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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).

Difficulty: Medium

Width varies directly with bending moment M
Width varies inversely with bending moment M
Width varies with shear force V only
Width varies with deflection y

Correct Answer: Width varies directly with bending moment M

Explanation:

Derivation
For 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.

Conclusion
Width must increase where bending moment is larger.

Final Answer
Width varies directly with bending moment M.

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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).

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