For a fixed-ended (both ends built-in) beam of length l carrying a central point load W, what is the maximum deflection?

Introduction / Context:
Beam deflection formulas are essential in stiffness checks and serviceability design. The fixed-ended beam provides higher stiffness than a simply supported beam because end rotations are restrained, reducing deflection.

Given Data / Assumptions:

• Beam is prismatic with constant E and I.
• Both ends are fixed (built-in) → zero slope and zero deflection at supports.
• Central point load W applied at midspan.

Concept / Approach:
Standard closed-form solutions (e.g., double integration, area-moment, or conjugate beam methods) give the maximum deflection at midspan for this case. Boundary conditions (zero rotations at ends) reduce the curvature distribution compared to a simply supported beam.

Step-by-Step Solution:

Verification / Alternative check:
Energy methods yield the same expression; numerous beam tables list W l^3 / (192 E I) for a fixed beam with a midspan point load.

Why Other Options Are Wrong:
W l^3 / (48 E I): Simply supported case.W l^3 / (96 E I) or / (384 E I): Do not match fixed-end boundary conditions for this loading.2 W l^3 / (3 E I): Dimensionally correct but not a standard result; orders of magnitude too large.

Common Pitfalls:
Confusing fixed-ended and simply supported cases; forgetting that built-in ends reduce rotations and deflection significantly.

Final Answer:

W l^3 / (192 E I)