A steel beam and an aluminium beam of identical dimensions are subjected to the same bending moment. The Young's modulus of steel is twice that of aluminium. Assuming both materials are linearly elastic and isotropic, which statement about their bending behaviour is correct?
- AThe steel beam will experience half the curvature of the aluminium beam because of curvatureCorrect
- BThe aluminium beam will have twice the bending stress
- CBoth beams will deflect equally
- DThe steel beam’s neutral axis shifts downward
Solution
Concept
Beam flexural curvature and elastic modulus relationship
Formula
κ = 1 / R = M / (E × I)
Step-by-step solution
1. From the beam bending formula: M / I = σ / y = E / R = E × κ, where κ = 1/R is the curvature. 2. Curvature is given by κ = M / (E × I). 3. The two beams have identical cross-sectional dimensions (same I) and carry the same bending moment M. 4. Therefore, curvature is inversely proportional to Young's modulus: κsteel / κalum = Ealum / Esteel = 1 / 2. 5. The steel beam experiences half the curvature of the aluminium beam. Answer: A
Explanation
Under pure bending, curvature κ = 1/R = M / (E·I). For steel and aluminium beams of identical dimensions (same I) carrying the same bending moment M, curvature is inversely proportional to Young's modulus E. Because Esteel = 2·Ealuminium, the steel beam experiences exactly half the curvature of the aluminium beam.
Why the other options are wrong
B) Bending stress σ = M·y/I depends solely on moment and cross-section geometry, so it is identical for both beams. C) Deflection is inversely proportional to EI, so the aluminium beam deflects twice as much. D) The neutral axis location depends purely on cross-sectional geometry.
Shortcut
Curvature κ = M / (EI) => Doubling E halves the curvature.
Common mistake
Assuming bending stress differs between materials; bending stress σ = M·y/I is independent of E.
Correct answer: A) The steel beam will experience half the curvature of the aluminium beam because of curvature