Propped cantilever under full uniformly distributed load A cantilever beam of length L carrying a uniformly distributed load over its full length is propped at the free end so that the free end remains level with the fixed end. The bending moment is zero at the free end and also at which approximate location along the beam?

Introduction / Context:
A propped cantilever combines features of both a fixed-free cantilever and a simple support at the free end. Under a uniformly distributed load (UDL), the internal bending moment diagram has zeros at locations where the curvature changes sign, known as points of contraflexure. This question probes the qualitative position of that second zero moment within the span.

Given Data / Assumptions:

• Beam length L; uniformly distributed load w over 0 to L.
• Fixed at x = 0; propped (vertical reaction) at x = L so that deflection at x = L is zero (same level as fixed end).
• Linear elastic behavior; small deflections.

Concept / Approach:

Because the tip is propped to zero deflection, bending moment vanishes at both ends: M(0) = 0 (due to compatibility for this statically indeterminate system) and M(L) = 0 at the prop. The UDL creates a hogging domain near the fixed support and a sagging domain closer to the free end with a transition (M = 0) inside the span. Classical analysis or standard charts place this interior zero approximately near the quarter point from the free end.

Step-by-Step Solution (outline):

Verification / Alternative check:

Standard propped-cantilever tables for a full-length UDL show a contraflexure near the quarter point from the free end, corroborating the qualitative result required by the MCQ.

Why Other Options Are Wrong:

• Fixed end and mid-span are not zero-moment points for this case.
• Three-quarters from free end corresponds to near the fixed region, which is not where the second zero occurs.
• L/3 is not supported by canonical solutions.

Common Pitfalls:

• Assuming the free end moment must be nonzero as in a basic cantilever; the prop alters boundary conditions.

Final Answer:

At about one-quarter of the length from the free end (≈ L/4).