Torsion - Non-Circular Prism Saint-Venant's Theory

[RFreund]

My questions stem from J.P. Den Hartog's book "Advanced Strength of Materials" (he has two good books if your trying to obtain a more intuitive feel for mechanics of materials) see attached. Just FYI - this is not a homework assignment yet it is not related to any work project either just trying to get my Mechanics of materials knowledge up.

In the 'prelude' to torsion of non-circular prisms, eh makes an argument of why a section normal to the applied torsion does not deform by stating that the such a distortion that shear stresses would appear parallel to the longitudinals and that no shear stress in the normal section is necessary. Then states that only the shear stress in the normal section can resist the torsion and the stresses shown in Fig 3 are useless for resisting a twisting torque.

I'm having a hard time trying to understand and visualize what he is saying. So my questions are:

1. What is Fig 3 representing? Is the grid surface the normal surface? What are the arrow representing in the figure to the right of the grid.

2. My guess at what he is saying: "Distortion of the normal surface would create a longitudinal shear stress" - Ok I get that but if you have a shear stress on the normal section there will be a complementary shear stress on the longitudinal, right?. "While no shear stress in a normal section is necessary" - because of the distortion this eliminates the shear stress (confused)? "The stresses of figure 3 are useless for resisting torque" - I thought these were shear stresses on the normal surface, why are the useless?

I feel like there is something basic that I'm just not getting for some reason.

Thanks in advance!


EIT

http://www.HowToEngineer.com

 

[Cockroach]

The grid represents element "cells" comprising the body. They can be normal to the surface throughout the body, in other words, you dice the body into cubic elements. The arrow to the right represents the net horizontal force experienced by the wall of these cubes. It is uniform throughout the depth of the body meaning, "plane elements remain in the plane". So such a force is uniform and invariant through the body.

I believe what he is saying is that because the outside surface is exposed to air, there is no shear. In other words, there is no stress in the outer fibers of the body. To my recollection this was the basic principle, Mohr's Circle captured this thought and always noted that a plane element before distortion remained plane during distortion. I think the idea here was that you can't have material popping out of the plane when subject to twisting, which would introduce nonlinear features to the model.

I would review Beers & Johnson or Shigley, who give it a better discussion. Your reference is good, but buddy glosses over essential points which I believe is throwing you for the spin.

Regards,
Cockroach

 

[RFreund]

Thanks for the response. I see what you are saying and I may try to sit down and draw a couple stress cubes. In the past I have taken St. Venant's theory 'for granted' basically that he assumes only shear stress on the surface normal to the applied torque will resist the torque. However I wanted try to understand why he made this assumption and why it is correct.
I think you are also correct in that a different advanced mechanics textbook maybe more helpful and then I may return to Den (buddy?) for a more joyous read.
Is there an 'advanced' mechanics of materials text that you would recommend?

Thanks again.


EIT

http://www.HowToEngineer.com

 

[CoryPad]

1. The left side of Figure 3 represents a generic body with one element experiencing shear as denoted by the arrows. It is not representative of the shaft outer surface since all of that would be experiencing shear. The right side of Figure 3 represents a plan view of the shaft. The arrows in the right side image show the direction of torque/force/deformation on the outer surface of the shaft.

2. What you have written is confusing so I won't try to decipher that. My comment on the caption of Figure 3 is that it is trying to say that if you section the shaft longitudinally (parallel to the broken line in the right side image), there are no shear stresses on these planes. The caption is not written well.

For a reference, Mechanical Behavior Materials by Dowling is good.

 

[GregLocock]

The torsional constant for re-entrant thin walled sections, even if they are closed, is extremely difficult to analyse by hand. It is an important property in the automotive world, and in the days gone by it was not unusual to fab sections up and measure it on a rig. With FEA it is easy to simulate that, and rather more importantly, simulate the boundary conditions correctly.

Cheers

Greg Locock


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