Simply supported beam with a central point load W For a prismatic simply supported beam of span L carrying a single concentrated load W at midspan, what is the maximum vertical deflection δ_max at the centre (assume linear elastic behaviour with modulus E and second moment of area I)?

Introduction / Context:
This question checks recall and correct application of the classic deflection formula for a simply supported beam subjected to a single central point load. Such closed-form results are used constantly in preliminary sizing and quick serviceability checks for beams in civil and mechanical engineering.

Given Data / Assumptions:

• Span between simple supports = L.
• Point load W acts at midspan.
• Material is linearly elastic with Young’s modulus E.
• Second moment of area about the bending axis = I (constant along the span).
• Small deflection (Euler–Bernoulli) theory; plane sections remain plane.

Concept / Approach:

The maximum deflection for this loading case occurs at midspan. It can be obtained from double integration of the elastic curve equation E I d²y/dx² = M(x), area-moment method, or standard tables. All methods give the same closed form: δ_max = W L^3 / (48 E I).

Step-by-Step Solution:

Verification / Alternative check:

Area-moment method: slope discontinuity areas lead to the same coefficient 1/48. Numerical checks for typical values confirm midspan deflection scales with L^3 and inversely with E I as expected physically.

Why Other Options Are Wrong:

• W L^3/(3 E I) and W L^3/(16 E I) greatly overestimate deflection; wrong coefficients.
• W L^2/(8 E I) has wrong length power (should be L^3).
• W L^4/(384 E I) corresponds to a different boundary condition (cantilever/UDL mix) and wrong dimension.

Common Pitfalls:

• Forgetting the 1/48 coefficient or mixing with UDL formula δ_max = 5 w L^4/(384 E I).
• Using section modulus instead of second moment of area for deflection problems.

Final Answer:

δ_max = W L^3 / (48 E I).