Question

The maximum deflection of a fixed beam of length \( l \) carrying a total load \( W \) being uniformly distributed over the entire length is

• A. \( \dfrac{Wl^3}{48EI} \)
• B. \( \dfrac{Wl^3}{96EI} \)
• C. \( \dfrac{Wl^3}{192EI} \)
• D. \( \dfrac{Wl^3}{384EI} \)

Hint:

Fixed beams have smaller deflection compared to simply supported beams under the same loading.

Answer 1

The Correct Option is C

Step 1: For a beam fixed at both ends under a uniformly distributed load \( w = W/l \), the fixed-end moments are \( M_A = M_B = \dfrac{wl^2}{12} \), found from the standard fixed-end moment formulas.
Step 2: Using the moment-area method with these end moments and the applied UDL, the deflection curve is symmetric with the maximum deflection occurring at mid-span.
Step 3: Solving the governing bending equation with the fixed-end boundary conditions and evaluating at mid-span gives the standard closed-form result:
\[ \delta_{\max} = \dfrac{wl^4}{384EI} = \dfrac{Wl^3}{192EI} \]

Answer 2

Another way to confirm this result is by recalling how standard structural handbooks tabulate beam deflection formulas by support and loading type. Checking each option:

1. \( \dfrac{Wl^3}{48EI} \): Handbooks list this coefficient only for a simply supported beam with a central point load, not for a fixed beam under a distributed load.
2. \( \dfrac{Wl^3}{96EI} \): This coefficient corresponds to the maximum deflection of a propped cantilever (one end fixed, one end simply supported) under a central point load in standard tables, a different support case altogether.
3. \( \dfrac{Wl^3}{192EI} \): Standard structural tables consistently list this exact coefficient for a beam fixed at both ends carrying a uniformly distributed load, matching the case described in the question.
4. \( \dfrac{Wl^3}{384EI} \): Handbooks list this coefficient for a beam fixed at both ends under a central point load, not a distributed load, so the loading type does not match.

Matching the support and loading conditions against standard handbook cases confirms the coefficient \( \dfrac{Wl^3}{192EI} \) applies here.

Therefore, the correct answer is \( \dfrac{Wl^3}{192EI} \).