For a simply supported beam, what is the ratio of the maximum mid-span deflection under a central point load W to that under a uniformly distributed load w over the whole span, assuming equal total load (W = wL)?

Introduction / Context:
Comparing deflections under different load patterns helps engineers judge serviceability. For the same total load, a concentrated load produces a different deflection shape and magnitude than a uniformly distributed load (UDL).

Given Data / Assumptions:

• Simply supported beam of span L.
• Case 1: Central point load W.
• Case 2: UDL of intensity w over length L with W = wL (equal total load).
• Material linear elastic with modulus E and constant second moment I.

Concept / Approach:
Use standard deflection formulas at mid-span:Point load at centre: δ_PL = W * L^3 / (48 * E * I).UDL over L: δ_UDL = 5 * w * L^4 / (384 * E * I).With W = wL, take the ratio δ_PL / δ_UDL.

Step-by-Step Solution:
δ_PL = W L^3 / (48 E I).δ_UDL = 5 w L^4 / (384 E I).Let W = wL → δ_PL / δ_UDL = [W L^3 / 48] / [5 (W/L) L^4 / 384] = (1/48) / (5/384) = 384 / 240 = 1.6.

Verification / Alternative check:
Numerical example with E = I = L = 1, W = 1 → w = 1; compute both deflections to confirm 1.6 ratio.

Why Other Options Are Wrong:

• 1.2, 0.8, 2.0: not consistent with standard closed-form solutions and equal-load assumption.

Common Pitfalls:

• Comparing deflections without equating total load (W ≠ wL).
• Using wrong mid-span formula or mixing cantilever and simply supported results.

Final Answer:
1.6