Cantilever with Uniformly Distributed Load (UDL) For a cantilever of length l carrying a uniformly distributed load w per unit length, what is the bending moment at the free end?

Introduction:
Bending moment diagrams (BMD) and shear force diagrams (SFD) are foundational tools in beam analysis. Recognizing key values at supports and free ends for standard loads is essential for quick design checks.

Given Data / Assumptions:

• Beam type: cantilever of length l.
• Loading: uniformly distributed load w per unit length over the entire span.
• Small deflection, linear elastic behavior; prismatic member.

Concept / Approach:
For a cantilever with UDL, the shear and moment vary with distance x measured from the free end (or from the fixed end, consistently). At the free end there is no support to provide a resisting couple. Hence, the bending moment must be zero at the free end, and it reaches its maximum magnitude at the fixed end.

Step-by-Step Solution:
Resultant of UDL: W_total = w * l acting at midspan from the free end.Shear at a section x from the free end: V(x) = w * x.Moment at the same section: M(x) = w * x^2 / 2 (when origin is at the free end).At x = 0 (free end): M(0) = 0, confirming zero bending moment.

Verification / Alternative check:
Compute the fixed end moment: M_fixed = w * l^2 / 2. The BMD varies quadratically from 0 at the free end to this maximum at the fixed end, consistent with theory.

Why Other Options Are Wrong:
w l / 4 and w l / 2: nonzero values contradict boundary condition at the free end.w l: also violates the free-end zero-moment condition and has wrong dimensions if units are not balanced.

Common Pitfalls:
Mixing up shear and moment boundary conditions; remember free end has zero moment and zero reaction.Placing the origin at the fixed end and forgetting M = 0 at the free end by boundary condition.

Final Answer:
zero