Hello,
is it possible to get the buckling mode of a spherical shell?
I have tried with a half of a perfect spherical shell,
with uniform pressure, but i dont’ get it work.
wbr dichtstoff
I’ve tried a few approaches and I get the best results with a full model supported by soft springs. Mode shapes are irregular but the critical pressure agrees well with Roark’s:
hello Calc_em,
thnx a lot for your calculatin.
you have applied sigma stress = 1,
these is pressure = stress * 2 * t / r = 0.0133 !?
i recognized, that for an ideal buckling load you need an ideal system.
the first model is a half sphere with shell elements,
boundary is transformed to cylindrical coordinate system.
pressure is applied with *DLOAD, Eall,P
but you see always the structure of the mesh: with shell or solid elements:
second model is an eighth sphere with solid elements, same mesh, from
expanding shell elements, boundaries ortho to the cut section,
pressure is applied with *DLOAD Eall, P1 and P2
how can i avoid the influence of the mesh structure for my model !?
how have you applied your boundaries and your springs?
I will try an eighth part of the shell with unstructured mesh,
like you. how does look your deformation?
how can i print my values in cgx without prnt se all!? 
thnx in advance, wbr dichtstoff
I applied unit an external pressure load (1 MPa) to the whole sphere in PrePoMax. I rarely use cgx.
Your results at least are nicely symmetric. My buckling patterns were highly irregular and varied depending on BCs. But the buckling factor is correct. Did you compare it with analytical result in your cases ?
I’ve tried several different approaches - 1/8th with symmetry BCs and full sphere with different supports - 3-2-1 method, point springs in 3 locations and springs on the entire surface. I also tested various stiffnesses of the springs. In some cases there was a significant rigid body motion but it’s a matter of choosing sufficiently high spring constant.
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i have fixed one calc. with solid elements and boundary 1-2-3.
It seems stable, but i have to check different geometries.
i will create a stp file and mesh it with netgen to get a perfect shpere and i’ll go on
to work with one model with shell elements
calc. stress 75 MPA
thnx to calc_em, i had a fallacy with the pressure!
so here are my scripts:
sphere-creating-solid.fbd
# written by dichtstoff
# buckling Spherical shell
# sigma critical according Theory of elastic stability
# amount of element with line division
valu lidiv 8
# radius of the sphere
valu sphrd 150
# sphv1 = sin 45.000000°
valu sphv1 0.70710678
# sphv2 = sin 35.264400° ???
valu sphv2 0.57735042
valu sphp1 * sphrd sphv1
valu sphp2 * sphrd sphv2
PNT base 0.00000 0.00000 0.00000
PNT D001 0.00000 0.00000 sphrd
PNT D002 0.00000 sphrd 0.00000
PNT D003 sphrd 0.00000 0.00000
PNT M001 sphp1 sphp1 0.00000
PNT M002 0.00000 sphp1 sphp1
PNT M003 sphp1 0.00000 sphp1
PNT MSPP sphp2 sphp2 sphp2
LINE ! D001 M003 base 1
LINE ! M003 D003 base 1
LINE ! D003 M001 base 1
LINE ! M001 D002 base 1
LINE ! D002 M002 base 1
LINE ! M002 D001 base 1
LINE ! M001 MSPP base 1
LINE ! M002 MSPP base 1
LINE ! M003 MSPP base 1
div all mult lidiv
GSUR A001 - BLEND + L001 + L006 + L008 - L009
GSUR A002 + BLEND + L002 + L003 + L007 - L009
GSUR A003 + BLEND + L004 + L005 + L008 - L007
seta part s A001 A002 A003
copy all all rot x 90
copy all all rot x 180
copy part all rot z 90
copy part all rot z 180
copy part all rot y 180
merg p all
merg l all
plot sa all
prnt se
ELTY all QU4
MESH all
prnt se
plot e all
view elem
send all abq
sphere-creating-solid.inp
**** written by dichtstoff
**** buckling Spherical shell
**** sigma critical according Theory of elastic stability
**** by S. Timoshenko 1936
**** sigma critical = E*t²/[r²(3(1-v^2))]
**** E = Young's modulus
**** v = Poisson's ratio
**** t = thickness of the shell
**** r = radius of the sphere
*HEADING
Model: sphere-creating-solid
*INCLUDE, INPUT=all.msh
*MATERIAL, NAME=steel
*ELASTIC
210000, 0.30
*DENSITY
7.85e-9
*SHELL SECTION, ELSET=Eall, MATERIAL=steel
1
*STEP
*STATIC, SOLVER=SPOOLES
*END STEP
read-sphere-creating-solid
# written by dichtstoff
# buckling Spherical shell
# sigma critical according Theory of elastic stability
read sphere-creating-solid.frd
read sphere-creating-solid.inp nom
seta sphsolqua e all
seta sphsolqua n all
seta nodes n all
enq nodes bound rec 0 _ _ 1
comp bound d
seta bound-x f bound
comp bound-x d
del se bound
enq nodes bound rec _ 0 _ 1
comp bound d
seta bound-y f bound
comp bound-y d
del se bound
enq nodes bound rec _ _ 0 1
comp bound d
seta bound-z f bound
comp bound-z d
del se bound
seta boundary se bound-x bound-y bound-z
seta fix-x se boundary
setr fix-x se bound-y bound-z
seta fix-y se boundary
setr fix-y se bound-x bound-z
seta fix-z se boundary
setr fix-z se bound-x bound-y
prnt se
plot e all
plus na fix-x t
plus na fix-y m
plus na fix-z n
send sphsolqua abq
send fix-x abq nam
send fix-y abq nam
send fix-z abq nam
send boundary abq nam
sphere-buckling-solid.inp
**** written by dichtstoff
**** buckling spherical shell
**** sigma critical according Theory of elastic stability
**** by S. Timoshenko 1936
**** sigma critical = E*t²/[r²(3(1-v^2))]
**** E = Young's modulus
**** v = Poisson's ratio
**** t = thickness of the shell
**** r = radius of the sphere
*HEADING
Model: buckling spherical shell
*INCLUDE, INPUT=sphsolqua.msh
*INCLUDE, INPUT=fix-x.nam
*INCLUDE, INPUT=fix-y.nam
*INCLUDE, INPUT=fix-z.nam
*INCLUDE, INPUT=boundary.nam
*MATERIAL, NAME=steel
*ELASTIC
210000, 0.30
*DENSITY
7.85e-9
*SOLID SECTION, MATERIAL=steel, ELSET=Esphsolqua
*STEP
*BUCKLE, SOLVER=SPOOLES
1
***STATIC, SOLVER=SPOOLES
*BOUNDARY
Nfix-x,1,1,0
Nfix-y,2,2,0
Nfix-z,3,3,0
*DLOAD
**Esphsolqua,GRAV,9810.,0.,1.,0
Esphsolqua,P1,-0.5
Esphsolqua,P2,+0.5
*NODE FILE,
U
*EL FILE,
S
*NODE PRINT, NSET=Nfix-x, TOTALS=ONLY
RF
*NODE PRINT, NSET=Nfix-y, TOTALS=ONLY
RF
*NODE PRINT, NSET=Nfix-z, TOTALS=ONLY
RF
*NODE PRINT, NSET=Nboundary, TOTALS=ONLY
RF
*END STEP
1 Like
Thanks for sharing. Obtaining the Buckling shape of the complete sphere is remarkable.
or Without neglecting second therm.
The formula is valid for complete perfect shells.
Slightly more accurate.
Which is your final element. ¿Solid or shell?
¿Is the final buckling shape like an elipse?
Hello Disla,
thank you for your interest and curiosity .
at the moment i work only with solid elements.
I have for half sphere solid and shell models.
I think these buckling shape is not correct!?
I think you get these because of the boundary of
the sphere !? Maybe the shape for global buckling
(sh)(c)ould be assyemetric with two waves!?
I don’t know how to get the buckling shape?
can you help me with that ?
wbr & thnx
Here you can find a good reference for analytical solution. Free access.
https://royalsocietypublishing.org/doi/epdf/10.1098/rspa.2016.0577
Be careful not to impose artificially a symmetry with your BC or you could be killing the expected Eigen shape or forcing other solutions to show up. (Like 1/4 or 1/2 of the model)
"Eigenvalue problem for the buckling of a perfect spherical shell subject to uniform pressure
has simultaneous axisymmetric and non-axisymmetric eigenmodes ".
If you want to compare with eigen shapes that has been published I would suggest you to solve the thickness for a given E,nu,R and n using expression (3.6).
Use it in your model and your solution should be a clean pattern with given number of waves. Don’t forget to respect the assumptions behind the analytical solution.
The largest the number of waves you consider, the finest your mesh must be to capture it.
Here examples with n=18
| Nº Waves | R/t | t [mm] | Pc |
|---|---|---|---|
| 18 | 103.49 | 1.93 | 23.7 |
This is my mesh density.R=200mm. S8R
*STEP
*BUCKLE,SOLVER=PARDISO
3,1E-9
*DSLOAD
PRESSURE_FACES_int,P,-500000
*DSLOAD
PRESSURE_FACES_out,P,500000
*NODE FILE,GLOBAL=YES
U
*END STEP
hello Disla,
can you show us the deformation of your sphere with your impact?
wbr
¿Not sure if I understand you?
There is no impact. This is the result of a Buckling step. A wavy patern similar to the ones that emerge in a cylinder.
Uniform membrane solution for the perfect spherical shell subject to uniform pressure p assuming short wavelengths relative to R.
you have these:
*DSLOAD
PRESSURE_FACES_int,P,-500000
*DSLOAD
PRESSURE_FACES_out,P,500000
so you get a deformation on your sphere for your prestress.
how does it looks like?
I can not reproduce your result, with ccx, but with other software.
the big difference is the deformation of the prestress.
what ever i do, i dont get a homogenic deformation.
it looks always like these, and i think these is the problem.
with other software, i have solid quadratic elements, half sphere,
same boundary, and i get the same result.
deformation looks homogenic, and not like these:
wbr & thnx
My deformation is on the last post. Solution is for n=18. You can count them. Expected Bucling is 23.7 and I get 23.49.
Use S8R and full sphere. Imposing symmetry boundary conditions you are removing non symmetric solutions. Mine is antisymmetric and would not show up if I were using half sphere.
According to formula 3.6 your sphere:
E=210000 MPa
R=150mm
t=1mm
Nu=0.3
Your solution should get a mixed pattern of around 22 waves. Your mesh density looks insufficient to capture that.
Try increasing slightly your thickness t=1.82235790mm. It should deliver 16 waves.
¿Could you post the picture of the solution you get with other software?
1 Like
By the way , your solution looks like the ones on stiffer spheres. Kind of chess board.
Your element could be to stiff.
Hello Disla,
i have for my sphere areas, for creating the mesh.
These ares influences the deformation and stress
for the buckling analysis. but i don’t know yet,
how to fix these with cgx.
the first picture are the areas with one element
for each area,
and the second one is the deformation with pressure:
with high amount of elements:
these is, why i want to see your deformation for the static analysis,
only pressure!? how can fix these?
I’m loosing interest to fix these.wasting to much time 
wbr and thnx
solid elements fixed only with cylindrical equal to ccx:
*Transform, Nset=Nboundary, Type=C
0, 0, 0, 1, 0, 0
*BOUNDARY
Nboundary,2,3,0
STATIC
There is a neat displaceemnt in Pole dirrection because I have a light spring there to remove rigid body. That is as expected. Appart from that Stresses show some stiffenning along the welds of the different sections but doesn’t seem to affect noticebly to the result. .
I’m not cgx user and I haven’t faced your issue. Maybe it is the source of the problem. ¿Have you tried meshing in Prepomax?
I will with a non structured quad mesh
You have set up a different problem or Abaqus. You have imposed an Axisymmetric BC. Then your solution is according to your BC …..Axisymmetric. That’s not the general solution of the free sphere under external pressure.
Seems your thikness is more close to 2mm that to 1mm. ¿Is that possible?
That’s appart from the cgs issue.
I’m using second order try elements now.
These is with R=200 and t=1.93 with 1 pressure,
for the BC i see there no conflict or contradiction
for the general solution of free shere under external pressure.
but i will take these in mind, and look for your solution, to reproduce
with ccx with full sphere.
for the first calc. i use always full geometry,
and then try to use BC with symmetry.
But here it helps a lot, makes it much easier to handle.
I don’t understand the problem with the static solution,
and the mesh with ccx. these is really a mess !?
maybe someone has some ideas to fix these.
but thnx a lot for your time and sharing your results.
wbr