Cantilever with triangular (linearly varying) load: A cantilever beam carries a load that varies linearly from zero at the free end to a maximum at the fixed end. The shape of the bending-moment diagram is a:

Introduction / Context:
Recognizing shear-force and bending-moment (BM) diagram shapes for standard load cases enables quick checks on analysis and design.

Given Data / Assumptions:

• Cantilever fixed at one end, free at the other.
• Load intensity w(x) increases linearly from 0 at the free end to w_max at the fixed end (triangular load).
• Prismatic member, small deflection, linear elasticity.

Concept / Approach:
The mathematical orders relate as: load w(x) is the derivative of shear V(x) with a sign, and shear is the derivative of moment M(x) with a sign. If w(x) is linear in x, then V(x) is quadratic, and M(x) is cubic in x.

Step-by-Step Solution:
Let x be the distance from the fixed end.Given w(x) ∝ x (linear) ⇒ V(x) = −∫ w(x) dx ⇒ V(x) is quadratic in x.M(x) = −∫ V(x) dx ⇒ M(x) is cubic in x.Therefore, the BM diagram is a cubic parabola.

Verification / Alternative check:
For UDL (constant w), BM is parabolic; increasing the order of w by one (to linear) increases the order of M by one (to cubic), confirming the shape.

Why Other Options Are Wrong:
Triangle/Rectangle correspond to shear diagrams in some cases, not BM here.Parabola applies to UDL, not linearly varying load.

Common Pitfalls:
Mixing up the roles of w, V, and M in differential relationships or reading the diagram orientation incorrectly.

Final Answer:
Cubic parabola.