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):
- Hollow_Sphere.jpg (42.19 KiB) Viewed 2198 times
The GMSH mesh generated in FEM WB has 3916 nodes and 1974 elements:
- Hollow_Sphere_Mesh.jpg (65.16 KiB) Viewed 2198 times
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.
- Collapse_Hollow_Sphere.png (33.19 KiB) Viewed 2198 times
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
