Integral for the calculation of torque

[laurajk]

Hello,
I found an integral to calculate the torque from the applied torsional shear stress, and I didn't find an explanation of how this integral is deviated. Where does it come from? Could someone explain?

T = ∫τ⋅r⋅dA = ∫τ⋅2πr⋅dr,
where T is the torque and τ the shear stress.

Thanks a lot!

 

[jrmichler]

integral is deviated.

I think you mean derived. And welcome to PF.

Do you understand how force is stress times area? Torque is similar, except that you need to include the distance from center. The torque contribution of an infinitesimal area is proportional to the distance from center, so a simple multiplication does not work. The integral is summing up the contribution of each infinitesimal area times its distance from the center. That's your first equation: Tau is stress, dA is area, and r is distance from center, and the integral adds it all up over the entire area.

Hope this helps.

 

[laurajk]

Hello!
Yes, it should be derived, of course :)
And thank you very much for your explanation, it's very helpful and I think, I got it now! :)