Beam of uniform strength with constant depth: how should the width vary? For a rectangular beam of uniform strength with constant depth d, keep maximum bending stress constant along the span. State how the beam width b must vary with bending moment M(x).
Mechanical Engineering
Strength of Materials
Difficulty: Medium
Choose an option
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AWidth varies directly with bending moment M
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BWidth varies inversely with bending moment M
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CWidth varies with shear force V only
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DWidth varies with deflection y
Answer
Correct Answer: Width varies directly with bending moment M
Explanation
DerivationFor rectangular section: Z = I/y = (b d^2)/6.Flexure formula: σ = M/Z. For uniform strength, σ = constant ⇒ M/Z = constant ⇒ Z ∝ M.With constant depth d, Z ∝ b d^2 ⇒ b ∝ Z ∝ M.
ConclusionWidth must increase where bending moment is larger.
Final AnswerWidth varies directly with bending moment M.