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-
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A.
ML/2EI
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B.
3ML/2EI
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C.
2ML/3EI
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D.
ML²/2EI
Correct Answer:
B. 3ML/2EI
Explanation:
To find the slope at the free end of the cantilever beam, we can use the Moment-Area Theorem, which states that the change in slope between two points is equal to the area of the M/EI diagram between those points.1. Bending Moment (M): For a cantilever beam with a moment M applied at the free end, the bending moment is constant throughout the entire length L.
2. M/EI Diagram: - From the fixed end to L/2: The flexural rigidity is EI, so the value is M/EI. The area of this segment is (M/EI) \* (L/2) = ML/2EI.
- From L/2 to the free end: The flexural rigidity is EI/2, so the value is M/(EI/2) = 2M/EI. The area of this segment is (2M/EI) \* (L/2) = ML/EI.3. Total Slope: Since the slope at the fixed end is zero, the slope at the free end is the sum of these two areas:
ML/2EI + ML/EI = ML/2EI + 2ML/2EI = 3ML/2EI.The correct answer is 3ML/2EI.
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