Cantilever fundamentals:\nWhat is the bending moment at the free end of a cantilever beam (no overhang beyond the free end)?

Introduction / Context:
A cantilever is fixed at one end and free at the other. The distribution of shear force and bending moment depends on the boundary conditions. Knowing that the bending moment at the free end is zero is a staple check in structural analysis and an immediate boundary condition for drawing moment diagrams.

Given Data / Assumptions:

• Beam: classic cantilever (one fixed, one free end).
• No overhang or external couple applied exactly at the free end.
• Loads may be present on the span, but end is moment-free.

Concept / Approach:
Bending moment represents internal couples balancing external actions. At a free end, there is no support to develop a counteracting couple; hence the internal bending moment must be zero at that section. This is a boundary condition independent of the specific loading along the span.

Step-by-Step Solution:

Verification / Alternative check:
For typical cases (end point load, UDL, triangular load), plotting M(x) shows M(0 at free end) = 0 and maximum magnitude at the fixed support, confirming the boundary condition.

Why Other Options Are Wrong:

• “Minimum but not zero” contradicts statics; the minimum value is exactly zero.
• “Maximum” occurs at the fixed end, not the free end.
• “Equal to load times span” applies to specific cases at the fixed end.
• “Indeterminate” is incorrect because boundary conditions determine this unambiguously.

Common Pitfalls:
Confusing shear boundary conditions with moment boundary conditions; at a free end both shear and moment are zero unless a point load or couple is applied at the tip.

Final Answer:
Zero