Axial deformation under self-weight vs. equivalent end load: For a uniform vertical bar, how does the elongation caused by its own weight compare with the elongation produced if the same total weight is applied as a single axial load at the free end?
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Aequal to
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Bhalf
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Cdouble
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Dquadruple
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Eone-third
Answer
Correct Answer: half
Explanation
Introduction / Context:Designers often compare axial deformation due to self-weight with the deformation caused by a lumped end load. Recognizing the correct relationship avoids overestimating or underestimating elongation in hanging members such as tie rods, cables, and tall columns.
Given Data / Assumptions:
- Uniform prismatic bar of length L, cross-sectional area A, density rho, and modulus E.
- Self-weight W = rho * g * A * L acts continuously along the length.
- Comparison is with a concentrated end load equal to W.
- Linear elasticity, small deformation, axial loading only.
Concept / Approach:Deformation is the integral of strain along the bar. For a concentrated end load W, the axial force is constant everywhere, giving a simple expression. For self-weight, axial force varies linearly from zero at the free end to W at the top, so the average internal force is W/2, leading to half the elongation of the same bar under end load W.
Step-by-Step Solution:
End load case: delta_end = (W * L) / (A * E)Self-weight case: internal force at distance x from free end is rho * g * A * xElongation: delta_sw = ∫(0→L) [rho * g * A * x / (A * E)] dx = (rho * g / E) * (L^2 / 2)But W = rho * g * A * L ⇒ delta_sw = (W * L) / (2 * A * E) = 0.5 * delta_endVerification / Alternative check:The self-weight produces a linearly varying force with an average of W/2, so elongation equals that from a constant force of W/2 applied along the length, confirming the half relation.
Why Other Options Are Wrong:
- Equal to/double/quadruple/one-third contradict the integral result and average-force reasoning.
Common Pitfalls:Using W instead of the varying internal force in the integral; forgetting that the free end carries zero axial force in the self-weight case.
Final Answer:half