Location of maximum bending moment using the shear-force relation In a loaded beam, at what location does the maximum (or minimum) bending moment occur with respect to the shearing force diagram?

Introduction / Context:
The relationship between shear force and bending moment is foundational in structural analysis. Designers routinely use the shear-force diagram (SFD) to locate critical points for the bending-moment diagram (BMD).

Given Data / Assumptions:

• Beam under transverse loads with well-behaved diagrams.
• Sign convention is consistent (sagging positive).

Concept / Approach:
The differential relationships are:
dM/dx = VdV/dx = wwhere M is bending moment, V is shear force, and w is the distributed load intensity. A stationary point (extreme) of M occurs where dM/dx = 0, i.e., where V = 0. In practice, this is where the SFD crosses the axis (changes sign).

Step-by-Step Solution:

Verification / Alternative check:
Sketch SFD and BMD for common cases (point load at midspan, uniform load). In each case, the peak of M aligns with a zero crossing of V.

Why Other Options Are Wrong:

• Maximum or minimum V values (options a and b) do not generally correspond to extrema of M.
• Any section with nonzero shear (option c) implies dM/dx ≠ 0, not an extremum.
• Midspan regardless of loading (option e) is only true for certain symmetric cases.

Common Pitfalls:
Confusing zeros at supports (M = 0 for simple supports) with internal maxima; always check V = 0 within the span.

Final Answer:
Where shear force changes sign (passes through zero)