Torsion of prismatic pieces

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Chapter VIII

• Torsion of prismatic pieces

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Torsion of prismatic piecesPure torsion
Solution by St-Venant
(pure torsion)
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Torsion of prismatic piecesPure torsion
Basic assumptions
Method to solve the problem
balance equations(volume surface)
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Torsion of prismatic piecesPure torsion
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Torsion of prismatic piecesPure torsion
! new idea if ? exists, then volume balance is
automatically satisfied
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Torsion of prismatic piecesPure torsion
initially ?(x,y) ? (assumption on the
displacements) new idea ?(x,y) ? (assumption on
the stresses)
THIS IS A COMPATIBILITY EQUATION !
(? is the stress function of Prandtl)
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Torsion of prismatic piecesPure torsion

• Balance equation (surface)
• Lateral surface // z

? is constant along C
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Torsion of prismatic piecesPure torsion

• Balance equation (surface)
• Lateral surfaces ? z

the solution is strictly exact only if both Tx
and Ty are distributed in the same way as txz
and tyz over the top and bottom faces
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Torsion of prismatic piecesPure torsion

• Average balance

if ?0 along C
if both Tx ? txz and Ty ? tyz over the top
and bottom faces,then their resulting forces
(translation) are equal to zero
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Torsion of prismatic piecesPure torsion

• Average balance

if ?0 along C
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Torsion of prismatic piecesPure torsion

• Multiply-connected sections

?0 along C1
inside C1
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Torsion of prismatic piecesPure torsion

• Analogy with membrane

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Torsion of prismatic piecesPure torsion

• Analogy with membrane