Problem_06_56 - Problem 6.56 Given Thin-hollow torsion member with uniform wall thickness t Find Show that the interior walls are stress free 2 ⋅

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Unformatted text preview: Problem 6.56 Given: Thin-hollow torsion member with uniform wall thickness t Find: Show that the interior walls are stress free. 2 ⋅ A1 = 2 ⋅ A2 = A3 ⌠ ⎮ 1 ⋅⎮ θi = 2 ⋅ G⋅ Ai ⎮ ⌡ t = constant q1 q i − q' t dL q3 1 b h b h θ1 = ⋅ ⎡q 1 ⋅ ⎛ + ⎞ + q1 − q 2 ⋅ + q 1 − q 3 ⋅ ⎤ ⎢⎜ ⎟ ⎥ 2 ⋅ G⋅ A1 ⋅ t ⎣ ⎝ 2 2⎠ 2 2⎦ ( ( ) ) ( ( ) ) q2 1 b h b h θ2 = ⋅ ⎡q 2 ⋅ ⎛ + ⎞ + q2 − q 1 ⋅ + q 2 − q 3 ⋅ ⎤ ⎢⎜ ⎟ ⎥ 2 ⋅ G⋅ A2 ⋅ t ⎣ ⎝ 2 2⎠ 2 2⎦ 1 h h θ3 = ⋅ ⎡q 3 ⋅ ( b + h ) + q3 − q 1 ⋅ + q 3 − q 2 ⋅ ⎤ ⎢ ⎥ 2 ⋅ G⋅ A3 ⋅ t ⎣ 2 2⎦ ( ) ( ) Set 1 θ1 = θ2 ⋅ ⎡q 1 ⋅ ⎛ ⎢⎜ b + h⎞ 1 b b h⎤ h b h ⋅ ⎡q 2 ⋅ ⎛ + ⎞ + ( q2 − q 1 ) ⋅ + ( q 2 − q 3 ) ⋅ ⎤ ⎟ + ( q1 − q2 ) ⋅ + ( q 1 − q 3) ⋅ ⎥ = ⎢⎜ ⎟ ⎥ 2⎠ 2 2⎦ 2 ⋅ G⋅ A2 ⋅ t ⎣ ⎝ 2 2⎠ 2 2⎦ q1⋅ ⎛ ⎜ b h 2 b + h⎞ b h b h ⎛b h⎞ ⎟ + ( q1 − q 2) ⋅ + ( q1 − q3 ) ⋅ = q2 ⋅ ⎜ + ⎟ + ( q2 − q1 ) ⋅ + ( q 2 − q 3) ⋅ 2 2⎠ 2 2 2⎠ 2 2 ⎝ + h 2 + b⎞ h h ⎛b h b h b⎞ ⎟ − q3⋅ = q2⋅ ⎜ + + + + ⎟ − q3⋅ 2⎠ 2 2 ⎝2 2 2 2 2⎠ q1 = q2 2 ⋅ G⋅ A1 ⋅ t Since ⎣ ⎝2 A1 = A2 q1⋅ ⎛ ⎜ ⎝2 + ⎝2 + b 2 Set 1 θ2 = θ3 ⋅ ⎡q 2 ⋅ ⎛ ⎢⎜ b 2 ⋅ G⋅ A2 ⋅ t Since ⎣ ⎝2 + h⎞ 1 b h⎤ h h ⋅ ⎡q 3 ⋅ ( b + h ) + ( q3 − q 1 ) ⋅ + ( q 3 − q 2 ) ⋅ ⎤ ⎟ + ( q2 − q1 ) ⋅ + ( q 2 − q 3) ⋅ ⎥ = ⎢ ⎥ 2⎠ 2 2⎦ 2 ⋅ G⋅ A3 ⋅ t ⎣ 2 2⎦ and q1 = q2 2 ⋅ A2 = A3 ⎡q ⋅ ⎛ b + h ⎞ + q2 − q ⋅ b + q − q ⋅ h⎤ = 1 ⋅ ⎡q ⋅ ( b + h) + q3 − q ⋅ h + q − q ⋅ h⎤ ⎢2⎜ ⎟( ( 2) 2 ( 2 3 ) 2⎥ 2 ⎢ 3 2) 2 ( 3 2 ) 2⎥ ⎣ ⎝2 2⎠ ⎦ ⎣ ⎦ q2⋅ h + 1 2 ⋅ b ⋅ q2 − b 2 + 1 2 ⋅ q3⋅ h = 1 2 ⋅ q3⋅ b + 3 4 ⋅ q3⋅ h + 1 4 ⋅ h ⋅ q3 − 1 2 ⋅ q2⋅ h q2⋅ ⎛ h + ⎜ ⎝ h⎞ h h⎞ ⎛b 3 ⎟ = q3⋅ ⎜ + ⋅ h + + ⎟ 2⎠ 4 2⎠ ⎝2 4 q2 = q3 = q1 Net shear flow on interior walls is zero. ...
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This note was uploaded on 01/11/2011 for the course MAE 3040 taught by Professor Mechanicsofsolids during the Spring '10 term at Utah State University.

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