Bending moment diagram shape – cantilever with tip load For a cantilever beam carrying a single concentrated load at its free end, what is the shape of the bending moment diagram along the span?

Introduction / Context:Recognizing standard shear force and bending moment diagram (SFD/BMD) shapes speeds up preliminary design and checking of members in frames, bridges, and machine elements.

Given Data / Assumptions:

• Cantilever with fixed end at the wall and free end at the tip.
• Single point load W applied at the free end.
• Linear elastic, small deflection theory.

Concept / Approach:The bending moment at a section located a distance x from the fixed end equals the end shear times the lever arm from the section to the free end: M(x) = −W * (L − x). This relation is linear in x, varying from −W L at the fixed end to 0 at the free end, which graphically is a straight line—hence a triangular diagram when plotted against the beam axis.

Step-by-Step Solution:Take section at distance x from fixed end; free end distance = L − x.Bending moment: M(x) = −W (L − x) → linear in x.At x = 0 (fixed end): M = −W L (maximum hogging).At x = L (free end): M = 0.A straight-line variation from −W L to 0 plots as a triangle.

Verification / Alternative check:The shear force is constant V(x) = −W along the entire span, whose integral gives a linear (triangular) bending moment diagram.

Why Other Options Are Wrong:Rectangle corresponds to constant bending moment, which does not occur here. Parabola or cubic parabola are associated with distributed loads or varying loads, not a single point load on a cantilever.

Common Pitfalls:Reversing sign conventions or plotting from the free end; always check boundary values: BMD must be zero at the free end and maximum at the fixed end for this case.

Final Answer:triangle