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Problem_06_56 - Problem 6.56 Given Thin-hollow torsion...

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q 1 q 2 q 3 Problem 6.56 Given: Thin-hollow torsion member with uniform wall thickness t Find: Show that the interior walls are stress free. 2 A 1 2 A 2 = A 3 = t constant = θ i 1 2 G A i L q i q' t d = θ 1 1 2 G A 1 t q 1 b 2 h 2 + q1 q 2 ( ) b 2 + q 1 q 3 ( ) h 2 + = θ 2 1 2 G A 2 t q 2 b 2 h 2 + q2 q 1 ( ) b 2 + q 2 q 3 ( ) h 2 + = θ 3 1 2 G A 3 t q 3 b h + ( ) q3 q 1 ( ) h 2 + q 3 q 2 ( ) h 2 + = Set θ 1 θ 2 = 1 2 G A 1 t q 1 b 2 h 2 + q1 q 2 ( ) b 2 + q 1 q 3 ( ) h 2 + 1 2 G A 2 t q 2 b 2 h 2 + q2 q 1 ( ) b 2 + q 2 q 3 ( ) h 2 + = Since A 1 A 2 = q 1 b 2 h 2 + q1 q 2 ( ) b 2 + q 1 q 3 ( ) h 2 + q 2 b 2 h 2 + q2 q 1 ( ) b 2 + q 2 q 3 ( ) h 2 + = q 1 b 2 h 2 + b 2 + h 2 + b 2 + q 3 h 2 q 2 b 2 h 2 + b 2 + h 2 + b 2 + q 3 h 2 = q 1 q 2 = Set θ 2 θ 3 = 1 2 G A 2 t q 2 b 2 h 2 + q2 q 1 ( ) b 2 + q 2 q 3 ( ) h 2 + 1 2 G A 3 t q 3 b h + ( ) q3 q 1 ( ) h 2 + q 3 q 2 ( ) h 2 + = Since
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