A cantilever beam of span L is subjected to a concentrated load P at the mid-span. If the flexural rigidity of beam is EI, then the slope at the free end of the beam will be
- APL² / (2EI)
- BPL² / (4EI)
- CPL² / (6EI)
- DPL² / (8EI)Correct
Solution
Concept
The slope at the free end of a cantilever beam carrying a concentrated load at mid-span is determined by considering the bending moment diagram and applying either the moment-area method or double integration method.
Formula
θ = P(L/2)² / (2EI) = PL² / (8EI)
Step-by-step solution
1. Consider a cantilever beam of length L with a concentrated load P acting at the mid-span (x = L/2). 2. The bending moment at any section between the fixed end and the mid-span (0 ≤ x ≤ L/2) is given by M(x) = -P(L/2 - x). 3. Between the mid-span and the free end (L/2 < x ≤ L), the bending moment is zero. 4. Using the moment-area theorem, the slope at the free end is equal to the area of the M/EI diagram between the fixed end and the mid-span. 5. Slope θ = (1/EI) × [0.5 × (L/2) × (-PL/2)] = PL² / (2EI).
Common mistake
Using the formula for a load at the free end (PL² / 2EI) instead of a load at the mid-span (PL² / 8EI).
Correct answer: D) PL² / (8EI)