Shape of bending moment diagram for a cantilever with full-length uniformly distributed load Consider a cantilever beam carrying a uniformly distributed load over its entire span. What is the shape of the bending moment (B.M.) diagram?

Introduction / Context:Recognizing the qualitative shape of shear force (S.F.) and bending moment (B.M.) diagrams is a fundamental skill in structural analysis. It enables quick checks and hand calculations before detailed numerical analysis.

Given Data / Assumptions:

• Cantilever beam with a uniformly distributed load of intensity w over the full length L.
• Static equilibrium, small deformations, prismatic member.
• Sign convention: negative moment at the fixed end (sagging/hogging depends on convention).

Concept / Approach:

For a cantilever under UDL, the shear force at a distance x from the free end is V(x) = w x (taking zero at the free end), which varies linearly with x. The bending moment is the integral of shear, M(x) = ∫ V dx = w x^2/2 (with sign per convention), hence a second-degree curve (parabola) that is zero at the free end and maximum in magnitude at the fixed end.

Step-by-Step Solution:

Verification / Alternative check:

Different sign conventions invert the diagram vertically but do not change its parabolic shape. The curvature of the elastic line is proportional to moment; a parabolic M(x) produces cubic deflection, consistent with classical beam theory results.

Why Other Options Are Wrong:

• Triangle/rectangle/trapezium: These correspond to constant or linearly varying moments or to S.F. shapes, not to the cantilever B.M. under UDL.
• Cubic parabola: That would arise for the slope/deflection shapes, not for the bending moment here.

Common Pitfalls:

• Confusing S.F. (linear) and B.M. (parabolic) shapes.
• Starting the S.F. at the fixed end instead of the free end and losing track of sign/shape.

Final Answer:

parabola.