Question: A pure torque does not have a point of application. For example, in the beam shown below, the reactions are same regardless of the torque placement:

[![enter image description here](https://i.stack.imgur.com/B7hD8.jpg)](https://i.stack.imgur.com/B7hD8.jpg)

But how do you apply torque in reality?
For example if you use a screwdriver or a double force the beam deformation is different based on where you apply the torque as shown below:

[![enter image description here](https://i.stack.imgur.com/SNklK.png)](https://i.stack.imgur.com/SNklK.png)

Another example, how do you calculate the reaction forces in the systems shown below:

[![enter image description here](https://i.stack.imgur.com/xdxUZ.png)](https://i.stack.imgur.com/xdxUZ.png)

So, does pure torque exist in reality?

Answer: 
> 
> A pure torque does not have a point of application.
> 
> 
> 

This is incorrect (as you show in your second example). A better statement is

*"The location of application of a pure torque doesn't matter in certain analyses, such as when balancing moments around any point of a rigid free object."*

The qualifiers are satisfied in your first example. But the beam in your second examples isn't a rigid free object, and the point of application of the torque does happen to matter.
I think that the heart of this paradox is that "pure torque" is in idealization (as is "rigid body") in fact "pure torques" do not exist (anything applying a torque applies finite forces at points that have a finite distance apart). What we are seeing with the latter example is that the idea of applying torque at a point is useful only to the degree that the rigid body approximation (e.g. r\_ij is fixed for all the elements comprising the body) is valid.

If the body can deform, then how it deforms is determined by where the forces act. You need to work with pressures and stress on the object in order to determine how it deforms (e.g. a hex nut that is stripped will not deform the same way as a phillips screw because the stresses on the object are different)