Question

A cantilever beam is subjected to a uniformly distributed load. At what point along the beam's length does the maximum deflection occur?

• A. At the fixed end
• B. At the midpoint of the beam
• C. At a quarter span from the free end
• D. At the free end

Hint:

For any cantilever configuration with downward loads, the fixed end has zero slope and zero deflection, while the free end always experiences the maximum deflection and maximum slope.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A cantilever beam is rigidly fixed at one end and unsupported (free) at the other end.
Boundary conditions dictate that displacement and slope are zero at the fixed end.

Step 2: Key Formula or Approach:
For a cantilever of length \(L\) under a uniformly distributed load \(w\), the deflection \(y(x)\) at any distance \(x\) from the fixed support is:
\[ y(x) = \frac{wx^2}{24EI} (6L^2 - 4Lx + x^2) \]

Step 3: Detailed Explanation:
Evaluating deflection at key locations:
- At the fixed support (\(x = 0\)):
\[ y(0) = 0 \] - At the free end (\(x = L\)):
\[ y(L) = \frac{wL^4}{8EI} \] Since the beam bends downward away from the fixed support, the deflection increases continuously from zero at the fixed end to its maximum value at the free end.

Step 4: Final Answer:
The correct option is 4, which corresponds to "At the free end".