A simply supported uniform beam under a full-length uniformly distributed load is propped at mid-span to keep the level there equal to the end supports. What is the net bending moment at the centre after propping?

Introduction / Context:
Propping a simply supported beam at mid-span changes internal force distribution. Using superposition and compatibility, we can determine the central reaction and the resulting bending moment at the centre.

Given Data / Assumptions:

• Span L, UDL intensity w over entire length.
• Prop at mid-span enforces zero deflection at the mid-point (same level as supports).
• Linear elastic behavior; superposition valid.

Concept / Approach:
Superpose two systems: (1) simply supported beam under UDL, (2) same beam under an upward point load R_p at mid-span (the prop reaction). Choose R_p such that net mid-span deflection is zero, then compute the net central moment.

Step-by-Step Solution:
Mid-span deflection due to UDL: δ_w = 5 w L^4 / (384 E I).Mid-span deflection due to upward point load R_p: δ_p = R_p L^3 / (48 E I) upward.Compatibility: δ_w = δ_p → R_p = (5/8) w L.Central bending moment due to UDL alone: M_w = w L^2 / 8.Central bending moment due to R_p (upward) alone: M_p = - R_p * L / 4 = - (5/8) w L * L / 4 = - 5 w L^2 / 32.Net M_center = M_w + M_p = w L^2 / 8 - 5 w L^2 / 32 = w L^2 / 32.

Verification / Alternative check:
Sign and magnitude are consistent with a reduced but non-zero sagging moment at the centre after propping.

Why Other Options Are Wrong:

• wL^2/8 is the unpropped moment; others do not follow from the compatibility-based reaction.

Common Pitfalls:

• Mistakes in the deflection formulas or forgetting the negative sign for the upward point load contribution.

Final Answer:
w L^2 / 32