I’m using FEAnalyst last vid (Dome shell buckling ) as reference. (Thank you for your excellent work).
Regardless the buckling factor values which are not the aim of the question, ccx delivers a first Buckling shape with zero buckling factor.
In my case I requested 12 modes, and 13 modes are displayed.
Surprisingly , when I open the .dat file to see if I can find the empty Buckling factor associated with that First shape, that mode is not listed.
From where is Prepomax extracting that buckling shape? That ghost Buckling shape is later the first one to emerge when the Nonlinear Buckling analysis is performed. Could that Buckling Factor value be found somewhere?
I haven’t included linear buckling analysis in that tutorial also because it’s unlikely to provide accurate results in the case of such highly nonlinear snap-through behavior.
Abaqus gives the following results for this case (with unit pressure load):
Yes, it happens to show the expected shape. But since the factor is 0 and Abaqus doesn’t show it at all, I would still treat it as erroneous and avoid linear buckling analyses in such cases. With snap-through, there’s a jump to a different configuration and significant change of the stiffness matrix for which linear buckling can’t account.
I have performed a frequency analysis with increasing prestress and First mode becomes purely imaginary at 4.58 MPa. It could be a rough estimate of our first Buckling Factor.
Do you know the book page where that Timoshenko formula is described. (4.07 MPa). I would like to see the assumption behind that formula. I can only find the Buckling load for a dome hinged roof with punctual load.
I have made some progress.
Model and material properties based on FEAnalist Youtube video.
1- I have solved the LBA for the perfect shell. Attached inp. Deviation is below 0.45%.
It requires special BC as the perfect shell can expand. Fixed BC doesn’t fullfill that condition.
Linear static solution is the Sphere pure contraction. In this case Buckling mode =0 is clearly shown in the solution.
First Buckling mode exact solution is as provided in Timoshenko pag 517 but I have included the commonly neglected second term to the most common expression.
This is just an exercise to verify the correct set up of BC before going to NLBA. LBA is known to fail in the prediction of real Buckling capacity of spherical shells. Anyway, the solution and a set up with known references is needed because final solution is based on reduction factors applied to this fundamental solution.
Yeah, actual snap-through buckling load (as shown by nonlinear analysis) is much lower than predicted by the methods assuming geometrical linearity (classic formula and LBA). But there are also more accurate analytical approaches to take that into account. Still, I would only trust a fully nonlinear analysis.
So much difference in that small BC difference !!!. (Perfect Spherical shell against Clamped Dome)
Your model nonlinear buckling load agrees really well with the upper curve from the set of curves collected in the following paper. (Open acces)
Design buckling pressure for thin spherical shells: Development and validation Alexander Yu. Evkin a , ∗, Olga V. Lykhachova
Result Qbar=0.59 (2.42 / 4.07 ) against curve (Qbar=0.6)
That curve doesn’t take into consideration initial imperfections or external perturbations so, although can match the FEM model, it’s not recommended for design purposes.
The value of Lambda=3.51 associated with your geometry is, by the way, in the lower boundary of the charts, even out from some of the studies.
