Required cross-sectional area at working stress For a prismatic tension member carrying a given load P, if the computed stress equals the specified working (allowable) stress, what can be said about the required cross-sectional area?
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AZero
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BInfinite
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CMaximum possible
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DMinimum that satisfies the allowable stress
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EIndeterminate without Poisson’s ratio
Answer
Correct Answer: Minimum that satisfies the allowable stress
Explanation
Introduction / Context:Allowable stress design sizes members so that actual stress does not exceed a prescribed limit. This question asks you to connect that concept to the required area for a given load.
Given Data / Assumptions:
- Axially loaded prismatic member.
- Working (allowable) stress = sigma_allow.
- Applied load = P (constant).
Concept / Approach:Stress is defined by sigma = P / A. To ensure sigma ≤ sigma_allow, the area must satisfy A ≥ P / sigma_allow. The minimum area that just satisfies the limit (and is therefore most economical in terms of area) is A_min = P / sigma_allow.
Step-by-Step Solution:
Start with sigma = P / A.Impose the design constraint sigma ≤ sigma_allow.Rearrange to A ≥ P / sigma_allow; choose A = P / sigma_allow for minimum compliant area.Verification / Alternative check:Using any smaller area violates the allowable stress; using any larger area is safe but not minimal for area economy.
Why Other Options Are Wrong:
- Zero or infinite area are physically meaningless for a finite load.
- Maximum area is not required; larger area is conservative but uneconomical.
- Poisson’s ratio is irrelevant for uniaxial allowable-stress sizing.
Common Pitfalls:Confusing working stress with ultimate or yield stress; forgetting that design often seeks minimum adequate size, not maximum.
Final Answer:Minimum that satisfies the allowable stress