Structural Analysis – Propped cantilever of span L under uniform load w (per unit length) With a prop at the free end ensuring zero deflection there, what is the bending moment at the fixed end?

Introduction / Context:
A propped cantilever is a statically indeterminate beam with a vertical prop at the free end. Under a uniformly distributed load (UDL), the prop reaction is chosen so that the deflection at the free end is zero. Determining the fixed-end moment requires compatibility (deflection) in addition to equilibrium. This problem tests the standard result and the reasoning behind it.

Given Data / Assumptions:

• Beam fixed at one end, propped vertically at the other (no settlement).
• Uniformly distributed load = w over entire span L.
• Linearly elastic behavior (E, I constant).
• Small deflection theory; superposition applies.

Concept / Approach:

Use the deflection compatibility at the free end: total deflection due to UDL and due to the (unknown) prop reaction must sum to zero. For a cantilever, end deflection from a UDL w is δ_w = w L^4 / (8 E I). End deflection from an upward force R at free end is δ_R = − R L^3 / (3 E I). Setting δ_w + δ_R = 0 gives R. Then compute the fixed-end moment from resultant actions on the cantilever (UDL minus the end reaction).

Step-by-Step Solution:

Verification / Alternative check:

Known closed-form solutions for a propped cantilever under UDL list R = 3 w L / 8 and M_A = w L^2 / 8, confirming the computed value.

Why Other Options Are Wrong:

w L^2 / 2 and w L^2 / 4 ignore the relieving effect of the prop. w L^2 / 12 and w L^2 / 16 are too small and do not satisfy compatibility with the computed prop reaction.

Common Pitfalls:

Forgetting to include the additive effect of the prop reaction on fixed-end moment, or mixing sign conventions. Also, some confuse the indeterminate propped cantilever with a simple cantilever.

Final Answer:

w L^2 / 8