Torsion [Part 1]
On the “Position of hinge“ post we discussed a little about connections to the beam of open section and how the main beam in such connection rotates around its axis.
It’s time to speak a little more about torsion, I think.
What are “torsion characteristics”, why are they “poor” for opened sections and what is the difference between opened and closed sections?
Why is torsion so complicated? Isn’t it just a torque moment?
What is “warping”? Is it the same? And what is “bimoment”?
How can IDEA StatiCa Member help to manage with torsion?
Experiment 1
Two cantilevers here, both twisted.
I want to ask you to guess what is the difference between them.
The farthest end is absolutely fixed.
The torsional moments of equal magnitude are applied on the free cantilever ends.
The cross-section, steel grade, length — everything is the same.
But there is one invisible detail that makes the cantilevers to have different deformed shape and different stresses magnitude and distribution.
Are you ready to see the answer?
This is the key. Right cantilever has a "Rigid support member" on its free end.
What is rigid support member?
This is a body with infinite stiffness. In IDEA StatiCa Member they can be added to the members’ ends and they are usually works as stiff firm supports. Like on the farthest end on these cantilvers.
But right here, in this exact example I used this in some special way — not for the supporting. Not for the external supporting, I would say. You can see on the picture supports for this rigid element are all turned completely off!
Once again — here we can see a support, that supports nothing?
Exactly!
1) This rigid element could restrain the cantilever from moving to X, Y or Z direction. But all three translations checkboxes are unchecked. So it doesn’t prevents the cantilever point to move to any direction.
2) This rigid elements could prevent rotation. But all three rotational checkboxes (Rx, Ry, Rz) are also unchecked!
So this “support” does completely nothing here?
Jo! Yes, it does something, we can see it on the very first picture. And I would say it plays a big role since the difference between two cantilevers is huge!
Warping
Here we can see the free ends of our cantilevers.
We can see the left cantilever is warping — the points of the edge cross-section are no more lying on the same plane.
And now it is obvious what exactly did the rigid body here.
The rigid body prevented the cross-section of the cantilever from warping.
So even though we unchecked all the external supports it still support the cantilever... internally.
Theory
Torsion is complicated. I have tried to explain it here as easy and short as I could.
If you want to know how to calculate the exact values of torsional stresses (be ready for third derivatives) or read more theory or see more practical examples or if you see my explanation unclear or untrustful…
in any case I recommend you the SCI’s guide “Design of steel beams in torsion” (P385), that is available for downloading on SteelConstruction.info.
Also AISC Design Guide 09 is a nice guide, but P385 is my favourite for this topic and free to read.
So, torsion consists of two components:
the free-to-warp torsion, or St. Venant torsion. This effect is only causes the shear stresses in the beam (τ).
the warping torsion, that produces longitudinal stresses (σ) and shear stresses (τ).
What is St Venant torsion again?
If you twisting a beam and every section of it has freedom to warp, then no normal stresses will occur in the beam. This is pure St Venant torsion.
And here is The Example. I mean, it is shown in any book that tries to explain torsion. A spherical cow beam with two equal torsional moments applied to opposite ends without any supports.
The additional support in the middle here doesn’t plays any role for stresses and deformations, but calms down the “singularity detected” error.
Stresses
Ux deformations
Now we see why St Venant’s torsion is also called “uniform torsion”.
The stresses remains mostly the same all along the beam. UX deformations diagram is also interesting — each edge of both flanges moved entirely by the same distance. So we can say that warping of every section is the same. Everything is the same.
But such a beautiful picture is rare. The usual type of torsion is warping torsion.
What is warping torsion again?
Ux deformations for cantilevers
This term is used for that type of torsion when something resist warping and warping becomes non-unified.
It must not be only an external support. As we saw above and as we will talk below — section can restrain itself from warping if additional elements occurs.
Look at the picture of the Ux deformations on the cantilevers from our experiment — here is how the warping torsion looks like. Each section warps by different magnitude of deformation.
It causes the additional stresses, both longitudinal and shear. But St Venant’s shear stresses are still there.
So for the state of warping torsional stress we need to consider both St Venant’s shear stresses and warping longitudinal and warping shear stresses.
As it is usual in mechanics — the more constraints your element have — the more complicated its state of stress, but the less are deformations of element.
Can we say that torsion of the left cantilever is st venants?
No. Both of these cantilevers here are under influence of warping torsion. To prove it I made some calculations according to the book I mentioned earlier.
Fortunately, both our cases are described there with scary huge equations (if you want, just write me, I can send you the .sm file):
Click on image to open it in full-size mode
According to these formulas I’ve plotted two charts. And how surprised I was comparing the equivalent stresses. They are equal!
Look at the legend, compare IDEA StatiCa’s colors to numbers on the plot. The “right” cantilever looks like it has less stresses, but the very edges of flanges acts really close to the analytical solution.
So the red and blue lines show the analytical components of equivalent stress.
Red one is showing us warping stresses. They comes to the very zero on the free end for the left beam like if it was bending moment for bending situation. But in 10 cm from this free edge the warping stresses are already appears.
For the right beam the warping component drawing the very funny picture with zero point in the middle. It is a point where warping stresses changes their sign, actually.
In points where the section is restrained against warping the St Venant stresses are equal to zero, but warping stresses has their highest values.
Conclusion to this example
Rigid element doesn’t add any external supports
But it creates some sort of internal support that significantly rearranges stresses and minimizes deformations
The very results analytical solution from P385 book and results of IDEA StatiCa Member are very close to each other, so we performed some accidental verification
Even if we twisting the cantilever with a free end — is is not St Venant’s torsion, warping is still plays a tangible role
Warping torsion is always there if you twisting the I-beam
Here I understood that this post is already too big, but I still have a lot to say and show.
So I decided to divide it to a series of posts.
In the next episodes
Bimoment — what does it mean?
What is 7th degree of freedom?
What about closed sections?
How to restrain beam from warping in real life?
…
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STILL HAVE QUESTIONS?
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