Torsion - Non-Circular Prism Saint-Venant's Theory
Torsion - Non-Circular Prism Saint-Venant's Theory
(OP)
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!
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!





RE: Torsion - Non-Circular Prism Saint-Venant's Theory
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
RE: Torsion - Non-Circular Prism Saint-Venant's Theory
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
www.HowToEngineer.com
RE: Torsion - Non-Circular Prism Saint-Venant's Theory
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.
RE: Torsion - Non-Circular Prism Saint-Venant's Theory
Cheers
Greg Locock
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