2
$\begingroup$

enter image description here

Which one deforms more when subjected to the same torque? I am having trouble thinking about this mathematically and with equations. Don't know what to do...

$\endgroup$
2
  • $\begingroup$ What are the boundary conditions and loads for this problem? $\endgroup$
    – nicoguaro
    Commented May 14, 2020 at 16:42
  • $\begingroup$ Maybe are you referring to a rod that is long compared with $D_0$ with a torque applied in the ends? $\endgroup$
    – nicoguaro
    Commented May 14, 2020 at 16:53

1 Answer 1

0
$\begingroup$

I will assume that you are talking about the twist of a slender bar.

In that case, you can consider that the torsional stiffness is written as

$$ \kappa_0 = J_0 G_0 + J_1 G_1 = \frac{\pi G_0 (r_1^4 -r_0^4)}{2} + \frac{\pi r_0^4\, G_1}{2}\, , $$

for one case, and

$$ \kappa_1 = J_0 G_1 + J_1 G_0 = \frac{\pi G_1 (r_1^4 -r_0^4)}{2} + \frac{\pi r_0^4\, G_0}{2}\, , $$

in the other.

Now, let us compare both of these

$$\kappa_r = \frac{\kappa_1}{\kappa_0} = \frac{G_1 r_1^4 - r_0^4 G_1 + G_0 r_0^4} {G_0 r_1^4 + r_0^4 G_1 -G_0 r_0^4}\, ,$$

and if we take the following limits,

\begin{align} \lim_{G_1 \rightarrow \infty} = \frac{r_1^4 - r_0^4}{r_0^4}\, ,\\ \lim_{G_1 \rightarrow 0} = \frac{r_0^4}{r_1^4 - r_0^4}\, , \end{align}

we see that the stiffness is higher when the external part is stiffer.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.