Solid Mechanics Question - Help

[Mapes]

Wow, thanks!

So in this question there's a little dilema.
As I understand, all I really care about is a(1,3) and a(3,1) because they are the sheer stress on the direction that I am looking for, right?

This approach let's you look at all the possible shear stresses to find out where the highest stress occurs. In other words, don't pick a certain location; instead, figure out where the shear stress is maximized. What's the largest shear stress term?

 

[dislect]

This approach let's you look at all the possible shear stresses to find out where the highest stress occurs. In other words, don't pick a certain location; instead, figure out where the shear stress is maximized. What's the largest shear stress term?

Got lost in the translation :) what is a shear stress term?
Do you mean that i should look at the entire matrix, for example a(2,1)= $-\cos\phi\sin\theta$
And in this question, $\sin\theta=\frac{4}{5}$ so a(2,1)=-0.8*$\cos\phi$ which is the max value I can get as long as $-\phi$
is 0 ? and so I get that the first point to break is on the the middle of the perimeter between A to B ?

 

[Mapes]

In the stress tensor (matrix) $\bold{\sigma}$, the terms that are off the diagonal are shear stress terms.

 

[dislect]

yeah i kinda guessed that 1 minute later and edited my post :S

 

[Mapes]

Great, I agree that $\bold{\sigma '_{21}}=\bold{\sigma '_{12}}=-\frac{4}{5}\tau_0\cos\phi$ has the largest magnitude over the entire joint (compare to $\bold{\sigma '_{13}}=-\frac{7}{25}\tau_0\sin\phi$ and $\bold{\sigma '_{23}}=-\frac{3}{5}\tau_0\cos\phi$). This corresponds to $\phi =0$ or $\pi$, which looks like points A and B on your diagram. (So you guessed correctly in your original solution!). But we get slightly different answers for the maximum permissable torque T; perhaps one of us has a calculation error.

EDIT: Fix typo; $\phi=\pi$, not $\pi/2$.

 

[dislect]

That's kind of far from what I got :\ Could you take a look and see if I did any critical mistakes in my calculation? Because I checked yours and its right, and I can't find where I got it wrong (plus I need to explain it to my friend)

 

[Mapes]

Actually, I can't follow your approach. I wanted you to solve in terms of the angles because that makes it easy to look at the answer and see if it works at extreme values (like joint angles of 0° and 90°). It's too difficult to follow many lines of someone else's numerical calculations.

 

[dislect]

So it's safe to say that the max tourqe is T=(5/8)*pi*(R^3)*Sigma0 (reminder:Sigma0 is the max shear stress)?

Mapes, thanks a lot for your help and supreme patience :) I know I've been a nag (and I probably will be).

 

[Mapes]

That looks good.

 

[dislect]

Oh and one last question I forgot to ask, how come there's a shear stress on a(3,1) or a(1,3), meaning in the direction of e3? I thought there was supose to be only on e2.

 

[Mapes]

I'm not sure what you mean. The matrix $\bold{a}$ is the rotation matrix, also known as the direction cosine matrix. It relates different coordinate systems. It doesn't represent stress.

 

[dislect]

I was relating to the Sigma' matrix, it has stress terms on a(3,1) and a(1,3)

 

[Mapes]

Well, doesn't the shear stress act in the 1-3 direction at points d and e in your diagram? These points correspond to $\phi=\pi/2$ and $3\pi/2$, which would correspond to $\sin\phi=1$ or $-1$.