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I am trying to compare the torsional stiffness of two injection moulded parts. I have applied torque to these parts and measured their resulting twist angle.

I have plotted the data for both parts with torque (Nm) on the y-axis and twist angle (deg) on the x-axis. Both lines are linear, but the gradients are different.

Wikipedia defines the torsional stiffness as:

$$\frac{GJ}{L} = \frac{T}{\theta}$$

and the torsional rigidity as:

$$GJ = \frac{TL}{\theta}$$

Where:

$\theta$ = angle of twist (deg/rad)

$T$ = torque (Nm)

$L$ = beam length (m)

$G$ = shear modulus (Pa)

$J$ = torsion constant

Therefore, it is my understanding that I can simply find the gradient of my results ($\frac{T}{\theta}$) in order to determine the torsional stiffness for both parts with units of $\text{Nm/deg}$. Logically, the higher value will be the 'stiffer' part in terms of how resistant they are to twisting under a torque load. I can calculate the torsional rigidity in the same way, as I have measured the part lengths.

However, I don't understand why torsional rigidity exists. As far as I understand it, torsional rigidity differs from torsional stiffness in that torsional rigidity is independent of the length of the beam. However, I cannot find any concrete confirmation of this.

Furthermore, I am unable to determine the torsional stiffness/rigidity empirically using the torsion constant, as the cross-section of the part is a complex shape. This means that I can only estimate the stiffness using experimental results.

Both parts are slightly different in length. I want to understand which part is 'stiffer' if I assumed both of them to be the same length. Which value would be the correct one to use in this instance?

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  • $\begingroup$ Have you looked at any sources like Timoshenko? If not, then you should. $\endgroup$
    – Solar Mike
    Commented Aug 27 at 6:51

2 Answers 2

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Torsional rigidity exists mainly for beams with constant sections, as it is basically a property of the cross section and material and is independent of length. I think this is very similar to axial stiffness/rigidity.

Due to complex shapes in your case, it makes more sense to use torsional stiffness. If you want to understand which part is stiffer, what do you mean by that? Like what functional property of the part is affected by this stiffness? Maybe if you are considering variations of the parts by "stretching" their models in the axial direction with all other dimensions remaining the same, rigidity might be better as it should remain very similar for all of the stretched version of one part.

One thing to be very careful about is angle units. What you refer to as torsion constant, $J$, is usually second polar moment of area for constant section beams, whose unit is length to 4th power. If you combine this with the remaining quantities, the angle should be just a dimensionless ratio, therefore $\text{rad}$. Otherwise, you need to smuggle in $\text{deg}$ into one of the quantities, making it non-standard.

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  • $\begingroup$ My aim is to understand which part is 'stiffer' in the sense of being resistant to a twisting load (torque). The parts are usually fixed on one end. When a torque is applied to the other end, the part twists - which is unwanted behaviour. So I'm trying to see which part twists less to understand which cross-section or material choice is more effective. Both parts have a similar cross-section. The problem is that one of the parts is shorter by at least 10 cm. This makes it difficult to compare the parts 'like for like', as of course the shorter one will naturally be more resistant to twisting. $\endgroup$
    – golfplatz
    Commented Aug 27 at 8:51
  • $\begingroup$ Thank you for the comment regarding the angle units. I will work with $\text{rad}$ to try and maintain the use of the standard units. $\endgroup$
    – golfplatz
    Commented Aug 27 at 8:56
  • $\begingroup$ Based on your comment, I am sure torsional stiffness is more suitable, because you really need just relation between the torque and the angle. You really seem to be interested in 'how much does it twist' not 'how much does it twist per unit length'. $\endgroup$
    – Tomáš Létal
    Commented Aug 27 at 9:50
  • $\begingroup$ In the context of the experiment, you are right that I am looking only at the relation between the torque and angle for the specific parts mentioned. However, I would still like to have an understanding of a 'corrected' stiffness if they were both a metre long for example. The problem is that the cross-sections are not constant, so I'm not sure torsional rigidity could be used here. $\endgroup$
    – golfplatz
    Commented Aug 27 at 10:49
  • $\begingroup$ In that case, you could use something like torsional rigidity, maybe you could call it equivalent torsional rigidity so that it is not confused with the torsional rigidity of constant section beams. $\endgroup$
    – Tomáš Létal
    Commented Aug 27 at 10:54
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Torsional rigidity is the rigidity of the entire bar, or shaft, or any prismatic member.

Torsional stiffness is the rigidity of one unit length of the bar. It is derived by dividing the torsional rigidity by the length L.

In the case of two shafts with the same torsional stiffness but one is twice as long as the other and both are exposed to the same torque, the longer shaft twists twice. Hence its torsional rigidity is half of the shorter shaft, even though their stiffness is the same.

By comparing the two equations, note that.

$$torsional / stiffness = \frac{GJ}{L} = \frac{T}{\theta} \ = \frac{torsional \ rigidity}{L} = \frac{\frac{TL}{\theta}}{L} $$

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