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?
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ABending moment is constant between those points
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BBending moment is zero between those points
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CBending moment varies linearly with position
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DBending moment varies parabolically with position
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EBending moment becomes infinite at the ends
Answer
Correct Answer: Bending moment is constant between those points
Explanation
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:
No load between the points → w = 0.Hence dV/dx = 0 → V is constant (often zero in unloaded segments of idealized problems).If V = 0 in that region → dM/dx = 0 → M is constant.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