Cantilever with uniformly distributed load (UDL): for a cantilever beam of length l carrying a UDL of w per unit length, what is the bending moment at the fixed end?

Introduction / Context:Knowing end moments for standard loading cases is essential for rapid design. A classic case is a cantilever with a uniformly distributed load (UDL), where the critical bending moment occurs at the fixed support.

Given Data / Assumptions:

• Cantilever span = l.
• UDL intensity = w (force per unit length).
• Static equilibrium; prismatic member.

Concept / Approach:For a cantilever, the support must balance the resultant of the distributed load. The resultant of a UDL equals w * l acting at the centroid of the load distribution, which is at l/2 from the support for a full-length UDL. The fixed-end moment equals the moment of this resultant about the support.

Step-by-Step Solution:

Verification / Alternative check:Derive from the shear diagram (linear from −w l to 0) and integrate to get a quadratic bending moment diagram peaking at w l^2 / 2 at the fixed end.

Why Other Options Are Wrong:w l / 4, w l / 2, w l: Missing the essential l^2 dependence of moment.w l^2 / 8: This corresponds to other common cases (e.g., simply supported with central point load) and is not applicable here.

Common Pitfalls:Confusing cantilever results with simply supported beam formulas; forgetting the location of the UDL resultant.

Final Answer:

w l^2 / 2