Difficulty: Medium
Correct Answer: w * l^2 / 6
Explanation:
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:
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:
Resultant load R = (1/2) * w * l.Distance of R from free end = 2l/3 → from fixed end = l − 2l/3 = l/3.Shear at fixed end V_fixed = R = (w * l) / 2.Bending moment at fixed end M_fixed = R * (l/3) = (w * l / 2) * (l/3) = w * l^2 / 6.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
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