# Question About Shear Stress Naming Convention

## Main Question or Discussion Point

Question About Shear Stress "Naming" Convention

I cannot seem to find this in any of my texts.

For a shaft of circular cross-section, how do you name the shear stress $\tau_{ij}$ caused by a Torque?

That is, how do you assign the indices i j ? Is it the plane that contains the cross-section?

That is the only way I can make any sense of it.

Silly question, but it is driving me nuts.

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Why would it be any different than normal?

FredGarvin

None of your text show a 3D infinitesimal element?

http://ic.ucsc.edu/%7Ecasey/eart150/Lectures/Stress/Fig3.6.jpg [Broken]
In this naming convention, the taus are the off diagonal elements.

I always remember it as...and this is just me...

$$\tau_{ij}$$ = in the "j" axis direction, perpendicular to "i" axis.

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cristo
Staff Emeritus

Since this is a cylindrical shaft, wouldn't it be easier to use cylindrical coordinates? That could be the origin of the OP's question...

Since this is a cylindrical shaft, wouldn't it be easier to use cylindrical coordinates? That could be the origin of the OP's question...
That doesn't change things because a differential element in cylindrical coordinates looks like a square.

FredGarvin

It really is the same thing, but if cylindrical coordinates are what one wants to deal with then how about this...

http://web.mse.uiuc.edu/courses/mse280/Handouts/Example_Stress_Cyl_Coord.pdf [Broken]

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Perhaps I am either wording this incorrectly, or I am more confused than I thought.

I was told by my instructor that when speaking of a shear stress $\tau_{ij}$ that i is the direction normal to the plane, and j is the direction of the shear.

That description doesn't make sense to me. But that might be because I am applying that definition to the entire cross section instead of just a differential element.

Let's take an example. If a torque T is applied clockwise to a cylindrical shaft whose longitudinal axis coincides with the z-axis. How do we name the shear stress that is induced?

If you look at the entire cross section, z is normal to the plane but in which direction do you say the shear acts? Does it even make sense to ask that question since it acts in ALL directions tangent to any radial distance?

If you look at a differential element at the top of the shaft then we can certainly assign a direction to the shear, i.e., to the "right" in the x-direction.

Is my question any clearer? Is that definition that I was given in naming tau correct? I think it only makes sense when talking about an element of the shaft.

Thanks

"That description doesn't make sense to me. But that might be because I am applying that definition to the entire cross section instead of just a differential element."

Stress states are defined for differential elements so that equilibrium is met.

You tried to define a single stress state for the entire cross sectional area. This resulted in some craziness that certainly didn't jive with your intuition.

Mapes
Homework Helper
Gold Member

Let's take an example. If a torque T is applied clockwise to a cylindrical shaft whose longitudinal axis coincides with the z-axis. How do we name the shear stress that is induced?

If you look at the entire cross section, z is normal to the plane but in which direction do you say the shear acts?
The torque induces a shear stress $\tau_{z\theta}=\tau_{\theta z}\propto T$, as cristo indicated.

"That description doesn't make sense to me. But that might be because I am applying that definition to the entire cross section instead of just a differential element."

Stress states are defined for differential elements so that equilibrium is met.

You tried to define a single stress state for the entire cross sectional area. This resulted in some craziness that certainly didn't jive with your intuition.
That's what I thought My confusion stemmed from my misinterpretation of the definition.