# Why does the torsion in a circular bar result in shear stress along the axial direction?

In the case of pure torsion, how does a differential area on cross-section of the cylinder with dx length undergo a shear force that is perpendicular to the cross-section ? I can understand that a shear force parallel to cross-section generates because if we divide the cylinder into many such disks of dx length then each disk rotates relative to each other, thus giving rise to such shear stress. But why the other one ?

If I haven't been able to make myself clear here's another way. On the 2D stress block, why is there a horizontal shear stress?

Static equilibrium dictates that $$\tau_{xy} = \tau_{yx}$$, $$\tau_{xz} = \tau_{zx}$$, and $$\tau_{yz} = \tau_{zy}$$.

You can easily see it in the figure you provided. For the second figure (square), sum the moments (with respect to one of the corners) caused by the shear stresses. The sum of the moments must be equal to zero because of the equilibrium. It will give you the equalities I typed above.

You can feel it intuitively as well. For example, if you try to rotate a square and somehow it doesn't, then you can be sure that there are opposing forces on the other sides that prevent the rotation.

EDIT: I will elaborate on my last sentence to be more clear.

• If you want to rotate an object and induce angular acceleration, you have to apply a moment to it.
• The object might accelerate (like a yoyo), or it may not (try to rotate a wall in your home).
• If it does not accelerate (angular acceleration), you can be sure that something is resisting your moment. So something is applying a moment back to you. This something can be anything, and you may not know what.
• Now, you twist the bar, but nothing accelerates. Everything is static. Think about that little square; it doesn't accelerate, right? Then you conclude that it must be $$\tau_{xy} = \tau_{yx}$$.
• But let's keep digging; why this little square doesn't want to accelerate and generate that stress? Because if it accelerates and doesn't resist you, it will break. It is not easy to break something because its molecules hold each other. So no material will give up and let you break it.

Basically, when you try to break something, it will resist you back due to its molecular connections. And these molecular bonds will fight you and prevent you from breaking them! That causes that stress.

• Yes I understand the explanation, supported by the equilibrium equations but this answer is more like why it should happen whereas I wanted something like why it does happen. I mean it explains the causality but not the reasons behind. Commented Mar 15, 2023 at 17:55
• Thanks for the edit. I found just the answer I was looking for. If I could, I would have voted, but my low rep doesn't allow that. Thanks again. Commented Mar 16, 2023 at 15:11