# Length change of rod under torsion force

1. Oct 13, 2008

### kernel2705

Hi I want to calculate the change in length of a cylinder under torsional force. (e. g. material = steel, initial length 1500 mm, diameter 25 mm, one end fixed, other end 450 Nm).

Can anyone point me to the proper formulae (Saint-Venant??) or data sheets.

Thanks

2. Oct 15, 2008

### kernel2705

I calculated the problem with FEM. The result was the zylinder gets (in average) longer by a fraction of a micrometer, at the circumference (approx. outer 10%) it gets shorter by about 10 % of the maximum elongation. The elongation profile of the cross section looks like an inverse parabola.
Do you agree with this result?
Is there an approximation formula for this problem (elongation as a function of zylinder length, radius, torque, elastic modulus)?
Regards

3. Oct 15, 2008

### Mapes

In classic linear elasticity theory, the elongation of a cylinder under torsional load is zero. Your FEM results suggest that the actual elongation, if nonzero, is negligible.

4. Oct 15, 2008

### kernel2705

Hi yes I know that according to Saint-Venant it is zero. But if you include 2nd order effects it isn't zero. From what I read depending on the shape of the cross section of the rod the rod can become shorter or longer when twisted. I also read that if the cross section is a circle (if the rod is a cylinder) it will elongate. But I cannot find a formula which would allow me to calculate the elongation - if only approximately - in numbers.
Regards

5. Oct 15, 2008

### minger

Roark says that:
Poynting, J.H.: Proc. R. Soc. Lond., Ser. A,vol 32, 1909; and vol 36, 1912

He gives a semi-way to get longitudinal stress in a narrow rectangle, but the term vanishes for a circular cross section.

6. Oct 22, 2008

### Mech_Engineer

How are you applying the torsion and fixed boundary conditions to the rod?

7. Oct 23, 2008

### kernel2705

Hi Minger thanks for that nice quote, I am now quite confident that the result is true.

@Mech Engineer:
I applied the torsion force averaged to (6) symmetrically spread out internal boundaries of about 1 cm^2 each at one end of the rod, the other end of the rod is fixed (i.e. circular boundary area fixed), all other boundaries are free.
Regards

Last edited: Oct 23, 2008