Q.4 The value of slope at fixed support of a cantilever beam of length L, flexural rigidity EI, subjected to a point load 'P' at its mid span, and another point load 'P' at its free end, is:
- AzeroCorrect
- BPL² / 2EI
- C3PL² / 4EI
- DPL² / 4EI
Solution
Concept
A fixed support stops both rotation and movement, so slope and deflection there are zero for any load.
Formula
At fixed support: θ = 0, y = 0
Step-by-step solution
1. A fixed support does not allow rotation or translation. 2. Boundary condition at the fixed end: slope θ = 0 and deflection y = 0. 3. This holds for any loading (point loads at mid span and free end). 4. So slope at the fixed support = 0.
Explanation
A fixed support holds the beam fully, so the beam cannot rotate there. So slope at the fixed end is always zero, whatever the loads are. Maximum slope occurs at the free end, not at the fixed end.
Why the other options are wrong
B: PL²/2EI is the free end slope for a single end load P, not the fixed end slope. C: The free end slope for these loads is PL²/2EI + PL²/8EI = 5PL²/8EI, so 3PL²/4EI does not match any point. D: PL²/4EI is not the slope at the fixed end, which must be zero.
Shortcut
Fixed end of a cantilever: slope = 0 and deflection = 0, always.
Common mistake
Calculating the free end slope instead of reading 'fixed support' in the question.
Correct answer: A) zero