a single central point load, identifying where the maximum bending moment occurs is fundamental. Designers use this to select section size, reinforcement, and to estimate deflection and stresses safely.

Given Data / Assumptions:

• Beam is simply supported at both ends.
• A single point load P acts at the midspan (center).
• Material is linearly elastic; self-weight ignored for clarity.

Concept / Approach:
The shear force diagram changes sign at the location of maximum bending moment. For a symmetric beam and centrally placed load, the symmetry dictates that the maximum moment occurs at the midspan under the load.

Step-by-Step Solution:

Verification / Alternative check:
From calculus, dM/dx = V. Setting V = 0 gives the extremum at midspan; checking neighbouring values confirms a maximum.

Why Other Options Are Wrong:
False or conditional statements contradict the shear sign-change rule and symmetry for a central point load on a simply supported beam.

Common Pitfalls:
Confusing with uniformly distributed loads (UDL) or off-center point loads; mixing simply supported with fixed-ended boundary conditions.

Final Answer:

True