Shear force sign change and bending moment: At a section where the shear force changes sign, what happens to the bending moment?

Introduction / Context:Shear force (V) and bending moment (M) are related by beam theory: dM/dx = V and dV/dx = w, where w is the load intensity. Recognizing the qualitative link between the sign of V and extrema of M is key in plotting diagrams quickly and accurately.

Given Data / Assumptions:

• Prismatic beam with standard loading.
• Shear force diagram (SFD) crosses the zero line (sign change) at a particular section.
• Continuum mechanics relations apply (beam is linearly elastic within the analysis region).

Concept / Approach:If dM/dx = V, then where V = 0, the slope of the bending moment diagram is zero, which indicates a local extremum (maximum or minimum) in M, provided the function is smooth there. Hence a sign change in V broadly marks a peak or valley in M, not necessarily a zero moment value.

Step-by-Step Solution:

Verification / Alternative check:Plot M for a simply supported beam with UDL: V crosses zero at midspan and M is maximum there. For a cantilever with point load at the tip, V does not change sign in span and M is monotonic—consistent with the rule.

Why Other Options Are Wrong:Zero moment occurs at supports of simply supported beams, not necessarily where V changes sign. Infinity/undefined is non-physical here. Constancy of M implies V = 0 everywhere in that region, which is not the case around a sign change.

Common Pitfalls:Assuming “zero shear implies zero moment.” The correct inference is an extremum of M, not necessarily M = 0.

Final Answer:It attains an extremum (maximum or minimum) at that section