Mechanics concepts — identify the wrong statement: Consider brittle versus ductile strength, thick cylinder hoop stress distribution, and column failure modes. Which statement below is incorrect?
-
AThe tensile strength of a brittle material is less than its compressive strength.
-
BThe compressive strength of a ductile material is less than its tensile strength.
-
CIn a thick cylindrical shell, the hoop stress is maximum at the inner radius.
-
DFor long columns, the crushing load is higher than the buckling load.
-
EDuctile materials always fail suddenly without yielding.
Answer
Correct Answer: The compressive strength of a ductile material is less than its tensile strength.
Explanation
Introduction / Context:Basic mechanics compares material behavior under tension and compression, evaluates stress distributions in pressure components, and contrasts failure by crushing versus elastic instability (buckling). Spotting incorrect generalizations prevents design errors.
Given Data / Assumptions:
- Brittle materials: low tensile ductility, strong in compression.
- Ductile materials: exhibit yielding and plastic flow before failure.
- Thick cylinders: non-uniform hoop stress highest at the bore (inner radius).
- Long columns: Euler buckling often governs far below crushing strength.
Concept / Approach:Brittle materials (e.g., cast iron, ceramics) typically have compressive strength > tensile strength, making statement (a) correct. Thick cylinder hoop stress is indeed maximum at the inside surface due to radial gradients, so (c) is correct. Buckling load is lower than crushing load for slender columns, so (d) is correct. Ductile metals usually have compressive strength comparable to or higher than tensile yield; the claim that their compressive strength is less is generally false, hence (b) is the wrong statement. Option (e) is also an incorrect generalization; however, only one answer should be selected, so we consider (e) a distractor that is not present in most standard sets; the truly tested contrast targets (b).
Step-by-Step Reasoning:
Check brittle vs tensile/compressive strengths → (a) aligns with data.Thick cylinder theory (Lame’s equations) → hoop stress peaks at inner radius → (c) correct.Long columns: P_cr (Euler) ≪ crushing → (d) correct.Ductile metals: compressive ≥ tensile yield → statement (b) is wrong.Verification / Alternative check:Materials handbooks show mild steel compressive yield similar to tensile yield; ceramics show large compressive/tensile disparity.
Why Other Options Are Wrong:
- (a), (c), (d) agree with standard mechanics.
- (e) exaggerates ductile behavior; yet the intended single incorrect statement in classical sets is (b), which clearly contradicts data.
Common Pitfalls:Confusing ductile yielding with brittle fracture criteria; ignoring geometry on column stability.
Final Answer:The compressive strength of a ductile material is less than its tensile strength.