Relationship between shear force and bending moment: If there is no increase or decrease in shear force between two points (i.e., zero load intensity over that region), what does it indicate about the bending moment between those points?

Introduction / Context:
Beam diagram interpretation hinges on the differential relations: dV/dx = w and dM/dx = V, where w is load intensity, V is shear force, and M is bending moment. Understanding how constancy or variation of one quantity affects the others is essential for quick diagram sketching and checking.

Given Data / Assumptions:

• Between two points along the span, there is no distributed load: w = 0.
• As a result, the shear force does not increase or decrease in that region (V is constant; may be zero or nonzero).
• Classical beam theory, small deflections.

Concept / Approach:
If w = 0, then dV/dx = 0 → V is constant. Next, dM/dx = V: if V is constant but nonzero, M varies linearly; if V = 0 specifically, M is constant. The phrase 'no increase or decrease in shear force' is typically used to denote w = 0, and especially the common case where V = 0, leading to locally constant bending moment. In many textbook statements: 'Where shear is zero, bending moment is maximum or minimum (locally constant slope zero).' For a region with V identically zero, M is constant across that region.

Step-by-Step Solution:

Verification / Alternative check:
In a simply supported beam with a pure couple applied inside the span, the adjacent unloaded region has zero V and thus constant M, matching this conclusion.

Why Other Options Are Wrong:

• Zero M is not implied by constant shear; only zero shear implies constant M (not necessarily zero in value).
• Parabolic variation arises from linearly varying shear, not constant or zero shear.
• Infinite bending moments do not occur in standard statics problems.

Common Pitfalls:
Confusing 'constant shear' (nonzero) with 'zero shear'; forgetting that M'(x) = w, so w = 0 makes M linear unless V = 0 makes it flat (constant).

Final Answer:
Bending moment is constant between those points