For a prismatic beam that carries no external load along its span, what is the resulting shape of the bending moment diagram over the entire length?

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:

  • No distributed or concentrated loads along the beam span.
  • Supports or end couples may exist, but there is no load intensity w(x) between them.
  • Beam is prismatic and behaves linearly elastically.



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:

  • Parabolic/cubical/circular: require varying shear or distributed load; not applicable when w(x) = 0.



Common Pitfalls:

  • Confusing the shape produced by a uniform load (parabolic M) with the unloaded span (linear M).



Final Answer:
linear

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