Bending moment–shear force relationship: the bending moment on a section is maximum at locations where the shear force is zero (i.e., where the SF diagram changes sign).

Introduction / Context:
Shear force (SF) and bending moment (BM) diagrams are linked by differential relationships. Identifying where BM attains maxima or minima is critical for section design and reinforcement placement.

Given Data / Assumptions:

• Beam under transverse loading.
• Linearly elastic behavior; small deflections.
• Continuous SF and BM functions except at point loads or couples.

Concept / Approach:
The relationship is dM/dx = V, where M is bending moment and V is shear force. Therefore, extremum of M occurs where dM/dx = 0, i.e., where V = 0. At those points the SF diagram crosses or touches the axis, often changing sign.

Step-by-Step Solution:

Verification / Alternative check:
For a simply supported beam with a central point load, SF changes sign at midspan and BM is maximum there. Similarly, under uniformly distributed load, SF passes through zero at midspan where BM peaks.

Why Other Options Are Wrong:
'Minimum' or 'maximum' SF does not directly determine BM extremum; the zero condition does.'Changing sign' is often true at the same location, but the defining condition is V = 0.

Common Pitfalls:
Confusing zero shear with zero moment points; forgetting that point couples create jumps in M without V being zero.

Final Answer:

zero