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Elasticity/Torsion of noncircular cylinders

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Torsion of Non-Circular Cylinders

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Torsion of a noncircular cylinder

About the problem

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  • Solution first found by St. Venant.
  • Tractions at the ends are statically equivalent to equal and opposite torques .
  • Lateral surfaces are traction-free.

Assumptions:

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  • An axis passes through the center of twist ( axis).
  • Each c.s. projection on to the plane rotates,but remains undistorted.
  • The rotation of each c.s. () is proportional to .

where is the twist per unit length.

  • The out-of-plane distortion (warping) is the same for each c.s. and is proportional to .

Find:

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  • Torsional rigidity ().
  • Maximum shear stress.

Solution:

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Displacements

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where is the warping function.

If (small strain),

Strains

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Therefore,

Stresses

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Therefore,

Equilibrium

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Therefore,

Internal Tractions

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  • Normal to cross sections is .
  • Normal traction .
  • Projected shear traction is .
  • Traction vector at a point in the cross section is tangent to the cross section.

Boundary Conditions on Lateral Surfaces

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  • Lateral surface traction-free.
  • Unit normal to lateral surface appears as an in-plane unit normal to the boundary .

We parameterize the boundary curve using

The tangent vector to is

The tractions and on the lateral surface are identically zero. However, to satisfy the BC , we need

or,

Boundary Conditions on End Surfaces

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The traction distribution is statically equivalent to the torque . At ,

Therefore,

From equilibrium,

Hence,

The Green-Riemann Theorem

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If and then

with the integration direction such that is to the left.

Applying the Green-Riemann theorem to equation (17), and using equation (16)

Similarly, we can show that . since .

The moments about the and axes are also zero.

The moment about the axis is

where is the torsion constant. Since , we have

If , then , the polar moment of inertia.

Summary of the solution approach

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  • Find a warping function that is harmonic. and satisfies the traction BCs.
  • Compatibility is not an issue since we start with displacements.
  • The problem is independent of applied torque and the material properties of the cylinder.
  • So it is just a geometrical problem. Once is known, we can calculate
    • The displacement field.
    • The stress field.
    • The twist per unit length.