Burst Pressure of a thick-walled Sphere
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Burst Pressure of a thick-walled Sphere
A nice verification example for fcFEM.
I modelled a thick walled sphere in PART WB using two concentric spheres, one of radius 100mm and the other 80mm. After that I sliced the sphere with two vertical planes to get vertices where I could apply support boundary conditions. Then I created 4 solids by first slicing apart the part in PART WB and subsequently exploding the 4 slices in DRAFT WB. The result is shown below (with one solid made invisible for clarity):
The GMSH mesh generated in FEM WB has 3916 nodes and 1974 elements:
The Cuthill McKee routine in fcFEM reduced the bandwidth of the stiffness matrix to 476 (which is a nice and small percentage of the number of nodes, i.e. 3916). The most intensive 10 steps of the 40 step analysis comprised 107 iterations and took 20 seconds to run. After each 10 steps a load-deflection curve is produced with a choice to add 10 steps or complete the analysis.
The load-displacement curve shows that fcFEM predicts a load factor of 8.930 at burst. This should be multiplied by the input pressure of 50MPa to yield a burst pressure of 446.5 MPa.
Now, can we trust this number? Well, a simple equilibrium consideration of a half-sphere , where the yield stress (Sig_y =1000 MPa) is acting on the material cross section PI*(R_out^2-R_in^2) and the pressure on the internal area of the cut (PI*R_in^2) would give a burst pressure P_ult / Sig_ult = (R_out/R_in)^2 - 1 = 1.25^2 - 1 = 0.5625. So P_ult = 0.5625 * Sig_y = 562.5 MPa.
Hmmm, so fcFEM under-predicts the burst pressure by 20% ?? If anything, it should over-predict the theoretical solution.
A bit of digging shows that the theoretical solution I use above is only valid for a thin-walled vessel; the right one for a thick-walled sphere is: P_ult / Sig_ult = 2 log (R_out/R_in) = 0.4463 - where log is the natural logarithm. With this the theoretical burst pressure becomes P_ult = 0.4463 * Sig_y = 446.3 MPa, which is 0.045% below the pressure predicted by fcFEM.
So in this case fcFEM over-predicts theory by 0.045% ... not bad
I modelled a thick walled sphere in PART WB using two concentric spheres, one of radius 100mm and the other 80mm. After that I sliced the sphere with two vertical planes to get vertices where I could apply support boundary conditions. Then I created 4 solids by first slicing apart the part in PART WB and subsequently exploding the 4 slices in DRAFT WB. The result is shown below (with one solid made invisible for clarity):
The GMSH mesh generated in FEM WB has 3916 nodes and 1974 elements:
The Cuthill McKee routine in fcFEM reduced the bandwidth of the stiffness matrix to 476 (which is a nice and small percentage of the number of nodes, i.e. 3916). The most intensive 10 steps of the 40 step analysis comprised 107 iterations and took 20 seconds to run. After each 10 steps a load-deflection curve is produced with a choice to add 10 steps or complete the analysis.
The load-displacement curve shows that fcFEM predicts a load factor of 8.930 at burst. This should be multiplied by the input pressure of 50MPa to yield a burst pressure of 446.5 MPa.
Now, can we trust this number? Well, a simple equilibrium consideration of a half-sphere , where the yield stress (Sig_y =1000 MPa) is acting on the material cross section PI*(R_out^2-R_in^2) and the pressure on the internal area of the cut (PI*R_in^2) would give a burst pressure P_ult / Sig_ult = (R_out/R_in)^2 - 1 = 1.25^2 - 1 = 0.5625. So P_ult = 0.5625 * Sig_y = 562.5 MPa.
Hmmm, so fcFEM under-predicts the burst pressure by 20% ?? If anything, it should over-predict the theoretical solution.
A bit of digging shows that the theoretical solution I use above is only valid for a thin-walled vessel; the right one for a thick-walled sphere is: P_ult / Sig_ult = 2 log (R_out/R_in) = 0.4463 - where log is the natural logarithm. With this the theoretical burst pressure becomes P_ult = 0.4463 * Sig_y = 446.3 MPa, which is 0.045% below the pressure predicted by fcFEM.
So in this case fcFEM over-predicts theory by 0.045% ... not bad
- DeepSOIC
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Re: Burst Pressure of a thick-walled Sphere
It's always nice to see people check fem for validity. Mistakes there can be difficult to spot, yet very costly if the thing you calculated and made fails.
Re: Burst Pressure of a thick-walled Sphere
Indeed @DeepSOIC. Even if a theoretical solution does not exist for the configuration or properties you are analysing, then some benchmark that comes close needs to be considered.
Re: Burst Pressure of a thick-walled Sphere
Hi HarryvL,
Nice outcome
Ist it possible to share your model? On there on hand it is always good to have examples to start from, and on there other hand we could use in the future to test if FC still works fine after making changes in the code.
BR,
HoWil
Re: Burst Pressure of a thick-walled Sphere
Sure @HoWil, I will add it to my repo with examples when I have access to my computer.
Re: Burst Pressure of a thick-walled Sphere
I Harry, thanks for the information and the pedagogical way to say it, as usual in your threads.
About benchmarking, do you plane to achieve some set of models in order to benchmark fcfem solver? Those days I was thinking about that, if it was possible to achieve a set of well known standard study case modeles and performe an automatic routine to solve each case and compare with theoretical results. The routine could do the same for each solver, calculix, etc...
It could increase the reliability and trustness of freecad FEM workbench for users.
I guess this kind of stress test is definied in some FEM codes, am I wrong? I don't remember...
Maybe this kind of validating procedure is already used by Bernd or you? I don't know.
I'm not a specialist about FEM, I can't realy help. Maybe I could modelise a set of models? Does it helps?
Re: Burst Pressure of a thick-walled Sphere
Hi Alex, for now I will focus on further development of fcFEM. I still have tons of ideas. Then I will post benchmarks along the way. If and when fcFEM gets fully integrated with FEM WB (which I am not good at) then the examples and benchmarks can be turned into unit tests. This is definitely not my field of expertise, but I am sure Bernd will come to the rescue at that point. Harry
Re: Burst Pressure of a thick-walled Sphere
FYI I remember now, I saw this kind of routine in Freelem software. After installing the soft, for the first run, a stress test routine was testing 126 typical study cases. Then the soft told to the user a result "126/126" tests succed. Something like that.
http://www.freelem.com/qualif/index.htm
http://www.freelem.com/qualif/index.htm
Re: Burst Pressure of a thick-walled Sphere
Yes OOFEM does the same.
Re: Burst Pressure of a thick-walled Sphere
A sphere is the best shape to contain pressure. So what if we tried to achieve the same functionality with a hollow cube?
Let's see what happens with a cube of the same internal volume and wall thickness:
The cube only has 58% of the burst strength of the sphere. Well, that is to say in a small deformation analysis. I am sure that if we would take the effect of bulging into account (see below) that the ultimate strength would be higher .,. this is something to come back to when I have that capability included in fcFEM.
Let's see what happens with a cube of the same internal volume and wall thickness:
The cube only has 58% of the burst strength of the sphere. Well, that is to say in a small deformation analysis. I am sure that if we would take the effect of bulging into account (see below) that the ultimate strength would be higher .,. this is something to come back to when I have that capability included in fcFEM.