Cantilever with linearly varying load (from zero at free end to w at fixed end): Bending moment at the fixed end (length l)?

Introduction / Context:
A cantilever under a triangular (uniformly varying) load is a common exam case. Knowing how to replace distributed loads with resultants and proper lines of action helps compute support shear and moment quickly.

Given Data / Assumptions:

• Cantilever of length l, fixed at one end, free at the other.
• Load intensity varies from 0 at free end to w at fixed end (linear variation).
• Linearly elastic behavior.

Concept / Approach:
Replace the UVL by its resultant and location. For a triangular load with peak w at the fixed end, the resultant equals area of triangle and acts at one-third of its base from the higher-intensity end (i.e., at l/3 from the fixed end, or 2l/3 from the free end).

Step-by-Step Solution:

Verification / Alternative check:
Check units (w in force/length, M in force*length). Also compare with UDL case (w * l^2 / 2) to see the smaller magnitude for triangular loading as expected.

Why Other Options Are Wrong:
w * l / 2 is shear, not moment; w * l and w * l^2 / 2 correspond to other load cases; w * l^3 / 3 has wrong dimensions for moment.

Common Pitfalls:
Placing the resultant at 2l/3 from the fixed end instead of from the free end; mixing up shear and moment formulas.

Final Answer:
w * l^2 / 6