Fixed beam with central point load: For a prismatic beam fixed at both ends and loaded by a central point load, where does the maximum vertical deflection occur?

Introduction / Context:
Serviceability checks require the location and magnitude of maximum deflection. For statically indeterminate beams such as fixed–fixed beams, symmetry provides powerful insight without full analysis.

Given Data / Assumptions:

• Beam is prismatic and fixed at both ends.
• Single concentrated load acts at midspan.
• Linear elastic behavior; small deflection assumptions hold.

Concept / Approach:
Due to geometric and loading symmetry, the slope at midspan is zero and the deflected shape is symmetric about the centreline. The maximum deflection occurs where the slope changes sign, i.e., at the centre under a central load.

Step-by-Step Solution:

Verification / Alternative check:
Closed-form solution gives y_max at midspan with magnitude y_max = (Pl^3) / (192E*I) for fixed–fixed central load, confirming the location.

Why Other Options Are Wrong:

• At the fixed ends: deflection is zero by boundary condition.
• At l/3: not the extremum for this loading.
• None of these: incorrect because the centre is correct.

Common Pitfalls:
Confusing fixed–fixed with simply supported beams where maximum deflection is also at midspan, but magnitudes differ.

Final Answer:
At the centre of the beam