So, rather far from the simulation results. However, if I make this assumption (from the paper on which Roark’s case is based - “On the Theory of Thin Elastic Toroidal Shells” by R.A. Clark):
For the section AO it may be shown that the maximum direct stress and the maximum bending stress occur at similar locations but are opposite in sign.
and add minus to the equation for σ_h (as in the article), then I get σ_vM = 13.26 MPa, so much closer to the expected result (9% error).
I even tried stress linearization to check the values and signs of both stresses. The results were not helpful.
But more importantly, the stretch is still off (by 17%).
I also tried symmetric and full models (with and without the straight tips), axisymmetric and 3D, as well as different BCs, but I haven’t managed to improve the accuracy of the results. Any ideas what else could help ?
The only analytical solution I have for the displacement is just the total stretch value from a formula. But I can try different BC configurations (so far none helped) later and share the screenshots of the corresponding deformed shapes.
I also suspect that. After all, if stiffness is too high, it’s almost always a fault of overconstraining BCs. Maybe I will try with symmetry in a 3D model and try to figure out the assumptions made by the author of the original paper.
Hint: The Stretch formula is linearly proportional to n. It suggest formula assumes all your sections Stretch the same. You can check if your set up suits that condition.
There is also a solved problem “A corrugated-steel tube” in Roarks which states: “The ends are rigidly fixed”. and later rafers to this same formulas. Ends are probably fixed.
The problem is not simple at all, and I have not yet been able to rule out any of the possible errors: mine, the formula’s, or ccx’s.
Assumptions or BC behind the formula are not clear too.
In my case, I am using Saint Venant’s principle to minimize the effects of boundary conditions. I’m using a 9-wave spring and focus on the central section.
By other hand, seems like according to the formula overall stretch should be the sum of the individual stretches of each section like Springs in series.
Note that your bellow doesn’t satisfy the t/b condition of less than 1/10.
According to the formula Stresses should not depend on “n” value and in ccx it does right now for me. (Specially Membrane) ¿?¿?
EDITED. Just noticed an unnexpected behaviour in ccx. Maybe my intuition fails but I would expect the bellow should srink in the middle section and not expand as it does. ¿?¿.
Yeah, I’ve been struggling with it for quite some time already and it’s one of the last few cases that I’ve never managed to validate analytically despite several attempts and apparent simplicity of the problem. I’ve tried using different kinds of models (full/symmetry, 2D/3D), various BC configurations and even solvers, but there’s still a significant discrepancy.
I only know some assumptions from the R.A. Clark’s paper, but they just mention the conditions at point B (and partially at point O) and in a rather enigmatic way (my interpretation is shared above).
I was also thinking about simulating only a small segment corresponding to the one shown in the figure from Clark’s paper attached before. Or a single corrugations. But either way, the expected BCs are not clear.
Why ? In my case t = 4 mm and b = 42 mm so t/b≈0.0952 < 0.1
Right, the paper says:
Away from the ends of the pipe the state of stress may be assumed to be periodic in the axial direction so that it is necessary to obtain a solution only for a portion of the shell such as AOB.
Are you using stress linearization in Mecway ? Does it confirm the assumption about the sign of the stress components ?
I have solved the problem with MECWAY as it has true shell elements (You probably did that with ABAQUS too). I have managed to get the periodic solution, and the Hoop Stress is fully uniform all around the 1/4 of the modeled section. (That’s a good indicator that the Symmetry BC is working properly. Mesh is pretty good quality too after some geometry adjustments. My best solution deviates up to a 15% in the worst case.
-From my point of view the formula assumptions are restrictive. Especially when it is stated that one only needs to solve for section OB because the OA is symmetric. I don’t think that’s the case for the bellow. That’s true for Lambda<<1. Yours is .525. Also recall this is an axisymmetric problem.
(You could maybe focus on the OB section when comparing to the formula).
-Axisymmetric shell problems normally disregard one of the principal stress components. In this case it is not negligible.
-Inner section is stiffer than the external one.
-When solving complex differential equations, one needs to assume certain conditions that narrow down the range of validity of the solution. Those assumptions sometimes are too strong and there is not an equivalent BC combination.
-Deflection (v) is not solved in the original paper. Only an order of magnitude is provided. Solution is given for the horizontal component .
Don’t read me grong. Obtaining an anlytical solution is remarkable considering the complexity of the problem and has probably been very useful for a long time. Applause to MrCLARK.
Maybe I’m wrong and someone finds the right set up that fulfill all the assumptions behind the formul a.
Can you show your BCs ? My stretch deviation is around 17%. For the stress, it’s ≈ 9%.
Right, I wonder how Roark’s got it. And maybe we should check the horizontal displacement too, then.
I should also add for other readers that my geometry had a flaw - the internal and external corrugations (semicircles) had different radii because I used offset in FreeCAD to draw this:
I’m now using this geometry which is closer to fullfill all the assumptions, at least in terms of Numerical values. There are still some uncertainties in the BC considered in the formula.
Y on the base, Symmetry on the sides and pressure on a thin line on top. Reactions are fine. I’m using full 5 waves so BC are far from the center section which is the one used for inspection.
Example 4: Corrugated pipe subject to axial load. Consider a pipe with a cross
section as shown in figure 8a subject to a uniform axial load. Away from the
ends of the pipe the state of stress may be assumed to be periodic in the axial
direction so that it is necessary to obtain a solution only for a portion of the
shell such as AOB.
bold marking is mine, but solution may not be accurate for such a reduced number of corrugations.
The far the BC are from your area of interest the smaller will be its impact on the result. That’s useful if there is some uncertainty on the BC of your problem. In this case it can be used just extending the number of corrugations.
Yeah, but I calculate it from these components (the last equation in my original post). I would have to do the transformation to get the components, but probably also perform linearization because it’s supposed to be meridional bending stress and hoop membrane stress, specifically.
Pure transformation to cylindrical CSYS on the results where I got U2 = 11.48 mm:
To whom it may concern/interest I modeled the Roark case with your (@Calc_em) dimensions and analysed in 2 1/2 D with Code-Aster. By 2 1/2 D I mean 1/4 sector (for ease of symmetry constraints application) and approached by shell elements on the mid-plane.
Axial displacement of the unconstrained, loaded end is 0,086mm and Von Mises stress of the highest loaded zones is 14MPa.
