# Convert translational stiffness to rotational stiffness of a screw

I have the formula for the translational stiffness which is : $k=\frac{A\cdot&space;E}{l}$

However, what I would like to find is the rotational stiffness. Is there any formulas to convert from one to an other knowing the pitch of the screw and the diameter ?

The translational stiffness is written as $$k_l = \frac{A\,E}{\ell},$$ where the stiffness $k_l$ is in $\left[\frac{N}{m}\right]$, the area $A$ is in $\left[m^2\right]$, the young's modulus is in $\left[\frac{N}{m^2}\right]$ and the length $\ell$ is in $\left[m\right]$.

$$\,$$

The rotational stiffness is written as $$k_r = \frac{G\,J}{\ell},$$ where the stiffness $k_r$ is in $\left[\frac{Nm}{rad}\right]$, the second moment of area (or torsion constant) $J$ is in $\left[m^4\right]$, the rigidity modulus is in $\left[\frac{N}{m^2}\right]$ and the length $\ell$ is in $\left[m\right]$.

$$\,$$

Assuming linear elasticity the following relation between the rigidity modulus and the young modulus holds $$G = \frac{E}{2(1+v)},$$ where $v$ is the poison ratio.

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The second moment of area (or torsion constant) for a circle is $$\iint\limits_{R} r^2\,dA = \int_0^{2\pi}\int_0^r r^2\left(r\,dr\,d\theta\right) = \int_0^{2\pi}\int_0^r r^3\,dr\,d\theta = \int_0^{2\pi} \frac{r^4}{4}\,d\theta = \frac{\pi}{2}r^4 = \frac{A\,r^2}{2}.$$

$$\,$$

Hence the rotational stiffness of a cylinder can be rewritten as function of $k_l$ $$k_r = \frac{\frac{E}{2(1+v)} \frac{A\,r^2}{2}}{\ell} = \frac{A\,E}{\ell} \frac{r^2}{4(1+v)} = k_l \, \frac{r^2}{4(1+v)}$$

This equation does not hold for any other shape!

When looking at a screw, there is a thread at the outside which has a significant effect on the rotational stiffness.

For example, leaving half the diameter open in the cylinder, the stiffness will only decrease $0.0625\%$ which is insignificant. Increasing the outer radius will increase the stiffness significant.

• The formula is how to calculate the rotational stiffness from the screw parameters. But what I'm trying to do is convert the translational stiffness into rotational stiffness. Without knowing G or J. – Crapsy Oct 5 '16 at 20:21
• I improved my answer. Is this what you meant? – useless-machine Oct 5 '16 at 21:23
• There is not any correlation between axial stiffness in a prismatic section and torque or rotational stiffness. J which is mass moment of inertia of the section is dependent on the cross section of the piece and weather it is hollow or solid! – kamran Oct 5 '16 at 21:27
• @kamran there is a correlation between them since I assumed linear elasticity and that the cylinder is solid. – useless-machine Oct 5 '16 at 21:44
• What if the bar is shaped like an "H" or anything other than a cylinder? The OP has not restricted the cross-section to cylinder. – kamran Oct 5 '16 at 22:10