A cantilever beam of length L has flexural rigidity EI up to length L/2 from the fixed end and EI/2 for the rest. It carries a moment M at the free end. The slope at the free end is given by-

Concept:

The slope at the free end of a cantilever beam with varying flexural rigidity is determined using the moment-area theorem.

Given:

• Flexural rigidity up to length L/2 from the fixed end: \( EI \)
• Flexural rigidity for the remaining length: \( EI/2 \)
• Moment applied at the free end: \( M \)

Calculation:

Using the moment-area theorem, the total slope at the free end is given by the sum of contributions from both segments.

For the first segment (0 to L/2) with flexural rigidity \( EI \):

\( \theta_1 = \frac{M (L/2)}{EI} \)

For the second segment (L/2 to L) with flexural rigidity \( EI/2 \):

\( \theta_2 = \frac{M (L/2)}{(EI/2)} = \frac{2M (L/2)}{EI} = \frac{ML}{EI} \)

Total slope at the free end:

\( \theta = \theta_1 + \theta_2 = \frac{ML}{2EI} + \frac{ML}{EI} \)

\( \theta = \frac{3ML}{2EI} \)

Final Answer: \( \frac{3ML}{2EI} \)