Standard maximum deflection formulas (small deflection, prismatic beams) Which of the following expressions for maximum deflection δ_max are correct?

Introduction / Context:
Deflection control is essential for serviceability. Memorizing the classic small-deflection formulas for common loading cases saves time during preliminary design and checks.

Given Data / Assumptions:

• Slender, prismatic beams obeying Euler–Bernoulli theory.
• Small deflections; linear material behavior.
• Loads as stated (point load W or uniformly distributed load w).

Concept / Approach:
The listed formulas are standard closed-form solutions obtained by integrating the differential equation E * I * d^2y/dx^2 = M(x) with appropriate boundary conditions for each case.

Step-by-Step Solution (outline):

Verification / Alternative check:
Cross-reference with standard tables (Roark, structural handbooks) or quick energy methods (Castigliano’s theorem) which yield identical results.

Why Other Options Are Wrong:
Only option e collects the correct set; each of a–d is independently correct, so “All of the above” is the best answer.

Common Pitfalls:
Mixing the constants (3 vs 8 vs 48 vs 384); remembering that cantilevers deflect more than simply supported beams for similar loading is a useful check.

Final Answer:
All of the above