Cantilever Beam under pure torsion - stress at fixed end?
Cantilever Beam under pure torsion - stress at fixed end?
(OP)
I'm looking at a cantilever beam with a pure torsion load applied at the free end. From the various references I've read, I understand that at the free end the stress is 100% torsional shear (Saint Venant's torsion). As you move towards the fixed end of the beam, the section stress is partly torsional shear stress and partly due to lateral shear as depicted in the picture below.

Because the flanges at the fixed end of the beam are restrained from warping the torsional stresses can't develop at that end. Instead, the applied torsional load is reacted by lateral shear forces in the flanges of a I-Beam for example, and also in-plane bending moments. I can visualize and understand this concept when looking at a I-Beam section, but what about something simpler like a slender rectangular section?
For a rectangular section of dimensions b and t, Saint Venant's shear stress due to pure torsion is calculated as 3T/(b*t^2). From my description above I understand this is the shear stress at the free end of the beam. What about the fixed end of the beam? For a rectangular section do we have the same shear stress, or is there zero shear stress? If so, how is the torsion load reacted at the fixed end?
Thanks.

Because the flanges at the fixed end of the beam are restrained from warping the torsional stresses can't develop at that end. Instead, the applied torsional load is reacted by lateral shear forces in the flanges of a I-Beam for example, and also in-plane bending moments. I can visualize and understand this concept when looking at a I-Beam section, but what about something simpler like a slender rectangular section?
For a rectangular section of dimensions b and t, Saint Venant's shear stress due to pure torsion is calculated as 3T/(b*t^2). From my description above I understand this is the shear stress at the free end of the beam. What about the fixed end of the beam? For a rectangular section do we have the same shear stress, or is there zero shear stress? If so, how is the torsion load reacted at the fixed end?
Thanks.






RE: Cantilever Beam under pure torsion - stress at fixed end?
RE: Cantilever Beam under pure torsion - stress at fixed end?
RE: Cantilever Beam under pure torsion - stress at fixed end?
RE: Cantilever Beam under pure torsion - stress at fixed end?
Dik
RE: Cantilever Beam under pure torsion - stress at fixed end?
>> torsional stresses you have is correct. Warping stresses are negligible.
So are you saying that the shear stress for a the rectangular cross-section beam under pure torsion would be 100% St Venant's shear stress (3T/(b*t^2) at the fixed end?
RE: Cantilever Beam under pure torsion - stress at fixed end?
RE: Cantilever Beam under pure torsion - stress at fixed end?
In practical, design office terms, yes. However, I believe that every non-circular thing warps to some degree.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Cantilever Beam under pure torsion - stress at fixed end?
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Cantilever Beam under pure torsion - stress at fixed end?
I'm coming to the conclusion that I'll need to examine a detailed FEA model of a beam under torsion and try to make some correlation or judgement on the reaction loads and stresses on the fixed end.
RE: Cantilever Beam under pure torsion - stress at fixed end?
RE: Cantilever Beam under pure torsion - stress at fixed end?
>> bar varies with the b/t ratio.
Yes, I understand that. I should have mentioned that the b/t ratio I'm using is >> 10, therefore the factor in that equation is 3.
RE: Cantilever Beam under pure torsion - stress at fixed end?
do you really have a long section like that ??
if you did have that section, I'd've thought that torsion (from a shear load) would be the least of your problems.
you show a pic of a beam loaded in shear (in fact shear through the shear center so no torsion) but the thread is "beam under pure torsion" ... which implies to me a beam loaded by torque ?
and, yes, I get that even a beam loaded through the shear centre can develop torsion, due to displacements, though the usual assumption is "small deflections".
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