Cantilever with end load — location of failure under increasing load: When a vertical cantilever beam carries a load at its free end, where does failure initiate as the load is increased?
Mechanical Engineering
Strength of Materials
Difficulty: Easy
Choose an option
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Aat the free end
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Bat the fixed end
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Cin the middle of the beam
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Dat a distance 2l/3 from the free end
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Euniformly along the span
Answer
Correct Answer: at the fixed end
Explanation
Introduction:Bending failure typically initiates where bending moment is maximum. For a cantilever with an end load, recognizing the bending moment distribution is key.Given Data / Assumptions:
- Cantilever of length l; point load at the free end.
- Linear elastic behavior up to failure.
Concept / Approach:Maximum bending moment occurs at the fixed support for a cantilever with end load. Extreme fiber stress is largest there, so yielding/crack initiation starts at the fixed end section.Step-by-Step Solution:
Shear diagram: constant along span; bending moment varies linearly from 0 at free end to Pl at fixed endMaximum σ_bending at fixed end: σ = Mc/ITherefore, failure initiates at the fixed endVerification / Alternative check:Comparative check of stress magnitudes along the beam confirms the fixed section is critical.Why Other Options Are Wrong:
- Free end / middle / 2l/3: bending moment is smaller away from the fixed support.
- Uniformly along the span: bending stresses are not uniform.
Common Pitfalls:Confusing maximum deflection (free end) with maximum bending stress (fixed end).Final Answer:
at the fixed end