Shear–Moment Relationship in Beams If the shear force V at a section of a beam is zero, what can be stated about the bending moment M at that same section?

Introduction / Context:
The fundamental differential relationship between shear force and bending moment is dM/dx = V. This means the slope of the bending moment diagram equals the shear force at that section. Understanding this directly connects where moments are extreme to where shear crosses zero, a vital concept for locating critical sections for design.

Given Data / Assumptions:

• Prismatic beam under static loading.
• Loads are piecewise continuous (no impulsive singularities at the section).
• Sign conventions are consistent.

Concept / Approach:

If V = 0 at a section, then dM/dx = 0 at that location. A zero derivative indicates an extremum (a local minimum or maximum) of the moment diagram, provided continuity holds and there is no point of non-differentiability due to a concentrated moment at the section.

Step-by-Step Solution:

Verification / Alternative check:

Check adjacent segments of the moment diagram: if M changes from increasing to decreasing, the extremum is a maximum; if from decreasing to increasing, it is a minimum.

Why Other Options Are Wrong:

• Zero: Only true if boundary conditions impose M = 0 (e.g., at a simply supported end), not generally valid when V = 0.
• Maximum or minimum alone: Without context, either could occur; hence the correct statement is “minimum or maximum.”

Common Pitfalls:

Ignoring concentrated moments which create jumps in M; misreading sign conventions and incorrectly identifying extrema locations.

Final Answer:

Minimum or maximum