When the shear force diagram passes through the zero axis, the corresponding bending moment at the same location will be...?

Question

BlackTom AI · Accepted Answer

Consider the relationship between the shear force diagram (SFD) and the bending moment diagram (BMD): the slope of the bending moment at any location equals the shear force at that location. With that in mind, we can evaluate each option.

Option a: 'equal to zero' — This would claim that the bending moment value is zero exactly where the shear crosses zero. While it is possible for the moment to be zero at certain points, the fact that the shear force is zero does not guarantee that the moment itself is zero at that exact location. The moment value depends on the accumulated bending effect from the loads along the beam up to that point, not solely on the instantaneous shear being zero. Therefore, this option is not necessarily true in general.

Option b: 'above the zero axis line, i.e., negative' — Saying the moment is negative simply because the shear crosses zero ignores the overall loading history and sign conventions. A zero shear point does not imply a fixed sign of the moment; the moment could be positive or negative depending on how the bending accumulates over the span. Hence this is not a reliable general statement.

Option c: 'below the zero axis line, i.e., positive' — This is the symmetric misinterpretation of option b: asserting a persistent sign (positive moment) based on a zero shear crossing. As with option b, the actual sign of the moment at the zero-shear location depends on the broader loading and can vary. Therefore this option is also not generally correct.

Option d: 'a local maximum or minimum' — This aligns with the fundamental relationship: M′(x) = V(x), where M is the bending moment and V is the shear force. When V(x) crosses zero and changes sign, the derivative of M changes sign, indicating a local extremum (maximum or minimum) of the bending moment. Even in cases where V crosses zero and then returns, the zero-crossing point corresponds to a stationary point on the M diagram, i.e., a local extremum. This option accurately describes the typical behavior of the BMD at a zero shear location.

In summary, the critical idea is that the slope of the moment diagram equals the shear force, so a zero crossing of shear indicates a stationary point on the moment diagram, which is a local maximum or minimum. The other options misinterpret the direct consequence of V = 0 on M at that point, neglecting the accumulated bending effects and sign conventions from the loading profile.

When the shear force diagram passes through the zero axis, the corresponding bending moment at the same location will be...?

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