Difficulty: Easy
Correct Answer: linear
Explanation:
Introduction / Context:
In structural analysis of beams, the relationships between load w(x), shear force V(x), and bending moment M(x) are fundamental. Recognizing how the absence or presence of loads affects the bending moment diagram is essential for quick checks and design sanity.
Given Data / Assumptions:
Concept / Approach:
The differential relations are:dV/dx = -w(x)dM/dx = V(x)With w(x) = 0 over the span, shear force is constant, and therefore bending moment varies linearly with x.
Step-by-Step Solution:
Set w(x) = 0 → dV/dx = 0 → V(x) = constant.Since dM/dx = V(x) and V is constant → M(x) is a linear function of x.Hence, the bending moment diagram is a straight line (which may be a horizontal line if V = 0 or an inclined straight line if V ≠ 0).
Verification / Alternative check:
Consider a span with only end moments (no shear). V(x) = 0, so M(x) = constant → the diagram is a straight horizontal line, confirming linearity.
Why Other Options Are Wrong:
Common Pitfalls:
Final Answer:
linear
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